Proof. This follows from combining the arguments of the proof of [40] to handle the sum together with the arguments of the proof of [41] to obtain that the \(k\)-dependent constant \(C_k\) of [40] can in fact be replaced by \(C^k\) for some constant \(C>0\). ◻

Proof. Let \(0<\alpha < 1\) and \(\delta > 0\). Set \(D(\delta,\alpha)=\delta^2\sum_{x \in D_{\delta}} d_{\delta}(x)^{-\alpha}\). With \(d_{D}:= {\operatorname{diam}}(D)\), we have that \[\label{ineq-D-alpha-N-l} D(\delta,\alpha) \leq \delta^2 \sum_{l=0}^{\log(\delta^{-1}d_{D})} \vert N_{l}(\partial D_{\delta}) \vert e^{-\alpha l}\delta^{-\alpha}\tag{3}\] where for \(l \in \mathbb{N}\), we have set \(N_{l}(\partial D_{\delta}) := \{ x \in D_{\delta}: e^{l}\delta \leq d_{\delta}(x) < e^{l+1}\delta \}\). By the condition 1 imposed on \((\partial D_{\delta})_{\delta}\) (see Section 1.0.1.1), we have that for any \(l \in \mathbb{N}\), \(\vert N_{l}(\partial D_{\delta}) \vert \leq C e^{l+1}\delta^{-1}\). Plugging this upper bound into the right-hand side of 3 , we obtain the lemma. ◻

Proof of Theorem 1. The proof of Theorem 1 consists of two steps. In Proposition 8 below, we first show tightness of the laws of \((\gamma_{\delta})_{\delta}\) under \((\mu_{\delta}^{\pm,\lambda,h})_{\delta}\) in the topology [topo_4], that is tightness of \((\nu_{\delta}^{(\pm,\lambda,h)})_{\delta}\). Then, in Proposition 9, we prove a characterization of the limiting law of any weakly convergent subsequence obtained as a consequence of the tightness of \((\gamma_{\delta})_{\delta}\) under \((\mu_{\delta}^{\pm,\lambda,h})\). This yields convergence in law of \((\gamma_{\delta})_{\delta}\) under \((\mu_{\delta}^{\pm,\lambda,h})_{\delta}\) in the topology [topo_4]. As explained in Section 1.0.1.3.2, this also implies convergence of \((\gamma_{\delta})_{\delta}\) under \((\mu_{\delta}^{\pm,\lambda,h})_{\delta}\) in the topologies [topo_1]–[topo_3] and the limiting laws in each topology agree. For later reference, let us denote by \(\nu_{\delta}^{\pm}\) the law of the interface under \(\mu_{\delta}^{\pm}\).

The key observation that will be used both to prove tightness and uniqueness in law is that \({\mathbb{P}}\)-almost surely, for each \(\delta > 0\), the law \(\nu_{\delta}^{\pm,\lambda,h}\) of \(\gamma_{\delta}\) under \(\mu_{\delta}^{\pm,\lambda, h}\) is absolutely continuous with respect to that of \(\gamma_{\delta}\) under \(\mu_{\delta}^{\pm}\) with Radon-Nikodym derivative given by \[F^{h}(\gamma_{\delta}):= \frac{{\operatorname{d}}\nu_{\delta}^{\pm,\lambda,h}}{{\operatorname{d}}\nu_{\delta}^{\pm}}(\gamma_{\delta}) = \frac{1}{\mathcal{Z}_{\delta}^{\pm,\lambda, h}} \mu_{\delta}^{\pm}\bigg[\exp\bigg(\delta^{\frac{7}{8}}\sum_{x \in D_{\delta}} \lambda(x)h_x\sigma_x \bigg) \bigg \vert \gamma_{\delta} \bigg].\] Tightness of the laws \((\nu_{\delta}^{\pm,\lambda,h})_{\delta}\) in the topology [topo_4] will in fact follow from the uniform (in \(\delta\)) boundedness of \((F^{h}(\gamma_{\delta}))_{\delta}\) in \(L^{1+\alpha}\), for any \(0<\alpha<1\). This is established in the proposition below.

We postpone the proof of Proposition 8 to Section 1.0.2.1 and now state our characterization of the limiting law of weakly convergent subsequences.

We will give the proof of Proposition 9 in Section 1.0.2.2. In view of Proposition 8 and Proposition 9, the convergence part of Theorem 1 follows from Prokhorov’s theorem while the almost sure absolute continuity of \(\nu_{W,\lambda}^{(D,a,b)}\) with respect to \(\nu_{\operatorname{SLE}_{3}}^{(D,a,b)}\) follows from Proposition 8. Finally, conformal covariance of \(\nu_{W,\lambda}^{(D,a,b)}\) is established in Proposition 17 below. ◻

Proof of Proposition 8. Let \(\alpha > 0\) be such that \(1+\alpha <2\). Let us first introduce some notations. For \(\delta > 0\) fixed, we let \(D_{L,\delta}\), respectively \(D_{R,\delta}\), be the connected component of \(D_{\delta} \setminus \gamma_{\delta}\) on the left, respectively right, of \(\gamma_{\delta}\). We also define the following two random renormalization factors: \[\label{definition-theta-L-R} \theta_{L}(\delta) := \exp\big(-\frac{1}{2}\delta^{\frac{7}{4}}\sum_{x \in D_{L,\delta}}\lambda(x)^2 \big), \quad \theta_{R}(\delta) := \exp\big(-\frac{1}{2}\delta^{\frac{7}{4}}\sum_{x \in D_{R,\delta}}\lambda(x)^2\big).\tag{9}\] Below, \(\mu_{\delta,D_{L}}^{+}\), respectively \(\mu_{\delta,D_{R}}^{-}\), will stand for the Ising measure in \(D_{L,\delta}\) with \(+1\) boundary conditions, respectively in \(D_{R,\delta}\) with \(-1\) boundary conditions. Finally, we introduce the two following random variables \[\label{definition-S-L-S-R} S_{L}^{h}(\delta):= \mu_{\delta,D_{L}}^{+}\bigg [ \exp\big(\delta^{\frac{7}{8}}\sum_{x \in D_{L,\delta}}\lambda(x)h_x\sigma_x\big) \bigg], \quad S_{R}^{h}(\delta):= \mu_{\delta,D_{R}}^{-} \bigg [ \exp\big(\delta^{\frac{7}{8}}\sum_{x \in D_{R,\delta}}\lambda(x)h_x\sigma_x\big) \bigg].\tag{10}\] By definition of \(\gamma_{\delta}\) and the Markov property of the Ising model, we have that, \({\mathbb{P}}\otimes \mu_{\delta}^{\pm}\)-almost surely, \[\label{decomposition-RN-interface} F^{h}(\gamma_{\delta}) = \frac{S_{L}^{h}(\delta)S_R^{h}(\delta)}{\mathcal{Z}_{\delta}^{\pm,\lambda, h}} \times \exp\bigg( \sum_{x \in V_L(\gamma_{\delta})}\delta^{\frac{7}{8}}\lambda(x)h_x - \sum_{x \in V_R(\gamma_{\delta})}\delta^{\frac{7}{8}}\lambda(x)h_x \bigg)\tag{11}\] where \(V_{L}(\gamma_{\delta})\), respectively \(V_{R}(\gamma_{\delta})\), is the set of vertices of \(D_{\delta}\) that are adjacent to \(\gamma_{\delta}\) on its left, respectively its right. Here, a vertex \(x \in D_{\delta}\) is said to be adjacent to \(\gamma_{\delta}\) if \(\gamma_{\delta}\) traverses an edge of the octagon whose center is \(x\). Let us set \[\label{definition-magnetization-interface} M^{h}(\gamma_{\delta}) := \sum_{x \in V_L(\gamma_{\delta})}\delta^{\frac{7}{8}}\lambda(x)h_x - \sum_{x \in V_R(\gamma_{\delta})}\lambda(x)\delta^{\frac{7}{8}}h_x.\tag{12}\] Fix some \(p > 1\) such that \((1+\alpha)p <2\) and let \(q>1\) be such that \(p^{-1}+q^{-1}=1\). Hölder’s inequality implies that \[\begin{align} \label{Holder-for-RN-curve} {\mathbb{E}}\otimes \mu_{\delta}^{\pm} \big[ F^{h}(\gamma_{\delta})^{1+\alpha} \big] \leq &{\mathbb{E}}\otimes \mu_{\delta}^{\pm} \big[\big( \theta_{L}(\delta)\theta_{R}(\delta) S_{R}^{h}(\delta)S_{L}^{h}(\delta) \big)^{p(1+\alpha)}\big]^{\frac{1}{p}}\times\bigg(\frac{\theta(\delta)}{\theta_D(\delta)}\bigg)^{1+\alpha} \nonumber \\ &\times {\mathbb{E}}\otimes \mu_{\delta}^{\pm} \bigg[\exp(q(1+\alpha)M(\gamma_{\delta}))(\tilde{\mathcal{Z}}_{\delta}^{\pm,\lambda, h})^{-q(1+\alpha)} \bigg(\frac{\theta_{D}(\delta)}{\theta_{R}(\delta)\theta_{L}(\delta)}\bigg)^{q(1+\alpha)} \bigg]^{\frac{1}{q}}, \end{align}\tag{13}\] where \(\theta(\delta)\) is defined as in 7 and where we have set \[\theta_D(\delta) := \exp\big( - \frac{\delta^{\frac{7}{4}}}{2} \sum_{x \in D_{\delta}} \lambda(x)^2 \big).\] Observe that \(\theta(\delta)/\theta_D(\delta) \to 1\) as \(\delta \to 0\): above, we introduced \(\theta_{D}(\delta)\) because the ratio \(\frac{\theta_{D}(\delta)}{\theta_{R}(\delta)\theta_{L}(\delta)}\) is easier to control that the ratio \(\frac{\theta(\delta)}{\theta_{R}(\delta)\theta_{L}(\delta)}\). In view of this and of 13 , Proposition 8 is a consequence of the following two claims, whose proofs are given just below. In the statement of these claims, the constants depend only on \(\lambda\) and \(D\).

 ◻

Proof of Claim 10. We first use the Fubini-Tonelli theorem and write \[\begin{align} &{\mathbb{E}}\otimes \mu_{\delta}^{\pm} \big[ (\theta_{R}(\delta)\theta_{L}(\delta)S_{R}^{h}(\delta)S_{L}^{h}(\delta))^{(1+\alpha)p}\big] \\ &= \mu_{\delta}^{\pm}\big[ \theta_{R}(\delta)^{(1+\alpha)p} \theta_{L}(\theta)^{(1+\alpha)p}{\mathbb{E}}\big[ S^{h}_R(\delta)^{(1+\alpha)p}S^{h}_L(\delta)^{(1+\alpha)p} \big]\big] \end{align}\] We now observe that the random variables \(S^{h}_R(\delta)\) and \(S^{h}_L(\delta)\) are \({\mathbb{P}}\)-independent as they are measurable with respect to the random variables \((h_x)_x\) in two disjoint domains. This implies that to control the above inner expectation, we must control \({\mathbb{E}}[S^{h}_R(\delta)^{(1+\alpha)p}]{\mathbb{E}}[S^{h}_L(\delta)^{(1+\alpha)p}]\). For this, we use the same expansion as in the proof of [21]. This yields that \(\mu_{\delta}^{\pm}\)-almost surely, \[\begin{align} &\theta_{R}(\delta)S_{R}^{h}(\delta) = \theta_{R}(\delta)\big(\prod_{x \in D_{R,\delta}} \cosh(\delta^{\frac{7}{8}}\lambda(x)h_x)\big) \sum_{I \subset D_{R,\delta}} \mu_{\delta,D_{R}}^{-}\big[ \prod_{x \in I} \sigma_x\big] \prod_{x\in I}\vartheta_{\delta}(x)\eta_{x} \tag{14}\\ &\theta_{L}(\delta)S_{L}^{h}(\delta) = \theta_{L}(\delta)\big(\prod_{x \in D_{L,\delta}} \cosh(\delta^{\frac{7}{8}}\lambda(x)h_x)\big) \sum_{I \subset D_{L,\delta}} \mu_{\delta,D_{L}}^{+}\big[ \prod_{x \in I} \sigma_x\big] \prod_{x\in I}\vartheta_{\delta}(x)\eta_{x} \tag{15} \end{align}\] where the sums are over subsets of \(D_{R,\delta}\) and \(D_{L,\delta}\) and where we have set \[\begin{align} \label{def-sigma-mu-eta} &\vartheta_{\delta}(x)^2 := {\mathbb{V}\!\operatorname{ar}}[\tanh(\delta^{\frac{7}{8}}\lambda(x) h_{x})] = \delta^{\frac{7}{4}}\lambda(x)^2 + O(\delta^{\frac{7}{2}}), \nonumber \\ &\eta_x := \vartheta_{\delta}(x)^{-1}\tanh(\delta^{\frac{7}{8}}\lambda(x)h_{x}). \end{align}\tag{16}\] Note in particular that the random variables \((\eta_x)_x\) are \({\mathbb{P}}\)-independent, have mean \(0\) and variance \(1\). Setting \[\begin{align} \label{def-Psi-R-Psi-L} &\Psi_{\delta,L}(\eta):= \sum_{I \subset D_{L,\delta}} \mu_{\delta,D_{L}}^{+}\big[ \prod_{x \in I} \sigma_{x} \big] \prod_{x \in I} \vartheta_{\delta}(x)\eta_x \\ &\Psi_{\delta,R}(\eta):= \sum_{J \subset D_{R,\delta}} \mu_{\delta,D_{R}}^{-} \big [ \prod_{x \in J} \sigma_{x} ] \prod_{x \in J} \vartheta_{\delta}(x)\eta_x, \nonumber \end{align}\tag{17}\] and choosing \(r, \tilde{r}>1\) such that \((1+\alpha)rp = 2\) and \(r^{-1} + \tilde{r}^{-1} = 1\), we obtain by Hölder’s inequality that \[\begin{align} &\mu_{\delta}^{\pm} \otimes {\mathbb{E}}\big[ \big( \theta_{R}(\delta)\theta_{L}(\delta) S_{R}^{h}(\delta) S_{L}^{h}(\delta) \big)^{(1+\alpha)p}\big] \\ &\leq \mu_{\delta}^{\pm}[{\mathbb{E}}[\Psi_{\delta,L}(\eta)^2]{\mathbb{E}}[\Psi_{\delta,R}(\eta)^2]]^{\frac{1}{r}} \mu_{\delta}^{\pm}\otimes {\mathbb{E}}\big[\big(\theta_{R}(\delta)\theta_{L}(\delta)\prod_{x \in D_{L,\delta} \cup D_{R,\delta}}\cosh(\delta^{\frac{7}{8}}\lambda(x)h_x) \big)^{(1+\alpha)p\tilde{r}}\big]^{\frac{1}{\tilde{r}}}. \end{align}\] The second expectation on the above right-hand side is controlled thanks to the following claim, whose proof is postponed to the end.

With Claim 12 in hands, it now remains control \(\mu_{\delta}^{\pm}[{\mathbb{E}}[\Psi_{\delta,L}(\eta)^2]{\mathbb{E}}[\Psi_{\delta,R}(\eta)^2]]\). We write \[\begin{align} \label{dvt-variance-expectation-psi} \mu_{\delta}^{\pm}[{\mathbb{E}}[\Psi_{\delta,L}(\eta)^2]{\mathbb{E}}[\Psi_{\delta,R}(\eta)^2]] = &\mu_{\delta}^{\pm}\big[{\mathbb{E}}[\Psi_{\delta,L}(\eta)]^2{\mathbb{E}}[\Psi_{\delta,R}(\eta)]^2\big] + \mu_{\delta}^{\pm}\big[{\mathbb{E}}[\Psi_{\delta,L}(\eta)]^2{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,R}(\eta))] \nonumber \\ &+ \mu_{\delta}^{\pm}\big[{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}(\eta)){\mathbb{E}}[\Psi_{\delta,R}(\eta)]^2\big] \nonumber \\ &+ \mu_{\delta}^{\pm}\big[{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}(\eta)){\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,R}(\eta))\big]. \end{align}\tag{18}\] Let us start by considering the first term on the right-hand side of 18 . It is easy to see that by \({\mathbb{P}}\)-independence of the \((h_x)_x\) (note that in the high-temperature expansion of the Ising model, points of \(D_{L,\delta}\) in the sets \(I \subset D_{L,\delta}\) cannot appear more than once in \(I\) and \(I\) is allowed to be empty), we have that \(\mu_{\delta}^{\pm}\)-almost surely, \[\begin{align} \label{expectation-Psi-L} {\mathbb{E}}[\Psi_{\delta,L}(\eta)] = \sum_{I \subset D_{L,\delta}} \mu_{\delta,D_{L}}^{+}[\prod_{x \in I}\sigma_x] {\mathbb{E}}[\prod_{x \in I}\vartheta_{\delta}(x)\eta_x]=1. \end{align}\tag{19}\] This yields that for any \(\delta > 0\), \[\label{prod-of-expectations-is-1} \mu_{\delta}^{\pm}\big[{\mathbb{E}}[\Psi_{\delta,L}(\eta)]^2{\mathbb{E}}[\Psi_{\delta,R}(\eta)]^2\big] =1.\tag{20}\] Let us now bound the product of variances in 18 . Using the same expansion and bounds as in the proof of [21], we obtain that for any \({\epsilon}>0\), \[\begin{align} &\mu_{\delta}^{\pm} \big[ {\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}(\eta)){\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,R}(\eta))\big] \\ &\leq \mu_{\delta}^{\pm}\bigg[ \big(\sum_{I \subset D_{R,\delta}} (1+{\epsilon})^{\vert I \vert}[\prod_{x \in I}\vartheta_{\delta}(x)^2]\mu_{\delta,D_{R}}^{-}\big[\prod_{x \in I}\sigma_x \big]^2\big)\big(\sum_{J \subset D_{L,\delta}} (1+{\epsilon})^{\vert J \vert}[\prod_{x \in J}\vartheta_{\delta}(x)^2]\mu_{\delta,D_{L}}^{+}\big[ \prod_{x \in J}\sigma_x \big]^2\big) \bigg]. \end{align}\] Let us denote by \(\mathcal{E}_{\delta}\) the \(\mu_{\delta}^{\pm}\)-expectation on the right-hand side above. To upper bound \(\mathcal{E}_{\delta}\), we can expand the sums, use Fubini-Tonelli theorem and the boundedness of \(\lambda\) to obtain that \[\mathcal{E}_{\delta} \leq \sum_{k,p \geq 0} \frac{(1+{\epsilon})^{k+p}M_{\lambda}^{k+p}\vartheta_{\delta}^{2(k+p)}}{p!k!} \mu_{\delta}^{\pm}\bigg[ \big(\sum_{x_1,\dots,x_p \in D_{R,\delta}}\mu_{\delta, D_{R}}^{-}[\prod_{n=1}^{p}\sigma_{x_{n}}]^2\big) \big(\sum_{x_1,\dots,x_k \in D_{L,\delta}}\mu_{\delta,D_{L}}^{+}[\prod_{j=1}^{k} \sigma_{x_{j}}]^2\big) \bigg]\] where \(\vartheta_{\delta}^{2}=\delta^{\frac{7}{4}}\) and \(M_{\lambda}\) is such that for any \(x \in D\), \(\lambda(x)^2 \leq M_{\lambda}\). We now must bound the \(\mu_{\delta}^{\pm}\)-expectations above. These bounds are provided by the following claim, whose proof is postponed to the end.

Claim 13 implies that there exists \(K_2>0\) such that for any \(\delta > 0\), \[\mu_{\delta}^{\pm}\big[{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}(\eta)){\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,R}(\eta))\big] \leq K_2.\] In view of this upper bound and of 18 and 20 , to conclude the proof of Claim 10, it remains to upper bound uniformly in \(\delta >0\) the terms \[\mu_{\delta}^{\pm}[{\mathbb{E}}[\Psi_{\delta,L}(\eta)]^2{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,R}(\eta))] \quad \operatorname{and} \quad \mu_{\delta}^{\pm}[{\mathbb{E}}[\Psi_{\delta,R}(\eta)]^2{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}(\eta))].\] The equality 19 yields that \[\mu_{\delta}^{\pm}[{\mathbb{E}}[\Psi_{\delta,L}(\eta)]^2{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,R}(\eta))] \leq \mu_{\delta}^{\pm}[{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}(\eta))].\] The expectation \(\mu_{\delta}^{\pm}[{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}(\eta))]\) can be bounded uniformly in \(\delta\) using similar arguments as those used to bound \(\mu_{\delta}^{\pm}[{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}(\eta)){\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,R}(\eta))]\). We leave the details to the reader. The term \(\mu_{\delta}^{\pm}[{\mathbb{E}}[\Psi_{\delta,R}(\eta)]^2{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}(\eta))]\) is treated similarly.

Putting everything together, we have obtained that there exists \(K>0\) such that for any \(\delta > 0\) \[{\mathbb{E}}\otimes \mu_{\delta}^{\pm} \big[ (\theta_{R}(\delta)\theta_{L}(\delta)S_{R}^{h}(\delta)S_{L}^{h}(\delta))^{(1+\alpha)p}\big] \leq K,\] which completes the proof of Claim 10. ◻

Proof of Claim 12. We establish the claim for \(N_{R}(\delta):=\theta_{R}(\delta)\prod_{x \in D_{R,\delta}}\cosh(\delta^{\frac{7}{8}}\lambda(x)h_x)\) since the proof for \(N_{L}(\delta)\) is the same, where we have defined \(N_{L}(\delta)\) as \(N_{R}(\delta)\) but with \(R\) replaced by \(L\).

Let first show that \(N_{R}(\delta)\) converges to \(1\) in \({\mathbb{P}}\otimes \mu_{\delta}^{\pm}\)-probability as \(\delta \to 0\). Observe that for any \(x \in D_{\delta}\), \[{\mathbb{E}}[\log \cosh(\delta^{\frac{7}{8}}\lambda(x)h_x)] = \frac{1}{2}\delta^{\frac{7}{4}}\lambda(x)^2 +O(\delta^{\frac{7}{2}}).\] Therefore, we have that \(\mu_{\delta}^{\pm}\)-almost surely, \[\begin{align} &\sum_{x \in D_{R,\delta}} {\mathbb{E}}[\log \cosh(\delta^{\frac{7}{8}}\lambda(x)h_x)] \nonumber \\ &= \sum_{x \in D_{R,\delta}} (\frac{\delta^{\frac{7}{4}}}{2}\lambda(x)^2+O(\delta^{\frac{7}{2}})) = -\log \theta_{R}(\delta) + \sum_{x \in D_{R,\delta}} O(\delta^{\frac{7}{2}}). \end{align}\] This yields that \({\mathbb{P}}\otimes \mu_{\delta}^{\pm}\)-almost surely, \[\begin{align} N_{R}(\delta) = \exp\bigg( \sum_{x \in D_{R,\delta}} \big[\log \cosh(\delta^{\frac{7}{8}}\lambda(x)h_x) - {\mathbb{E}}[\log \cosh (\delta^{\frac{7}{8}}\lambda(x)h_x)] + O(\delta^{\frac{7}{2}})\big] \bigg). \end{align}\] Setting \(x \in D_{\delta}\), \(W_{x}^{\delta}:= \log \cosh(\delta^{\frac{7}{8}}\lambda(x)h_x) -{\mathbb{E}}[\log \cosh(\delta^{\frac{7}{8}}\lambda(x)h_x)]\), we thus obtain that \({\mathbb{P}}\otimes \mu_{\delta}^{\pm}\)-almost surely, \[\exp\big( \sum_{x \in D_{R,\delta}} W_{x}^{\delta} - O(\delta^{\frac{3}{2}}{\operatorname{vol}}(D)) \big) \leq N_{R}(\delta) \leq \exp\big( \sum_{x \in D_{R,\delta}} W_{x}^{\delta} + O(\delta^{\frac{3}{2}}{\operatorname{vol}}(D) \big).\] Moreover, by Chebychev’s inequality and \({\mathbb{P}}\)-independence of the \((W_{x}^{\delta})_{x \in D_{\delta}}\), \[\begin{align} {\mathbb{P}}\otimes \mu_{\delta}^{\pm}\big[ \vert \sum_{x \in D_{R,\delta}} W_{x}^{\delta} \vert \geq {\epsilon}\big] &\leq {\epsilon}^{-2} {\mathbb{E}}\otimes \mu_{\delta}^{\pm}\big[\big(\sum_{x \in D_{R,\delta}} W_{x}^{\delta} \big)^2\big]\\ &= {\epsilon}^{-2}\mu_{\delta}^{\pm}\big[ \sum_{x \in D_{R,\delta}} {\mathbb{V}\!\operatorname{ar}}(W_{x}^{\delta})\big] \\ &\leq {\epsilon}^{-2}\mu_{\delta}^{\pm}\big[\sum_{x \in D_{R,\delta}} O(\delta^{\frac{7}{2}})\big] \leq {\epsilon}^{-2}O({\operatorname{vol}}(D)\delta^{\frac{3}{2}}). \end{align}\] This shows that \(\sum_{x \in D_{R,\delta}} W_{x}^{\delta}\) converges to \(0\) as \(\delta \to 0\) in \({\mathbb{P}}\otimes \mu_{\delta}^{\pm}\)-probability, which implies that \(N_{R}(\delta)\) converges to \(1\) in \({\mathbb{P}}\otimes \mu_{\delta}^{\pm}\)-probability as \(\delta \to 0\).

Finally, to prove that \(N_{\delta}(R)\) is uniformly bounded in \(L^q\) for any \(q \geq 1\), we can argue as in the last part of the proof of [21]. This completes the proof of Claim 12. ◻

Proof of Claim 13. To prove the claim, we are going to use Lemma 2 to upper bound \(\mu_{L,\delta}^{+}[\prod_{j=1}^{k} \sigma_{x_{j}}]\) and \(\mu_{R,\delta}^{-}[\prod_{j=1}^{k} \sigma_{x_{j}}]\). However, one can see that the upper bound on these quantities provided by Lemma 2 deteriorates as \(\min_{i \neq j} \vert x_i - x_j \vert\) gets smaller. As this produces a divergence that must be treated carefully, let us introduce the following sets. Let \(k \in \mathbb{N}\). For \(\delta > 0\), we define \(B_{k}(\delta):= \{(x_1,\dots,x_k) \in D_{\delta}^{k}: \exists i \neq j, \vert x_i - x_j \vert \leq 4\delta \}\) and set \[\label{def-good-subsets} \tilde{D}_{\delta}^{k} := D_{\delta}^k \setminus B_{k}(\delta), \quad \tilde{D}_{Q,\delta}^k := D_{Q,\delta}^{k} \setminus B_k(\delta),\tag{21}\] for \(Q=L,R\). \(\tilde{D}_{\delta}^{k}\), \(\tilde{D}_{Q,\delta}^k\), \(Q=L,R\), are the “good" sets on which points are sufficiently far from each other. Let us also define the following random variables \[\begin{align} &F_{L}(k,\delta) = \vartheta_{\delta}^{2k}\sum_{(x_1,\dots,x_k) \in \tilde{D}_{L,\delta}^k} \mu_{L,\delta}^{+} \big[\prod_{j=1}^{k} \sigma_{x_{j}} \big]^2 \quad \operatorname{and} \\ &B_{L}(k,\delta) = \vartheta_{\delta}^{2k}\sum_{(x_1,\dots,x_k) \in B_k(\delta) \cap D_{L,\delta}^k} \mu_{L,\delta}^{+} \big[\prod_{j=1}^{k} \sigma_{x_{j}} \big]^2. \end{align}\] \(F_{R}(k,\delta)\) and \(B_{R}(k,\delta)\) are defined similarly but with \(L\) replaced by \(R\) and \(+\) replaced by \(-\). The first step of the proof of Claim 13 consists in showing that there exists \(C>0\) such that for any \(k, p \in \mathbb{N}\) and any \(\delta >0\), \[\label{upper-bound-F-L-F-R} \mu_{\delta}^{\pm}[F_{L}(k,\delta)F_{R}(p,\delta)] \leq C^{k+p}k^{\frac{k}{2}}p^{\frac{p}{2}}.\tag{22}\] We will then deal with the contributions arising from \(B_{L}(k,\delta)\) and \(B_{R}(p,\delta)\) by proving that there exists \(\tilde{C} > 0\) such that for any \(k,p \in \mathbb{N}\) and any \(\delta > 0\), \[\begin{align} \label{bounds-B-R-B-L-F-R-F-L} &\mu_{\delta}^{\pm}[F_{L}(k,\delta)B_{R}(p,\delta)] \leq \tilde{C}^{k+p}\delta^{\frac{7}{4}} k^{\frac{k}{2}}p^{\frac{p}{2}}, \quad \mu_{\delta}^{\pm}[F_{R}(p,\delta)B_{L}(k,\delta)] \leq \tilde{C}^{k+p}\delta^{\frac{7}{4}} k^{\frac{k}{2}}p^{\frac{p}{2}}, \nonumber \\ &\mu_{\delta}^{\pm}[B_{R}(p,\delta)B_{L}(k,\delta)] \leq \tilde{C}^{k+p}\delta^{\frac{7}{4}} k^{\frac{k}{2}}p^{\frac{p}{2}}. \end{align}\tag{23}\] Claim 13 will then follow from combining 22 and 23 .

Let us start by showing 22 . The first step is to use Lemma 2 together with Cauchy-Schwarz inequality with respect to \(\mu_{\delta}^{\pm}\) to obtain that \[\mu_{\delta}^{\pm}[F_{L}(k,\delta)F_{R}(p,\delta)] \leq D_{L}(k,\delta)D_{R}(p,\delta)\] where we have set \[\begin{align} D_{L}(k,\delta) = \delta^{2k}\sum_{(x_1,\dots,x_k) \in \tilde{D}_{\delta}^k} \big[\prod_{j=1}^{k} L(x_j)\big]^{-\frac{1}{4}} \mu_{\delta}^{\pm}\big[\mathbb{I}_{(x_1,\dots,x_k) \in D_{L,\delta}^k} \big[\prod_{j=1}^{k}d_{\delta,\infty}(x_j)\big]^{-\frac{1}{2}} \big]^{\frac{1}{2}} \end{align}\] and where \(D_{R}(p,\delta)\) is defined analogously but with \(L\) replaced by \(R\). Above, for \(Q \in \{L,R\}\) and \(x \in D_{Q,\delta}\), we have defined \(d_{\delta,\infty}(x_j):= {\operatorname{dist}}(x,\partial D_{Q,\delta})\), where \({\operatorname{dist}}\) is the Euclidean distance. The upper bound 22 will follow immediately once we have shown that there exists \(C>0\) such that for any \(k, p \in \mathbb{N}\) and any \(\delta >0\), \[\label{upper-bound-D-L-D-R} D_{L}(k,\delta) \leq C^kk^{\frac{k}{2}}, \quad \operatorname{and} \quad D_{R}(p,\delta) \leq C^pp^{\frac{p}{2}}.\tag{24}\] Since the arguments are the same for both \(D_{L}(k,\delta)\) and \(D_{R}(p,\delta)\), we are going to show 24 for \(D_{L}(k,\delta)\) only (24 for \(D_{R}(p,\delta)\) actually follows from symmetry). To upper bound \(D_{L}(k,\delta)\), we introduce, for \(0 \leq r \leq k\), the sets \[\mathcal{L}(r,k) := \{(x_1,\cdots,x_k) \in D_{\delta}^k: \forall j \leq r, L(x_j) \leq d_{\delta}(x_j), \quad \forall j \geq r+1, L(x_j) > d_{\delta}(x_j) \}\] where for \(x \in D_{\delta}\), \(d_{\delta}(x):= {\operatorname{dist}}(x,\partial D_{\delta})\), \({\operatorname{dist}}\) being the Euclidean distance. Note that if \((x_1,\dots,x_k)\) belongs to \(\mathcal{L}(r,k)\), then in the upper bound of Lemma 2 on \(k\)-spin correlations, we can replace \(L(x_j) \wedge d_{\delta}(x_j)\) by \(L(x_j)\) if \(j \leq r\) and by \(d_{\delta}(x_j)\) if \(j \geq r+1\). This is why introducing the sets \(\mathcal{L}(r,k)\) will turn out to be useful. Using these sets, we can write that \[\begin{align} \label{decomposition-D-L-k-delta} D_{L}(k,\delta) &= \sum_{r=0}^{k} \binom{k}{r} \delta^{2k}\sum_{(x_1,\dots,x_k) \in \tilde{D}_{\delta}^k} \mathbb{I}_{\mathcal{L}(r,k)} \big[\prod_{j=1}^{k} L(x_j)\big]^{-\frac{1}{4}} \mu_{\delta}^{\pm}\big[\mathbb{I}_{(x_1,\cdots,x_k)\in D_{L,\delta}^k} \big[\prod_{j=1}^{k}d_{\delta,\infty}(x_j)\big]^{-\frac{1}{2}} \big]^{\frac{1}{2}}\nonumber \\ &=: \sum_{r=0}^{k} \binom{k}{r} I(r,k;\delta) \end{align}\tag{25}\] where we recall the definition 21 of \(\tilde{D}_{\delta}^{k}\). Above, the binomial coefficient \(\binom{k}{r}\) accounts for the number of ways to choose the coordinates of the points for which \(L(x_j) \leq d_{\delta}(x_j)\). Let us start with the term \(r=0\) in 25 . For \((x_1,\dots,x_k) \in \tilde{D}_{\delta}^{k} \cap \mathcal{L}(0,k)\), we have that \[\begin{align} &\mu_{\delta}^{\pm}\big[\mathbb{I}_{(x_1,\dots,x_k) \in D_{L,\delta}^k} \big[\prod_{j=1}^{k}d_{\delta,\infty}(x_j)\big]^{-\frac{1}{2}} \big]\\ &\leq \sum_{l_1=0}^{\log(\delta^{-1}d_{\delta}(x_1))} \dots \sum_{l_k=0}^{\log(\delta^{-1}d_{\delta}(x_k))} \prod_{j=1}^{k} (e^{l_j}\delta)^{-\frac{1}{2}} \mu_{\delta}^{\pm} \bigg[ \bigcap_{j=1}^{k} \{ x_j \in N_{l_{j}}(\gamma_{\delta}) \}\bigg] \end{align}\] where for \(l \in \mathbb{N}\), we have set \[N_{l_{j}}(\gamma_{\delta}) := \{ x \in D_{\delta}: e^{l}\delta \leq {\operatorname{dist}}(x,\gamma_{\delta}) \leq e^{l+1}\delta \}.\] By Lemma 40, we obtain that there exist \(C, \delta_0 > 0\) such that for any \(0 < \delta < \delta_0\) and any \(k \in \mathbb{N}\), \[\begin{align} &\mu_{\delta}^{\pm}\big[\mathbb{I}_{(x_1,\dots,x_k) \in D_{L,\delta}^k} \big[\prod_{j=1}^{k}d_{\delta,\infty}(x_j)\big]^{-\frac{1}{2}} \big]\\ &\leq \sum_{l_1=0}^{\log(\delta^{-1}d_{\delta}(x_1))} \dots \sum_{l_k=0}^{\log(\delta^{-1}d_{\delta}(x_k))} \prod_{j=1}^{k} (e^{l_j}\delta)^{-\frac{1}{2}} \bigg(\prod_{j=1}^{k}\frac{e^{l+1}\delta}{d_{\delta}(x_j)}\bigg)^{\frac{5}{8}-\rho}\\ &\leq C^k \prod_{j=1}^{k} d_{\delta}(x_j)^{-\frac{1}{2}}. \end{align}\] Plugging this upper bound in the definition of \(I(0,k;\delta)\) and using Lemma 3 and Lemma 4 to upper bound the resulting sum, we find that for some \(\tilde{C}>0\), \[\label{upper-bound-I-0-k-delta} I(0,k;\delta) \leq \tilde{C}^k k^{\frac{k}{4}}.\tag{26}\] Let us now turn to the terms with \(1 \leq r \leq k\) in 25 . For fixed \((x_{1},\dots,x_{k}) \in \tilde{D}_{\delta}^{k} \cap \mathcal{L}(r,k)\), we define the event, for \(0 \leq m \leq r\), \[\mathcal{G}(r,m) := \{\forall 1 \leq j \leq m, d_{\delta,\infty}(x_{j}) > L(x_j), \quad \forall m+1 \leq j \leq r, d_{\delta,\infty}(x_j) \leq L(x_j) \}.\] The events \((\mathcal{G}(r,m))_m\) will be useful to estimate the \(\mu_{\delta}^{\pm}\)-expectation of \(\prod_j d_{\delta,\infty}(x_j)\) appearing in 25 . Indeed, we will use Lemma 40 to upper bound this expectation and introducing the events \((\mathcal{G}(r,m))_m\) will allow us to upper bound the random variable \(d_{\delta,\infty}(x)^{-1/2}\) by \(L(x_j)^{-1/2}\) for points that are close to each other and for which the estimate provided by Lemma 40 is thus not good. We also let \(M(x_j)=L(x_j)\) if \(m+1 \leq j \leq r\) and \(M(x_j) = d_{\delta}(x_j)\) if \(j \geq r+1\). For each \(1 \leq r \leq k\), we then have that \[\begin{align} \label{decomposition-I-r-k-delta} I(r,k;\delta) = \sum_{m=0}^{r} \binom{r}{m} \delta^{2k}\sum_{(x_1,\dots,x_k) \in \tilde{D}_{\delta}^k} &\mathbb{I}_{\mathcal{L}(r,k)} \big[\prod_{j=1}^{k} L(x_j)\big]^{-\frac{1}{4}} \nonumber \\ &\mu_{\delta}^{\pm}\big[\mathbb{I}_{(x_1,\dots,x_k) \in D_{L,\delta}^k} \mathbb{I}_{\mathcal{G}(r,m)}\big[\prod_{j=1}^{k}d_{\delta,\infty}(x_j)\big]^{-\frac{1}{2}} \big]^{\frac{1}{2}}. \end{align}\tag{27}\] Moreover, for each \(1 \leq r \leq k\), \(0 \leq m \leq r\) and \((x_1,\dots,x_k) \in \tilde{D}_{\delta}^k \cap \mathcal{L}(r,k)\), we have that, using Lemma 40 and a reasoning similar to the one below 25 , \[\begin{align} &\mu_{\delta}^{\pm}\big[\mathbb{I}_{(x_1,\dots,x_k) \in D_{L,\delta}^k} \mathbb{I}_{\mathcal{G}(r,m)}\big[\prod_{j=1}^{k}d_{\delta,\infty}(x_j)\big]^{-\frac{1}{2}}\big] \\ &\leq \big[\prod_{j=1}^{m} L(x_j)\big]^{-\frac{1}{2}} \mu_{\delta}^{\pm}\big[\mathbb{I}_{(x_1,\dots,x_k) \in D_{L,\delta}^k} \mathbb{I}_{\mathcal{G}(r,m)}\big[\prod_{j=m+1}^{k}d_{\delta,\infty}(x_j)\big]^{-\frac{1}{2}}\big] \\ &\leq \big[\prod_{j=1}^{m} L(x_j)\big]^{-\frac{1}{2}} \sum_{l_{m+1}=0}^{\log(\delta^{-1}M(x_{m+1}))} \dots \sum_{l_{k}=0}^{\log(\delta^{-1}M(x_k))} \begin{aligned}[t]&\big[\prod_{j=1}^{k} e^ l\delta]^{-\frac{1}{2}} \bigg(\prod_{j=m+1}^{r}\frac{e^{(l_j+1)}\delta}{L(x_j)}\bigg)^{\frac{5}{8}-\rho}\\ &\times \bigg(\prod_{j=r+1}^{k}\frac{e^{(l_j+1)}\delta}{d_{\delta}(x_j)}\bigg)^{\frac{5}{8}-\rho}\end{aligned}\\ &\leq C^{k-m}\big[\prod_{j=1}^{r}L(x_j)\big]^{-\frac{1}{2}}\big[\prod_{j=r+1}^{k}d_{\delta}(x_j) \big]^{-\frac{1}{2}}. \end{align}\] Using this to upper bound the right-hand side of 27 , we obtain that for each \(1 \leq r \leq k\), \[\begin{align} I(r,k;\delta) &\leq \sum_{m=0}^{r} \binom{r}{m} \delta^{2k}\sum_{(x_1,\dots,x_k) \in \tilde{D}_{\delta}^k} \mathbb{I}_{\mathcal{L}(r,k)} [\prod_{j=1}^{k} L(x_j)]^{-\frac{1}{4}} [\prod_{j=1}^{r} L(x_j)]^{-\frac{1}{4}} [\prod_{j=r+1}^{k} d_{\delta}(x_j)]^{-\frac{1}{4}}\\ &\leq C^k\sum_{m=0}^{r} \binom{r}{m} \delta^{2k}\sum_{(x_1,\dots,x_k) \in \tilde{D}_{\delta}^k} \mathbb{I}_{\mathcal{L}(r,k)} [\prod_{j=1}^{k} L(x_j)]^{-\frac{1}{2}}[\prod_{j=r+1}^{k} d_{\delta}(x_j)]^{-\frac{1}{4}} \end{align}\] where the last inequality follows from rescaling \(D_{\delta}\) by \({\operatorname{diam}}(D)\). To upper bound the sums appearing on the right-hand side above, one can use Hölder’s inequality with \(\alpha =\frac{3}{2}\) for the term \([\prod_{j=1}^{k} L(x_j)]^{-\frac{1}{2}}\) and \(\tilde{\alpha}=3\) for the term \([\prod_{j=r+1} d_{\delta}(x_j)]^{-\frac{1}{4}}\) and then apply Lemma 3 and Lemma 4. We then obtain that for each \(1 \leq r \leq k\), \[I(r,k;\delta) \leq \tilde{C}^k k^{\frac{k}{2}}.\] Going back to the decomposition 25 , this upper bound together with the inequality 26 for the \(r=0\) term shows that the inequality 24 indeed holds, which, as explained above, implies the inequality 22 .

To complete the proof of Claim 13, it remains to prove the inequalities 23 . Let us give the details for the term \(\mu_{\delta}^{\pm}[F_{L}(k,\delta)B_{R}(p,\delta)]\), as the upper bounds on the other terms follow from similar arguments. Let \(k,p \in \mathbb{N}\). We have that \[\begin{align} &\mu_{\delta}^{\pm}\big[F_{L}(k,\delta)B_{R}(p,\delta)\big]\\ &= \vartheta_{\delta}^{2(k+p)} \sum_{\substack{(x_{p+1},\dots,x_{p+k}) \in \tilde{D}_{\delta}^{k} \\ (x_1,\dots,x_p) \in B_{p}(\delta)}} \mu_{\delta}\big[ \mathbb{I}_{\substack{(x_1,\dots,x_p) \in D_{R,\delta}\\ (x_{p+1},\dots,x_{k+p}) \in D_{L,\delta}}} \mu_{L,\delta}^{+}[\prod_{j=p+1}^{k+p}\sigma_{x_{j}}]^2 \mu_{R,\delta}^{-}[\prod_{j=1}^{p}\sigma_{x_{j}}]^2 \big]. \end{align}\] For \(2 \leq r \leq p\), we define the following subset of \(B_{p}(\delta)\): \[\mathcal{C}(r,p) := \{(x_1,\dots,x_p) \in D_{\delta}^p: \forall 1 \leq j \leq r, L(x_j) \leq 4\delta,\quad \forall j \geq r+1, L(x_j) > 4\delta \}.\] The sets \(\mathcal{C}(r,p)\) will be useful to isolate the “bad", i.e., potentially diverging, contributions of points close to each other in the upper bound on \(k\)-point spin correlations provided by Lemma 2. We then have that \[\begin{align} \label{decomposition-F-L-B-R} \mu_{\delta}^{\pm}\big[F_{L}(k,\delta)B_{R}(p,\delta)\big] =\sum_{r=2}^{p} \binom{p}{r} J(r,p,k;\delta) \end{align}\tag{28}\] where we have set \[\begin{align} \label{def-J-r-p-k} &J(r,p,k;\delta) \nonumber \\ &:= \vartheta_{\delta}^{2(k+p)} \sum_{\substack{(x_{p+1},\dots,x_{p+k}) \in \tilde{D}_{\delta}^{k} \\ (x_1,\dots,x_p) \in B_{p}(\delta)}} \mathbb{I}_{(x_1,\dots,x_p)\in \mathcal{C}(r,p)} \mu_{\delta}^{\pm}\big[ \mathbb{I}_{\substack{(x_1,\dots,x_p) \in D_{R,\delta}\\ (x_{p+1},\dots,x_{k+p}) \in D_{L,\delta}}} \mu_{L,\delta}^{+}[\prod_{j=p+1}^{k+p}\sigma_{x_{j}}]^2 \mu_{R,\delta}^{-}[\prod_{j=1}^{p}\sigma_{x_{j}}]^2 \big]. \end{align}\tag{29}\] For \((x_{1},\dots,x_{r}) \in \mathcal{C}(r,p)\), denote by \((x_{i_{1}},\dots, x_{i_{m}}) \subset (x_1,\dots,x_r)\) a choice of points such that for any \(n \neq m\), \(\vert x_{i_{n}} - x_{i_{m}} \vert > 4\delta\). We denote by \(\mathcal{M}\) this set of coordinates. In the special case \(r=2\), simply choose one of the two points that are distance less than \(4\delta\) from each other. Note also that \(m \leq r/2\). For \((x_1,\dots,x_{k+p}) \in \mathcal{C}(r,p) \times \tilde{D}_{\delta}^k\), Lemma 2 yields that \[\begin{align} &\mu_{\delta}^{\pm}\big[ \mathbb{I}_{\substack{(x_1,\dots,x_p) \in D_{R,\delta} \nonumber \\ (x_{p+1},\dots,x_{k+p}) \in D_{L,\delta}}} \mu_{L,\delta}^{+}[\prod_{j=p+1}^{k+p}\sigma_{x_{j}}]^2 \mu_{R,\delta}^{-}[\prod_{j=1}^{p}\sigma_{x_{j}}]^2 \big] \nonumber \\ &\leq \delta^{\frac{(k+p-r+m)}{4}}\big[\prod_{j=p+1}^{k+p} L(x_j;X[p+1,k+p])\big]^{-\frac{1}{4}} \nonumber \\ &\times \big[\prod_{j \in \mathcal{M} \cup \{r+1,\dots, p \}} L(x_j;X[(r+1,p)\cup \mathcal{M}])\big]^{-\frac{1}{4}} \nonumber \\ &\times \mu_{\delta}^{\pm}\big[\mathbb{I}_{\substack{(x_{r+1},\dots,x_p) \in D_{L,\delta}^k \\ (x_{r+1}, \dots, x_{p})}} \big[\prod_{j\in \mathcal{M} \cup \{r+1,\dots,k+p\}} d_{\delta,\infty}(x_j)\big]^{-\frac{1}{4}}\big]. \label{upper-bound-for-J-r-p} \end{align}\tag{30}\] Above, we have set \(X[p+1,k+p]=(x_{p+1},\dots,x_{k+p})\) and \(X[(r+1,p) \cup \mathcal{M}]\) is the set of points \((x_j)_j\) with index \(j\) in the set \([r+1,p] \cup \mathcal{M}\). For a subset \(P\) of indices in \([1,\dots,k+p]\), the quantity \(L(x_j; X[P])\) is then defined as \[L(x_j; X[P]) := \frac{1}{4} \min_{p \in P, p \neq j} \vert x_p - x_j \vert.\] Plugging the upper bound 30 in the definition 29 of \(J(r,p,k;\delta)\) and using the same arguments as above to control the sum, we find that for each \(2 \leq r \leq p\), \[\begin{align} J(r,p,k;\delta) \leq \sum_{m=1}^{\lfloor \frac{r}{2} \rfloor} \binom{r}{m} \delta^{2m} \delta^{\frac{7(r-m)}{4}}C^{k+p-r+m} k^{\frac{k}{2}} (p-r+m)^{\frac{(p-r+m)}{2}}. \end{align}\] Using the decomposition 28 , this shows that \[\mu_{\delta}^{\pm}[F_{L}(k,\delta)B_{R}(p,\delta)] \leq \tilde{C}^{k+p}\delta^{\frac{7}{4}} k^{\frac{k}{2}}p^{\frac{p}{2}}.\] As explained above, this inequality completes the proof of Claim 13. ◻

Proof of Claim 11. We use Hölder’s inequality twice to obtain that \[\begin{align} &{\mathbb{E}}\otimes \mu_{\delta}^{\pm} \bigg[ \exp\big(q(1+\alpha)M^{h}(\gamma_{\delta})\big)\big(\tilde{\mathcal{Z}}_{\delta}^{\pm, h}\big)^{-q(1+\alpha)}\bigg(\frac{\theta_{D}(\delta)}{\theta_L(\delta)\theta_{R}(\delta)}\bigg)^{q(1+\alpha)} \bigg]\\ &\leq {\mathbb{E}}\otimes \mu_{\delta}^{\pm} \bigg[ \exp\big(\alpha_1 M^{h}(\gamma_{\delta})\big)\bigg]^{\frac{1}{\alpha_1}} {\mathbb{E}}\otimes \mu_{\delta}^{\pm} \bigg[ \big(\tilde{\mathcal{Z}}_{\delta}^{\pm, \lambda, h}\big)^{-\alpha_2}\bigg]^{\frac{1}{\alpha_2}} {\mathbb{E}}\otimes \mu_{\delta}^{\pm} \bigg[ \bigg(\frac{\theta_{D}(\delta)}{\theta_L(\delta)\theta_{R}(\delta)}\bigg)^{\alpha_3}\bigg]^{\frac{1}{\alpha_3}} \end{align}\] for some \(\alpha_1, \alpha_2, \alpha_3 > 0\). [21] shows that the second expectation on the right-hand side above is uniformly bounded in \(\delta\). On the other hand, Claim 14 below implies that the first expectation on the right-hand side above is uniformly bounded. To conclude the proof of the claim, it suffices to observe that \(\mu_{\delta}^{\pm}\)-almost surely, \[\bigg(\frac{\theta_{D}(\delta)}{\theta_L(\delta)\theta_{R}(\delta)}\bigg)^{\alpha_3} = \exp \big(-\frac{\alpha_3\delta^{\frac{7}{4}}}{2}\sum_{x \in V(\gamma_{\delta})} \lambda(x)^2 \big),\] which implies that the \({\mathbb{P}}\otimes \mu_{\delta}^{\pm}\)-expectation of \((\theta_D(\delta)/\theta_{L}(\delta)\theta_{R}(\delta))^{\alpha_3}\) is bounded by \(1\), uniformly in \(\delta\). This completes the proof of Claim 11. ◻

Proof of Claim 14. Observe first that by symmetry of the law of \((h_x)_{x}\) under \({\mathbb{P}}\), we can assume that \(q \in (0,\infty)\). Thus, let \(q \in (0,\infty)\). By definition of \(\gamma_{\delta}\), \(\mu_{\delta}^{\pm}\)-almost surely, \(V_{L}(\gamma_{\delta}) \cap V_{R}(\gamma_{\delta}) = \emptyset\). Therefore, since the random variables \((h_{x})_{x}\) are \({\mathbb{P}}\)-independent, we have that \[{\mathbb{E}}\otimes \mu_{\delta}^{\pm} \big[ \exp(qM^{h}(\gamma_{\delta})) \big] = \mu_{\delta}^{\pm}\big[ \exp(\frac{q^2}{2}\delta^{\frac{7}{4}}\sum_{x \in V(\gamma_{\delta})} \lambda(x)^2) \big]\] where we used Fubini-Tonelli theorem and where \(V(\gamma_{\delta}) = V_{L}(\gamma_{\delta}) \cup V_{R}(\gamma_{\delta})\) denotes the set of vertices that are neighboring \(\gamma_{\delta}\). Letting \(0 < M_{\lambda}<\infty\) be such that for any \(x \in D\), \(\lambda(x)^2 \leq M_{\lambda}\), we obtain the upper bound \[{\mathbb{E}}\otimes \mu_{\delta}^{\pm} \big[ \exp(qM^{h}(\gamma_{\delta})) \big] \leq \mu_{\delta}^{\pm}\big[ \exp(\frac{q^2M_{\lambda}}{2}\delta^{\frac{7}{4}}\vert V(\gamma_{\delta}) \vert ) \big].\] Expanding the exponential on the right-hand side above and using the Fubini-Tonelli theorem, we obtain that \[{\mathbb{E}}\otimes \mu_{\delta}^{\pm} \big[ \exp(qM^{h}(\gamma_{\delta})) \big] \leq \sum_{k \geq 0} \frac{2^{-k}q^{2k}M_{\lambda}^{k}\delta^{\frac{7k}{4}}}{k!} \mu_{\delta}^{\pm}[\vert V(\gamma_{\delta}) \vert^k].\] To show that the sum over \(k\ge1\) on the right-hand side converges to \(0\) as \(\delta \to 0\), we are going to use the dominated convergence theorem. To do so, we will prove the following two claims.

The proof of these claims is given just below. Assuming them for a moment, observe that the sequence \((C^kk^{(\frac{5}{8}-\rho)k})_{k \geq 0}\) of upper bounds given by Claim 16 is summable in \(k\) once divided by (\(k!\)). Therefore, as explained above, Claim 14 now follows from the dominated convergence theorem. ◻

Proof of Claim 15. Let \(k \in \mathbb{N} \setminus \{0\}\) and \(0 < \rho \ll 1\). We define the following two subsets of \(D_{\delta}^{k}\): \[\begin{align} &B_{k}(4\delta):= \{(x_1,\dots,x_{k}) \in D_{\delta}^{k}: \exists i \neq j, \vert x_i - x_j \vert \leq 4\delta\}\\ &C_{k}(\delta) := \{(x_1,\dots,x_k) \in D_{\delta}^{k}: \exists j, {\operatorname{dist}}(x_j,\partial D_{\delta}) \leq c\delta\}. \end{align}\] where \(c>1\). With these definitions, we have that \[\begin{align} \mu_{\delta}^{\pm}[\vert V(\gamma_{\delta}) \vert^k] &= \sum_{(x_1,\dots,x_k) \in D_{\delta}^k} \mu_{\delta}^{\pm}\bigg[ \bigcap_{j=1}^{k} \{ x_j \in V(\gamma_{\delta}) \} \bigg] \nonumber \\ &\leq \sum_{(x_1,\dots,x_k) \in D_{\delta}^k} \mu_{\delta}^{\pm}\bigg[ \bigcap_{j=1}^{k} \{ d(x_j, \gamma_{\delta}) \leq \delta \} \bigg] \nonumber\\ &= \sum_{(x_1,\dots,x_k) \in D_{\delta}^k \setminus (B_{k}(4\delta) \cup C_{k}(\delta))} \mu_{\delta}^{\pm}\bigg[ \bigcap_{j=1}^{k} \{ d(x_j, \gamma_{\delta}) \leq \delta \} \bigg] \nonumber \\ &+ \sum_{(x_1,\dots,x_k) \in B_{k}(4\delta) \cup C_{k}(\delta)} \mu_{\delta}^{\pm}\bigg[ \bigcap_{j=1}^{k} \{ d(x_j, \gamma_{\delta}) \leq \delta \} \bigg] \nonumber\\ &=: I_{1}(k,\delta) + I_{2}(k,\delta). \label{ineq-k-moment--I-1-I-2} \end{align}\tag{31}\] Let us first show that \[\label{cvg-to-0-I-1} \lim_{\delta \to 0} \delta^{\frac{7k}{4}}I_1(k,\delta) = 0.\tag{32}\] To shorten notations, we set \(G(k,\delta) := D_{\delta}^k \setminus (B_{k}(4\delta) \cup C_{k}(\delta))\). Using Lemma 40 to upper bound the probability that \(\gamma_{\delta}\) visits the \(\delta\)-neighborhood of \(x_{1}, \dots, x_k\), we obtain that there exist \(C, \delta_0 >0\) such that for any \(0 < \delta < \delta_0\) and any \(k\), \[\begin{align} \label{ineq-I-1-I-tilde-1} I_1(k,\delta) &\leq \delta^{(\frac{5}{8}-\rho)k}\sum_{(x_1,\dots,x_k) \in G(k,\delta)} \big[ \prod_{j=1}^{k} L(x_j) \wedge d_{\delta}(x_j) \big]^{-\frac{5}{8}+\rho} =:\delta^{(\frac{5}{8}-\rho)k} \tilde{I}_{1}(k,\delta). \end{align}\tag{33}\] To upper bound \(\tilde{I}_{1}(k,\delta)\), by symmetry of the summand, we may fix the coordinates of points such that \(L(x_j) \wedge d_{\delta}(x_j) = d_{\delta}(x_j)\). For \(0 \leq r \leq k\), we set \[\begin{align} G(r,k;\delta):= \Big\{&(x_1,\dots,x_k) \in G(k,\delta): \forall 1 \leq j \leq r,\\ &L(x_j) \wedge d_{\delta}(x_j) = d_{\delta}(x_j), \forall j \geq r+1, L(x_j) \wedge d_{\delta}(x_j) = L(x_j) \Big\}. \end{align}\] We then have that \[\begin{align} \delta^{(\frac{5}{8}-\rho)k}\tilde{I}_{1}(k,\delta) &= \delta^{(\frac{5}{8}-\rho)k}\sum_{r=0}^{k} \binom{k}{r} \sum_{(x_1,\dots,x_k) \in G(r,k;\delta)} \big[\prod_{j=1}^{r}d_{\delta}(x_j)\big]^{-\frac{5}{8}+\rho} \prod_{j=r+1}^{k}L(x_j)\big]^{-\frac{5}{8}+\rho}\\ &\leq \delta^{(\frac{5}{8}-\rho)k}\sum_{r=0}^{k} \binom{k}{r} \sum_{(x_1,\dots,x_r) \in G(r,\delta)} \big[\prod_{j=1}^{r}d_{\delta}(x_j)\big]^{-\frac{5}{8}+\rho} f_{\delta}(x_1,\dots,x_r) \end{align}\] where we have set \[f_{\delta}(x_1,\dots,x_r) := \sum_{(x_{r+1},\dots,x_{k})} \mathbb{I}_{G(k,\delta)}(x_1,\dots,x_k) \prod_{j=r+1}^{k} L(x_j; x_1,\dots,x_k)^{-\frac{5}{8}+\rho}\] and where we wrote \(L(x_j; x_1,\dots,x_k)\) instead of just \(L(x_j)\) to stress that in the definition of \(L\), the minimum is taken over points in \((x_1,\dots,x_k)\) and not only over points in \((x_{r+1},\dots,x_k)\). To upper bound \(f_{\delta}(x_1,\dots,x_r)\), we can use the same arguments as in [40] and [41]. We leave the details to the reader and simply note that the important point to notice is that in the course of the proof, one must count the number of a certain type of graphs and that this number is upper bounded by the number of such graphs appearing when summing \(\prod_{j=1}^{k}L(x_j)^{-\frac{5}{8}+\rho}\) over \(G(k,\delta)\) instead of \(\prod_{j=r+1}^{k}L(x_j)^{-\frac{5}{8}+\rho}\) over \((x_{r+1},\dots,x_k)\) with \((x_1,\dots,x_r)\) fixed. This observation allows us to upper bound \(f_{\delta}(x_1,\dots,x_r)\) uniformly in \((x_1,\dots,x_r)\) and we obtain that for any \((x_1,\dots,x_r)\), \[f_{\delta}(x_1,\dots,x_r) \leq \delta^{-2(k-r)}c^kk^{(\frac{5}{8}-\rho)k}\] for \(c>0\) a constant depending on \(D\) only. This in turn yields that \[\begin{align} \delta^{(\frac{5}{8}-\rho)k}\tilde{I}_{1}(k,\delta) \leq \delta^{(\frac{5}{8}-\rho)k} c^kk^{(\frac{5}{8}-\rho)k} \sum_{r=0}^{k} \binom{k}{r} \delta^{-2(k-r)}\sum_{(x_1,\dots,x_r) \in G(r,\delta)} \big[\prod_{j=1}^{r}d_{\delta}(x_j)\big]^{-\frac{5}{8}+\rho}. \end{align}\] Using Lemma 4 to upper bound the sum on the right-hand side above, we get that for some constant \(C>0\) depending on \(D\) only, \[\label{final-bound-I-1-k-delta} \delta^{(\frac{5}{8}-\rho)k}\tilde{I}_{1}(k,\delta) \leq \delta^{(\frac{5}{8}-\rho)k} \delta^{-2k}c^kk^{(\frac{5}{8}-\rho)k} \sum_{r=0}^{k} \binom{k}{r} C^r \leq \delta^{(\frac{5}{8}-\rho)k}\delta^{-2k}\tilde{C}^kk^{(\frac{5}{8}-\rho)k}.\tag{34}\] Plugging this upper bound into 33 then shows that the convergence 32 indeed holds. Going back to the inequality 31 , we thus see that to conclude the proof of the claim, it remains to prove that \[\label{cvg-to-0-I-2} \lim_{\delta \to 0} \delta^{\frac{7k}{4}} I_2(k,\delta) = 0.\tag{35}\] To do so, we write \[\begin{align} I_2(k,\delta) &= \sum_{(x_1,\dots,x_k) \in B_{k}(4\delta) \setminus C_k(\delta) } {\mathbb{P}}\bigg[ \bigcap_{j=1}^{k} \{ d(x_j, \gamma_{\delta}) \leq \delta \} \bigg] \nonumber \\ &+ \sum_{(x_1,\dots,x_k) \in B_{k}(4\delta) \cap C_k(\delta) } {\mathbb{P}}\bigg[ \bigcap_{j=1}^{k} \{ d(x_j, \gamma_{\delta}) \leq \delta \} \bigg] \nonumber \\ &=: I_2^{B}(k,\delta) + I_2^{C}(k,\delta). \label{decomposition-I-2} \end{align}\tag{36}\] Let us first establish an upper bound on \(I_2^{B}(k,\delta)\). We decompose \(I_2^{B}(k,\delta)\) by first choosing the number and the coordinates of the points that are such that \(L(x_j) \leq 4\delta\). For \(2 \leq m \leq k\), we set \[B(m,k;\delta) := \{ (x_1,\dots,x_k) \in D_{\delta}^k: \forall 1 \leq j \leq m, L(x_j) \leq 4\delta, \forall j \geq m+1, L(x_j) > 4\delta \}.\] Then, if \((x_1,\dots,x_k) \in B(m,k;\delta)\), we count the number \(b\) of squares of side-length \(4\delta\) that are necessary to cover these points. In each of these squares, we choose a “representative” for the square among the points in \((x_1,\dots,x_m)\) that belong to this square. Observe that we can choose the representatives \((x_{i_1},\dots,x_{i_b})\) of the \(b\) squares in such a way that that for any \(p \neq n\), \(\vert x_{i_{n}} - x_{i_{p}} \vert > 4\delta\). Let us denote by \(\mathcal{B}(m)\) the set of coordinates of the representatives of the \(b\) squares and define \(\mathcal{S}(m,k):= \mathcal{B}(m) \cup \{m+1,\dots,k\}\). With this decomposition, we obtain that \[\begin{align} I_2^{B}(k,\delta) &= \sum_{m=2}^{k} \binom{k}{m} \sum_{b=\lfloor \frac{m}{16} \rfloor +1}^{\lfloor \frac{m}{2} \rfloor} \sum_{(x_1,\dots,x_k)\in D_{\delta} \setminus C_{k}(\delta)}\mathbb{I}_{B(m,k;\delta)} \mathbb{I}_{\vert \mathcal{B}(m)\vert=b} {\mathbb{P}}\bigg[ \bigcap_{j=1}^{k} \{ d(x_j, \gamma_{\delta}) \leq \delta \} \bigg]\nonumber \\ &\leq \sum_{m=2}^{k} \binom{k}{m} \sum_{b=\lfloor \frac{m}{16} \rfloor +1}^{\lfloor \frac{m}{2} \rfloor} \sum_{(x_1,\dots,x_k)\in D_{\delta} \setminus C_{k}(\delta)} \mathbb{I}_{B(m,k;\delta)} \mathbb{I}_{\vert \mathcal{B}(m)\vert=b} {\mathbb{P}}\bigg[ \bigcap_{j \in \mathcal{S}(m,k)} \{ d(x_j, \gamma_{\delta}) \leq \delta \} \bigg] \nonumber \\ &\leq \sum_{m=2}^{k} \binom{k}{m} \sum_{b=\lfloor \frac{m}{16} \rfloor +1}^{\lfloor \frac{m}{2} \rfloor} \delta^{(\frac{5}{8}-\rho)(k-m+b)}I_{2}^{B}(b,m,k,\delta) \label{ineq-decomposition-I-2-B} \end{align}\tag{37}\] where we have set \[I_{2}^{B}(b,m,k,\delta) := \sum_{(x_1,\dots,x_k)\in D_{\delta} \setminus C_{k}(\delta)} \mathbb{I}_{B(m,k;\delta)} \mathbb{I}_{\vert \mathcal{B}(m)\vert =b} \big[ \prod_{j \in \mathcal{S}(m,k)} L(x_j; \mathcal{S}(m,k)) \wedge d_{\delta}(x_j)\big]^{-\frac{5}{8}+\rho}\] and where the last inequality follows from Lemma 40. Above, for \(j \in \mathcal{S}(m,k)\), we have also defined \[L(x_j; \mathcal{S}(m,k)) = \frac{1}{4} \min_{p \in \mathcal{S}(m,k), p \neq j } \vert x_p - x_j \vert.\] To upper bound \(I_{2}^{B}(b,m,k,\delta)\), we use Hölder’s inequality with \(\alpha, \tilde{\alpha} >1\) such that \(\alpha(\frac{5}{8}-\rho)<1\), \(\tilde{\alpha}(\frac{5}{8}-\rho)<2\) and \(\alpha^{-1}+\tilde{\alpha}^{-1} = 1\). This yields that \[\begin{align} I_{2}^{B}(b,m,k,\delta) \leq &\bigg( \sum_{(x_1,\dots,x_k)\in D_{\delta} \setminus C_{k}(\delta)} \mathbb{I}_{B(m,k;\delta)} \mathbb{I}_{\vert \mathcal{B}(m)\vert =b} \bigg)^{\frac{1}{\tilde{\alpha}}} \\ &\times \bigg( \delta^{-2(k-m+b)} \sum_{(x_{1},\dots,x_{k-m+b}) \in G(k-m+b,\delta)} \big[ \prod_{j=1}^{k-m+b}L(x_j) \wedge d_{\delta}(x_j) \big]^{-\alpha(\frac{5}{8}-\rho)} \bigg)^{\frac{1}{\alpha}} \end{align}\] where we recall that \(G(k-m+b,\delta) = D_{\delta}^{k-m+b} \setminus (B_{k-m+b}(4\delta) \cup C_{k-m+b}(\delta))\). Since \(\alpha(\frac{5}{8}-\rho)<1\), the sum on the right-hand side above can be upper bounded using the same arguments as those which led to the upper bound 34 . We thus obtain that \[\begin{align} I_{2}^{B}(b,m,k,\delta) \leq \delta^{-\frac{2}{\tilde{\alpha}}(k-m+b)} \delta^{-\frac{2}{\alpha}k} C^{k-m+b}(k-m+b)^{(\frac{5}{8}-\rho)(k-m+b)} \end{align}\] Multiplying both sides by \(\delta^{(\frac{5}{8}-\rho)(k-m+b)}\) and observing that \(\frac{2}{\tilde{\alpha}}(m-b) - (\frac{5}{8}-\rho)(m-b)>0\) by our choice of \(\tilde{\alpha}\) and since \(m-b \geq m/2 \geq 1\), we thus see that \[I_{2}^{B}(b,m,k,\delta) \leq \delta^{(\frac{5}{8}-\rho)k}\delta^{-2k}C^{k-m+b}(k-m+b)^{(\frac{5}{8}-\rho)(k-m+b)}.\] Plugging this upper bound into 37 then yields that \[\label{final-bound-I-2-B} I_2^{B}(k,\delta) \leq \tilde{C}^{k}\delta^{(\frac{5}{8}-\rho)k}\delta^{-2k} C^k k^{(\frac{5}{8}-\rho)k}.\tag{38}\] To conclude the proof of 35 , in view of 36 , it now remains to find an appropriate upper bound on \(I_2^{C}(k,\delta)\). To do so, we first fix the coordinates of the points in \(C_{k}(\delta)\) and write \[\begin{align} I_2^{C}(k,\delta) &= \sum_{r=1}^{k} \binom{k}{r} \sum_{\substack{(x_1,\dots,x_r) \in C_1(\delta)^r \\ (x_{r+1}, \dots,x_k) \in D_{\delta}^k \setminus C_k(\delta)}} {\mathbb{P}}\bigg[ \bigcap_{j=1}^{k} \{ d(x_j, \gamma_{\delta}) \leq \delta \} \bigg]\\ &\leq \sum_{r=1}^{k} \binom{k}{r} {\operatorname{Area}}(C_{1}(\delta)^r) \sum_{(x_{r+1},\dots,x_{k}) \in D_{\delta}^{k-r} \setminus C_{k-r}(\delta)} {\mathbb{P}}\bigg[ \bigcap_{j=r+1}^{k} \{ d(x_j, \gamma_{\delta}) \leq \delta \} \bigg]. \end{align}\] Decomposing \(D_{\delta}^{k-r} \setminus C_{k-r}(\delta)\) as \(D_{\delta}^{k-r} \setminus (B_{k-r}(\delta) \cup C_{k-r}(\delta)) \cup B_{k-r}(\delta) \setminus C_{k-r}(\delta)\), we see that the inner sum above can upper bounding using the upper bounds 34 and 38 that we obtained in the first part of the proof. This yields that \[\begin{align} I_2^{C}(k,\delta) \leq \sum_{r=1}^{k} \binom{k}{r} \delta^{-r} \delta^{-2(k-r)} \delta^{(\frac{5}{8}-\rho)(k-r)}C^{k-r}(k-r)^{(\frac{5}{8}-\rho)(k-r)} \leq c^k \delta^{-2k} \delta^{(\frac{5}{8}-\rho)k}\delta^{(\frac{3}{8}+\rho)r} k^{(\frac{5}{8}-\rho)k}. \end{align}\] Together with the decomposition 36 and the upper bound 38 , this implies that \[\label{final-bound-I-2-k-delta} I_2(k,\delta) \leq \tilde{c}^k \delta^{-2k} \delta^{(\frac{5}{8}-\rho)k}k^{(\frac{5}{8}-\rho)k}\tag{39}\] which concludes the proof of the convergence 35 and thus completes the proof of the claim. ◻

Proof of Claim 16. This directly follows from the upper bound on \(\delta^{\frac{7k}{4}}{\mathbb{E}}[\vert V(\gamma_{\delta}) \vert^k]\) obtained in the course of the proof of Claim 15, see in particular 34 and 39 . ◻

Proof of Proposition 9. To prove this proposition, we will also need to consider convergence of the random variables \((W_{\delta}^{h}(\phi):=\delta\sum_{x \in D_{\delta}}h_{x}\phi(x))_{\phi}\) for functions \(\phi: \mathbb{R}^2 \to \mathbb{R}\) that are square-integrable. We refer the reader to [20] for details on the topology in which convergence takes place and background material on white noise.

Recall the decomposition 11 of the Radon-Nikodym derivative \(F^h(\gamma_{\delta})\) of \(\nu_{\delta}^{\pm,\lambda,h}\) with respect to \(\nu_{\delta}^{\pm}\) and the definitions of \(\theta_{L}(\delta), \theta_{R}(\delta), S_{L}^{h}(\delta)\) and \(S_{R}^{h}(\delta)\) given in 9 and 10 . In view of this decomposition, the first step to prove Proposition 9 is to prove the following lemma.

Lemma 6 will be shown just below. Assuming this result, the second step of the proof of Proposition 9 is to take into account the contribution of \(\exp(M^{h}(\gamma_{\delta}))\) and of \((\theta_{L}(\delta)\theta_{R}(\delta))^{-1}\theta_{D}(\delta)\) to the limit of \(F^h(\gamma_{\delta})\). However, this contribution in fact vanishes in the limit, as implied by Claim 14 and its proof. Indeed, this claim shows that the Laplace transform of \(M^{h}(\gamma_{\delta})\) under \({\mathbb{P}}\otimes \mu_{\delta}^{\pm}\) converges to \(1\), which implies that \(M^{h}(\gamma_{\delta})\) converges in law to \(0\), and therefore that, since \(x \to \exp(x)\) is continuous, \(\exp(M^{h}(\gamma_{\delta}))\) converges in law to \(1\). A similar reasoning, using again (the proof of) Claim 14, applies to \((\theta_{L}(\delta)\theta_{R}(\delta))^{-1}\theta_{D}(\delta)\) once one notices that this random variable is in fact equal to \(\exp(-\frac{\delta^{\frac{7}{4}}}{2}\sum_{x \in V(\gamma_{\delta})}\lambda(x)^2)\). This concludes the proof of Proposition 9. ◻

Proof of Lemma 6. The starting point of the proof is to observe that \(\theta_L(\delta)S_{L}^{h}(\delta)\), \(\theta_{R}(\delta)S_{R}^{h}(\delta)\) and \(\tilde{\mathcal{Z}}_{\delta}^{\pm,\lambda,h}\) can be expressed in terms of a polynomial chaos decomposition, see 15 , 14 and 8 . Moreover, recall that by Claim 12, the prefactors \(\theta_j(\delta)\prod_{x \in D_{j,\delta}} \cosh(\delta^{\frac{7}{8}}\lambda(x)h_x)\), \(j=L,R\), converge to \(1\) in probability and are uniformly bounded in \(L^p\), for any \(p \geq 1\). Similarly, as shown in [21], the prefactor \(\theta(\delta)\prod_{x \in D_{\delta}} \cosh(\delta^{\frac{7}{8}}\lambda(x)h_x)\) appearing in the chaos expansion 8 of \(\tilde{\mathcal{Z}}_{\delta}^{\pm,\lambda, h}\) converges to \(1\) in any \(L^p\), \(p \geq 1\). Therefore, it suffices to show the lemma with \(\theta_{L}(\delta)S_{L}^{h}(\delta)\) and \(\theta_{R}(\delta)S_{R}^{h}(\delta)\) replaced by \(\Psi_{\delta,L}(\eta)\) and \(\Psi_{\delta,R}(\eta)\) as defined in 17 and with \(\tilde{\mathcal{Z}}_{\delta}^{\pm,\lambda,h}\) replaced by \[\begin{align} \Psi_{\delta}^{\pm}(\eta) := \sum_{I \subset D_{\delta}} \mu_{\delta}^{\pm}[\prod_{x \in I}\sigma_{x}] \prod_{x \in I} \vartheta_{\delta}(x)\eta_x \end{align}\] where \(\eta_x\) and \(\vartheta_{\delta}(x)\) are as in 16 . Then, as in [21], we first truncate the chaos expansions \(\Psi_{\delta}(\eta)\), \(\Psi_{\delta,L}(\eta)\) and \(\Psi_{\delta,R}(\eta)\) to order \(\ell\), that is we restrict the sums in the definitions of \(\Psi_{\delta}^{\pm}(\eta)\), \(\Psi_{\delta,L}(\eta)\) and \(\Psi_{\delta,R}(\eta)\) to subsets \(I\) of size at most \(\ell\). Denote by \(\Psi_{\delta,L}^{\leq \ell}(\eta)\), \(\Psi_{\delta,R}^{\leq \ell}(\eta)\) and \(\Psi_{\delta}^{\pm,\leq \ell}(\eta)\) these truncated expansions. The first step is to show that \[\begin{align} &\lim_{\ell \to \infty} \lim_{\delta \to 0} {\mathbb{E}}\otimes\mu_{\delta}^{\pm}\big[\vert \Psi_{\delta,j}^{\leq \ell}(\eta) - \Psi_{\delta,j}(\delta)\vert^2\big] = 0, \quad j=L,R \tag{40} \\ &\lim_{\ell \to \infty} \lim_{\delta \to 0} {\mathbb{E}}\otimes\mu_{\delta}^{\pm}\big[\vert \Psi_{\delta}^{\pm,\leq \ell}(\eta) - \Psi_{\delta}^{\pm}(\eta) \vert^2\big] =0 . \tag{41} \end{align}\] The second step is to linearize the noise: namely, in the definitions of \(\Psi_{\delta,L}^{\leq \ell}(\eta)\), \(\Psi_{\delta,R}^{\leq \ell}(\eta)\) and \(\Psi_{\delta}^{\pm,\leq \ell}(\eta)\), we replace \(\tanh(\delta^{\frac{7}{8}}\lambda(x)h_x)\) with \(\delta^{\frac{7}{8}}\lambda(x)h_x\), \(\vartheta_{\delta}(x)^2\) with \(\delta^{\frac{7}{4}}\lambda(x)^2\) (in the definition 16 of \(\eta\)). Denote by \(\Xi_{\delta,L}^{\leq \ell}(h)\), \(\Xi_{\delta,R}^{\leq \ell}(h)\) and \(\Xi_{\delta,D}^{\pm,\leq \ell}(h)\) the random variables thus obtained. The second step then consists in establishing that \[\begin{align} &\lim_{\ell \to \infty} \lim_{\delta \to 0} {\mathbb{E}}\otimes \mu_{\delta}^{\pm}\big[ \vert \Psi_{\delta,j}^{\leq \ell}(\eta) - \Xi_{\delta,j}^{\leq \ell}(h) \vert^2\big] = 0, \quad j=L,R \tag{42} \\ &\lim_{\ell \to \infty} \lim_{\delta \to 0} {\mathbb{E}}\otimes \mu_{\delta}^{\pm}\big[ \vert \Psi_{\delta}^{\pm,\leq \ell}(\eta) - \Xi_{\delta,D}^{\pm,\leq \ell}(h) \vert^2]= 0. \tag{43} \end{align}\] In the third step, we replace the noise variables in \(\Xi_{\delta,L}^{\leq \ell}(h)\), \(\Xi_{\delta,R}^{\leq \ell}(h)\) and \(\Xi_{\delta,D}^{\pm, \leq \ell}(h)\) with their white noise approximation: each random variable \(h_x\) is replaced by \(w_{\delta}(x):= 2\delta^{-1}\int_{S_{\delta}(x)}W(dz)\) where \(S_{\delta}(x)\) is the box centered at \(x\) with side length \(\delta\) and \(W\) the limiting white noise. Denote by \(\Theta_{\delta,L}^{W,\leq \ell}\), \(\Theta_{\delta,R}^{W,\leq \ell}\) and \(\Theta_{\delta, D}^{\pm,W,\leq \ell}\) the random variables thus obtained. Let us also define, for \(\phi \in L^2(\mathbb{R}^2)\), \(W_{\delta}(\phi):= \delta \sum_{x \in \delta\mathbb{Z}^2} w_{\delta}(x)\phi(x)\). We then show that for any \(\ell, m \in \mathbb{N}\), any \(\phi_{1}, \dots, \phi_{m} \in L^{2}(\mathbb{R}^2)\) as \(\delta \to 0\), \[\label{cvg-white-noise-approximation} ((W_{\delta}(\phi_j))_{j=1}^{m}, \Theta_{\delta, D}^{\pm,W,\leq \ell}, \Theta_{\delta, L}^{W,\leq \ell}, \Theta_{\delta,R}^{W,\leq \ell}) \to ((W(\phi_j))_{j=1}^{m}, \mathcal{Z}_{D}^{\pm,W, \leq \ell}, \mathcal{Z}_{L}^{+,W, \leq \ell}, \mathcal{Z}_{R}^{-,W, \leq \ell})\tag{44}\] where the convergence is in \(L^2({\mathbb{P}}\otimes \nu_{\operatorname{SLE}_{3}}^{(D,a,b)})\).

The last step of the proof is to prove that for any \(\ell,j \in \mathbb{N}\), any \(\phi_1,\dots,\phi_m \in L^2(\mathbb{R}^2)\) and any bounded function \(g: \mathbb{R}^{m+3} \to \mathbb{R}\) whose first three derivatives are bounded, \[\begin{align} \lim_{\delta \to 0} \bigg \vert &{\mathbb{E}}\otimes \mu_{\delta}^{\pm}[g((W^{h}_{\delta}(\phi_j))_{j=1}^{m}, \Xi_{\delta,L}^{\leq \ell}(h), \Xi_{\delta,R}^{\leq \ell}(h), \Xi_{\delta,D}^{\pm, \leq \ell}(h))] \nonumber \\ &- {\mathbb{E}}\otimes \mu_{\delta}^{\pm}[g((W_{\delta}(\phi_j))_{j=1}^{m}, \Theta_{\delta,L}^{W,\leq \ell}, \Theta_{\delta,R}^{W,\leq \ell}, \Theta_{\delta,D}^{\pm,W,\leq \ell})] \bigg \vert = 0. \label{cvg-theta-vs-Xi} \end{align}\tag{45}\] Putting everything together, this concludes the proof of Proposition 9. The convergence results 40 –45 are established just below. ◻

Proof of 40 and 41 . Let us prove 40 for \(j=L\), as the convergences 40 for \(j=R\) and 41 follow from the same arguments. Let \(\ell \in \mathbb{N}\). We have that \[\begin{align} \Psi_{\delta,L}(\eta) - \Psi_{\delta,L}^{\leq \ell}(\eta) = \sum_{I \subset D_{L,\delta}, \vert I \vert \geq \ell+1}\mu_{\delta, D_{L}}^{+}[\prod_{x \in I} \sigma_x ] \prod_{x \in I} \vartheta_{\delta}(x)\eta_x =: \Psi_{\delta,L}^{> \ell}(\eta). \end{align}\] We now write \({\mathbb{E}}\otimes \mu_{\delta}^{\pm}[\vert \Psi_{\delta,L}^{> \ell}(\eta) \vert^2] = \mu_{\delta}^{\pm}[{\mathbb{E}}[\Psi_{\delta,L}^{> \ell}(\eta)]^2]+ \mu_{\delta}^{\pm}({\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}^{> \ell}(\eta)))\). The term \(\mu_{\delta}^{\pm}[{\mathbb{E}}[\Psi_{\delta,L}^{> \ell}(\eta)]^2]\) converges to \(0\) as \(\delta \to 0\) and then \(\ell \to \infty\) since only the empty interval contributes to this expectation, see the discussion around 20 . On the other hand, using the same bound as in the proof of [21], we have that for any \({\epsilon}> 0\), \[\begin{align} \mu_{\delta}^{\pm}[{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}^{> \ell}(\eta))] \leq \mu_{\delta}^{\pm}\bigg[\sum_{I \subset D_{L,\delta}, \vert I \vert > \ell} (1+{\epsilon})^{\vert I \vert} [\prod_{x \in I}\vartheta_{\delta}(x)^{2}] \mu_{\delta,D_{L}}^{+}\big[\prod_{x \in I} \sigma_x \big]^2 \bigg ]. \end{align}\] To prove that \(\mu_{\delta}^{\pm}[{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}^{> \ell}(\eta))]\) converges to \(0\) as \(\delta \to 0\) and then \(\ell \to \infty\), it thus suffices to show that this is the case of the \(\mu_{\delta}^{\pm}\)-expectation of the sum on the above right-hand side. But in fact, the expectation of each term in this sum can be bounded using the same arguments as those used to prove Claim 13 (taking \(j=0\) everywhere). This implies that the expectation of this sum can be upper bounded uniformly in \(\delta\) by the rest of order \(> \ell\) of an absolutely convergent series. This in turn yields that \(\mu_{\delta}^{\pm}[{\mathbb{V}\!\operatorname{ar}}(\Psi_{\delta,L}^{> \ell}(\eta))]\) converges to \(0\) as \(\delta \to 0\) and then \(\ell \to \infty\) and concludes the proof of 40 . ◻

Proof of 42 and 43 . Let us prove 42 for \(j=L\), as the convergences 42 for \(j=R\) and 43 follow from the same arguments. Thanks to the bounds of Claim 13 on the \(\mu_{\delta}^{\pm}\)-expectation of \(\sum_{I: \vert I \vert = k}\vartheta_{\delta}^{2k}\mu_{\delta,L}^{+}[\prod_{x \in I} \sigma_x]^2\) (where \(\mu_{\delta,L}^{+}[\prod_{x \in I} \sigma_x]\) is the analogue of \(\psi_{\delta}(I)\) in the notation of [21]) and thanks to Theorem 3 applied to \(\mu_{\delta,L}^{+}[\prod_{x \in I} \sigma_x]\), this in fact follows from the same arguments as the ones used to prove [21]. Details are left to the reader. ◻

Proof of 44 . Let \(\ell \in \mathbb{N}\) and let us first show that \({\mathbb{E}}\otimes \mu_{\delta}^{\pm}\big[\vert \Theta_{L}^{W,\leq \ell}(\delta) - \mathcal{Z}_{L}^{+,W,\leq \ell} \vert^2 \big] \to 0\) as \(\delta \to 0\). This in fact follows from the same arguments as those used in Step 1 of the proof of [20]. Indeed, applying the same decomposition as in [20] and using the bounds of Claim 13 on the \(\mu_{\delta}^{\pm}\)-expectations to upper bound the terms that we get from this decomposition, we obtain that \({\mathbb{E}}\otimes \mu_{\delta}^{\pm}\big[\vert \Theta_{L}^{W,\leq \ell}(\delta) - \mathcal{Z}_{L}^{+,W,\leq \ell} \vert^2 \big]\) indeed converges to \(0\) as \(\delta \to 0\). A similar reasoning can be used to show that \({\mathbb{E}}\otimes \mu_{\delta}^{\pm}\big[\vert \Theta_{R}^{W,\leq \ell}(\delta) - \mathcal{Z}_{R}^{-,W,\leq \ell} \vert^2 \big]\) and \({\mathbb{E}}\otimes \mu_{\delta}^{\pm}\big[\vert \Theta_{D}^{W,\leq \ell}(\delta) - \mathcal{Z}_{D}^{\pm,W,\leq \ell} \vert^2 \big]\) converge to \(0\) as \(\delta \to 0\). This proves 44 . ◻

Proof of 45 . Let \(\ell, m \in \mathbb{N}\), \(\phi_1, \dots, \phi_m \in L^2(\mathbb{R}^2)\), and let \(g: \mathbb{R}^{m+3} \to \mathbb{R}\) be a bounded function whose first three derivatives are bounded. To show 45 , we first apply the multivariate Lindeberg’s principle of [21], see also Step 3 of the proof of [20], to obtain that for any \({\epsilon}> 0\), \(\mu_{\delta}^{\pm}\)-almost surely, \[\begin{align} \label{upper-bound-via-Lindeberg} &\vert {\mathbb{E}}[\begin{aligned}[t]&g((W^{h}_{\delta}(\phi_j))_{j=1}^{m}, \Xi_{L}^{h,\leq \ell}(\delta), \Xi_{R}^{h,\leq \ell}(\delta), \Xi_{D}^{\pm, h,\leq \ell}(\delta))] \\ &- {\mathbb{E}}[g((W^{h}_{\delta}(\phi_j))_{j=1}^{m}, \Theta_{L}^{h,\leq \ell}(\delta), \Theta_{R}^{h,\leq \ell}(\delta), \Theta_{D}^{\pm, h,\leq \ell}(\delta))] \vert\end{aligned} \nonumber \\ &\leq C_{g,\ell}\sum_{j=1}^{m}{\mathbb{V}\!\operatorname{ar}}(W_{\delta}(\phi_j))\max_{x \in D_{\delta}} \big(\operatorname{Inf}_{x}[W_{\delta}(\phi_j)])^{\frac{1}{2}} + \tilde{C}_{g,\ell}C_{D}^{({\epsilon},\ell)}(\Psi;\delta)\big(\max_{x \in D_{\delta}}\operatorname{Inf}_{x}[\Psi_{\delta}^{({\epsilon}),\leq \ell}]\big)^{\frac{1}{2}} \nonumber \\ &+ C_{g,\ell}'C_{L}^{({\epsilon},\ell)}(\Psi;\delta)\big(\max_{x \in D_{L,\delta}}\operatorname{Inf}_{x}[\Psi_{\delta,L}^{({\epsilon}),\leq \ell}])^{\frac{1}{2}}+C_{g,\ell}'' C_{R}^{({\epsilon},\ell)}(\Psi;\delta)\big(\max_{x \in D_{R,\delta}}\operatorname{Inf}_{x}[\Psi_{\delta,R}^{({\epsilon}),\leq \ell}])^{\frac{1}{2}} \end{align}\tag{46}\] where \(C_{g,\ell}, \tilde{C}_{g,\ell}, C_{g,\ell}', C_{g,\ell}'' > 0\) are constant depending on \(g\) and \(\ell\) only. Above, we have set \[\begin{align} & \operatorname{Inf}_{x}[W_{\delta}(\phi_j)] := \delta \phi_j(x)\\ &C_{D}^{({\epsilon},\ell)}(\Psi;\delta):= \sum_{I \subset D_{\delta}, \vert I \vert \leq \ell} (1+{\epsilon})^{\vert I \vert} \vartheta_{\delta}^{2\vert I \vert}\mu_{\delta}^{\pm}[\prod_{x \in I} \sigma_x]^2 \\ &\operatorname{Inf}_{x \in D_{\delta}}[\Psi_{\delta}^{({\epsilon}),\leq \ell}] := \sum_{I \subset D_{\delta}: \vert I \vert \leq \ell, x \in I} (1+{\epsilon})^{\vert I \vert} \vartheta_{\delta}^{2\vert I \vert}\mu_{\delta}^{\pm}[\prod_{z \in I} \sigma_{z}]^2 \end{align}\] and for \(j=R,L\) and \(x \in D_{j,\delta}\), we have set \[\begin{align} &C_{j}^{({\epsilon},\ell)}(\Psi;\delta) = \sum_{I \subset D_{j,\delta}, \vert I \vert \leq \ell} (1+{\epsilon})^{\vert I \vert} [\prod_{x \in I}\vartheta_{\delta}(x)^{2}]\mu_{\delta,j}^{\xi_{j}}[\prod_{x \in I} \sigma_x]^2 \\ &\operatorname{Inf}_{x}[\Psi_{\delta,j}^{({\epsilon}),\leq \ell}] := \sum_{I \subset D_{j,\delta}: \vert I \vert \leq \ell. x \in I} (1+{\epsilon})^{\vert I \vert} [\prod_{z \in I}\vartheta_{\delta}(z)^{2}]\mu_{\delta,j}^{\xi_{j}}[\prod_{z \in I} \sigma_{z}]^2. \end{align}\] It is easy to see that for each \(j=1,\dots,m\), \({\mathbb{V}\!\operatorname{ar}}(W_{\delta}(\phi_j))\) is uniformly bounded in \(\delta\) while \(\operatorname{Inf}_{x}[W_{\delta}(\phi_j)] \to 0\) as \(\delta \to 0\). Therefore, the first summand on the right-hand side of 46 converges to \(0\) as \(\delta \to 0\). Moreover, the fact that \(C_{D}^{({\epsilon},\ell)}(\Psi;\delta)\big(\max_{x} \operatorname{Inf}_{x \in D_{\delta}}[\Psi_{\delta}^{({\epsilon}),\leq \ell}]\big)^{\frac{1}{2}}\) also vanishes as \(\delta \to 0\) is proved in Step 2 of the proof of [20]. Thus, to conclude the proof of 45 , it thus suffices to show that the \(\mu_{\delta}^{\pm}\)-expectations of the two last summands on the right-hand side of 46 converge to \(0\) as \(\delta \to 0\). Let us prove this for the first one, as the proof for the second one is identical. We first use the Cauchy-Schwarz inequality to get that \[\mu_{\delta}^{\pm}[C_{L}^{({\epsilon},\ell)}(\Psi;\delta)\big(\max_{x \in D_{L,\delta}}\operatorname{Inf}_{x}[\Psi_{\delta,L}^{({\epsilon}),\leq \ell}])^{\frac{1}{2}}]^2 \leq \mu_{\delta}^{\pm}[C_{L}^{({\epsilon},\ell)}(\Psi;\delta)^2]\mu_{\delta}^{\pm}[\max_{x \in D_{L,\delta}} \operatorname{Inf}_{x}\Psi_{\delta,L}^{({\epsilon}),\leq \ell}].\] The term \(\mu_{\delta}^{\pm}[C_{L}^{({\epsilon},\ell)}(\Psi;\delta)^2]\) can be bounded uniformly in \(\delta\): when expanding the square as the product of two sums, the \(\mu_{\delta}^{\pm}\)-expectation of each summand can be bounded using arguments similar to those used to prove Claim 13. In fact, replacing “R" with”L" in the statement of Claim 13, one can see that the corresponding inequality can be proved by repeating the proof of Claim 13 by replacing “R" with”L" everywhere. It thus now remains to show that \(\mu_{\delta}^{\pm}[\max_{x \in D_{L,\delta}} \operatorname{Inf}_{x} [\Psi_{\delta,L}^{({\epsilon}),\leq \ell}]]\) converges to \(0\) as \(\delta \to 0\). To do so, we are going to use the dominated convergence theorem. Indeed, it is easy to see that almost surely, \(\max_{x \in D_{L,\delta}} \operatorname{Inf}_{x} [\Psi_{\delta,L}^{({\epsilon}),\leq \ell}] \to 0\) as \(\delta \to 0\). On the other hand, since for any \(x \in D_{L,\delta}\), \(\operatorname{Inf}_{x} [\Psi_{\delta,L}^{({\epsilon}),\leq \ell}] \geq 0\), we have that \(\mu_{\delta}^{\pm}\)-almost surely, \[\begin{align} \max_{x \in D_{L,\delta}} \operatorname{Inf}_{x} [\Psi_{\delta,L}^{({\epsilon}),\leq \ell}] &\leq \sum_{x \in D_{L,\delta}} \sum_{I \subset D_{L,\delta}: \vert I \vert \leq \ell, x \in I} (1+{\epsilon})^{\vert I \vert} [\prod_{z \in I}\vartheta_{\delta}(z)^{2}]\mu_{\delta,L}^{+}[\prod_{z \in I} \sigma_{z}]^2\\ &= \ell \sum_{I \subset D_{L,\delta}: \vert I \vert \leq \ell} (1+{\epsilon})^{\vert I \vert} [\prod_{z \in I}\vartheta_{\delta}(z)^{2}]\mu_{\delta,L}^{+}[\prod_{z \in I} \sigma_{z}]^2. \end{align}\] Claim 13 (applied with no “right" terms) implies that the \(\mu_{\delta}^{\pm}\)-expectation of the random variable on the right-hand side above is uniformly bounded in \(\delta\). Therefore, the dominated convergence theorem yields that \(\mu_{\delta}^{\pm}[\max_{x \in D_{L,\delta}} \operatorname{Inf}_{x} [\Psi_{\delta,L}^{({\epsilon}),\leq \ell}]] \to 0\) as \(\delta \to 0\). As explained before, this concludes the proof of 45 . ◻

Proof. Proposition 17 is proved by using the explicit expression ?? for the Radon-Nikodym derivative \(F_{W,\lambda}^{(D,a,b)}\) of \(\nu_{W,\lambda}^{(D,a,b)}\) with respect to \(\nu_{\operatorname{SLE}_{3}}^{(D,a,b)}\). Indeed, since SLE\(_3\) is conformally invariant in law, it suffices to prove that \(F_{W,\lambda}^{(D,a,b)}\) transforms as claimed in Proposition 17. To do so, we first observe that by [20], \(\mathcal{Z}_{D}^{\pm,W,\lambda}\) has the same law as \(\mathcal{Z}_{\tilde{D}}^{\pm,W,\tilde{\lambda}}\) where \(\tilde{\lambda}\) is as in the statement of the proposition. Next, let us look at how the term \(S_{L}^{W,\lambda}\) transforms. If \(W\) denotes white noise on \(\mathbb{R}^2\), then the processes \((\int_{D_{L}}f(z)W(dz))_{f \in L^2(D)}\) and \((\int_{\tilde{D}_{L}}f(\phi^{-1}(z))\vert (\phi^{-1})'(z) \vert W(dz))_{f \in L^2(D)}\) have the same law, since there are both centered Gaussian processes with the same covariance. Here, we have set \(\tilde{D}_{L}:=\phi(D_{L})\). This extends to an equality in law for multiple integrals, that is, for any \(k \in \mathbb{N}\), \[\int_{D_{L}^k} f(z_{1},\dots,z_{k})\prod_{j=1}^{k}W(dz_j) \overset{(d)}{=}\int_{\tilde{D}_{L}^k} f(\phi^{-1}(z_{1}),\dots,\phi^{-1}(z_{k}))\prod_{j=1}^{k}\vert (\phi^{-1})'(z_j) \vert W(dz_j).\] It follows from this and the conformal covariance of the functions \((f_{L}^{(+,k)})_{k}\) in Theorem 3 that \(S_{L}^{W,\lambda}\) has the same law as \[1 + \sum_{k \geq 0} \frac{C_{\sigma}^k}{k!}\int_{\tilde{D}_{L}} [\prod_{j=1}^{k}\vert (\phi^{-1})'(z_j) \vert^{-\frac{1}{8}}] f_{\tilde{D}_{L}}^{(+,k)}(z_1,\dots,z_k) \prod_{j=1}^{k} \vert (\phi^{-1})'(z_j) \vert \lambda(\phi^{-1}(z_j)) W(dz_j).\] A similar equality in law holds for \(S_{R}^{W,\lambda}\). Since by conformal covariance of SLE\(_3\), \(\tilde{D}_{L}\) and \(\tilde{D}_{R}:=\phi(D_{R})\) have the same law as \(D_{L}\) and \(D_{R}\), the proof of Proposition 17 is complete. ◻

Proof of Theorem 2. The first step of the proof is to show the tightness of \[(\delta^{-1}\sum_{x \in \Lambda_{\delta}}h_x\mathbb{I}_{S_{\delta}(x)}, \nu_{\delta}^{\operatorname{oloop},h})_{\delta}.\] The topology in which tightness of \(\delta^{-1}\sum_{x \in \Lambda_{\delta}}h_x\mathbb{I}_{S_{\delta}(x)}\) holds is as described at the beginning of Proposition 9 while tightness of \((\nu_{\delta}^{\operatorname{oloop}})_{\delta}\) is going to be proved in the metric space \((\mathcal{X},d_{\mathcal{X}})\) described in Section 1.0.1.4.1. Tightness of \(\delta^{-1}\sum_{x \in \Lambda_{\delta}}h_x\mathbb{I}_{S_{\delta}(x)}\) in fact follows from standard arguments, and we refer the reader to [20] for details. As for tightness of \((\nu_{\delta}^{\operatorname{oloop},h})_{\delta}\), it is established by the following lemma.

Lemma 7 will be shown just below. With this lemma in hand, it now remains to prove the singularity of any convergent subsequence of \((\nu_{\delta}^{\operatorname{oloop},h})_{\delta}\). In order to do so, for \(\iota > 0\), define \(M = \lfloor \iota L \rfloor\) and partition \(\Lambda_{L}\) into \(2M \times 2M\) boxes \(B_{1}, \dots, B_{n}\). Denote by \(u_{i}\) the center of the box \(B_{i}\), so that \(B_{i} = \Lambda_{M}(u_i)\) and define the annulus \(A_{M}(u_i):= \Lambda_{M/2}(u_i) \setminus \Lambda_{M/4}(u_i)\). For \(i=1,\dots,n\), set \[X_{i} := \mathbb{I}_{\{\text{there exists an outermost loop in A_{M}(u_i)}\}}.\] For \(\tau \in \{0,1\}^n\), we then define the event \(\mathcal{E}_{\tau}:=\{(X_1,\dots, X_n) = \tau\}\). To prove singularity of the law of any of the subsequential limits of \((\nu_{\delta}^{\operatorname{oloop,h}})_{\delta}\) with respect to the law \(\nu_{\operatorname{CLE}_3}^{\Lambda_{L}}\) of CLE\(_3\) in \(\Lambda\), we will show the following proposition.

Proposition 18 will be proved just below. Observe that \(\frac{1}{2}\sum_{\tau \in \{0,1\}^n} \vert \nu_{\operatorname{CLE}_{3}}^{\Lambda}(\mathcal{E}_{\tau}) - \nu^{W}(\mathcal{E}_{\tau}) \vert\) is in fact the total variation distance between the law of \((X_1,\dots,X_n)\) under \(\nu_{\operatorname{CLE}_3}^{\Lambda}\) and the law of \((X_1,\dots,X_n)\) under \(\nu^{W}\). Thus, singularity of the law of any of the subsequential limits of \((\nu_{\delta}^{\operatorname{oloop,h}})_{\delta}\) with respect to \(\nu_{\operatorname{CLE}_3}^{\Lambda}\) follows directly from Proposition 18. This completes the proof of Theorem 2. ◻

Proof of Lemma 7. The proof is similar to what is done in [21]. As random measures probability measures on the complete and separable metric space \((\mathcal{X},d_{\mathcal{X}})\) (see Lemma 5), tightness of \((\nu_{\delta}^{\operatorname{oloop},h})_{\delta}\) will be established once we show that for any \({\epsilon}> 0\), there exists a compact set \(K \subset \mathcal{X}\) such that \[\label{condition-tightness} \sup_{\delta \in(0,1]} {\mathbb{E}}[\nu_{\delta}^{\operatorname{oloop},h}(K^c)] \leq {\epsilon}.\tag{47}\] This follows from the tightness criterion for random probability measures of [45]. So let us prove 47 . Fix \(\delta > 0\) and observe that \(\nu_{\delta}^{\operatorname{oloop},h}\) has the following Radon-Nikodym derivative with respect to \(\nu_{\delta}^{\operatorname{oloop}}\), the law of the outermost spin loops under \(\mu_{\delta}^{+}\): \[\frac{{\operatorname{d}}\nu_{\delta}^{\operatorname{oloop},h}}{{\operatorname{d}}\nu_{\delta}^{\operatorname{oloop}}}(\Gamma_{\delta}) = \frac{1}{\mathcal{Z}_{\delta}^{+,h}}\mu_{\delta}^{+}\bigg[\exp\bigg(\delta^{\frac{7}{8}}\sum_{x \in \Lambda_{\delta}} h_x\sigma_x\bigg) \big \vert \Gamma_{\delta} \bigg].\] Above, \(\Gamma_{\delta}\) denotes the collection of outermost spin loops. To prove 47 , we first observe that for any compact set \(K \subset \mathcal{X}\) and any \(\eta > 0\), \[{\mathbb{E}}[\mu_{\delta}^{+,h}[K^{c}]] \leq {\mathbb{P}}[\tilde{{\mathcal{Z}}}_{\delta}^{+,h} \leq \eta] + {\mathbb{E}}[\mathbb{I}_{\tilde{{\mathcal{Z}}}_{\delta}^{+,h} > \eta}\mu_{\delta}^{+,h}[K^c]]\] where \(\tilde{{\mathcal{Z}}}_{\delta}^{+,h}\) is defined as in 5 (with \(\lambda \equiv 1\) here). Since the limit in law of \(\tilde{{\mathcal{Z}}}_{\delta}^{+,h}\) is \({\mathbb{P}}\)-positive almost surely by [21], for any \({\epsilon}>0\), we can choose \(\eta >0\) such that \[\limsup_{\delta \to 0} {\mathbb{P}}[\tilde{{\mathcal{Z}}}_{\delta}^{+,h} \leq \eta] \leq \frac{1}{2}{\epsilon}.\] To establish 47 , it is therefore enough to upper bound \({\mathbb{E}}[\mu_{\delta}^{+,h}[K^c]\mathbb{I}_{\tilde{{\mathcal{Z}}}_{\delta}^{+,h} > \eta}]\). To do so, we note that for any compact set \(K \subset \mathcal{X}\), \[\begin{align} {\mathbb{E}}[\mu_{\delta}^{+,h}[K^c]\mathbb{I}_{\tilde{{\mathcal{Z}}}_{\delta}^{+,h} > \eta}] &= {\mathbb{E}}\bigg[\frac{\theta(\delta)}{\tilde{{\mathcal{Z}}}_{\delta}^{+,h}}\mathbb{I}_{\tilde{ {\mathcal{Z}}}_{\delta}^{+,h} > \eta} \mu_{\delta}^{+}\bigg[\mathbb{I}_{K^c}\mu_{\delta}^{+}\bigg[ \exp\bigg(\delta^{\frac{7}{8}}\sum_{x \in \Lambda_{\delta}} h_x\sigma_x\bigg) \vert \Gamma_{\delta} \bigg] \bigg] \bigg]\\ &= {\mathbb{E}}\bigg[\frac{\theta(\delta)}{\tilde{{\mathcal{Z}}}_{\delta}^{+,h}}\mathbb{I}_{\tilde{{\mathcal{Z}}}_{\delta}^{+,h} > \eta} \mu_{\delta}^{+}\bigg[\mu_{\delta}^{+}\bigg[\mathbb{I}_{K^c}\exp\bigg(\delta^{\frac{7}{8}}\sum_{x \in \Lambda_{\delta}} h_x\sigma_x\bigg) \vert \Gamma_{\delta} \bigg] \bigg] \bigg]\\ &\leq \frac{1}{\eta}\mu_{\delta}^{+}\bigg[ \mathbb{I}_{K^c}\theta(\delta){\mathbb{E}}\bigg[ \exp\bigg(\delta^{\frac{7}{8}}\sum_{x \in \Lambda_{\delta}}h_x\sigma_x\bigg) \bigg] \bigg] \end{align}\] where the second equality follows from the measurability of \(K^c\) with respect to \(\sigma(\Gamma_{\delta})\). Now, by [21], \(\theta(\delta){\mathbb{E}}[\exp(\delta^{\frac{7}{8}}\sum h_x\sigma_x)]\) converges to \(1\) as \(\delta \to 0\), uniformly in \((\sigma_x)_x\). Therefore, we obtain that \[{\mathbb{E}}[\mu_{\delta}^{+,h}[K^c]\mathbb{I}_{\tilde{{\mathcal{Z}}}_{\delta}^{+,h} > \eta}] \leq \frac{2}{\eta}\mu_{\delta}^{+}[K^c].\] A direct consequence of [25] (see Theorem 6 in Section 1.0.1.4.1) is that the law of \((\Gamma_{\delta})_{\delta}\) under \((\mu_{\delta}^{+})_{\delta}\) is tight in \((\mathcal{X}, d_{\mathcal{X}})\). Hence, for any \({\epsilon}> 0\), there exists a compact set \(K\) such that for any \(\delta > 0\), \(\mu_{\delta}^{+}[K^c]\leq \frac{1}{4}\eta{\epsilon}\). This establishes 47 and concludes the proof of Lemma 7. ◻

Proof of Proposition 18. To prove Proposition 18, let us first show that \[\begin{align} \label{observable-as-limit-discrete-observable} \frac{1}{2}{\mathbb{E}}\bigg[ \sum_{\tau \in \{0,1\}^n} \vert \nu_{\operatorname{CLE}_{3}}^{\Lambda}(\mathcal{E}_{\tau}) - \nu^{W}(\mathcal{E}_{\tau}) \vert \bigg] = \lim_{k \to \infty} \frac{1}{2}{\mathbb{E}}\bigg[ \sum_{\tau \in \{0,1\}^n} \vert \nu_{\delta_{k}}^{\operatorname{oloop}}(\mathcal{E}_{\tau}^{\delta_{k}}) - \nu_{\delta_{k}}^{\operatorname{oloop},h}(\mathcal{E}_{\tau}^{\delta_{k}}) \vert \bigg] \end{align}\tag{48}\] where \(\mathcal{E}_{\tau}^{\delta_{k}}\) is defined analogously to \(\mathcal{E}_{\tau}\), but for the collection of outermost loops \(\Gamma_{\delta_{k}}\) in \(D_{\delta_{k}}\) and with respect to the annuli \((A_{M,\delta_{k}}(u_i))_i:= (A_{M}(u_i) \cap \delta_{k} \mathbb{Z}^2)_i\). The equality 48 is in fact a consequence of the following lemmas.

The proofs of Lemma 8 and Lemma 9 are given just below. Lemma 8 shows continuity of the crossing events. This in particular implies that \(\nu_{\delta_{k}}^{\operatorname{oloop}}(\mathcal{E}_{\tau}^{\delta_{k}})\) converges to \(\nu_{\operatorname{CLE}_3}^{\Lambda}(\mathcal{E}_{\tau})\) as \(k \to \infty\). Indeed, for each \(i\), one can divide \(A_{M}^{\delta_k}(u_i)\) into eight boxes \((B_{M}^{\delta_{k},j}(u_i))_j\) such that for a loop in \(A_m^{\delta_k}(u_i)\) to touch \(\partial A_m^{\delta_k}(u_i)\) without crossing \(\partial A_m^{\delta_k}(u_i)\), there must be a box \(B_m^{\delta_k,j}(u_i)\) such that one of the two events denoted \(\mathcal{C}_{\operatorname{out}}^{\xi}(B_m^{\delta_k, j}(u_i))\setminus \mathcal{C}_{\operatorname{in}}^{\xi}(B_m^{\delta_k, j}(u_i))\), \(\xi = \pm\), in the proof of Lemma 8 occurs. But Lemma 8 shows that these events have \(\nu_{\delta_k}^{\operatorname{oloop}}\)-probability converging to \(0\) as \(k \to \infty\). Therefore, using the absolutely continuity of \({\mathbb{P}}[\nu^{W}[\cdot]]\) with respect to \(\nu_{\operatorname{CLE}_3}^{\Lambda}\) from Lemma 9, we obtain that for any \(\tau \in \{0,1\}^n\), \[\begin{align} \lim_{k \to \infty} &{\mathbb{E}}\big[\vert \nu_{\delta_{k}}^{\operatorname{oloop}}[(X_1^{\delta_k},\dots,X_n^{\delta_k}) = \tau \big] - \nu_{\delta_{k}}^{\operatorname{oloop},h}[(X_1^{\delta_k},\dots,X_n^{\delta_k}) = \tau]\vert \big] \\ =& {\mathbb{E}}[\vert \nu_{\operatorname{CLE_3}}^{\Lambda}[(X_1,\dots,X_n) = \tau] - \nu^{W}[(X_1,\dots,X_n) = \tau] \vert]. \end{align}\] Since the sum over \(\tau \in \{0,1\}^n\) in 48 is a finite sum, the limit 48 follows from the above equality.

With the equality 48 at hand, it now suffices to prove that there exists \(c > 0\) and \(\delta_0\) such that for any \(0 < \delta < \delta_0\) and any \(\iota > 0\), \[\frac{1}{2}{\mathbb{E}}\bigg[ \sum_{\tau \in \{0,1\}^n} \vert \nu_{\delta_{k}}^{\operatorname{oloop}}(\mathcal{E}_{\tau}^{\delta_{k}}) - \nu_{\delta_{k}}^{\operatorname{oloop},h}(\mathcal{E}_{\tau}^{\delta_{k}}) \vert \bigg] \geq 1-c^{-1}\iota^{c}.\] This inequality is established by Theorem 28 in Section 2.0.3 in Part 2. There, \(N\) plays the role of \(\delta_{k}^{-1}\) and the underlying domain \(\Lambda_{\delta}\) is denoted by \(\Lambda_{N}\) of side-length \(2N\). Besides, in Theorem 28, the laws of the loops under \(\mu_{\Lambda_{N},0}^{+}\) and under \(\mu_{\Lambda_{N},{\epsilon}h}^{+}\) are the same as under \(\nu_{\delta}^{\operatorname{oloop}}\) and \(\nu_{\delta}^{\operatorname{oloop},h}\). This completes the proof of Theorem 2. ◻

Proof of Lemma 8. We only discuss the event \(\mathcal{C}^+(Q)\), since the proof for \(\mathcal{C}^-(Q)\) is identical.

Let \(\mathcal{X}\) be the space of loop collections equipped with the metric \(d_{\mathcal{X}}\) defined in [25]. By the convergence theorem of [25] for critical Ising interfaces, along the subsequence \((\delta_k)\) under consideration we have \[\nu_{\delta_k}^{\mathrm{oloop}} \Longrightarrow \nu_{\mathrm{CLE}_3}^{\Lambda} \qquad\text{weakly on }(\mathcal{X},d_{\mathcal{X}}).\] It is therefore enough to prove that \(\mathcal{C}^+(Q)\) is a \(\nu_{\mathrm{CLE}_3}^{\Lambda}\)-continuity event, namely \[\nu_{\mathrm{CLE}_3}^{\Lambda}\bigl(\partial \mathcal{C}^+(Q)\bigr)=0.\] Indeed, the desired convergence then follows immediately from the Portmanteau theorem.

We now prove the continuity statement. For \(\eta>0\), define an inner approximation \(\mathcal{C}^+_{\mathrm{in},\eta}(Q)\) of \(\mathcal{C}^+(Q)\) as the event that there exists an oriented loop \(\gamma\) and a connected component \(\alpha\) of \(\gamma\cap Q\) which joins the two prescribed opposite sides of \(Q\), has the correct orientation, and satisfies the following \(\eta\)-robustness conditions: the two endpoints of \(\alpha\) lie at a distance of at least \(\eta\) from the four corners of \(Q\), and \(\alpha\), away from small \(\eta\)-neighbourhoods of its endpoints, stays at a distance of at least \(\eta\) from \(\partial Q\). Similarly, define an outer approximation \(\mathcal{C}^+_{\mathrm{out},\eta}(Q)\) as the event that there exists an oriented loop whose trace has an \(\eta\)-neighbourhood containing a path in \(Q\) joining the two prescribed opposite sides with the correct orientation.

Then \[\mathcal{C}^+_{\mathrm{in},\eta}(Q) \subset \mathcal{C}^+(Q) \subset \mathcal{C}^+_{\mathrm{out},\eta}(Q).\] Moreover, \(\mathcal{C}^+_{\mathrm{in},\eta}(Q)\) is open in \((\mathcal{X},d_{\mathcal{X}})\). Indeed, an \(\eta\)-robust crossing remains a crossing under sufficiently small uniform perturbations of the corresponding loop. Likewise, \(\mathcal{C}^+_{\mathrm{out},\eta}(Q)\) is closed, since an \(\eta\)-approximate crossing is preserved under limits of loop collections in \(d_{\mathcal{X}}\).

Hence, by weak convergence, \[\liminf_{k\to\infty} \nu_{\delta_k}^{\mathrm{oloop}}[\mathcal{C}^+(Q)] \ge \nu_{\mathrm{CLE}_3}^{\Lambda}[\mathcal{C}^+_{\mathrm{in},\eta}(Q)]\] and \[\limsup_{k\to\infty} \nu_{\delta_k}^{\mathrm{oloop}}[\mathcal{C}^+(Q)] \le \nu_{\mathrm{CLE}_3}^{\Lambda}[\mathcal{C}^+_{\mathrm{out},\eta}(Q)].\] It remains to let \(\eta\downarrow0\).

We claim that \[\lim_{\eta\downarrow0} \nu_{\mathrm{CLE}_3}^{\Lambda} \bigl[ \mathcal{C}^+_{\mathrm{out},\eta}(Q) \setminus \mathcal{C}^+_{\mathrm{in},\eta}(Q) \bigr] =0.\] Indeed, if a loop collection belongs to \(\mathcal{C}^+_{\mathrm{out},\eta}(Q)\setminus \mathcal{C}^+_{\mathrm{in},\eta}(Q)\), then some macroscopic CLE\(_3\) loop produces a crossing of \(Q\) which is not stable under an \(O(\eta)\)-perturbation. This can happen only if, at scale \(\eta\), one of the following degenerate configurations occurs near \(\partial Q\): the loop hits an \(O(\eta)\)-neighbourhood of a corner of \(Q\); the loop has a near-tangential contact with one of the sides of \(Q\); or two distinct pieces of the loop approach each other and the boundary of \(Q\) within distance \(O(\eta)\) in a way that changes the crossing topology.

Each of these alternatives implies a multi-arm event from scale \(\eta\) to a macroscopic scale. By the standard arm estimates for critical Ising interfaces, equivalently for their CLE\(_3\) scaling limit as obtained in Benoist–Hongler, the probability of such a degenerate event tends to zero as \(\eta\downarrow0\). Therefore \[\nu_{\mathrm{CLE}_3}^{\Lambda}\bigl(\partial \mathcal{C}^+(Q)\bigr)=0.\]

Letting \(\eta\downarrow0\) in the preceding liminf and limsup estimates gives \[\lim_{k\to\infty} \nu_{\delta_k}^{\mathrm{oloop}}[\mathcal{C}^+(Q)] = \nu_{\mathrm{CLE}_3}^{\Lambda}[\mathcal{C}^+(Q)].\] This proves the claim for \(\mathcal{C}^+(Q)\). The proof for \(\mathcal{C}^-(Q)\) is identical. ◻

Proof of Lemma 9. The proof is similar to that of the absolutely continuity part of [21]. Indeed, recall that for fixed \(\delta_k > 0\), we have that \[\frac{{\operatorname{d}}\nu_{\delta_{k}}^{\operatorname{oloop},h}}{{\operatorname{d}}\nu_{\delta_{k}}^{\operatorname{oloop}}}(\Gamma_{\delta_{k}}) = \frac{\theta(\delta_{k})}{\theta(\delta_{k})\mathcal{Z}_{\delta_{k}}^{+,h}}\mu_{\delta_{k}}^{+}\bigg[ \exp\bigg( \delta^{\frac{7}{8}}\sum_{x \in \Lambda_{\delta_{k}}} h_x\sigma_x \bigg) \vert \Gamma_{\delta_{k}}\bigg].\] Let \(A\) be an event in the Borel \(\sigma\)-algebra of \((\mathcal{X}, d_{\mathcal{X}})\). We have that \[\begin{align} {\mathbb{E}}\big[ \theta(\delta_k)\mathcal{Z}_{\delta_{k}}^{+,h} \nu_{\delta_{k}}^{\operatorname{oloop},h}[A] \big] &= \theta(\delta_{k}) {\mathbb{E}}\bigg[ \mu_{\delta_{k}}^{+}\bigg[ \mathbb{I}_{A} \mu_{\delta_{k}}^{+}\bigg[ \exp\bigg( \delta_{k}^{\frac{7}{8}}\sum_{x \in \Lambda_{\delta_{k}}}h_x\sigma_x \bigg) \vert \Gamma_{\delta_{k}}\bigg] \bigg] \bigg]\\ &= \theta(\delta_{k}) {\mathbb{E}}\bigg[ \mu_{\delta_{k}}^{+}\bigg[ \mathbb{I}_{A} \exp\bigg( \delta_{k}^{\frac{7}{8}}\sum_{x \in \Lambda_{\delta_{k}}}h_x\sigma_x \bigg) \bigg]\bigg] \\ &=\theta(\delta_{k}) \mu_{\delta}^{+}\bigg[ \mathbb{I}_{A} {\mathbb{E}}\big[ \exp\big( \delta_{k}^{\frac{7}{8}}\sum_{x \in \Lambda_{\delta_{k}}}h_x\sigma_x \big) \big] \bigg]\\ &=\theta(\delta_{k}) \mu_{\delta}^{+}\big[ \mathbb{I}_{A} \exp(\frac{\delta_k^{\frac{7}{4}}}{2} \vert \Lambda_{\delta_{k}} \vert) \big] \end{align}\] which proves that \[\lim_{\delta_{k} \to 0} {\mathbb{E}}\big[ \theta(\delta_k)\mathcal{Z}_{\delta_{k}}^{+,h} \nu_{\delta_{k}}^{\operatorname{oloop},h}[A] \big] = \nu_{\operatorname{CLE}_3}^{\Lambda}[A].\] On the other hand, by weak convergence of \(\nu_{\delta_{k}}^{\operatorname{oloop},h}\) to \(\nu^{W}\) and of \(\theta(\delta_{k})\mathcal{Z}_{\delta_{k}}^{+,h}\) to \(\mathcal{Z}_{\Lambda}^{+,W}\), we have that \[\lim_{\delta_{k} \to 0} {\mathbb{E}}\big[ \theta(\delta_k)\mathcal{Z}_{\delta_{k}}^{+,h} \nu_{\delta_{k}}^{\operatorname{oloop},h}[A] \big] = {\mathbb{E}}[\mathcal{Z}_{\Lambda}^{+,W}\nu^{W}[A]].\] Since \(\mathcal{Z}_{\Lambda}^{+,W} > 0\) \({\mathbb{P}}\)-almost surely by [21], for any measurable event \(F\) with \(\nu_{\operatorname{CLE}_3}^{\Lambda}[F]=0\), we can find a set \(E_F\) with \({\mathbb{P}}[W \in E_F] = 1\) such that \(\nu^{W}[F]=0\) for all \(W \in E_F\). Therefore, \({\mathbb{E}}[\nu^{W}[F]] = 0\), and thus \({\mathbb{E}}[\nu^{W}[\cdot]]\) is absolutely continuous with respect to \(\nu_{\operatorname{CLE}_3}^{\Lambda}\), which completes the proof of Lemma 9. Let us remark as in [21] that this does not contradict the claim that almost surely, \(\nu^{W}\) is singular with respect to \(\nu_{\operatorname{CLE}_3}^{\Lambda}\) because \(\cap E_{F}\) will not be a measurable event with positive probability. ◻

Proof of Lemma 10. The proof is similar to the proof of Lemma 7 and details are left to the reader. ◻

Proof of Proposition 19. Tightness of \((\Gamma_{\delta})_{\delta}\) under \((\mu_{\delta}^{+,h})_{\delta}\) in the topology described in Section 1.0.1.4.1 is established by Lemma 10. On the other hand, tightness of \((\delta^{-1}\sum_{x \in D_{\delta}}h_{x}\mathbb{I}_{S_{\delta}(x)})_{\delta}\) in the topology described in Lemma 6 follows from standard arguments, see [20]. We immediately get from this that \((\delta^{-1}\sum_{x \in D_{\delta}}h_{x}\mathbb{I}_{S_{\delta}(x)})_{\delta}, \nu_{\delta}^{\operatorname{loop},h})_{\delta}\) is tight. To prove Proposition 19, it thus only remains to prove the singularity of any subsequential limit. Let \((\nu_{\delta_k}^{\operatorname{loop},h})_{k}\) be a convergent subsequence. Below, for each \(k\), we abbreviate \(\Gamma_{\delta_k}\) by \(\Gamma_{k}\) and \(\mu_{\delta_{k}}^{+}\), respectively \(\mu_{\delta_{k}}^{+,h}\), by \(\mu_{k}^{+}\), respectively \(\mu_{k}^{+,h}\). We first observe that \(\mu_{\delta}^{+}\)-almost surely, for any \(x \in D_k\), \[\sigma_{x} = (-1)^{\vert \Gamma_{k}(x) \vert}\] where \(\Gamma_{k}(x)\) denotes the set of loops in \(\Gamma_k\) that surround \(x\). This equality also holds \({\mathbb{P}}\otimes \mu_{\delta}^{+,h}\)-almost surely. Once this observation has been made, singularity of the limiting law of \((\nu_{\delta_{k}}^{\operatorname{loop},h})_k\) with respect to nested CLE\(_3\) in \(D\) can be established using a similar strategy as that used to show that the law of the magnetization field of the \(2d\) near-critical RFIM is singular with respect to that of the magnetization field of the \(2d\) critical Ising model established in [21]. Given this similarity, let us now only briefly explain how the proof can be adapted, leaving the details to the reader. Below, we keep the same notations as in the proof of [21].

The families of random variables \((\Phi_{i,j}^{k,N})_{i,j}\) and \((\Phi_{i,j}^{k,N,m})_{i,j}\) are defined as in [21] but by expressing them using the random variables \(((-1)^{\vert \Gamma_{k}(x) \vert})_{x \in D_k}\). That is, for each \(k, N, m \in \mathbb{N}\), we set \[\begin{align} \Phi_{i,j}^{k,N} := \sum_{x \in B_{i,j}^{N}} \delta_{k}^{\frac{15}{8}} (-1)^{\vert \Gamma_{k}(x) \vert}, \quad \Phi_{i,j}^{k,N,m} := 2^{-m}\lfloor 2^m\Phi_{i,j}^{k,N} \rfloor \end{align}\] where \(B_{i,j}^{N} := (\frac{i-1}{N}, \frac{i}{N}) \times (\frac{j-1}{N}, \frac{j}{N})\). The families of random variables \((W_{i,j}^{k,N})_{i,j}\) and \((W_{i,j}^{k,N,m})_{i,j}\) are defined as in [21] and the \(\sigma\)-algebras \(\mathcal{F}_{N}^{k}\) and \(\mathcal{F}_{N,m}^{k}\) are constructed using the families \((\Phi_{i,j}^{k,N}, W_{i,j}^{k,N,m})_{i,j}\) and \((\Phi_{i,j}^{k,N,m}, W_{i,j}^{k,N,m})_{i,j}\), respectively. We then have that \[\begin{align} Q_{N}^{k} := {\mathbb{E}}\nu_{\delta_{k}}^{\operatorname{loop}} \bigg[ \frac{{\operatorname{d}}\nu_{\delta_k}^{\operatorname{loop},h}}{{\operatorname{d}}\nu_{\delta_k}^{\operatorname{loop}}}(\Gamma_{k}) \vert \mathcal{F}_{N}^{k}\bigg] &= {\mathbb{E}}\mu_{k}^{+} \bigg[ \frac{{\operatorname{d}}\mu_{k}^{+,h}}{{\operatorname{d}}\mu_{k}^{+}}(\Gamma_k)\vert \mathcal{F}_{N}^{k}\bigg]\\ &= {\mathbb{E}}\mu_{k}^{+} \bigg[ \frac{\theta(\delta_{k})}{\tilde{{\mathcal{Z}}}_{\delta_{k}}^{+,h}} \mu_{k}^{+}\bigg[ \exp\bigg(\delta_{k}^{\frac{7}{8}}\sum_{x \in D_k} h_x\sigma_x \bigg) \vert \Gamma_{k} \vert \bigg] \vert \mathcal{F}_{N}^{k} \bigg] \\ &= {\mathbb{E}}\mu_{k}^{+} \bigg[ \frac{\theta(\delta_{k})}{\tilde{{\mathcal{Z}}}_{\delta_{k}}^{+,h}} \mu_{k}^{+}\bigg[ \exp\bigg(\delta_{k}^{\frac{7}{8}}\sum_{x \in D_k} h_x(-1)^{\vert \Gamma_k(x) \vert} \bigg) \vert \Gamma_{k} \vert \bigg] \vert \mathcal{F}_{N}^{k} \bigg]\\ &= {\mathbb{E}}\mu_{k}^{+}\bigg[ \frac{\theta(\delta_{k})}{\tilde{{\mathcal{Z}}}_{\delta_{k}}^{+,h}} \exp\bigg(\delta_{k}^{\frac{7}{8}}\sum_{x \in D_k} h_x(-1)^{\vert \Gamma_k(x) \vert} \bigg) \vert \mathcal{F}_{N}^{k} \bigg]. \end{align}\] Similarly, \[\begin{align} Q_{N,m}^{k} &:= {\mathbb{E}}\nu_{\delta_{k}}^{\operatorname{loop}} \bigg[ \frac{{\operatorname{d}}\nu_{\delta_k}^{\operatorname{loop},h}}{{\operatorname{d}}\nu_{\delta_k}^{\operatorname{loop}}}(\Gamma_{k}) \vert \mathcal{F}_{N,m}^{k}\bigg]\\ &={\mathbb{E}}\mu_{k}^{+} \bigg[ \frac{{\operatorname{d}}\mu_{k}^{+,h}}{{\operatorname{d}}\mu_{k}^{+}} (\Gamma_k)\vert \mathcal{F}_{N,m}^{k}\bigg] = {\mathbb{E}}\mu_{k}^{+}\bigg[ \frac{\theta(\delta_{k})}{\tilde{{\mathcal{Z}}}_{\delta}^{+,h}} \exp\bigg(\delta^{\frac{7}{8}}\sum_{x \in D_k} h_x(-1)^{\vert \Gamma_k(x)\vert} \bigg) \vert \mathcal{F}_{N,m}^{k} \bigg]. \end{align}\] As in [21], to prove singularity of the limiting law of \((\Gamma_{k})_{k}\) under \((\mu_{k}^{+,h})_k\) with respect to nested CLE\(_3\) in \(D\), it suffices to show that \[\lim_{N \to \infty} \lim_{m \to \infty} \lim_{k \to \infty} {\mathbb{E}}\mu_{k}^{+}[(Q_{N,m}^{k})^{\frac{1}{2}}] = 0.\] This can be shown exactly as [21], by simply replacing the random variables \((\sigma_x)_x\) by \(((-1)^{\vert \Gamma_k(x) \vert})_x\) everywhere. We leave the details to the reader. ◻

Proof. Set \(M=\iota \mathsf r^{3\kappa}N\) and for a box \(\mathsf{C}\), write \(S_\gamma(\mathsf{C}):=\sum_{x\in\mathsf{C}}{\operatorname{dist}}(x,\gamma)^{-1/4}.\) We introduce the following partition procedure. We first partite \(\Lambda_N\) into \(M\)-boxes. For each box \(\mathsf{B}\), we have two possibilities

• If \(\sum_{x\in \mathsf{B}}{\operatorname{dist}}(x,\gamma)^{-\frac{1}{4}}\ge M^{\frac{7}{4}}\), then we divide \(\mathsf{B}\) into two small boxes \(\mathsf{C}_1\) and \(\mathsf{C}_2\) such that \[\max\Big\{S_\gamma(\mathsf{C}_1),S_\gamma(\mathsf{C}_2)\Big\}\le (\frac{1}{2}+\kappa)\cdot S_\gamma(\mathsf{B}).\label{amwpqdbs}\tag{65}\]

• If \(S_\gamma(\mathsf{B})< M^{\frac{7}{4}}\), then we turn to the next box.

We next explain how we achieve [eq:32definition32of32the32partition32procedure32in32sle] in each step. Suppose that a current box \(\mathsf{B}\) satisfies \(S_\gamma(\mathsf{B})\ge M^{7/4}.\) Assume without loss of generality that \[\mathsf{B}=[1,a]\times[1,b], \qquad 1\le a\le b\le M.\] For \(0\le c\le b\), set \[f_1(c)=S_\gamma(\mathsf{C}_1(c)), \qquad f_2(c)=S_\gamma(\mathsf{C}_2(c)).\] When \(c\) increases by one, only one horizontal slice is moved from \(\mathsf{C}_2\) to \(\mathsf{C}_1\). Since \({\operatorname{dist}}(\cdot,\gamma)\ge1\), the contribution of such a slice is at most \(M\). Hence \[|f_1(c+1)-f_1(c)|\le M, \qquad |f_2(c+1)-f_2(c)|\le M.\] As \(f_1(0)=0\) and \(f_2(b)=0\), the minimizing choice of \(d\) gives \[|f_1(d)-f_2(d)|\le 2M.\] Therefore, \[\max\{S_\gamma(\mathsf{C}_1(d)),S_\gamma(\mathsf{C}_2(d))\} \le \frac{1}{2}S_\gamma(\mathsf{B})+M.\] Since \(S_\gamma(\mathsf{B})\ge M^{7/4}\), and \(M\) is sufficiently large, we have \(M\le \kappa S_\gamma(\mathsf{B})\). Thus \[\max\{S_\gamma(\mathsf{C}_1(d)),S_\gamma(\mathsf{C}_2(d))\} \le \Big(\frac{1}{2}+\kappa\Big)S_\gamma(\mathsf{B}). \label{eq:32balanced32subdivision}\tag{66}\] Similarly, \[\min\{S_\gamma(\mathsf{C}_1(d)),S_\gamma(\mathsf{C}_2(d))\} \ge \Big(\frac{1}{2}-\kappa\Big)M^{7/4}. \label{eq:32lower32bound32for32child32box}\tag{67}\]

The procedure terminates after finitely many steps, since each subdivision strictly decreases the longer side length of the box being subdivided, and all side lengths are positive integers.

Let \(\mathfrak B\) be the collection of final boxes which are not original \(M\)-boxes. By 67 , every box in \(\mathfrak B\) has \(S_\gamma\)-mass at least \((\frac{1}{2}-\kappa)M^{7/4}\). Since the boxes are disjoint, \[|\mathfrak B| \le \frac{S(\gamma)}{(\frac{1}{2}-\kappa)M^{7/4}} \le C_1\iota^{-7/4}\mathsf r^{-2-\frac{21}{4}\kappa}.\] The number of original \(M\)-boxes is at most \(\Big(\frac{N}{M}\Big)^2 = \iota^{-2}\mathsf r^{-6\kappa}.\) Therefore, \[n \le |\mathfrak B|+\Big(\frac{N}{M}\Big)^2 \le C_1\iota^{-7/4}\mathsf r^{-2-\frac{21}{4}\kappa} + \iota^{-2}\mathsf r^{-6\kappa} \le \frac{C_2}{\iota^2 \mathsf r^{2+\frac{21}{4}\kappa}}.\] It remains to prove 63 . By the stopping rule, every final box \(\mathsf{B}_i\) satisfies \(S_\gamma(\mathsf{B}_i)<M^{7/4}.\) Moreover, since \(\mathsf{B}_i\) is contained in an \(M\)-box, \[\sum_{x\in\mathsf{B}_i}{\operatorname{dist}}(x,\partial\mathsf{B}_i)^{-1/4} \le C_3M^{7/4}.\] Using \[{\operatorname{dist}}(x,\gamma\cup\partial\mathsf{B}_i)^{-1/4} \le {\operatorname{dist}}(x,\gamma)^{-1/4} + {\operatorname{dist}}(x,\partial\mathsf{B}_i)^{-1/4},\] we obtain \[\sum_{x\in\mathsf{B}_i} {\operatorname{dist}}(x,\gamma\cup\partial\mathsf{B}_i)^{-1/4} \le (1+C_3)M^{7/4}.\] Choosing \(c_2\) large enough proves 63 . ◻

Proof. Equation 74 follows immediately from 73 . Hence, it suffices to establish 73 . The argument is similar to those in [20] and [22].

We proceed by induction on \(k\). The case \(k=1\) follows directly from 72 . Assume that 73 holds for \(k=j\); we now prove it for \(k=j+1\). Throughout the proof, we adopt the notation from [22].

Fix \(J=\{x_1,x_2,\cdots,x_j\}\), let \[\Omega_0=\{y\in\Lambda\setminus J:{\operatorname{dist}}(y,\partial\Lambda)={\operatorname{dist}}(y,J\cup S)\},\] \[\Omega_i=\{y\in(\Lambda\setminus J)\setminus(\cup_{r=1}^{i-1}\Omega_r) :{\operatorname{dist}}(y,x_i)={\operatorname{dist}}(y,J\cup S)\}, \text{ for }i=1,2,\cdots,j.\] Then \(\Omega_0,\cdots,\Omega_j\) is a partition of \(\Lambda\setminus J\). First we get from [22] and 72 that \[\label{eq:32induction32from32FS321} \sum_{x\in \Omega_0}F(S, J\cup\{x\})^{8s}\le 2^{3/2} \sum_{x\in \Omega_0}\frac{F(S, J)^{8s}}{({\operatorname{dist}}(x,S))^{s}}\le C_1tN^{2-s}F(S, J)^{8s}.\tag{75}\] Furthermore, for \(1\le i\le j\) and \(x\in \Omega_i\), we get from [22] that \[\label{eq:32k32point32function32on32i32region} \sum_{x\in \Omega_i}F(S, J\cup\{x\})^{8s}\le C_2 N^{s}\cdot |\Omega_i|^{1-s}F(S, J)^{8s}.\tag{76}\] With 75 and 76 , the rest of the proof is the same as [22]. ◻

Proof. The proof of Lemma 14 follows from [22]. In the sequel (see Lemma 19), we will establish a significantly stronger version that controls the spin average under certain events that are neither increasing nor decreasing. This extension goes beyond the results of [22] and requires a self-contained argument. Therefore, we omit the proof of Lemma 14 here. ◻

Proof. The proof for \(\mathsf{B}_i\) follows directly from combining Lemma 14 and [22], so we omit it here. Combining with 63 , we extend the result to \(\mathsf{B}_i^-\) and \(\mathsf{B}_i^+.\) ◻

Proof of Lemma 12. We first prove the case for \(\mathsf{a}_i^+\). Similar to 59 , we apply chaos expansion to \(\mathcal{Z}^{+}_{{\epsilon}H^{i}}(\Lambda^+)\) with respect to \(\mathcal{Z}^{+}_{{\epsilon}H^{i-1}}(\Lambda^+)\) and get \[\mathsf{a}_i^+=\sum_{k=1}^{\infty}\sum_{I\subset\mathsf{B}_i^+,|I|=k}\langle\sigma^I\rangle^+_{i-1}\prod_{x\in I}\tanh({\epsilon}\beta_c h_x).\label{eq:32chaos32expansion32for32ai32in32sle}\tag{78}\] Combining Lemmas 13 and 14 with 63 , we get that \[\begin{align} \sum_{|I|=k,I\subset\mathsf{B}_i^+}\big(\langle\sigma^I\rangle^+_{i-1}\big)^2\le C_{1}^{k} \sum_{|I|=k,I\subset\mathsf{B}_i^+}F(\partial\mathsf{B}_i^+,I)^2\le C_{2}^{k}M^{\frac{7k}{4}}.\label{eq:32second32moment32in32adding32external32field} \end{align}\tag{79}\] Combined with [22], we get for any \(k\ge 1\) that \[\begin{align} {\mathbb{P}}\Big(\big|\sum_{|I|=k,I\subset\mathsf{B}_i^+}\langle\sigma^I\rangle^+_{i-1}\prod_{x\in I}\tanh({\epsilon}\beta_c h_x)\big|\ge \frac{1}{2}({\epsilon}M^{\frac{7}{8}})^{\frac{k}{2}} \Big)\le C_{3}^{-1}\exp(-C_{3} \big({\epsilon}M^{\frac{7}{8}}\big)^{-\frac{1}{2}}\cdot (k!)^{\frac{3}{8k}}).\nonumber \end{align}\]Taking union bound over \(1\le k\le |\mathsf{B}_i^+|\) and noting that \(\sum_{k=1}^{\infty}\frac{1}{2} ({\epsilon}M^{\frac{7}{8}})^{\frac{k}{2}}\le \frac{1}{2}\), we get that \[\begin{align} &{\mathbb{P}}\Big(\big|\sum_{k=1}^{\infty}\sum_{|I|=k,I\subset\mathsf{B}_i^+}\langle\sigma^I\rangle^+_{i-1}\prod_{x\in I}\tanh({\epsilon}\beta_c h_x)\big|\ge \frac{1}{2} \Big)\nonumber\\\le~&\sum_{k=1}^{\infty} {\mathbb{P}}\Big(\big|\sum_{|I|=k,I\subset\mathsf{B}_i^+}\langle\sigma^I\rangle^+_{i-1}\prod_{x\in I}\tanh({\epsilon}\beta_c h_x)\big|\ge \frac{1}{2} ({\epsilon}M^{\frac{7}{8}})^{\frac{k}{2}} \Big)\nonumber\\\le~& \sum_{k=1}^{\infty}C_{3}^{-1}\exp\Big(-C_{3} \big({\epsilon}M^{\frac{7}{8}}\big)^{-\frac{1}{2}}\cdot (k!)^{\frac{3}{8k}}\Big)\le C_{4}^{-1}\exp\Big(-C_{4}\big({\epsilon}M^{\frac{7}{8}}\big)^{-\frac{1}{2}}\Big).\nonumber \end{align}\]Thus we finish the proof of 69 for \(\mathsf{a}_i^+\). Next, we turn to the proof of 70 . By the Cauchy-Schwarz inequality and 79 , we get that \[\begin{align} &{\mathbb{E}}\Big|\sum_{|I|=k,I\subset\mathsf{B}_i^+}\langle\sigma^I\rangle^+_{i-1}\prod_{x\in I}\tanh({\epsilon}\beta_c h_x)\Big|\le\sqrt{{\mathbb{E}}\Big(\sum_{|I|=k,I\subset\mathsf{B}_i^+}\langle\sigma^I\rangle^+_{i-1}\prod_{x\in I}\tanh({\epsilon}\beta_c h_x)\Big)^2}\nonumber\\=& \sqrt{\sum_{|I|=|J|=k,I,J\subset\mathsf{B}_i^+}\langle\sigma^I\rangle^+_{i-1}\langle\sigma^J\rangle^+_{i-1}\prod_{x\in I\cap J}{\mathbb{E}}\left(\tanh({\epsilon}\beta_c h_x)\right)^2\prod_{x\in I\Delta J}{\mathbb{E}}\tanh({\epsilon}\beta_c h_x)} \nonumber\\ \le&\sqrt{\sum_{|I|=k,I\subset\mathsf{B}_i^+}\left(\langle\sigma^I\rangle^+_{i-1}\right)^2\cdot \left({\epsilon}^2\beta_c^2\right)^k}\stackrel{\eqref{eq:32second32moment32in32adding32external32field}}{\le} C_{5}{\epsilon}^{k}M^{\frac{7k}{8}},\nonumber \end{align}\] where the second inequality uses the facts that the expectation is not 0 only if \(I\Delta J=\emptyset\) and \((\tanh u)^2\leq u^2\). Summing over \(1\le k\le |\mathsf{B}_i^+|\), we get that \[{\mathbb{E}}|\mathsf{a}_i^+|\le \sum_{k=1}^{\infty} {\mathbb{E}}\Big|\sum_{|I|=k,I\subset\mathsf{B}_i^+}\langle\sigma^I\rangle^+_{i-1}\prod_{x\in I}\tanh({\epsilon}\beta_c h_x)\Big|\le \sum_{k=1}^{\infty} C_{5}{\epsilon}^{k}M^{\frac{7k}{8}}\le \frac{1}{3}\nonumber\]where we chose \(\iota>0\) to be small enough in the last inequality.

The proof for \(\mathsf{a}_i^-\) is the same, so we omit further details. The proof for \(\mathsf{a}_i\) follows from the special case \(\gamma=\emptyset\). ◻

Proof of Proposition 25. Without loss of generality, we may assume \(\iota>0\) is small enough. We first prove ?? . Recalling the definition of \(\mathsf{a}_i^+, \mathsf{a}_i^-\) from 68 and applying 70 recursively, we get that \[{\mathbb{E}}(\max\{\mathsf Z^+,\mathsf Z^-\}) ={\mathbb{E}}\Big(\max\{\prod_{i=1}^n(1+\mathsf{a}_i^+),\prod_{i=1}^n(1+\mathsf{a}_i^-)\}\Big)\le 2^{n}.\nonumber\] Combined with 64 , we obtain \[2^n \leq \exp\big(c_2\ln 2\cdot \mathsf r^{-2-\frac{21}{4}\kappa}\cdot\iota^{-2}\big) \leq \exp\Big(c_2\ln 2\cdot \Big(\frac{S(\gamma)}{N^{7/4}}\Big)^{1+3\kappa} \cdot\iota^{-2}\Big),\] where the second inequality follows from the relation \(\mathsf r=\frac{N^{8/7}}{\sqrt{S(\gamma)}}\le 1.\) This completes the proof of ?? . Next, we turn to the proof of ?? . Let \(\mathcal{H}_i\) denote the collection of external fields such that \[\max\{|\mathsf{a}_i|,|\mathsf{a}_i^+|,|\mathsf{a}_i^-|\}\le \frac{1}{2}.\] In addition, let \(\mathcal{H}=\cap_{i=1}^n\mathcal{H}_i\). Then by 69 we get that \[{\mathbb{P}}(\mathcal{H})\ge 1- n\cdot c_3^{-1}\exp\Big(-c_3\big({\epsilon}M^{\frac{7}{8}}\big)^{-\frac{1}{2}}\Big)\ge 1- C_1^{-1}\exp\Big(-C_1\big(\frac{S(\gamma)}{N^{\frac{7}{4}}}\big)^{\frac{21\kappa}{32}}\iota^{-\frac{1}{4}}\Big)\label{eq:32good32external32field32given32interface32in32sle32continuity}\tag{80}\] where the last inequality follows from Lemma 11 and the fact that \(M=\iota \mathsf r^{3\kappa}N,\) \(\mathsf r=\frac{\sqrt{S(\gamma)}}{N^{\frac{7}{8}}}\). For any \(h\in \mathcal{H}\), since \(|\log(1+a)|\leq 2|a|\) for all \(|a|\leq \frac{1}{2},\) we have \[\begin{align} &\max\{|\log(\mathsf Z)|,|\log(\mathsf Z^+)|,|\log(\mathsf Z^-)|\}\nonumber\\\le~&\max\{\sum_{i=1}^n |\log(1+\mathsf{a}_i)|,\sum_{i=1}^n|\log(1+\mathsf{a}_i^+)|,\sum_{i=1}^n\log|(1+\mathsf{a}_i^-)|\}\nonumber\\\le~& 2\sum_{i=1}^n\big(\max\{|\mathsf{a}_i|,|\mathsf{a}_i^+|,|\mathsf{a}_i^-|\}\big) \le n.\nonumber\label{eq:32sle32RN32derivative32convert32to32ai32lower32bound} \end{align}\tag{81}\] Combined with 80 and Lemma 11, we complete the proof of ?? . The proof for ?? follows from the special case \(\gamma=\emptyset.\) ◻

Proof of Theorem 21, Inequality 56 . We first control the probability of \(\mathcal{A}_0(\iota)\). Note that \[\frac{\mu^{\pm}_{\Lambda_N,{\epsilon}h}(\gamma)}{\mu^{\pm}_{\Lambda_N,0}(\gamma)}=\frac{\mathsf Z^+\cdot \mathsf Z^-}{\mathsf Z}.\] Thus, if \((h,\gamma)\) satisfies \(\max\{|\log(\mathsf Z)|,|\log(\mathsf Z^+)|,|\log(\mathsf Z^-)|\}\le \iota^{-3.1}\), then \((h,\gamma)\in \mathcal{A}_0(\iota)\).

For any \(\gamma\) such that \(\frac{S(\gamma)}{N^{\frac{7}{4}}}\le \iota^{-1}\), we get from Proposition 25 that \[\begin{align} {\mathbb{P}}\Big[\max\{|\log(\mathsf Z)|,|\log(\mathsf Z^+)|,|\log(\mathsf Z^-)|\}\ge \iota^{-3.1}\Big]\le \exp\left(-C_1\iota^{-\frac{1}{4}}\right).\label{eq:32good32external32field32in32sle32pure32rn32derivative} \end{align}\tag{83}\] Combining with Lemma 15, we get that \[{\mathbb{P}}\otimes\mu^{\pm}_{\Lambda_N,0}\big(\mathcal{A}_0(\iota)^c\big)\le \mu^{\pm}_{\Lambda_N,0}\bigg(\frac{S(\gamma)}{N^{\frac{7}{4}}}> \iota^{-1}\bigg)+\exp\left(-C_1\iota^{-\frac{1}{4}}\right)\le \exp\left(-C_2\iota^{-\frac{1}{4}}\right).\label{eq:32product32measure32bound32of32rn32derivative32in32sle32pure}\tag{84}\] Note that for any \(h\notin \mathcal{H}_0(\iota)\), we have \[\mu^{\pm}_{\Lambda_N,0}(\mathcal{A}_0(\iota)^c\mid h)\ge \exp\left(-\iota^{-0.1}\right).\nonumber\]Applying Markov inequality to 84 , we get that \[\begin{align} {\mathbb{P}}\big(\mathcal{H}_0(\iota)\big)\ge 1-\frac{\exp\left(-C_2\iota^{-\frac{1}{4}}\right)}{\exp\left(-\iota^{-0.1}\right)}\ge 1-\exp\left(-C_3\iota^{-\frac{1}{4}}\right).\nonumber \end{align}\]Thus, we complete the proof of 56 . ◻

Proof. The idea is quite straightforward. Lemma 15 already provides a large deviation estimate for atypical interfaces under the measure \(\mu^{\pm}_{\Lambda_N,0}\). Therefore, to obtain the corresponding estimate under \(\mu^{\pm}_{\Lambda_N,{\epsilon}h}\), it suffices to control the Radon–Nikodym derivative \(\frac{\mu^{\pm}_{\Lambda_N,{\epsilon}h}(\gamma)}{\mu^{\pm}_{\Lambda_N,0}(\gamma)}\) for interfaces satisfying \(N^{-7/4}S(\gamma)\in (2^{k-1},2^k].\) Recall that \(\frac{\mu^{\pm}_{\Lambda_N,{\epsilon}h}(\gamma)}{\mu^{\pm}_{\Lambda_N,0}(\gamma)}=\frac{\mathsf Z^+(\gamma)\cdot \mathsf Z^-(\gamma)}{\mathsf Z}\) and that \(\mathsf Z\) is independent of \(\gamma.\)

We begin by establishing a lower bound for \(\mathsf Z\). Let \(\mathcal{H}^*(\iota)\) denote the collection of external field such that \(\mathsf Z\ge \exp(-c_1\iota^{-2}).\) For any \(\gamma\) such that \({S(\gamma)}{N^{-7/4}}\in (2^{k-1},2^k]\), we compute from Proposition 25 and the fact that \(\mathsf Z^+(\gamma)\) and \(\mathsf Z^-(\gamma)\) are independent under \({\mathbb{P}}\) that \[\begin{align} {\mathbb{E}}\Big[\frac{\mu^{\pm}_{\Lambda_N,{\epsilon}h}(\gamma)}{\mu^{\pm}_{\Lambda_N,0}(\gamma)}\mathbb{I}_{\mathcal{H}^*(\iota)}\Big]&\le {\mathbb{E}}[\mathsf Z^+(\gamma)]\cdot {\mathbb{E}}[\mathsf Z^-(\gamma)]\cdot \exp(c_1\iota^{-2})\nonumber\\\le~& \exp\Big( 2c_1(\frac{S(\gamma)}{N^{\frac{7}{4}}})^{1+3\kappa}\cdot \iota^{-2}\Big)\cdot \exp(c_1\iota^{-2})\le \exp\Big( C_12^{(1+3\kappa)k}\cdot \iota^{-2}\Big).\nonumber \end{align}\] For \(k\ge \lfloor-\log_2\iota\rfloor\), summing over \(\gamma\) and choosing \(\kappa_1=\kappa\) in Lemma 15, we obtain that \[\begin{align} \sum_{\gamma:\frac{S(\gamma)}{N^{7/4}}\in (2^{k-1},2^k]}{\mathbb{E}}\Big[\mu^{\pm}_{\Lambda_N,{\epsilon}h}(\gamma)\mathbb{I}_{\mathcal{H}^*(\iota)}\Big] &= \sum_{\gamma:\frac{S(\gamma)}{N^{7/4}}\in (2^{k-1},2^k]}{\mathbb{E}}\Big[\frac{\mu^{\pm}_{\Lambda_N,{\epsilon}h}(\gamma)}{\mu^{\pm}_{\Lambda_N,0}(\gamma)}\mathbb{I}_{\mathcal{H}^*(\iota)}\Big]\cdot \mu^{\pm}_{\Lambda_N,0}(\gamma)\nonumber\\ \le~& \sum_{\gamma:\frac{S(\gamma)}{N^{7/4}}\in (2^{k-1},2^k]}\exp\Big( C_12^{(1+3\kappa)k}\cdot \iota^{-2}\Big)\cdot \mu^{\pm}_{\Lambda_N,0}(\gamma)\nonumber\\ \le~& \exp\Big( C_12^{(1+3\kappa)k}\cdot \iota^{-2}\Big)\cdot C_2\exp\big(-C_2^{-1}2^{(4-\kappa)(k-1)}\big)\nonumber\\ \le~& C_3\exp\big(-C_3^{-1}2^{(4-\kappa)k} \big).\nonumber \end{align}\] Applying the Markov inequality then yields that \[\begin{align} {\mathbb{P}}\big(\mathcal{H}_k^{\mathrm{aty}}(\iota)^c\cap\mathcal{H}^*(\iota)\big) &\le \sum_{\gamma:\frac{S(\gamma)}{N^{7/4}}\in (2^{k-1},2^k]}{\mathbb{E}}\Big[\mu^{\pm}_{\Lambda_N,{\epsilon}h}(\gamma)\mathbb{I}_{\mathcal{H}^*(\iota)}\Big]\cdot e^{2^{k}}\nonumber\\\le~& C_3\exp\big(-C_3^{-1}2^{(4-\kappa)k}\big)\cdot e^{2^{k}}\le C_4\exp\big(-C_4^{-1}2^{(4-\kappa)k}\big).\label{eq:32good32external32for32the32numerator32in32interface32dimension32with32external32field} \end{align}\tag{85}\] Summing 85 over \(k\ge \lfloor-\log_2\iota\rfloor\) and combining with Proposition 25, we get that \[\begin{align} {\mathbb{P}}\big(\bigcap_{k=\lfloor-\log_2\iota\rfloor}^{\infty}\mathcal{H}^{\mathrm{aty}}_k(\iota)\big)&\ge 1-{\mathbb{P}}(\mathcal{H}^*(\iota)^c)-\sum_{k=\iota^{-1}}^{\infty}{\mathbb{P}}\big(\mathcal{H}_k^{\mathrm{aty}}(\iota)^c\cap\mathcal{H}^*(\iota)\big)\nonumber\\&\ge 1-\exp(-C_5\iota^{-\frac{1}{4}})-\sum_{k=\lfloor-\log_2\iota\rfloor}^{\infty}C_4\exp\big(-C_4^{-1}2^{(4-\kappa)k}\big)\nonumber\\&\ge 1-\exp(-C_6\iota^{-\frac{1}{4}}).\label{eq:32good32external32field32for32interface32with32external32field} \end{align}\tag{86}\] Thus we finish the proof of Lemma 16. ◻

Proof of Theorem 21, Inequality 57 . With Lemma 16 in place of Lemma 15, the proof is the same as that of 56 . ◻

Proof. Let \(\mathfrak{D}^k\) denote the collection of \(\frac{N}{2^k}\) boxes. For any \(\mathsf{D}\in\mathfrak{D}^k\), let \(\mathcal{E}_{\mathsf{D}}\) denote the event that the interface intersects \(\mathsf{D}\). Note that for any integer \(d\ge 0\), if \(\mathcal{E}_{\mathsf{D}}\) occurs, then there exists a plus crossing and a minus crossing in the annulus \(d\mathsf{D}\setminus \mathsf{D}\). Here \(d \mathsf{D}\) denotes the box with the same center as \(\mathsf{D}\) and has side length \(\frac{dN}{2^{k-1}}\) (note that an \(\frac{N}{2^k}\)-box is a translate of \([-\frac{N}{2^k}, \frac{N}{2^k}]^{ 2}\), and thus have side length \(\frac{N}{2^{k-1}}\)). By [56], we get that for any configuration \(\xi\) on \(\Lambda_N\setminus d\mathsf{D}\), \[\mu^\xi_{d\mathsf{D}, 0}(\mathcal{E}_\mathsf{D})\le C_1 (d+1)^{-\frac{5}{8}+\kappa_2}.\label{eq:32interface32two32arm32bound}\tag{90}\] Here we use the convention that \(\mu^\xi_{d\mathsf{D}, 0}(\mathcal{E}_\mathsf{D})=1\) if \(d=0\). Next, we compute the moments of \(T_k\). For any integer \(r> 0\), we compute that \[\langle (T_k)^r\rangle_{\Lambda_N,0}^{\pm}=\sum_{\mathsf{D}_i\in\mathfrak{D}^k,1\le i\le r}\mu^{ \pm}_{\Lambda_N, 0}(\cap_{i=1}^r\mathcal{E}_{\mathsf{D}_i}).\label{eq:32Tk32moment32expansion}\tag{91}\] Fix a sequence of boxes \(\mathsf{D}_1,\cdots, \mathsf{D}_{r}\in \mathfrak D^k\). For any \(1\le i\le r\), let \[d_i=\Big\lfloor\frac{\min\left\{\min_{j\neq i}\left\{{\operatorname{dist}}(\mathsf{D}_i,\mathsf{D}_j)\right\},{\operatorname{dist}}(\mathsf{D}_i,\partial\Lambda_N)\right\}}{\frac{N}{2^{k-1}}}\Big\rfloor\] be the box distance of \(\mathsf{D}_i\) to the boundary and other box in \(\{\mathsf{D}_1,\mathsf{D}_2,\cdots,\mathsf{D}_r\}\). Since \(d_i\) may vanish, we use \(d_i+1\) throughout. Then we get from DMP and 90 that \[\mu^\pm_{\Lambda_N, 0}(\cap_{i=1}^r\mathcal{E}_{\mathsf{D}_i})\le \prod_{i=1}^{r}C_1 (d_i+1)^{-\frac{5}{8}+\kappa_2}.\nonumber\]Plugging into 91 we get that\[\begin{align} \langle (T_k)^r\rangle_{\Lambda_N,0}^{\pm}\le C_1^r \sum_{\mathsf{D}_i\in\mathfrak{D}^k,1\le i\le r}\prod_{i=1}^{r}C_1 (d_i+1)^{-\frac{5}{8}+\kappa_2}.\label{eq:32moment32bound32for32Tk32in32interface321} \end{align}\tag{92}\] Note that \(\mathfrak D^k\) can be viewed as a coarse-grained graph that is isomorphic to \(\Lambda_{2^k}\). Moreover, \[\sum_{\mathsf D\in \mathfrak D_k}(\Big\lfloor\frac{{\operatorname{dist}}(\mathsf{D},\partial\Lambda_N)}{\frac{N}{2^{k-1}}}\Big\rfloor+1)^{-\frac{5}{8}+\kappa_2}\leq 64\times (2^k)^{\frac{11}{8}+\kappa_2}.\] Applying Lemma 13 with \(s=\frac{5}{8}-\kappa_2, \Lambda=\mathfrak D_k, S=\partial\mathfrak D_k, t=64\) and combining with 92 , we get that \[\begin{align} \langle (T_k)^r\rangle_{\Lambda_N,0}^{\pm}\le C_2^r 2^{(\frac{11}{8}+\kappa_2)rk}\cdot (r!)^{\frac{5}{8}-\kappa_2}.\nonumber \end{align}\] Combined with the Holder’s inequality, it yields that for any integer \(m\in\{1,2,3,4\}\), \[\begin{align} \langle (T_k)^{r+\frac{m}{5}}\rangle_{\Lambda_N,0}^{\pm}&\le \Big( \big(\langle (T_k)^{r}\rangle_{\Lambda_N,0}^{\pm}\big)^{5-m}\cdot \big(\langle (T_k)^{r+1}\rangle_{\Lambda_N,0}^{\pm}\big)^{m}\Big)^{\frac{1}{5}}\nonumber\\&\le C_2^{r+1} 2^{(\frac{11}{8}+\kappa_2)(r+\frac{m}{5})k}\cdot \big((r+1)!\big)^{\frac{5}{8}-\kappa_2}.\label{eq:32moment32bound32for32Tk32in32interface} \end{align}\tag{93}\]

Now we are ready to compute the exponential moment of \((tT_k)^{\frac{8}{5}}\) with \(t\le C\cdot 2^{-(\frac{11}{8}+\kappa_2)k}\) for some small enough \(C>0\). By 93 , we get that \[\begin{align} \langle \exp\big((tT_k)^{\frac{8}{5}}\big)\rangle_{\Lambda_N,0}^{\pm}=\sum_{r=0}^{\infty}\frac{t^{\frac{8r}{5}}}{r!}\cdot \langle (T_k)^{\frac{8r}{5}} \rangle_{\Lambda_N,0}^{\pm}\le \sum_{r=0}^{\infty}\frac{t^{\frac{8r}{5}}}{r!}\cdot C_2^{\lceil\frac{8r}{5}\rceil} 2^{(\frac{11}{8}+\kappa_2)\lceil\frac{8r}{5}\rceil k}\cdot\big( \lceil\frac{8r}{5}\rceil!\big)^{\frac{5}{8}-\kappa_2}.\nonumber \end{align}\]Combined with the fact that \(\lceil\frac{8r}{5}\rceil!\le (\frac{8r}{5})^{\frac{8r}{5}}\) and \(r!\ge (\frac{r}{e})^{r}\), it yields that \[\begin{align} \langle \exp\big((tT_k)^{\frac{8}{5}}\big)\rangle_{\Lambda_N,0}^{\pm}&\le \sum_{r=0}^{\infty}\frac{t^{\frac{8r}{5}}}{(\frac{r}{e})^{r}}\cdot C_2^{\lceil\frac{8r}{5}\rceil} 2^{(\frac{11}{8}+\kappa_2)\lceil\frac{8r}{5}\rceil k}\cdot (\frac{8r}{5})^{(1-\frac{8\kappa_2}{ 5})r}.\nonumber\\&\le \sum_{r=0}^{\infty} C_3^{\lceil\frac{8r}{5}\rceil} (t2^{(\frac{11}{8}+\kappa_2)k})^{\lceil\frac{8r}{5}\rceil} \le C_4\label{eq:32super32exponential32moment32for32Tk32in32interface} \end{align}\tag{94}\] where we used the fact that \(t\le C2^{-(\frac{11}{8}+\kappa_2)k}\) and chose \(C>0\) to be small enough in the last inequality. Now we are ready to prove 89 . Let \(t=C\cdot 2^{-(\frac{11}{8}+\kappa_2)k}\) and apply the Markov inequality, we get that \[\begin{align} \mu^{\pm}_{\Lambda_N,0}(T_k\ge \lambda\cdot 2^{\frac{7k}{4}-\kappa_2 k})&\le \langle \exp\big((tT_k)^{\frac{8}{5}}\big)\rangle_{\Lambda_N,0}^{\pm}\cdot \exp\big(-(t\cdot \lambda\cdot 2^{\frac{7k}{4}-\kappa_2 k})^{\frac{8}{5}} \big)\nonumber\\&= C_5\exp\big(-C_6 \lambda^{\frac{8}{5}} 2^{\frac{3-16\kappa_2}{5}k} \big). \end{align}\]Thus we complete the proof of Lemma 17. ◻

Proof of Lemma 15. By choosing \(c_7>0\) large enough, we can, without loss of generality, assume \(\lambda>0\) is large enough. Let \(\kappa_2=\frac{5\kappa_1}{64}\) and let \(k_0=\lfloor4\log_2(\lambda)\rfloor-100>0\). From the trivial bound that \(T_k\leq (2^{k})^2\) we get that \[\sum_{k=1}^{k_0}T_k\cdot (\frac{N}{2^k})^{\frac{7}{4}}\le \sum_{k=1}^{k_0} N^{\frac{7}{4}}\cdot 2^{\frac{k}{4}}\le N^{\frac{7}{4}}2^{\frac{k_0}{4}}\cdot \frac{1}{1-2^{-\frac{1}{4}}}\le N^{\frac{7}{4}}\frac{\lambda}{4}.\nonumber\]

Combining with 88 and the fact that \[\sum_{k=k_0+1} 2^{\frac{7k}{4}-\kappa_2 k+1}\cdot (\frac{N}{2^k})^{\frac{7}{4}}\le N^{\frac{7}{4}}\cdot \frac{1}{2C_1}\] for some \(C_1>0\), we get that \[\mu^{\pm}_{\Lambda_N,0 }\Big(S(\gamma)\ge N^{\frac{7}{4}}\lambda\Big)\le \sum_{k=k_0+1}^{\infty}\mu^{\pm}_{\Lambda_N,0}(T_k\ge C_1\lambda\cdot 2^{\frac{7k}{4}-\kappa_2 k}).\label{eq:32upper32bound32for32small32levels32in32large32devitaion32of32interface}\tag{95}\] Applying Lemma 17 to 95 , we get that \[\begin{align} \mu^{\pm}_{\Lambda_N,0 }\Big(S(\gamma)\ge N^{\frac{7}{4}}\lambda\Big)&\le \sum_{k=k_0+1}^{\infty}C_2\exp\big(-C_3 \lambda^{\frac{8}{5}} 2^{\frac{3-16\kappa_2}{5}k} \big)\le C_4\exp\big(-C_3 \lambda^{\frac{8}{5}} 2^{\frac{3-16\kappa_2}{5}k_0} \big)\nonumber\\&\le C_4\exp\big(-C_5 \lambda^{4-\frac{64\kappa_2}{5}})\nonumber \end{align}\]where we use the fact that \(k_0=\lfloor4\log_2(\lambda)\rfloor-100\) and \(\kappa_2=\frac{5\kappa_1}{64}\) in the last inequality. ◻

Proof of Lemma 18. Recall from Section 2.0.1.3 that, under the Edwards–Sokal coupling, the measure \(\mu^{+}_{\Lambda_N,0}\) corresponds to the wired FK measure \(\phi^{\mathrm{w}}_{\Lambda_N,0}\). Therefore, throughout this proof, whenever we work with FK configurations, we shall freely identify \(\mu^+_{\Lambda_N,0}\) with \(\phi^{\mathrm{w}}_{\Lambda_N,0}\).

The proof proceeds in two steps. We first use the FK measure to construct, in many annuli \(\Lambda_{M/2}(u_i)\setminus\Lambda_{M/4}(u_i)\), an FK circuit which is connected to \(\partial\Lambda_N\). Such a circuit will later serve as the plus circuit. Then, inside each region enclosed by such a circuit, we use the DMP, CBC, and RSW estimates to construct, with uniformly positive probability, an inner FK circuit which could be assigned the minus spin. This will produce a plus-minus circuit in \(\Lambda_{M/2}(u_i)\setminus\Lambda_{M/4}(u_i)\).

For any \(M\)-box \(\Lambda_M(u_i)\), let \(X_i^1\) be the indicator of the event that there exists an FK circuit in the annulus \(\Lambda_{M/2}(u_i)\setminus\Lambda_{5M/12}(u_i)\) which is connected to \(\partial\Lambda_N\). For any \(\tau\in\{0,1\}^n\), define \[\widetilde{\mathcal{E}}_\tau=\{\omega:X_i^1(\omega)=\tau_i,\;1\le i\le n\},\] where \(\omega\) denotes the FK edge configuration under \(\phi^{\mathrm{w}}_{\Lambda_N,0}\). We shall use the following claim, whose proof is postponed until the end of the proof of the lemma.

By Claim 31, with high probability there are at least \(C_1n^{\frac{15}{16}-\alpha}\) indices \(i\) for which \(X_i^1=1\). We next show that, for each such index, the annulus \(\Lambda_{M/2}(u_i)\setminus\Lambda_{M/4}(u_i)\) contains a plus-minus circuit with probability bounded below uniformly.

Indeed, explore the FK configuration on \(\Lambda_N\setminus\big(\cup_{i=1}^n\Lambda_{M/2}(u_i)\big)\), and then explore each annulus \(\Lambda_{M/2}(u_i)\setminus\Lambda_{5M/12}(u_i)\) until the values of \(X_i^1\) are determined. If \(X_i^1=1\), let \(\Lambda_i\) be the region enclosed by the outermost FK circuit in \(\Lambda_{M/2}(u_i)\setminus\Lambda_{5M/12}(u_i)\) which is connected to \(\partial\Lambda_N\). Since the outer boundary condition is plus, the boundary of \(\Lambda_i\) carries the plus boundary condition. It remains to construct, inside \(\Lambda_i\), a minus circuit surrounding \(\Lambda_{M/4}(u_i)\).

Let \(X_i^2\) be the indicator of the event that there exists a dual FK circuit in \(\Lambda_{5M/12}(u_i)\setminus\Lambda_{M/3}(u_i)\), and let \(X_i^3\) be the indicator of the event that there exists an FK circuit in \(\Lambda_{M/3}(u_i)\setminus\Lambda_{M/4}(u_i)\). Let \(\Xi^2\subset\{0,1\}^{E(\Lambda_{5M/12}(u_i)\setminus\Lambda_{M/3}(u_i))}\) denote the set of configurations on \(\Lambda_{5M/12}(u_i)\setminus\Lambda_{M/3}(u_i)\) for which \(X_i^2=1\). Then \[\sum_{\omega\in \Xi^2}\phi^{\mathrm{w}}_{\Lambda_i,0}(\omega) \ge \phi^{\mathrm{w}}_{\Lambda_{5M/12}(u_i)\setminus\Lambda_{M/3}(u_i),0}(X_i^2=1). \label{eq:32X232lower32bound32in32small32region}\tag{97}\] Combining the domain Markov property, comparison between boundary conditions, and RSW estimates, we obtain that \[\begin{align} \phi_{\Lambda_i,0}^{\mathrm{w}}(X_i^2X_i^3=1) &\stackrel{\mathrm{DMP}}{\ge} \sum_{\omega\in \Xi^2}\phi^{\mathrm{w}}_{\Lambda_i,0}(\omega)\, \phi^{\mathrm{w}}_{\Lambda_i,0}(X_i^3=1\mid \omega) \nonumber\\ &\stackrel{\mathrm{CBC}}{\ge} \sum_{\omega\in \Xi^2}\phi^{\mathrm{w}}_{\Lambda_i,0}(\omega)\, \phi^{\mathrm{f}}_{\Lambda_{M/3}(u_i)\setminus\Lambda_{M/4}(u_i),0}(X_i^3=1) \nonumber\\ &\stackrel{\eqref{eq:32X232lower32bound32in32small32region}}{\ge} \phi^{\mathrm{w}}_{\Lambda_{5M/12}(u_i)\setminus\Lambda_{M/3}(u_i),0}(X_i^2=1)\, \phi^{\mathrm{f}}_{\Lambda_{M/3}(u_i)\setminus\Lambda_{M/4}(u_i),0}(X_i^3=1) \nonumber\\ &\stackrel{\mathrm{RSW}}{\ge} C_2 . \label{eq:32FK32circuit32connected32to32boundary32in32small32region} \end{align}\tag{98}\]

On the event \(\{X_i^2X_i^3=1\}\), let \(\gamma\) be the outermost FK circuit in \(\Lambda_{M/3}(u_i)\setminus\Lambda_{M/4}(u_i)\), and let \(X_i^4\) be the event that \(\gamma\) is assigned the minus spin. Since \(X_i^2=1\), the circuit \(\gamma\) is not connected to \(\partial\Lambda_i\). Hence, under the Edwards–Sokal coupling, \[\phi_{\Lambda_i,0}^{\mathrm{w}}(X_i^4=1\mid X_i^2X_i^3=1)=\frac{1}{2} .\] Together with 98 , this gives \[\phi_{\Lambda_i,0}^{\mathrm{w}}(X_i^2X_i^3X_i^4=1)\ge C_3 . \label{eq:32FK32circuit32connected32to32boundary32in32small32region321}\tag{99}\]

If \(X_i^1X_i^2X_i^3X_i^4=1\), then there exists a plus circuit in \(\Lambda_{M/2}(u_i)\setminus\Lambda_{M/4}(u_i)\), which is connected to \(\partial\Lambda_N\), and there also exists a minus circuit in the same annulus. Hence, an outermost plus-minus circuit exists in \(\Lambda_{M/2}(u_i)\setminus\Lambda_{M/4}(u_i)\), and therefore \(X_i=1\).

Consequently, by the DMP and 99 , for any \(I\subset\{1,\ldots,n\}\), conditioned on \(X_i^1=1\) for all \(i\in I\), the family \(\{X_i\}_{i\in I}\) stochastically dominates a Bernoulli process with density \(C_3\). Choose \(C_4\le C_1C_3/10\). Combining Hoeffding’s inequality with Claim 31, we get \[\begin{align} \sum_{|\tau|< C_4n^{\frac{15}{16}-\alpha}} \phi^{\mathrm{w}}_{\Lambda_N,0}(\mathcal{E}_\tau) &\le \sum_{|\tau|< C_1n^{\frac{15}{16}-\alpha}} \phi^{\mathrm{w}}_{\Lambda_N,0}(\widetilde{\mathcal{E}}_\tau) +2\exp\big(-C_5n^{\frac{15}{16}-\alpha}\big) \nonumber\\ &\le C_6^{-1}\exp\big(-C_6n^{\frac{144\alpha-7}{112}}\big), \end{align}\] where the last inequality also used the fact that \(\frac{15}{16}-\alpha>\frac{144\alpha-7}{112}\) for \(\alpha<\frac{1}{4}.\) This completes the proof of Lemma 18. ◻

Proof of Claim 31. We choose \(\tilde{N}=n^{-\alpha^\prime} N\) with \(\alpha^\prime\in (0,1)\) to be defined later. Without loss of generality, we can assume \(N\in\tilde{N}\mathbb{Z}.\) Partition \(\Lambda_N\) into boxes of side length \(\widetilde{N}\). We call a box in this partition a boundary \(\widetilde{N}\)-box if it intersects \(\partial\Lambda_N\). We want to lower-bound the number of \(M\)-boxes \(\mathsf{B}_i\) in a boundary \(\tilde{N}\)-box such that \(X_i^1=1.\) For a boundary \(\tilde{N}\)-box \(\mathsf{D}=\Lambda_{\tilde{N}}(v)\), let \(I_{\mathsf{D}}\) denote the collection of indices \(i\) such that \(\Lambda_{M}(u_i)\subset \Lambda_{\tilde{N}/2}(v)\). For any \(i\in I_{\mathsf{D}}\) let \(\eta_{\mathsf{D}}\) be the boundary condition on \(\partial \mathsf{D}\) where it is wired on \(\partial \mathsf{D}\cap\partial\Lambda_N\) and is free on \(\partial \mathsf{D}\setminus\partial\Lambda_N\). In addition, for any \(i\in I_{\mathsf{D}}\), let \(\tilde{X}^1_i\) denote the event that there exists an FK circuit in \(\Lambda_{M/2}(u_i)\setminus\Lambda_{5M/12}(u_i)\) and let \(\tilde{X}^2_i\) denote the event that \(\partial\Lambda_{5M/12}(u_i)\) is connected to the boundary \(\partial \mathsf{D}\cap\partial\Lambda_N\). Note that \(\tilde{X}^1_i\cdot\tilde{X}^2_i\le X_i^1\), thus we get that \[\sum_{i\in I_{\mathsf{D}}}X_i^1\ge \sum_{i\in I_{\mathsf{D}}}\tilde{X}^1_i\cdot\tilde{X}^2_i.\label{eq:32FK32circuit32connected32to32boundary321}\tag{100}\] By FKG and RSW, we get that \[\phi_{\mathsf{D},0}^{\eta_{\mathsf{D}}}(\tilde{X}^1_i\cdot\tilde{X}^2_i=1)\ge \phi_{\mathsf{D},0}^{\eta_{\mathsf{D}}}(\tilde{X}^1_i=1)\cdot \phi_{\mathsf{D},0}^{\eta_{\mathsf{D}}}(\tilde{X}^2_i=1)\ge C_2(\frac{\tilde{N}}{M})^{-\frac{1}{8}}.\nonumber\label{eq:32FK32circuit32connected32to32boundary322}\tag{101}\] Summing over \(i\in I_{\mathsf{D}}\), we get that \[\langle\sum_{i\in I_{\mathsf{D}}}\tilde{X}^1_i\cdot\tilde{X}^2_i\rangle_{\mathsf{D},0}^{\eta_{\mathsf{D}}}\ge \frac{C_2}{4}(\frac{\tilde{N}}{M})^{\frac{15}{8}}.\label{eq:32FK32circuit32connected32to32boundary32final}\tag{102}\] In addition, by CBC and 50 , we get that \[\phi_{\mathsf{D},0}^{\eta_{\mathsf{D}}}(\tilde{X}^1_i\cdot\tilde{X}^2_i=1)\le \phi_{\mathsf{D},0}^{\eta_{\mathsf{D}}}(\tilde{X}^2_i=1)\le \phi^{\mathrm{w}}_{\Lambda_{\tilde{N}/2}}(\partial\Lambda_{\tilde{N}/2}\leftrightarrow\partial \Lambda_{M})\le C_3(\frac{\tilde{N}}{M})^{-\frac{1}{8}}.\nonumber\] Summing over \(i,j\in I_{\mathsf{D}}\), we compute that \[\Big\langle\big(\sum_{i\in I_{\mathsf{D}}}\tilde{X}^1_i\cdot\tilde{X}^2_i\big)^2\Big\rangle_{\mathsf{D},0}^{\eta_{\mathsf{D}}}=\sum_{i,j\in I_{\mathsf{D}}}\phi_{\mathsf{D},0}^{\eta_{\mathsf{D}}}(\tilde{X}^1_i\cdot\tilde{X}^2_i\cdot \tilde{X}^1_j\cdot\tilde{X}^2_j=1)\le C_3(\frac{\tilde{N}}{M})^{\frac{31}{8}}.\label{eq:32FK32circuit32connected32to32boundary323}\tag{103}\] Let \(\Phi\) be the product measure of \(\phi_{\mathsf{D},0}^{\eta_{\mathsf{D}}}\) over all the boundary \(\tilde{N}\)-box. By 102 , 103 and Bernstein inequality, we get that \[\begin{align} \Phi\Big(\sum_{i=1}^n\tilde{X}^1_i\cdot\tilde{X}^2_i\le \frac{C_2}{2}\cdot(\frac{\tilde{N}}{M})^{\frac{15}{8}}\cdot\frac{N}{\tilde{N}}\Big)&\le \exp\Big(- \frac{C_4(\frac{\tilde{N}}{M})^{\frac{15}{4}}\cdot(\frac{N}{\tilde{N}})^2}{C_5(\frac{\tilde{N}}{M})^{\frac{31}{8}}\cdot \frac{N}{\tilde{N}}+C_6(\frac{\tilde{N}}{M})^{\frac{15}{8}}\cdot\frac{N}{\tilde{N}}\cdot (\frac{\tilde{N}}{M})^{2}}\Big)\nonumber\\&\le\exp\Big(-C_8 n^{\frac{18\alpha^\prime-1}{16}}\Big).\nonumber \end{align}\] Combined with the fact that \(\phi_{\Lambda_N,0}^{\mathrm{w}}\) stochastically dominates \(\Phi\) and \(\tilde{X}^1_i\cdot\tilde{X}^2_i\) is increasing, it yields that \[\phi_{\Lambda_N,0}^{\mathrm{w}}\Big(\sum_{i=1}^n\tilde{X}^1_i\cdot\tilde{X}^2_i\le \frac{C_2}{2}\cdot(\frac{\tilde{N}}{M})^{\frac{15}{8}}\cdot\frac{N}{\tilde{N}}\Big)\le \exp\Big(-C_3 n^{\frac{18\alpha^\prime-1}{16}}\Big).\label{eq:32good32circuit32boundary32box32number32deviation}\tag{104}\] Note that \((\frac{\tilde{N}}{M})^{\frac{15}{8}}\cdot\frac{N}{\tilde{N}}=n^{\frac{15-14\alpha^\prime}{16}}\), thus we can choose \(\alpha^\prime=\frac{8\alpha}{7}\). ◻

Proof. The proof follows directly from combining Lemma 19 and [22], so we omit it here. ◻

Proof of Proposition 32. Let \(\mathcal{F}_i\) be the \(\sigma\)-field generated by the external fields on \(\cup_{j=1}^i \mathsf{B}_j\). Similar to 59 , we first perform the chaos expansion with respect to \(\mu_{\Lambda_N,{\epsilon}H^{i-1}}^+\): \[\mathsf{a}_i=\frac{\sum_{k=1}^{\infty}\sum_{I\subset \mathsf{B}_i,|I|=k}\big(\langle\sigma^I \mathbb{I}_{\mathcal{E}_\tau}\rangle_{i-1}-\langle\sigma^I \rangle_{i-1}\cdot \langle \mathbb{I}_{\mathcal{E}_\tau}\rangle_{i-1}\big)\cdot\prod_{x\in I}\tanh({\epsilon}\beta_c h_x)}{\mu^+_{\Lambda_N,{\epsilon}H^{i-1}}(\mathcal{E}_\tau)\cdot[1+ \sum_{k=1}^{\infty}\sum_{I\subset \mathsf{B}_i,|I|=k}\langle\sigma^I \rangle_{i-1}\cdot\prod_{x\in I}\tanh({\epsilon}\beta_c h_x)]}.\label{eq:32chaos32expansion32for32cle32singularity}\tag{106}\] We next decompose the right-hand side of 106 into several terms. Writing \(H={\mathbb{E}}\tanh^2({\epsilon}\beta_c h_x)\), we define \[\begin{align} \mathsf{b}_i&=\sum_{x\in \mathsf{B}_i}\big(\langle\sigma_x \mid {\mathcal{E}_\tau}\rangle_{i-1}-\langle\sigma_x \rangle_{i-1}\big)\cdot {\epsilon}\beta_c h_x,\tag{107}\\ \mathsf{c}_{i,1}&=\sum_{\{x, y\}\subset \mathsf{B}_i}\big(\langle\sigma_x\sigma_y \mid {\mathcal{E}_\tau}\rangle_{i-1}-\langle\sigma_x\sigma_y \rangle_{i-1}\big)\cdot \tanh({\epsilon}\beta_c h_x)\tanh({\epsilon}\beta_c h_y),\tag{108}\\ \mathsf{c}_{i,2}&=\sum_{x, y\in \mathsf{B}_i}\big(\langle\sigma_x \mid {\mathcal{E}_\tau}\rangle_{i-1}-\langle\sigma_x \rangle_{i-1}\big)\cdot\langle\sigma_y \rangle_{i-1}\cdot \big(-\tanh({\epsilon}\beta_c h_x)\tanh({\epsilon}\beta_c h_y)+H\cdot \mathbb{I}_{x=y}\big) ,\tag{109}\\ \mathsf{d}_i&=-\sum_{x\in \mathsf{B}_i}\big(\langle\sigma_x \mid {\mathcal{E}_\tau}\rangle_{i-1}-\langle\sigma_x \rangle_{i-1}\big)\cdot\langle\sigma_x \rangle_{i-1}\cdot H .\tag{110} \end{align}\] We will prove [item:32bi], [item:32ci], [item:32di] and [item:32ai-bi-ci] for \(\mathsf{b}_i\), \(\mathsf{c}_i=\mathsf{c}_{i,1}+\mathsf{c}_{i,2}\) and \(\mathsf{d}_i\) in the following Lemmas. ◻

Proof. Recall that \(u_i\) is the center of the \(M\)-box \(\mathsf{B}_i\). Conditioned on \(\mathcal{F}_{i-1}\), it is clear that \(\mathsf{b}_i\) is a mean 0 Gaussian variable, so it suffices to control its variance \[{\mathbb{E}}\mathsf{b}_i^2={\epsilon}^2\beta_c^2\sum_{x\in \mathsf{B}_i}\big(\langle\sigma_x \mid{\mathcal{E}_\tau}\rangle_{i-1}-\langle\sigma_x \rangle_{i-1}\big)^2.\label{eq:32second32moment32of32bi32chaos32expansion}\tag{111}\] The upper bound of the right-hand side of 111 follows directly from combining Lemmas 19 and 14 with Lemma 13. Therefore, it suffices to derive the lower bound for the right-hand side of 111 . From now on, we assume that \(\tau_i=1\). By DMP and CBC, we get that \[|\langle\sigma_x \rangle_{i-1}|\le \langle\sigma_x\rangle_{\Lambda_{M}(u_i),0}^+ .\label{eq:32chaos32expansion32first32moment32pure32bound}\tag{112}\] Next we want to lower-bound \(|\langle\sigma_x \mid{\mathcal{E}_\tau}\rangle_{i-1}|\). The rough idea is that, when \(\tau_i=1\), the existence of a plus-minus interface in \(\Lambda_{M/2}(u_i)\setminus\Lambda_{M/4}(u_i)\) leads to a substantial effect on the spin average in \(\Lambda_{M/4}(u_i)\). More precisely, let \(\gamma^i\) be a plus-minus circuit in \(\mathsf{B}_i\) and let \(\Gamma^i\) be the region enclosed by \(\gamma^i\). Let \(\sigma^i\) be a configuration on \(\Lambda_N\setminus\Gamma^i\). We say a configuration \(\sigma\in\{-1,1\}^{\Lambda_N}\) is an extension of \(\sigma^i\) if \(\sigma_x=\sigma^i_x,~\forall x\in \Lambda_N\setminus\Gamma^i\). Next, we show that the events \(X_j\) are measurable under \(\sigma^i\). Let \(\mathcal{C}\) be the boundary plus cluster in \(\sigma\) and let \(\mathcal{C}^i\) be the boundary plus cluster in \(\sigma^i\). Note that if \(\gamma^i\) is an outermost interface, then for any extension \(\sigma\) of \(\sigma^i\), we have \(\mathcal{C}|_{\Lambda_N\setminus\Gamma^i}=\mathcal{C}^i\) and thus \(X_j(\sigma)=X_j(\sigma^i)\) for any \(1\le j\le n\). Thus we define a pair \((\gamma^i,\sigma^i)\) to be valid if \(\gamma^i\) is an outermost interface and for any \(1\le j\le n\), \(X_j(\sigma^i)=\tau_j\). Let \(\Sigma_{\tau,i}\) denote the collection of valid pairs \((\gamma^i,\sigma^i)\). Note that \(H^{i-1}_x=0\) for \(x\in \mathsf{B}_i\), then we calculate for any \(x\in \Lambda_{M/4}(u_i)\) \[\langle\sigma_x \mid{\mathcal{E}_\tau}\rangle_{i-1}=\sum_{(\gamma^i,\sigma^i)\in \Sigma_{\tau,i}}\frac{\mu^+_{\Lambda_N,{\epsilon}H^{i-1}}(\sigma^i)}{\mu^+_{\Lambda_N,{\epsilon}H^{i-1}}(\mathcal{E}_\tau)}\cdot \langle\sigma_x \rangle^-_{\Gamma^i,0}.\label{eq:32DMP32for32outermost32loop32bi}\tag{113}\] Recall the definition of \(X_i\), we get from \(X_i=1\) that \(\Gamma^i\subset\Lambda_{M/2}(u_i)\). Thus by CBC, we get that \(-\langle\sigma_x \rangle^-_{\Gamma^i,0}\ge \langle\sigma_x \rangle^+_{\Lambda_{M/2}(u_i),0}\ge 0\). Plugging into 113 , we get that for any \(x\in \Lambda_{M/4}(u_i)\) \[|\langle\sigma_x \mid{\mathcal{E}_\tau}\rangle_{i-1}|\ge \sum_{(\gamma^i,\sigma^i)\in \Sigma_{\tau,i}}\frac{\mu^+_{\Lambda_N,{\epsilon}H^{i-1}}(\sigma^i)}{\mu^+_{\Lambda_N,{\epsilon}H^{i-1}}(\mathcal{E}_\tau)}\cdot \langle\sigma_x \rangle^+_{\Lambda_{M/2}(u_i),0}=\langle\sigma_x \rangle^+_{\Lambda_{M/2}(u_i),0}\label{eq:32chaos32expansion32first32moment32conditional32bound}\tag{114}\] where we used the equality \(\sum_{(\gamma^i,\sigma^i)\in \Sigma_{\tau,i}}\mu^+_{\Lambda_N,{\epsilon}H^{i-1}}(\sigma^i)=\mu^+_{\Lambda_N,{\epsilon}H^{i-1}}(\mathcal{E}_\tau)\) in the last equation. Combining 112 with 114 , we get that for any \(x\in \Lambda_{M/4}(u_i)\), \[\big|\langle\sigma_x \mid{\mathcal{E}_\tau}\rangle_{i-1}-\langle\sigma_x \rangle_{i-1}\big|\ge\big|\langle\sigma_x \mid{\mathcal{E}_\tau}\rangle_{i-1}\big|-\big|\langle\sigma_x \rangle_{i-1}\big|\ge \langle\sigma_x \rangle^+_{\Lambda_{M/2}(u_i),0}-\langle\sigma_x \rangle^+_{\Lambda_{M}(u_i),0}\label{eq:32one32arm32increment32decay321}\tag{115}\] Recalling the Edwards–Sokal coupling, we obtain that \[\langle\sigma_x \rangle^+_{\Lambda_{M/2}(u_i),0}-\langle\sigma_x \rangle^+_{\Lambda_{M}(u_i),0}=\phi^\mathrm{w}_{\Lambda_{M/2}(u_i),0}\big(x\leftrightarrow\partial\Lambda_{M/2}(u_i)\big)-\phi^\mathrm{w}_{\Lambda_{M}(u_i),0}\big(x\leftrightarrow\partial\Lambda_{M}(u_i)\big).\label{eq:32one32arm32increment32decay322}\tag{116}\] For any region \(\Lambda_{M}(u_i)\setminus\Lambda_{M/2}(u_i)\subset\Lambda\), let \(\phi^{\mathrm{w},\mathrm{w}}_{\Lambda,0}\) denote the measure with wired boundary condition on \(\partial\Lambda_{M}(u_i)\) and \(\partial\Lambda_{M/2}(u_i)\). By CBC, we get that \[\begin{align} &\phi^\mathrm{w}_{\Lambda_{M}(u_i),0}\big(x\leftrightarrow\partial\Lambda_{M}(u_i)\big)\le \phi^{\mathrm{w},\mathrm{w}}_{\Lambda_{M}(u_i),0}\big(x\leftrightarrow\partial\Lambda_{M}(u_i)\big)\nonumber\\ \le~& \phi^\mathrm{w}_{\Lambda_{M/2}(u_i),0}\big(x\leftrightarrow\partial\Lambda_{M/2}(u_i)\big)\cdot \phi^{\mathrm{w},\mathrm{w}}_{\Lambda_{M}(u_i)\setminus\Lambda_{M/2}(u_i),0}(\partial\Lambda_{M}(u_i)\leftrightarrow\partial\Lambda_{M/2}(u_i))\nonumber\\ \le~& (1-C_1)\phi^\mathrm{w}_{\Lambda_{M/2}(u_i),0}\big(x\leftrightarrow\partial\Lambda_{M/2}(u_i)\big)\label{eq:32one32arm32increment32decay323} \end{align}\tag{117}\] for some \(C_1\in(0,1)\), where we use the RSW estimate in the last inequality. Combining 115 , 116 and 117 with 50 yields that \[\big|\langle\sigma_x \mid{\mathcal{E}_\tau}\rangle_{i-1}-\langle\sigma_x \rangle_{i-1}\big|\ge C_1 \phi^\mathrm{w}_{\Lambda_{M/2}(u_i),0}\big(x\leftrightarrow\partial\Lambda_{M/2}(u_i)\big)\stackrel{\eqref{eq:32FK32one32arm32event32exponent}}{\ge }C_2 M^{-\frac{1}{8}}. \label{eq:32one32arm32increment32decay32final}\tag{118}\] Plugging 118 into 111 , we get the desired lower bound for \({\mathbb{E}}b_i^2\). ◻

Proof. Note that conditioned on \(\mathcal{F}_{j}\) for \(j<i\), both \(\mathsf{c}_{i,1}\) and \(\mathsf{c}_{i,2}\) have mean \(0\). Then we have that, for any \(j<i,\) \[{\mathbb{E}}(\mathsf{c}_i\mathsf{c}_j)={\mathbb{E}}({\mathbb{E}}(\mathsf{c}_i\mathsf{c}_j\mid \mathcal{F}_j))={\mathbb{E}}({\mathbb{E}}(\mathsf{c}_i\mid \mathcal{F}_j)\mathsf{c}_j)=0.\] As a consequence, \({\mathbb{E}}\big(\sum_{i=1}^{n}\mathsf{c}_i\big)^2= {\mathbb{E}}\sum_{i=1}^{n}\big(\mathsf{c}_i\big)^2.\)

We first control the second moment of \(\sum_{i=1}^{n}\mathsf{c}_{i,1}\). Note that \(H={\mathbb{E}}\tanh^2({\epsilon}\beta_c h_x)\le C_1{\epsilon}^2\) and thus \[\begin{align} {\mathbb{E}}\big(\sum_{i=1}^{n}\mathsf{c}_{i,1}\big)^2=\sum_{i=1}^{n}{\mathbb{E}}\mathsf{c}_{i,1}^2\le C_1^{ 2}{\epsilon}^4\sum_{i=1}^{n}\sum_{\{x, y\}\subset \mathsf{B}_i}\Big(\langle\sigma_x\sigma_y\mid \mathcal{E}_\tau\rangle_{i-1}-\langle\sigma_x\sigma_y \rangle_{i-1}\rangle_{i-1}\Big)^2.\label{eq:32c1132upper32bound32expansion} \end{align}\tag{119}\] Applying Lemmas 19 and 14 with Lemma 13, we get that \[\sum_{\{x, y\}\subset \mathsf{B}_i}\Big(\langle\sigma_x\sigma_y\mid \mathcal{E}_\tau\rangle_{i-1}-\langle\sigma_x\sigma_y \rangle_{i-1}\Big)^2\le \sum_{\{x, y\}\subset \mathsf{B}_i} 4c_5^2 F(\bar{\partial}\mathsf{B}_i,\{x,y\})^2\le C_2M^{\frac{7}{2}}.\label{eq:32c1132upper32bound32for32each32box}\tag{120}\] Summing 120 over \(1\le i\le n\) and combining with 119 , we obtain that \[\begin{align} {\mathbb{E}}\big(\sum_{i=1}^{n}\mathsf{c}_{i,1}\big)^2\le \sum_{i=1}^{n}C_1^2C_2{\epsilon}^4 M^{\frac{7}{2}}=C_1^2C_2{\epsilon}^4 M^{\frac{7}{2}}n .\label{eq:32c1132upper32bound32final} \end{align}\tag{121}\] We then turn to the upper bound on \({\mathbb{E}}\big(\sum_{i=1}^{n}\mathsf{c}_{i,2}\big)^2\). We calculate \[\begin{align} {\mathbb{E}}\big(\sum_{i=1}^{n}\mathsf{c}_{i,2}\big)^2=\sum_{i=1}^{n}{\mathbb{E}}(\mathsf{c}_{i,2})^2\le C_1^{ 2}{\epsilon}^4\sum_{i=1}^{n}\sum_{x\neq y\in \mathsf{B}_i}\Big(\big(\langle\sigma_x\mid \mathcal{E}_\tau\rangle_{i-1}-\langle\sigma_x \rangle_{i-1}\big)\cdot\langle\sigma_y \rangle_{i-1}\Big)^2.\label{eq:32c1232upper32bound32expansion} \end{align}\tag{122}\] Applying Lemmas 19 and 14 with 13, we get that \[\begin{align} &\sum_{x\neq y\in \mathsf{B}_i}\Big(\big(\langle\sigma_x\mid \mathcal{E}_\tau\rangle_{i-1}-\langle\sigma_x \rangle_{i-1}\big)\cdot\langle\sigma_y \rangle_{i-1}\Big)^2\nonumber\\\le~& \sum_{x\neq y\in \mathsf{B}_i} 4c_5^2 F(\bar{\partial}\mathsf{B}_i,\{x\})^2\cdot F(\partial\mathsf{B}_i,\{y\})^{ 2}\le C_3M^{\frac{7}{2}}.\label{eq:32c1232upper32bound32for32each32box} \end{align}\tag{123}\] Summing 123 over \(1\le i\le n\) and combining with 122 yields that \[\begin{align} {\mathbb{E}}\big(\sum_{i=1}^{n}\mathsf{c}_{i,2}\big)^2\le \sum_{i=1}^{n}C_1^2C_3{\epsilon}^4 M^{\frac{7}{2}}=C_1^2C_3{\epsilon}^4 M^{\frac{7}{2}}n .\label{eq:32c1232upper32bound32final} \end{align}\tag{124}\] Combining 121 and 124 with the fact that \({\mathbb{E}}(\mathsf{c}_i^2)\leq 2{\mathbb{E}}(\mathsf{c}_{i,1}^2 )+2{\mathbb{E}}(\mathsf{c}_{i,2}^2 )\), we complete the proof of Lemma 21. ◻

Proof. Note that \(H={\mathbb{E}}\tanh^2({\epsilon}\beta_c h_x)\le C_0{\epsilon}^2.\) Thus we have \[|\mathsf{d}_i|\le C_0{\epsilon}^2\cdot\sum_{x\in \mathsf{B}_i}\Big|(\langle\sigma_x\mid \mathcal{E}_\tau\rangle_{i-1}-\langle\sigma_x \rangle_{i-1})\cdot\langle\sigma_x \rangle_{i-1}\Big|.\] Combined with Lemma 19 and the Cauchy-Schwarz inequality, it yields that \[\mathsf{d}_i^2\le C_0^2{\epsilon}^4|\mathsf{B}_i|\sum_{x\in \mathsf{B}_i} \big(\langle\sigma_x \rangle_{i-1} \big)^2\cdot F(\bar{\partial}\mathsf{B}_i,\{x\})^2.\label{eq:32di32upper32bound32step321}\tag{127}\] To control the right-hand side of 127 , we shall use the following claim, whose proof is deferred until after the proof of the lemma.

Plugging ?? into 127 , we get that \[\begin{align} &{\mathbb{E}}\mathsf{d}_i^2\le C_0^2C_{1}{\epsilon}^4M^2\sum_{x\in \mathsf{B}_i} \Big(\big({\operatorname{dist}}(\mathsf{B}_i,\partial\Lambda_N)\vee M\big)^{-\frac{1}{4}+\kappa_{5}}M^{-\kappa_5}+{\operatorname{dist}}(x,\partial\Lambda_N)^{-\frac{1}{4}}\Big)\cdot F(\bar{\partial}\mathsf{B}_i,\{x\})^2\nonumber\\ \le~& C_{2}{\epsilon}^4M^{\frac{15}{4}-\kappa_5}\big({\operatorname{dist}}(\mathsf{B}_i,\partial\Lambda_N)\vee M\big)^{-\frac{1}{4}+\kappa_{5}}+C_2{\epsilon}^4M^2\sum_{x\in \mathsf{B}_i}{\operatorname{dist}}(x,\partial\Lambda_N)^{-\frac{1}{4}}\cdot F(\bar{\partial}\mathsf{B}_i,\{x\})^2\label{eq:32di32upper32bound32step322} \end{align}\tag{128}\] where the last inequality follows from Lemma 13. Note that for any integer \(r\geq 0,\) the number of boxes \(\mathsf{B}_i\) such that \({\operatorname{dist}}(\mathsf{B}_i,\partial\Lambda_N)=rM\) is upper-bounded by \(\frac{4N}{M}\) and that \(\big({\operatorname{dist}}(\mathsf{B}_i,\partial\Lambda_N)\vee M\big)^{-\frac{1}{4}+\kappa_5}\) decreases in \({\operatorname{dist}}(\mathsf{B}_i,\partial\Lambda_N)\). Thus, summing over \(1\le i\le n\), we obtain that \[\begin{align} \sum_{i=1}^{n} \big({\operatorname{dist}}(\mathsf{B}_i,\partial\Lambda_N)\vee M\big)^{-\frac{1}{4}+\kappa_{5}}&\le (M)^{-\frac{1}{4}+\kappa_{5}}\cdot \frac{4N}{M}+\sum_{r=1}^{N/M}(rM)^{-\frac{1}{4}+\kappa_{5}}\cdot \frac{4N}{M}\nonumber \\ &\le C_3(\frac{N}{M})^{\frac{3}{4}+\kappa_{5}}\cdot M^{-\frac{1}{4}+\kappa_{5}}\cdot \frac{N}{M}.\label{eq:32di32upper32bound32final32for32first32moment32summation} \end{align}\tag{129}\] To treat the second term in 128 , we apply the Cauchy-Schwarz inequality and Lemma 13 to get that \[\begin{align} \sum_{i=1}^n\sum_{x\in \mathsf{B}_i}{\operatorname{dist}}(x,\partial\Lambda_N)^{-\frac{1}{4}}\cdot F(\bar{\partial}\mathsf{B}_i,\{x\})^2&\le \sqrt{\sum_{i=1}^n\sum_{x\in \mathsf{B}_i}{\operatorname{dist}}(x,\partial\Lambda_N)^{-\frac{1}{2}}\cdot \sum_{i=1}^n\sum_{x\in \mathsf{B}_i} F(\bar{\partial}\mathsf{B}_i,\{x\})^4}\nonumber\\ &\le \sqrt{C_4N^{\frac{3}{2}}}\cdot \sqrt{C_4M^{\frac{3}{2}}\cdot n}=C_4M^{\frac{3}{2}}\cdot\iota^{-\frac{7}{4}}.\label{eq:32di32upper32bound32final32for32first32moment32summation322} \end{align}\tag{130}\]

Plugging 129 and 130 into 128 and noting that \(M=\iota N\), we finish the proof of 125 . Next, we prove 126 . Applying the Cauchy-Schwarz inequality, we obtain that \[|\sum_{i=1}^n \mathsf{d}_i|^2\le H^2\sum_{i=1}^n\sum_{x\in \mathsf{B}_i}\big(\langle\sigma_x \mid{\mathcal{E}_\tau}\rangle_{i-1}-\langle\sigma_x \rangle_{i-1}\big)^2\cdot \sum_{i=1}^n\sum_{x\in \mathsf{B}_i}\big(\langle\sigma_x \rangle_{i-1}\big)^2.\nonumber\] Combined with the fact that \(\sigma_i^2={\epsilon}^2\beta_c^2\sum_{x\in \mathsf{B}_i}\big(\langle\sigma_x \mid{\mathcal{E}_\tau}\rangle_{i-1}-\langle\sigma_x \rangle_{i-1}\big)^2\) and \(H\le C_{ 0}{\epsilon}^2\), it yields that \[\begin{align} |\sum_{i=1}^n \mathsf{d}_i|^2&\le C_{ 5}^2{\epsilon}^2\sum_{i=1}^n\sigma_i^2\cdot \sum_{i=1}^n\sum_{x\in \mathsf{B}_i}\big(\langle\sigma_x \rangle_{i-1}\big)^2.\label{eq:32sum32of32di32upper32bound32using32variance32sum} \end{align}\tag{131}\] Summing ?? over \(x\in \mathsf{B}_i\) and \(1\le i\le n\), we get from 129 that \[\begin{align} {\mathbb{E}}\sum_{i=1}^n\sum_{x\in \mathsf{B}_i}\big(\langle\sigma_x \rangle_{i-1}\big)^2&~\le\sum_{i=1}^n C_6\big({\operatorname{dist}}(\mathsf{B}_i,\partial\Lambda_N)\vee M\big)^{-\frac{1}{4}+\kappa_{5}}M^{2-\kappa_{5}}+C_1\sum_{i=1}^n\sum_{x\in\mathsf{B}_i}{\operatorname{dist}}(x,\partial\Lambda_N)^{-\frac{1}{4}}\nonumber\\&\stackrel{\eqref{eq:32di32upper32bound32final32for32first32moment32summation}}{\le}C_6C_3(\frac{N}{M})^{\frac{3}{4}+\kappa_{5}}\cdot M^{-\frac{1}{4}+\kappa_{5}}\cdot \frac{N}{M}\cdot M^{2-\kappa_5}+C_7N^{\frac{7}{4}}\nonumber\\&\le C_8N^{\frac{7}{4}+\kappa_{5}}M^{-\kappa_{5}}.\nonumber \end{align}\] Combined with the Markov inequality and the fact that \(M=\iota N\), it yields that \[{\mathbb{P}}\Big(\sum_{i=1}^n\sum_{x\in \mathsf{B}_i}\big(\langle\sigma_x \rangle_{i-1}\big)^2\ge N^{\frac{7}{4}}\iota^{-2\kappa_{5}})\le C_9\iota^{\kappa_{5}}.\label{eq:32spin32expectation32square32summation32with32external32field}\tag{132}\] Combining 131 and 132 , we finish the proof of 126 . ◻

Proof of Claim 33. Let \(\hat{N}=\big({\operatorname{dist}}(\mathsf{B}_i,\partial\Lambda_N)\vee M\big)^{1-4\kappa_{5}}M^{4\kappa_{5}}\) and let \(\hat{\mathsf{B}}_i\) be the box with length \(\hat{N}\) and whose center is the same as that of \(\mathsf{B}_i\). Let \(\mathsf{D}=(\cup_{j=1}^{i-1} \mathsf{B}_j)\cap\hat{\mathsf{B}}_i\) and let \(\hat{H}^{i-1}\) be the external field such that for any vertex \(z\in\Lambda_N\), \(\hat{H}^{i-1}_z=H^{i-1}_z\cdot \mathbb{I}_{z\notin \mathsf{D}}\). We can apply chaos expansion to \(\langle\sigma_x \rangle_{i-1}\) with respect to the measure \(\mu^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}\). \[\begin{align} \langle\sigma_x \rangle_{i-1}=\frac{\langle\sigma_x \rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}+\sum_{k=1}^{\infty}\sum_{|I|=k,I\subset\mathsf{D}}\langle\sigma^I\sigma_x\rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}\cdot\prod_{z\in I}\tanh({\epsilon}\beta_c h_z)}{1+\sum_{k=1}^{\infty}\sum_{|I|=k,I\subset\mathsf{D}}\langle\sigma^I\rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}\cdot\prod_{z\in I}\tanh({\epsilon}\beta_c h_z)}.\label{eq:32chaos32expansion32for32expectation32in32cle32singular} \end{align}\tag{133}\] Let \[\begin{align} \Phi_k(h)&=\sum\limits_{\substack{|I|=k,\\I\subset\mathsf{D}}}\langle\sigma^I\sigma_x\rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}\cdot\prod_{z\in I}\tanh({\epsilon}\beta_c h_z), and\Psi_k(h)=\sum\limits_{\substack{|I|=k,\\I\subset\mathsf{D}}}\langle\sigma^I\rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}\cdot\prod_{z\in I}\tanh({\epsilon}\beta_c h_z) . \end{align}\] Then we get from a similar result as in Corollary 1 with \(\mathsf D\) replacing \(\mathsf{B}_i\) (see also [22] for detailed calculations) that \[\label{eq:32upper32bound32on32k32point32function32to32be32big} \begin{align} {\mathbb{P}}\Big(|\Phi_k(h)|>({\epsilon}\hat{N}^{\frac{7}{8}})^{k/2}\cdot \langle\sigma_x \rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}\Big)&\le C_2^{-1}\exp\Big(-C_2\sqrt{{\epsilon}^{-1} \hat{N}^{-\frac{7}{8}}}\times(k!)^{\frac{3}{8k}}\Big),\\and {\mathbb{P}}\Big(|\Psi_k(h)|>({\epsilon}\hat{N}^{\frac{7}{8}})^{k/2}\Big)&\le C_2^{-1}\exp\Big(-C_2\sqrt{{\epsilon}^{-1} \hat{N}^{-\frac{7}{8}}}\times(k!)^{\frac{3}{8k}}\Big). \end{align}\tag{134}\] Note that \(\sum_{k=1}^{\infty}({\epsilon}\hat{N}^{\frac{7}{8}})^{k/2}\le C_3\sqrt{{\epsilon}\hat{N}^{\frac{7}{8}}}\) for some constant \(C_3>0\). Moreover, we may assume that \(C_3<2\) by choosing \(\iota>0\) small enough. Thus, we conclude by 134 that \[\begin{align} &{\mathbb{P}}(|\sum_{k=1}^{\infty}\Psi_k(h)| > 2\sqrt{{\epsilon}\hat{N}^{\frac{7}{8}}})\le \sum_{k=1}^{\infty}{\mathbb{P}}(|\Psi_k(h)|>({\epsilon}\hat{N}^{\frac{7}{8}})^{k/2})\nonumber\\ \stackrel{\eqref{eq:32upper32bound32on32k32point32function32to32be32big}}{\le}& \sum_{k=1}^{\infty}C_2^{-1}\exp\Big(-C_2\sqrt{{\epsilon}^{-1} \hat{N}^{-\frac{7}{8}}}\times(k!)^{\frac{3}{8k}}\Big)\le C_4^{-1}\exp(-C_4\sqrt{{\epsilon}^{-1} \hat{N}^{-\frac{7}{8}}}).\label{eq:32upper32bound32on32Psi95k32to32be32big} \end{align}\tag{135}\] Similarly, we can get the upper bound for \(\sum_{k=1}^{\infty}\Phi_k(h)\) as follows: \[\begin{align} &{\mathbb{P}}(|\sum_{k=1}^{\infty}\Phi_k(h)| > 2\sqrt{{\epsilon}\hat{N}^{\frac{7}{8}}}\cdot \langle\sigma_x \rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}})\le \sum_{k=1}^{\infty}{\mathbb{P}}(|\Phi_k(h)|>({\epsilon}\hat{N}^{\frac{7}{8}})^{k/2}\cdot \langle\sigma_x \rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}})\nonumber\\ \stackrel{\eqref{eq:32upper32bound32on32k32point32function32to32be32big}}{\le}& \sum_{k=1}^{\infty}C_2^{-1}\exp\Big(-C_2\sqrt{{\epsilon}^{-1} \hat{N}^{-\frac{7}{8}}}\times(k!)^{\frac{3}{8k}}\Big)\le C_4^{-1}\exp(-C_4\sqrt{{\epsilon}^{-1} \hat{N}^{-\frac{7}{8}}}).\label{eq:32upper32bound32on32Phi95k32to32be32big} \end{align}\tag{136}\] Let \(\mathcal{H}(x)\) denote the collection of external fields on \(\mathsf{D}\) such that \(|\sum_{k=1}^{\infty}\Psi_k(h)| \le 2\sqrt{{\epsilon}\hat{N}^{\frac{7}{8}}}\) and \(|\sum_{k=1}^{\infty}\Phi_k(h)| \le 2\sqrt{{\epsilon}\hat{N}^{\frac{7}{8}}}\cdot \langle\sigma_x \rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}\). For any \(h\in \mathcal{H}(x)\), we get from 133 that \[|\langle\sigma_x \rangle_{i-1}-\langle\sigma_x \rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}|\le C_5\sqrt{{\epsilon}\hat{N}^{\frac{7}{8}}} \cdot |\langle\sigma_x \rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}|.\label{eq:32expectation32upper32bound32in32cle32singular}\tag{137}\] By 135 and 136 , we obtain that \[{\mathbb{P}}\big(\mathcal{H}(x)\big)\ge 1-2 C_3^{-1}\exp(-C_3\sqrt{{\epsilon}^{-1} \hat{N}^{-\frac{7}{8}}}).\label{eq:32good32external32field32in3232the32upper32bound32of32di}\tag{138}\] Combined with 137 , it yields that \[\begin{align} {\mathbb{E}}|\langle\sigma_x \rangle_{i-1}|^2\le 2 C_4^{-1}\exp(-C_4\sqrt{{\epsilon}^{-1} \hat{N}^{-\frac{7}{8}}})+(1+C_5\sqrt{{\epsilon}\hat{N}^{\frac{7}{8}}})\cdot |\langle\sigma_x \rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}|^2.\label{eq:32expectation32upper32bound32in32cle32singular32final} \end{align}\tag{139}\] Recall that \(\hat{N}=\big({\operatorname{dist}}(\mathsf{B}_i,\partial\Lambda_N)\vee M\big)^{1-4\kappa_5}M^{4\kappa_5}\). Combining with DMP and 50 , we get that \[\begin{align} |\langle\sigma_x \rangle^+_{\Lambda_N,{\epsilon}\hat{H}^{i-1}}|&\le c~{\operatorname{dist}}(x,\partial\hat{\mathsf{B}}_i)^{-\frac{1}{8}}\le c\big(\hat{N}-M+{\operatorname{dist}}(x,\partial\Lambda_N)\big)^{-\frac{1}{8}}\le C_5\hat{N}^{-\frac{1}{8}}+C_5{\operatorname{dist}}(x,\partial\Lambda_N)^{-\frac{1}{8}}\nonumber\\&\le C_6\big({\operatorname{dist}}(\mathsf{B}_i,\partial\Lambda_N)\vee M\big)^{-\frac{1-4\kappa_{5}}{8}}M^{-\frac{\kappa_{5}}{2}}+C_6{\operatorname{dist}}(x,\partial\Lambda_N)^{-\frac{1}{8}}. \nonumber \end{align}\] Plugging into 139 and noting that \({\epsilon}\hat{N}^{\frac{7}{8}}\le (\frac{N}{M})^{-\frac{7}{2}\kappa_{5}}\), we finish the proof of ?? . ◻

Proof. For notational clarity, let \[\begin{align} A^*_{i,r}&=\sum_{k=r}^{\infty}\sum\limits_{I\subset \mathsf{B}_i,|I|=k}\big(\langle\sigma^I \mid {\mathcal{E}_\tau}\rangle_{i-1}-\langle\sigma^I \rangle_{i-1}\big)\cdot\prod_{x\in I}\tanh({\epsilon}\beta_c h_x);\\ A_{i,r}&=\sum_{k=r}^{\infty}\sum\limits_{I\subset \mathsf{B}_i,|I|=k}\langle\sigma^I \rangle_{i-1}\cdot\prod_{x\in I}\tanh({\epsilon}\beta_c h_x). \end{align}\] Then 106 becomes \(\mathsf{a}_i=\frac{A^*_{i,1}}{1+A_{i,1}}.\) Define \[\begin{align} \mathsf{e}_{i,1} &:= \sum_{x\in \mathsf{B}_i} \big( \langle\sigma_x \mid \mathcal{E}_\tau\rangle_{i-1} -\langle\sigma_x\rangle_{i-1} \big) \left[ \tanh({\epsilon}\beta_c h_x) -{\epsilon}\beta_c h_x \right],\qquad \mathsf{e}_{i,2} := \frac{A^*_{i,1}}{1+A_{i,1}}(A_{i,1})^2,\\ \mathsf{e}_{i,3}&:=A^*_{i,3}, \qquad \mathsf{e}_{i,4} := A^*_{i,2}(-A_{i,1}+A_{i,2}), \qquad \mathsf{e}_{i,5}:=-A^*_{i,1}A_{i,2}. \end{align}\] By the definitions of \(\mathsf{b}_i,\mathsf{c}_{i,1},\mathsf{c}_{i,2}\) and \(\mathsf{d}_i\), we have that \[\mathsf{b}_i+\mathsf{e}_{i,1} = A^*_{i,1}-A^*_{i,2}, \qquad \mathsf{c}_{i,1} = A^*_{i,2}-A^*_{i,3}, \qquad \mathsf{c}_{i,2}+\mathsf{d}_i =-\big(A^*_{i,1}-A^*_{i,2}\big) \big(A_{i,1}-A_{i,2}\big).\] Using \(\mathsf{c}_i=\mathsf{c}_{i,1}+\mathsf{c}_{i,2}\) and \[\frac{A^*_{i,1}}{1+A_{i,1}} -A^*_{i,1} +A^*_{i,1}A_{i,1} = \frac{A^*_{i,1}}{1+A_{i,1}}(A_{i,1})^2 ,\] we obtain that \[\mathsf{a}_i-\mathsf{b}_i-\mathsf{c}_i-\mathsf{d}_i =\mathsf{e}_{i,1}+ \frac{A^*_{i,1}}{1+A_{i,1}} -\big(A^*_{i,1}-A^*_{i,3}\big) +\big(A^*_{i,1}-A^*_{i,2}\big) \big(A_{i,1}-A_{i,2}\big) =\sum_{k=1}^5 \mathsf{e}_{i,k}.\] We will show below that all the terms on the right-hand side are small with high probability.

We first control the term \(\mathsf e_{i,1}\). Note that for any \(x\), \(h_x\) is a standard normal variable, thus \({\mathbb{E}}\big|\tanh({\epsilon}\beta_c h_x)-{\epsilon}\beta_c h_x\big|\le C_1{\epsilon}^{3}\) for some constant \(C_1\ge 0\). Therefore, we get that \[\begin{align} {\mathbb{E}}|\mathsf{e}_{i,1}|\le \sum_{x\in \mathsf{B}_i}2{\mathbb{E}}\big|\tanh({\epsilon}\beta_c h_x)-{\epsilon}\beta_c h_x\big|\le C_2{\epsilon}^{3}M^2.\nonumber \end{align}\]Applying Markov inequality and recalling \({\epsilon}=N^{-\frac{7}{8}}, M=\iota N\), we get that \[{\mathbb{P}}\Big(|\mathsf{e}_{i,1}|\ge ({\epsilon}M^{\frac{7}{8}})^{2.7} \Big)\le \frac{C_2{\epsilon}^{3}M^2}{({\epsilon}M^{\frac{7}{8}})^{2.7}}\le C_2\iota^{-\frac{29}{80}} N^{-\frac{5}{8}}.\label{eq:32upper32bound32for32the32first32error32term}\tag{140}\]

Then we control the terms \(e_{i,k}\) for \(2\le k\le 5\). Applying Corollary 2 with \(\kappa_4=0.1\) to \(A^*_{i,1}\) and \(A_{i,1}\) yields that \[\begin{align} {\mathbb{P}}\big(|A^*_{i,k}|\ge C_3 ({\epsilon}M^{\frac{7}{8}})^{0.9k}\big)&\le C_4^{-1}\exp\big(-C_4({\epsilon}^{-1} M^{-\frac{7}{8}})^{0.1}\big),\tag{141}\\{\mathbb{P}}\big(|A_{i,k}|\ge C_3 ({\epsilon}M^{\frac{7}{8}})^{0.9k}\big)&\le C_4^{-1}\exp\big(-C_4({\epsilon}^{-1} M^{-\frac{7}{8}})^{0.1}\big). \tag{142} \end{align}\] On the event that \(A_{i,k}\le C_1 ({\epsilon}M^{\frac{7}{8}})^{0.9k}\) and \(A^*_{i,k}(x)\le C_1 ({\epsilon}M^{\frac{7}{8}})^{0.9k}\) for \(1\le k\le 3\), it is straightforward that \(e_{i,k}\le C_3 ({\epsilon}M^{\frac{7}{8}})^{2.7}\) for \(2\le k\le 5\). Therefore, we get from 141 and 142 that \[{\mathbb{P}}\Big(|\sum_{k=2}^5\mathsf{e}_{i,k}|\ge 4C_3 ({\epsilon}M^{\frac{7}{8}})^{2.7} \Big)\le 6C_4^{-1}\exp\big(-C_4({\epsilon}^{-1} M^{-\frac{7}{8}})^{0.1}\big).\label{eq:32upper32bound32for32the32second32error32term}\tag{143}\] The desired result thus follows from combining 140 and 143 and choosing \(N>0\) large enough depending on \(\iota\). ◻

Proof. The proof follows directly from combining 106 and Corollary 2, and we omit further details here. ◻

Proof. We want to compute the exponential moment of \(\sum_{i=1}^{n}(\frac{4\mathsf{b}_i+\sigma_i^2-2\mathsf{b}_i^2}{16})\) and then apply the Markov inequality. Recall the definition of \(\mathcal{F}_i\) in Proposition 32. We first compute \({\mathbb{E}}\big(e^{\frac{4\mathsf{b}_i+\sigma_i^2-2\mathsf{b}_i^2}{16}}\mid \mathcal{F}_{i-1}\big)\). Note that \(\mathsf b_i\) is a Gaussian variable with mean \(0\) and variance \(\sigma_i^2\). Thus we calculate that \[\begin{align} {\mathbb{E}}\big(e^{\frac{4\mathsf{b}_i+\sigma_i^2-2\mathsf{b}_i^2}{16}}\mid \mathcal{F}_{i-1}\big)&=\frac{1}{\sqrt{2\pi \sigma_i^2}}\int_{-\infty}^{\infty}e^{\frac{4x+\sigma_i^2-2x^2}{16}}\cdot e^{-\frac{x^2}{2\sigma_i^2}}dx=\frac{1}{\sqrt{1+\sigma_i^2/4}}\cdot e^{\frac{\sigma_i^2}{32+8\sigma_i^2}+\frac{\sigma_i^2}{16}}.\nonumber \end{align}\] Note that \(f(u)=-\log(1+u/4)/2+\frac{u}{32+8u}+\frac{u}{16}\) is decreasing for \(u\in[0,1] .\) By Proposition 32, we get that \(\sigma_i^2\le c_2{\epsilon}^2M^{\frac{7}{4}}\le 1.\) Therefore we get that \(f(\sigma_i^2)\le f(0)=0\) and thus we have that \[{\mathbb{E}}\big(e^{\frac{4\mathsf{b}_i+\sigma_i^2-2\mathsf{b}_i^2}{16}}\mid \mathcal{F}_{i-1}\big)\le 1.\label{eq:32recursive32relation32for32bi32exponential32bound321}\tag{144}\] In addition, if \(\tau_i=1\), then we have \(\sigma_i^2\ge c_3 {\epsilon}^2M^{\frac{7}{4}}\). Therefore we get that \(f(\sigma_i^2)\le f(c_3 {\epsilon}^2M^{\frac{7}{4}})\le -C_1{\epsilon}^2M^{\frac{7}{4}}\) and hence we have that \[{\mathbb{E}}\big(e^{\frac{4\mathsf{b}_i+\sigma_i^2-2\mathsf{b}_i^2}{16}}\mid \mathcal{F}_{i-1}\big)\le \exp(-C_1{\epsilon}^2M^{\frac{7}{4}}).\label{eq:32recursive32relation32for32bi32exponential32bound322}\tag{145}\] Combining 144 and 145 , we get that \[{\mathbb{E}}\exp\Big(\sum_{i=1}^{n}(\frac{4\mathsf{b}_i+\sigma_i^2-2\mathsf{b}_i^2}{16})\Big)\le \exp(-C_1{\epsilon}^2M^{\frac{7}{4}}|\tau|).\nonumber\]Choosing \(c_7=\min\{1,C_1\}/4\) and applying the Markov inequality to \(\exp\Big(\sum_{i=1}^{n}(\frac{4\mathsf{b}_i+\sigma_i^2-2\mathsf{b}_i^2}{16})\Big)\), we complete the proof of Lemma 24. ◻

Proof. Fix a good configuration \(\tau\). Applying the Markov inequality to \(\sum_{i=1}^{n}\mathsf{c}_i\) and combining with Proposition 32 [item:32ci], we get that \[\begin{align} {\mathbb{P}}(\mathcal{C}\mid \tau)&\ge 1-{\mathbb{E}}\left[(\sum_{i=1}^{n}\mathsf{c}_i)^2\right]\ge 1-c_3{\epsilon}^4M^{\frac{7}{2}}\cdot n= 1-c_3\iota^{\frac{3}{2}}.\label{eq:32good32external32field32for32ci321} \end{align}\tag{146}\] Combined with Proposition 32 [item:32di], [item:32ai-bi-ci], Lemma 24 and Corollary 3, it yields that \[\label{eq:32good32external32field32for32cle32singularity321} {\mathbb{P}}(\mathcal{A}\cap\mathcal{B}\cap\mathcal{C}\cap\mathcal{D}\mid \tau)\ge 1-C_2\iota^{\kappa}.\tag{147}\] Next, we control \((\mathcal{A}^*)^c\cap\mathcal{A}\). To this end, we first bound \({\mathbb{E}}\Big|\sum_{i=1}^{n}(\mathsf{a}_i^2-\mathsf{b}_i^2)\mathbb{I}_{\mathcal{A}} \Big|\) and then apply the Markov inequality. By the Cauchy-Schwarz inequality, we obtain that \[\begin{align} {\mathbb{E}}\Big|\sum_{i=1}^{n}(\mathsf{a}_i^2-\mathsf{b}_i^2)\mathbb{I}_{\mathcal{A}}\Big|&\le {\mathbb{E}}\Big|\sum_{i=1}^{n}2\mathsf{b}_i(\mathsf{a}_i-\mathsf{b}_i)\mathbb{I}_{\mathcal{A}}\Big|+{\mathbb{E}}\Big[\sum_{i=1}^{n}(\mathsf{a}_i-\mathsf{b}_i)^2\mathbb{I}_{\mathcal{A}}\Big]\nonumber\\&\le 2\sqrt{{\mathbb{E}}\sum_{i=1}^{n}\mathsf{b}_i^2\cdot {\mathbb{E}}\Big[\sum_{i=1}^{n}(\mathsf{a}_i-\mathsf{b}_i)^2\mathbb{I}_{\mathcal{A}}\Big]}+{\mathbb{E}}\Big[\sum_{i=1}^{n}(\mathsf{a}_i-\mathsf{b}_i)^2\mathbb{I}_{\mathcal{A}}\Big].\label{eq:32expansion32for32the32second32moment32of32ai} \end{align}\tag{148}\] Recalling the definition of \(\mathcal{A}\), we get from the Cauchy-Schwarz inequality and Proposition 32 that \[\begin{align} {\mathbb{E}}\Big[\sum_{i=1}^{n}(\mathsf{a}_i-\mathsf{b}_i)^2\mathbb{I}_{\mathcal{A}}\Big]&\le 3{\mathbb{E}}\sum_{i=1}^{n}\mathsf{c}_i^2+3{\mathbb{E}}\sum_{i=1}^{n}\mathsf{d}_i^2+ 3\sum_{i=1}^{n}{\mathbb{E}}\Big[(\mathsf{a}_i-\mathsf{b}_i-\mathsf{c}_i-\mathsf{d}_i)^2\mathbb{I}_{\mathcal{A}}\Big]\nonumber\\&\le c_4{\epsilon}^4M^{\frac{7}{2}}n+c_4{\epsilon}^4M^{\frac{7}{2}}\cdot \iota^{-\frac{7}{4}-\kappa_{1}}+3({\epsilon}^2M^{\frac{7}{4}})^{2.7}\cdot n\le C_3\iota^{1.5}.\label{eq:32second32moment32bound32for32ai} \end{align}\tag{149}\] Plugging 149 into 148 and combining with Lemma 20 yields that \[\begin{align} {\mathbb{E}}\Big|\sum_{i=1}^{n}(\mathsf{a}_i^2-\mathsf{b}_i^2)\mathbb{I}_{\mathcal{A}}\Big|&\le 2\sqrt{{\epsilon}^2M^{\frac{7}{4}}n\cdot C_3\iota^{1.5}}+C_3\iota^{1.5}\le C_4\sqrt{\iota}.\label{eq:32second32moment32bound32for32ai32final} \end{align}\tag{150}\] Applying the Markov inequality to 150 and combining with 147 , we get that \[\label{eq:32good32external32field32for32cle32singularity} {\mathbb{P}}(\mathcal{A}\cap\mathcal{A}^*\cap\mathcal{B}\cap\mathcal{C}\cap\mathcal{D}\mid \tau)\ge 1-C_3\iota^{\kappa}.\tag{151}\] Thus we complete the proof of Lemma 25. ◻

Proof of Theorem 28. Note that for any external field, we have that \[\frac{1}{2}\sum_{\tau\in\{0,1\}^n}\Big|\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau)-\mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)\Big|=1-\sum_{\tau\in\{0,1\}^n}\min\{\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau),\mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)\}.\label{eq:32total32variation32expansion}\tag{152}\] So it suffices to prove that \[{\mathbb{E}}\sum_{\tau\in\{0,1\}^n}\min\{\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau),\mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)\}\le C_1\iota^{\kappa}. \label{eq:32total32variation32distance32formula}\tag{153}\] We first split the set of configurations \(\tau\) according to Lemma 18:\[\begin{align} &{\mathbb{E}}\sum_{\tau\in\{0,1\}^n}\min\{\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau),\mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)\}\nonumber\\\le ~& {\mathbb{E}}\sum_{|\tau|\ge c_1n^{\frac{15}{16}-\alpha}}\min\{\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau),\mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)\}+\sum_{|\tau|< c_1n^{\frac{15}{16}-\alpha}}\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau)\nonumber\\\le~& {\mathbb{E}}\sum_{|\tau|\ge c_1n^{\frac{15}{16}-\alpha}}\min\{\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau),\mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)\}+c_1^{-1}\exp(-c_1n^{\frac{144\alpha-7}{112}}).\label{eq:32total32variation32expansion321} \end{align}\tag{154}\] Now we fix a good configuration \(\tau\), i.e., \(|\tau|\ge c_1n^{\frac{15}{16}-\alpha}\). (Recall that \(\alpha=\frac{7}{144}+10^{-5}\).) Recall the events defined in Lemma 25. We expand \(\log\Big(\frac{\mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)}{\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau)}\Big)\) as follows: \[\begin{align} &\log\Big(\frac{\mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)}{\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau)}\Big)= \sum_{i=1}^{n}\log(1+\mathsf{a}_i)\nonumber\\\le~& \sum_{i=1}^{n}\mathsf{b}_i-\sum_{i=1}^{n} \frac{ \mathsf{b}_i^2}{2}+\Big|\sum_{i=1}^{n}\mathsf{c}_i\Big|+\Big|\sum_{i=1}^{n}\mathsf{d}_i\Big| +\sum_{i=1}^{n}\Big|\mathsf{a}_i-\mathsf{b}_i-\mathsf{c}_i-\mathsf{d}_i\Big|+\Big|\sum_{i=1}^{n}\frac{\mathsf{a}_i^2-\mathsf{b}_i^2}{2}\Big|\nonumber\\&+\Big|\sum_{i=1}^{n}\big(\log(1+\mathsf{a}_i)-\mathsf{a}_i+\frac{\mathsf{a}_i^2}{2}\big)\Big|.\label{eq:32log32rn32derivative32expansion32singularity} \end{align}\tag{155}\] Recall the definitions in Definition 4. It suffices to control the term \(\Big|\sum_{i=1}^{n}\big(\log(1+\mathsf{a}_i)-\mathsf{a}_i+\frac{\mathsf{a}_i^2}{2}\big)\Big|\) in the right-hand of 155 . Assume that \((h,\tau)\in \mathcal{A}\cap\mathcal{A}^*\cap\mathcal{B}\cap\mathcal{C}\cap\mathcal{D}\). Then we get that \[\label{eq:32good32external32field32configuration32pair32bound} \begin{align} &\sum_{i=1}^{n}(\mathsf{b}_i-\frac{\sf\mathsf{b}_i^2}{2})\le -c_7{\epsilon}^2M^{\frac{7}{4}}|\tau|-\sum_{i=1}^n\frac{\sigma_i^2}{4} ,~~~~\Big|\mathsf{a}_i-\mathsf{b}_i-\mathsf{c}_i-\mathsf{d}_i\Big|\le ({\epsilon}M^{\frac{7}{8}})^{2.7},\forall 1\le i\le n,\\ &|\sum_{i=1}^{n}(\mathsf{a}_i^2-\mathsf{b}_i^2)|\le 1,~~~~~|\sum_{i=1}^{n}\mathsf{c}_i|\le 1,~~~~~|\sum_{i=1}^{n}\mathsf{d}_i|\le \sqrt{\iota^{-2\kappa}\cdot \sum_{i=1}^{n}\sigma_i^2}. \end{align}\tag{156}\] Noting that \(|\ln(1+x)-x+\frac{x^2}{2}|\le \frac{x^3}{3}\) and \(\mathsf{a}_i\le ({\epsilon}M^{\frac{7}{8}})^{0.9}\), we obtain that \[\begin{align} \Big|\sum_{i=1}^{n}\big(\log(1+\mathsf{a}_i)-\mathsf{a}_i+\frac{\mathsf{a}_i^2}{2}\big)\Big|\le \sum_{i=1}^{n} \frac{|\mathsf{a}_i^3|}{3}\le ({\epsilon}M^{\frac{7}{8}})^{2.7} n\le 1.\label{eq:32log32rn32derivative32expansion32ai32third32moment32bound} \end{align}\tag{157}\] Plugging 156 and 157 into 155 yields that \[\begin{align} \log\Big(\frac{\mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)}{\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau)}\Big)&\le -c_7{\epsilon}^2M^{\frac{7}{4}}|\tau|-\sum_{i=1}^n \frac{\sigma_i^2}{4}+3+({\epsilon}M^{\frac{7}{8}})^{2.7}\cdot n+\iota^{-\kappa}\sqrt{\sum_{i=1}^{n}\sigma_i^2}\nonumber\\&\le -C_2{\epsilon}^2M^{\frac{7}{4}}n^{\frac{15}{16}-\alpha}-\sum_{i=1}^n \frac{\sigma_i^2}{4}+3+({\epsilon}M^{\frac{7}{8}})^{2.7}\cdot n+\iota^{-\kappa}\sqrt{\sum_{i=1}^{n}\sigma_i^2}.\label{eq:32log32RN32lower32bound321} \end{align}\tag{158}\] Note that \(\iota>0\) is chosen small enough and \(\sum_{i=1}^n\sigma_i^2\ge {\epsilon}^2M^{\frac{7}{4}}|\tau|\), we get that \(\sum_{i=1}^n \frac{\sigma_i^2}{4}\ge \iota^{-\kappa}\sqrt{\sum_{i=1}^{n}\sigma_i^2}\). Plugging into \(n=(\frac{N}{M})^{2}=\iota^{-2}\) and \({\epsilon}M^{\frac{7}{8}}=\iota^{\frac{7}{8}}\) into 158 , we get that \[\log\Big(\frac{\mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)}{\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau)}\Big)\le -C_3 \iota^{-\frac{1}{8}+2\alpha}.\label{eq:32log32RN32lower32bound}\tag{159}\] Now we are ready to upper-bound the left-hand side of 153 . Summing 159 over all good configurations, we get that \[\begin{align} & {\mathbb{E}}\sum_{|\tau|\ge c_1n^{\frac{15}{16}-\alpha}}\min\{\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau),\mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)\}\nonumber\\\le~& {\mathbb{P}}\otimes\mu^+_{\Lambda_N,0}\Big(\big(\mathcal{A}\cap\mathcal{A}^*\cap\mathcal{B}\cap\mathcal{C}\cap\mathcal{D}\big)^c\Big)+\sum_{(h,\tau)\in \mathcal{A}\cap\mathcal{A}^*\cap\mathcal{B}\cap\mathcal{C}\cap\mathcal{D}}{\mathbb{P}}(h)\cdot \mu^+_{\Lambda_N,{\epsilon}h}(\mathcal{E}_\tau)\nonumber\\ \stackrel{\eqref{eq:32log32RN32lower32bound}}{\le}&{\mathbb{P}}\otimes\mu^+_{\Lambda_N,0}\Big(\big(\mathcal{A}\cap\mathcal{A}^*\cap\mathcal{B}\cap\mathcal{C}\cap\mathcal{D}\big)^c\Big)+\sum_{(h,\tau)\in \mathcal{A}\cap\mathcal{A}^*\cap\mathcal{B}\cap\mathcal{C}\cap\mathcal{D}}{\mathbb{P}}(h)\cdot e^{-C_3\iota^{-\frac{1}{8}+\alpha}}\times\mu^+_{\Lambda_N,0}(\mathcal{E}_\tau)\nonumber\\ \stackrel{\eqref{eq:32good32external32field32for32cle32singularity}}{\le}&C_4\iota^{\kappa}+e^{-C_4 \iota^{-\frac{1}{8}+2\alpha}}.\label{eq:32final32calculation32in32cle32singulartiy} \end{align}\tag{160}\] Combining with 152 and 154 , we finish the proof of Theorem 28 by noting that the right-hand side of 160 goes to \(0\) as \(\iota\) goes to \(0\). ◻

Proof of Lemma 26. Let \(\xi\in \Sigma_M^c\) be a configuration on \((\Lambda_M\setminus\partial\Lambda_M)^c\). After conditioning on \(\xi\) together with the event \(\mathcal{E}_\tau\), the relevant objects are as follows:

• \(\mathcal{P}_0\) is the set of plus spins that are connected to the boundary through \(\Lambda_M^c\).

• \(\mathcal{P}_1\) is the set of plus spins that should not be connected to the boundary.

• \(\mathcal{U}\) is the collection of boundary traces of clusters that are supposed to be connected to the boundary, but fail to be so through \((\Lambda_M\setminus\partial\Lambda_M)^c\). Consequently, they must be connected to \(\mathcal{P}_0\) inside \(\Lambda_M\setminus\partial\Lambda_M\).

• \(\mathcal{X}\) denotes the event that there exist both a plus circuit and a \(*\)-minus circuit in the annulus; the plus circuit is connected to \(\mathcal{P}_0\), and hence may be viewed as the outermost such circuit relevant to our construction.

Since we condition on \(\mathcal{E}_\tau\), it follows that for every \(U\in \mathcal{U}\), the set \(U\) is plus-connected to \(\mathcal{P}_0\) inside \(\Lambda_M\setminus\partial\Lambda_M\), while \(\mathcal{P}_0\) is not plus-connected to \(\mathcal{P}_1\). In other words, the event \(\mathcal{Y}=\mathcal{Y}^{-}\cap \mathcal{Y}^{+}\) occurs. Thus, after conditioning on the exterior configuration, the conditioning on \(\mathcal{E}_\tau\) is encoded by the event \(\mathcal{Y}\) inside \(\Lambda_M\). Moreover, by the definition of \(\mathcal{X}\), the event \(\mathcal{R}\) corresponds to the case \(\tau_i=0\), whereas the event \(\mathcal{S}\) corresponds to the case \(\tau_i=1\). ◻

Proof of Lemma 19. Recall the discussion in Lemma 26. For any boundary condition \(\xi\in\Sigma_M^c\), if \(\tau_i=0,\) then conditioned on the boundary condition \(\xi\), we get that \(\mathcal{E}_\tau\) is equivalent to \(\mathcal{R}\). Thus we calculate \[\label{eq:32conditioning32expansion32in32DHX32rare32event} \begin{align} \langle\sigma^I\mid \mathcal{R}\rangle^{\xi}_{\Lambda_M,0}&=\frac{\langle\sigma^I\cdot R\mid \xi\rangle^{+}_{\Lambda_N,{\epsilon}\hat{h}^i}}{\mu^{+}_{\Lambda_N,{\epsilon}\hat{h}^i}(\mathcal{R}\mid \xi)}=\frac{\langle\sigma^I\cdot \mathbf{1}_{\mathcal{E}_{\tau}}\mid \xi\rangle^{+}_{\Lambda_N,{\epsilon}\hat{h}^i}}{\mu^{+}_{\Lambda_N,{\epsilon}\hat{h}^i}({\mathcal{E}_{\tau}}\mid \xi)}=\frac{\langle\sigma^I\cdot \mathbf{1}_{\mathcal{E}_{\tau}}\mid \xi\rangle^{+}_{\Lambda_N,{\epsilon}\hat{h}^i}\cdot \mu^+_{\Lambda_N,{\epsilon}\hat{h}^i}(\xi)}{\mu^{+}_{\Lambda_N,{\epsilon}\hat{h}^i}({\mathcal{E}_{\tau}}\cap \xi)}. \end{align}\tag{162}\] Summing ?? over all possible boundary conditions \(\xi\) and combining with 162 , we get that \[\begin{align} c^{|I|} F(\bar\partial\Lambda_M,I)&\ge \sum_{\xi\in\Sigma_M^c}\mu^+_{\Lambda_N,{\epsilon}\hat{h}^i}(\xi\mid\mathcal{E}_{\tau})\cdot \langle\sigma^I\mid \mathcal{R}\rangle^{\xi}_{\Lambda_M,0}\nonumber\\&=\sum_{\xi\in\Sigma_M^c}\frac{\mu^+_{\Lambda_N,{\epsilon}\hat{h}^i}(\xi\cap\mathcal{E}_{\tau})}{\mu^+_{\Lambda_N,{\epsilon}\hat{h}^i}(\mathcal{E}_{\tau})}\cdot \frac{\langle\sigma^I\cdot \mathbf{1}_{\mathcal{E}_{\tau}}\mid \xi\rangle^{+}_{\Lambda_N,{\epsilon}\hat{h}^i}\cdot \mu^+_{\Lambda_N}(\xi)}{\mu^{+}_{\Lambda_N,{\epsilon}\hat{h}^i}({\mathcal{E}_{\tau}}\cap \xi)}\nonumber\\&=\sum_{\xi\in\Sigma_M^c}\frac{\langle\sigma^I\cdot \mathbf{1}_{\mathcal{E}_{\tau}}\mid \xi\rangle^{+}_{\Lambda_N,{\epsilon}\hat{h}^i}\cdot \mu^+_{\Lambda_N,{\epsilon}\hat{h}^i}(\xi)}{\mu^+_{\Lambda_N,{\epsilon}\hat{h}^i}(\mathcal{E}_{\tau})}=\langle\sigma^I\mid \mathcal{E}_{\tau}\rangle^{+}_{\Lambda_N,{\epsilon}\hat{h}^i}.\nonumber \end{align}\]Thus we finish the proof of Lemma 19 in the case \(\tau_i=0\). The case \(\tau_i=1\) follows similarly by replacing \(\mathcal{R}\) with \(\mathcal{S}\). ◻

Proof. The proof is similar to [22], and we follow the notation used there. List \(\partial \Lambda_{M_1+1}\) as \(\{x_1,x_2,\cdots,x_n\}\). For each \(x\), if \(\sigma_x=-1\), let \(\mathbf{C}_x = \emptyset\) and \(\mathbf{B}_x=\{x\}\); if \(\sigma_x=1\), let \(\mathbf{C}_x\) be the plus cluster containing \(x\) restricted to \(\Lambda_{M_1,M_2}:=\Lambda_{M_2}\setminus\Lambda_{M_1}\). In addition, define \[\mathbf{B}_x=\partial_{\mathrm{ext}}\mathbf{C}_x\cap\Lambda_{M_1,M_2},\] where \(\partial_{\mathrm{ext}}\mathbf{C}_x=\{u\notin \mathbf{C}_x:\exists\, v\in \mathbf{C}_x\text{ such that }\{u,v\}\in E\}\) is the exterior boundary of \(\mathbf{C}_x.\) We say a cluster \(\mathbf{C}_{x_i}\) is a plus crossing cluster if it connects \(\partial \Lambda_{M_2}\) and \(\partial \Lambda_{M_1+1}\).

Define \(\tau=\min\{i: \mathbf{C}_{x_i} \text{ is a plus crossing cluster}\}.\) If there is no plus crossing cluster, we set \(\tau=\infty\). Define \[\mathbf{B}=\bigcup_{i=1}^{{\min\{\tau,n\}}} \mathbf{B}_{x_i}, \quad \mathbf{C}=\bigcup_{i=1}^{{\min\{\tau,n\}}} \mathbf{C}_{x_i}.\] Note that although \(\partial_{\mathrm{ext}} \mathbf{C}\cap\Lambda_{M_1,M_2}\subset\mathbf{B}\), the inclusion may be strict (recall that \(\mathbf{B}_x=\{x\}\) if \(\mathbf{C}_x=\emptyset\)). Note also that the points in \(\mathbf{B}\) have spin \(-1\). Let \(\mathcal{S}(M_1,M_2)\) denote the event that there exists a plus spin cluster connecting \(\partial \Lambda_{M_2}\) and \(\partial \Lambda_{M_1+1}\) but containing no circuit in \(\Lambda_{M_1,M_2}\). In addition, for any deterministic set \(\mathsf B,\mathsf C\subset\Lambda_{M_1,M_2},\) let \(\mathcal{D}(M_1,M_2,\Lambda_{M_1,M_2}\setminus( \mathsf B\cup \mathsf C))\) denote the event that there exists a plus spin cluster connecting \(\partial \Lambda_{M_2}\) and \(\partial \Lambda_{M_1+1}\) in \(\Lambda_{M_1,M_2}\setminus( \mathsf B\cup \mathsf C)\).

We claim that, on the event \(\mathtt{Con_2}(M_1,M_2)\cap\{\mathbf{B}=\mathsf B,\;\mathbf{C}=\mathsf C\},\) both \(\mathcal{S}(M_1,M_2)\) and \(\mathcal{D}\bigl(M_1,M_2,\\\Lambda_{M_1,M_2}\setminus(\mathsf B\cup \mathsf C)\bigr)\) occur. We first show that \(\mathcal{S}(M_1,M_2)\) occurs. Suppose, to the contrary, that \(\mathcal{S}(M_1,M_2)\) does not occur. Since \(\mathtt{Con_2}(M_1,M_2)\) occurs, there exists at least one plus crossing cluster in \(\Lambda_{M_1,M_2}\). By the failure of \(\mathcal{S}(M_1,M_2)\), every plus crossing cluster contains a circuit in \(\Lambda_{M_1,M_2}\). Let \(\mathcal{C}\) be one such cluster. The circuit contained in \(\mathcal{C}\) separates \(\partial\Lambda_{M_1+1}\) from \(\partial\Lambda_{M_2}\). Hence every path connecting \(\partial\Lambda_{M_1+1}\) to \(\partial\Lambda_{M_2}\) must intersect \(\mathcal{C}\). In particular, every plus crossing intersects \(\mathcal{C}\), and therefore belongs to the same plus cluster as \(\mathcal{C}\). Thus, there is only one plus crossing cluster, contradicting the occurrence of \(\mathtt{Con_2}(M_1,M_2)\). This proves that \(\mathcal{S}(M_1,M_2)\) occurs.

It remains to show that \(\mathcal{D}\bigl(M_1,M_2,\Lambda_{M_1,M_2}\setminus(\mathsf B\cup \mathsf C)\bigr)\) occurs. Since \(\mathtt{Con_2}(M_1,M_2)\) occurs, we have \(\tau\leq n\). By the definition of \(\tau\), the cluster \(\mathbf{C}_{x_\tau}\) is the first plus cluster, in the prescribed ordering of \(\partial\Lambda_{M_1+1}\), which connects \(\partial\Lambda_{M_1+1}\) to \(\partial\Lambda_{M_2}\). On the event \(\{\mathbf{B}=\mathsf B,\;\mathbf{C}=\mathsf C\}\), the set \(\mathsf C\) consists of all plus clusters explored up to and including this first plus crossing cluster, while \(\mathsf B\) contains their exterior boundary in \(\Lambda_{M_1,M_2}\). Since \(\mathtt{Con_2}(M_1,M_2)\) occurs, there is another plus crossing cluster, distinct from \(\mathbf{C}_{x_\tau}\). This cluster is disjoint from \(\mathsf C\) and cannot intersect \(\mathsf B\), since all points of \(\mathsf B\) have spin \(-1\). Therefore, this additional plus crossing is contained in \(\Lambda_{M_1,M_2}\setminus(\mathsf B\cup \mathsf C).\) Consequently, \(\mathcal{D}\bigl(M_1,M_2,\Lambda_{M_1,M_2}\setminus(\mathsf B\cup \mathsf C)\bigr)\) occurs.

In conclusion, we have the following relations \[\begin{align} \mathtt{Con_2}(M_1,M_2)\subset\bigcup\limits_{\mathsf B,\mathsf C\subset \Lambda_{M_1,M_2}} \{\mathbf{B}=\mathsf B, \mathbf{C}=\mathsf C, \tau\leq n\}&\cap \mathcal{D}({M_1,M_2}, \Lambda_{M_1,M_2}\setminus( \mathsf B\cup \mathsf C)),\nonumber\\ \text{and }\bigcup\limits_{ \mathsf B,\mathsf C\subset \Lambda_{M_1,M_2}} \{\mathbf{B}=\mathsf B, \mathbf{C}=\mathsf C, \tau\leq n\}&\subset\mathcal{D}(M_1,M_2,\Lambda_{M_1,M_2}).\nonumber \end{align}\] We point out that in the unions above, some of the events of the form \(\{\mathbf{B}=\mathsf B, \mathbf{C}=\mathsf C, \tau\leq n\}\) are empty. In light of this, we say a pair of deterministic sets \((\mathsf B,\mathsf C)\) is valid if the event \(\{\mathbf{B}=\mathsf B, \mathbf{C}=\mathsf C, \tau\leq n\}\) is not empty, and there does not exist a connected circuit \(\gamma\) in \(\mathsf C\) that separates the inner and outer boundary of \(\Lambda_{M_1,M_2}\). Then we get that\[\begin{align} \mathtt{Con_2}(M_1,M_2)\subset\bigcup\limits_{\text{valid }\mathsf B,\mathsf C\subset \Lambda_{M_1,M_2}} \{\mathbf{B}=\mathsf B, \mathbf{C}=\mathsf C, \tau\leq n\}&\cap \mathcal{D}({M_1,M_2}, \Lambda_{M_1,M_2}\setminus( \mathsf B\cup \mathsf C)),\tag{165}\\ \text{and }\bigcup\limits_{\text{valid }\mathsf B,\mathsf C\subset \Lambda_{M_1,M_2}} \{\mathbf{B}=\mathsf B, \mathbf{C}=\mathsf C, \tau\leq n\}&\subset\mathcal{S}(\Lambda_{M_1,M_2}).\tag{166} \end{align}\] In addition, for each valid pair \((\mathsf B, \mathsf C)\), the event \(\{\mathbf{B}=\mathsf B, \mathbf{C}=\mathsf C, \tau\leq n\}\) is equivalent to \(\{\mathsf B\subset \mathsf{Plus}^c, \mathsf C\subset \mathsf{Plus}\}\), where \(\mathsf{Plus}\) denotes the set of plus spins.

Then we calculate that \[\begin{align} &\mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}(\mathtt{Con_2}(M_1,M_2))\stackrel{\eqref{eq:32inclusion32relation32in32BK321}}{\le}\sum_{\text{valid }\mathsf B,\mathsf C\subset \Lambda_{M_1,M_2}}\mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}(\sigma|_{\mathsf B}=-1,\sigma|_{\mathsf C}=1)\nonumber\\ &\times\mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}\Big(\mathcal{D}(M_1,M_2,\Lambda_{M_1,M_2}\setminus(\mathsf B\cup \mathsf C))~\Big|~ \sigma|_{\mathsf B}=-1,\sigma|_{\mathsf C}=1\Big). \label{decomposition-of-exploration} \end{align}\tag{167}\] To control the right-hand side of 167 , we claim that the following inequalities hold for any valid pair \(\mathsf{B},\mathsf{C}\), \[\begin{align} \mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}\Big(\mathcal{S}(M_1,M_2)\Big)&\le C_1(\frac{M_2}{M_1})^{-\frac{1}{8}},\tag{168}\\ \mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}\Big(\mathcal{D}\big(M_1,M_2, \Lambda_{M_1,M_2}\setminus( \mathsf B\cup \mathsf C)\big)\Big)&~\Big|~ \sigma|_{\mathsf B}=-1,\sigma|_{\mathsf C}=1)\le C_2(\frac{M_2}{M_1})^{-\frac{1}{8}}.\tag{169} \end{align}\] We first show how we proceed assuming 168 and 169 . Plugging 168 and 169 into 167 , we get that \[\begin{align} \mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}(\mathtt{Con_2}(M_1,M_2))&\stackrel{\eqref{eq:32FK32dual32path32one32arm32after32exploration}}{\le}\sum_{\text{valid }\mathsf B,\mathsf C\subset \Lambda_{M_1,M_2}}\mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}(\sigma|_{\mathsf B}=-1,\sigma|_{\mathsf C}=1)\times C_2(\frac{M_2}{M_1})^{-\frac{1}{8}}\nonumber\\ &\stackrel{(*)}{\le} \mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}\Big(\mathcal{S}(M_1,M_2)\Big)\times C_2(\frac{M_2}{M_1})^{-\frac{1}{8}} \stackrel{\eqref{eq:32FK32dual32path32one32arm}}{\le}C_1C_2(\frac{M_2}{M_1})^{-\frac{1}{4}}. \label{eq:32alternating32weak32four32arm32final} \end{align}\tag{170}\] In \((*)\), we used 166 and the fact that for valid pair \(\mathsf B,\mathsf C\subset \Lambda_{M_1,M_2}\), the event \(\{\sigma|_{\mathsf B}=-1,\sigma|_{\mathsf C}=1\}\) is equivalent to \(\{\mathbf{B}=\mathsf B, \mathbf{C}=\mathsf C, \tau\leq n\}\). Next, we turn to the proofs of 168 and 169 . ◻

Proof of 168 . Recall that \(\mathcal{S}(M_1,M_2)\) denotes the event that there exists a plus spin cluster connecting \(\partial \Lambda_{M_2}\) and \(\partial \Lambda_{M_1+1}\) but containing no circuit in \(\Lambda_{M_1,M_2}\). Let \(C_1>0\) be a sufficiently large constant to be chosen later, and let \(M_3=C_1M_1\) and \(M_4=\frac{M_2}{C_1}\). We further restrict that \(M_4>M_3\) (otherwise \(M_2<C_1^2M_1\) and Lemma 28 holds for free by taking \(c_8>C_1^8.\)). In order to prove 168 , we first apply Corollary 5 to remove the dependence on the boundary conditions \(\xi,\nu.\) By Corollary 5, we obtain that \[\begin{align} \mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}\Big(\mathcal{S}(M_1,M_2)\Big)&~~~\le \mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}\Big(\mathcal{S}(M_3,M_4)\Big)\nonumber\\&\stackrel{\text{Cor }~\ref{cor:32spatial32mixing32of32Ising32model}}{\le} \Big(\frac{1}{1-2cC_1^{-\frac{1}{8}}}\Big)\mu^{-,-}_{\Lambda_{M_1,M_2},0}\Big(\mathcal{S}(M_3,M_4)\Big).\label{eq:32boundary32elimination32in32BK} \end{align}\tag{171}\] Let \(\mathcal{D}^*(M_3,M_4)\) denote the event that there exists an FK dual path in \(\Lambda_{M_3,M_4}\) connecting \(\partial\Lambda_{M_3+1}\) and \(\partial\Lambda_{M_4}\). If \(\mathcal{D}^*(M_3,M_4)\) does not occur, then there exists an FK cluster \(\mathcal{C}\) containing a circuit in \(\Lambda_{M_3,M_4}\) and thus any spin crossing in \(\Lambda_{M_3,M_4}\) will intersect \(\mathcal{C}\). Thus, every plus crossing cluster in \(\Lambda_{M_3,M_4}\) contains a circuit in \(\Lambda_{M_3,M_4}\). Therefore, we get that \[\mathcal{S}(M_3,M_4)\subset\mathcal{D}^*(M_3,M_4).\label{eq:32Ising32crossing32change32to32FK32crossing}\tag{172}\] Combined with duality, we get that \[\begin{align} \mu^{-,-}_{\Lambda_{M_1,M_2}}\Big(\mathcal{S}(M_3,M_4)\Big)\le \phi^{\mathrm{w},\mathrm{w}}_{\Lambda_{M_1,M_2}}\Big(\mathcal{D}^{*}(M_3,M_4)\Big)&=\phi^{\mathrm{f},\mathrm{f}}_{\Lambda_{M_1,M_2}}\Big(\partial\Lambda_{M_3+1}\leftrightarrow\partial\Lambda_{M_4}\Big) .\label{eq:32FK32dual32one32arm32with32free32boundary} \end{align}\tag{173}\] Let us now upper-bound the right-hand side of 173 . The FKG and the CBC properties yield that \[\begin{align} \phi^{\mathrm{f}}_{\Lambda_{M_2}}(o\leftrightarrow \partial\Lambda_{M_4})&~\ge \phi^{\mathrm{f}}_{\Lambda_{M_2}}\big(o\leftrightarrow \partial\Lambda_{M_3},~\partial\Lambda_{M_3+1}\leftrightarrow\partial\Lambda_{M_4}\big)\nonumber\\&\stackrel{\mathrm{FKG}}{\ge} \phi^{\mathrm{f}}_{\Lambda_{M_2}}\big(o\leftrightarrow \partial\Lambda_{M_3}\big)\cdot \phi^{\mathrm{f}}_{\Lambda_{M_2}}\big(\partial\Lambda_{M_3+1}\leftrightarrow\partial\Lambda_{M_4}\big)\nonumber\\&\stackrel{\mathrm{CBC}}{\ge }\phi^{\mathrm{f}}_{\Lambda_{M_3}}\big(o\leftrightarrow \partial\Lambda_{M_3}\big)\cdot \phi^{\mathrm{f},\mathrm{f}}_{\Lambda_{M_1,M_2}}\big(\partial\Lambda_{M_3+1}\leftrightarrow\partial\Lambda_{M_4}\big).\label{eq:32FK32annulus32crossing32probability32step32one} \end{align}\tag{174}\] Plugging the one-arm event estimate 50 into 174 , we get that \[\phi^{\mathrm{f},\mathrm{f}}_{\Lambda_{M_1,M_2}}\big(\partial\Lambda_{M_3+1}\leftrightarrow\partial\Lambda_{M_4}\big)\le \frac{C_2M_4^{-\frac{1}{8}}}{C_3M_3^{-\frac{1}{8}}}=\frac{C_2}{C_3}(\frac{M_4}{M_3})^{-\frac{1}{8}}.\label{eq:32FK32dual32one32arm32with32free32boundary321}\tag{175}\] Combining 171 with 173 and 175 and then choosing \(C_1>0\) large enough such that \(2cC_1^{-\frac{1}{8}}\le \frac{1}{2}\), we finish the proof of 168 . ◻

Proof of 169 . Recall that \(\mathcal{D}(M_1,M_2,\Lambda_{M_1,M_2}\setminus( \mathsf B\cup \mathsf C))\) denotes the event that there exists a plus spin cluster connecting \(\partial \Lambda_{M_2}\) and \(\partial \Lambda_{M_1+1}\) in \(\Lambda_{M_1,M_2}\setminus( \mathsf B\cup \mathsf C)\). To prove 169 , we first fix a valid pair \(\mathsf B,\mathsf C\subset \Lambda_{M_1,M_2}\). Similar to the proof of 168 , we choose \(M_3=C_1M_1\) and \(M_4=\frac{M_2}{C_1}\) for some \(C_1>0\) to be chosen later and further restrict that \(M_4>M_3\). Since \(\mathcal{D}\big(M_3,M_4,\Lambda_{M_3,M_4} \setminus(\mathsf{B}\cup\mathsf{C})\big)\) is an increasing event, the DMP and CBC properties give that \[\begin{align} &\mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}\Big(\mathcal{D}\big(M_1,M_2, \Lambda_{M_1,M_2}\setminus( \mathsf B\cup \mathsf C)\big)~\Big|~ \sigma|_{\mathsf B}=-1, \sigma|_{\mathsf C}=1\Big)\nonumber\\\stackrel{\text{DMP}}{=}&\mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}\Big(\mathcal{D}\big(M_1,M_2, \Lambda_{M_1,M_2}\setminus( \mathsf B\cup \mathsf C)\big)~\Big|~ \sigma|_{\mathsf B}=-1\Big)\nonumber\\\stackrel{\text{CBC}}{\le}&\mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}\Big(\mathcal{D}\big(M_3,M_4,\Lambda_{M_3,M_4}\setminus(\mathsf{B}\cup\mathsf{C})\big)~\Big|~\sigma|_{\mathsf B\cap \Lambda_{M_3,M_4}}=-1\Big)\label{eq:32FK32dual32one32arm32CBC32to32a32smaller32region}. \end{align}\tag{176}\] Note that both \(\mathcal{D}\big(M_3,M_4,\Lambda_{M_3,M_4}\setminus(\mathsf{B}\cup\mathsf{C})\big)\) and \(\sigma_{\mathsf B\cap \Lambda_{M_3,M_4}}=-1\) are supported on \(\Lambda_{M_3,M_4}\). Corollary 5 implies that \[\begin{align} &\mu^{\nu,\xi}_{\Lambda_{M_1,M_2}}\Big(\mathcal{D}\big(M_3,M_4,\Lambda_{M_3,M_4}\setminus(\mathsf{B}\cup\mathsf{C})\big)~\Big|~\sigma|_{\mathsf B\cap \Lambda_{M_3,M_4}}=-1\Big)\nonumber\\\le~&\frac{1}{1-2cC_1^{-\frac{1}{8}}} \mu^{-,-}_{\Lambda_{M_{1},M_{2}}}\Big(\mathcal{D}\big(M_3,M_4,\Lambda_{M_3,M_4}\setminus(\mathsf{B}\cup\mathsf{C})\big)~\Big|~\sigma|_{\mathsf B\cap \Lambda_{M_3,M_4}}=-1\Big).\label{eq:32FK32dual32one32arm32CBC32to32a32smaller32region322} \end{align}\tag{177}\] As in the proof of 168 , we define \(\mathcal{D}^*\big(M_3,M_4,\Lambda_{M_3,M_4} \setminus(\mathsf{B}\cup\mathsf{C})\big)\) to be the event that there exists a FK dual path connecting \(\partial\Lambda_{M_3+1}\) and \(\partial\Lambda_{M_4}\) in \(\Lambda_{M_3,M_4}\setminus(\mathsf{B}\cup\mathsf{C}).\) By the same argument as in 172 , we get that under minus boundary condition on \(\mathsf B\cap \Lambda_{M_3,M_4}\), \[\mathcal{D}\big(M_3,M_4,\Lambda_{M_3,M_4} \setminus(\mathsf{B}\cup\mathsf{C})\big)\subset\mathcal{D}^*\big(M_3,M_4,\Lambda_{M_3,M_4} \setminus(\mathsf{B}\cup\mathsf{C})\big).\nonumber\] Thus, we have that \[\begin{align} &\mu^{-,-}_{\Lambda_{M_{1},M_{2}}}\Big(\mathcal{D}\big(M_3,M_4,\Lambda_{M_3,M_4}\setminus(\mathsf{B}\cup\mathsf{C})\big)~\Big|~\sigma|_{\mathsf B\cap \Lambda_{M_3,M_4}}=-1\Big)\nonumber\\\le~& \phi^{\mathrm{w},\mathrm{w}}_{\Lambda_{M_{1},M_{2}}\setminus(\mathsf{B}\cup\mathsf{C})}\Big(\mathcal{D}^*\big(M_3,M_4,\Lambda_{M_3,M_4}\setminus(\mathsf{B}\cup\mathsf{C})\big)\Big)\label{eq:32FK32dual32one32arm32CBC32to32a32smaller32region323}. \end{align}\tag{178}\] By duality and the CBC property of the FK measure, we get that \[\begin{align} &\phi^{\mathrm{w},\mathrm{w}}_{\Lambda_{M_1,M_2}\setminus(\mathsf{B}\cup\mathsf{C})}\Big(\mathcal{D}^*\big(M_3,M_4,\Lambda_{M_3,M_4}\setminus(\mathsf{B}\cup\mathsf{C})\big)\Big)\nonumber= \phi^{\mathrm{f},\mathrm{f}}_{\Lambda_{M_1,M_2}\setminus(\mathsf{B}\cup\mathsf{C})}\Big(\partial\Lambda_{M_4}\leftrightarrow\partial\Lambda_{M_3+1}\Big)\\\stackrel{\text{CBC}}{\le }&\phi^{\mathrm{f},\mathrm{f}}_{\Lambda_{M_{1},M_{2}}}\Big(\partial\Lambda_{M_4}\leftrightarrow\partial\Lambda_{M_3+1}\big)\Big)\stackrel{\eqref{eq:32FK32dual32one32arm32with32free32boundary321}}{\le} C_4(\frac{M_4}{M_3})^{-\frac{1}{8}}.\label{eq:32FK32dual32one32arm32CBC32to32a32smaller32region324} \end{align}\tag{179}\] Combining 176 , 177 and 178 with 179 , we get 169 . ◻

Proof. We first deal with the case that \(\mathcal{W}=\mathcal{X}^+.\) As explained before Lemma 28, we want to construct a \(\mathtt{Con_2}\) event to control the probability on the left-hand side of 180 . Since \(\mathcal{C}_x\) is local, we can assume that \(\mathcal{C}_x\cap\partial\Lambda_{2^t}(x)=\emptyset\) and \(\mathcal{C}_x\cap\partial\Lambda_{2^{t-1}}(x)\neq\emptyset\), for some integer \(t\leq \lceil\log_2(d(x)/10)\rceil\). Let \(\sigma\in\{-1,1\}^{\Lambda_M}\) denote the configuration associated with \(s_x=1\) and \(\hat{\sigma}\in\{-1,1\}^{\Lambda_M}\) denote the configuration associated with \(s_x=-1\), where \(s_x\) is the sign of the cluster \(\mathcal{C}_x\). Therefore we have \[\sigma|_{ \Lambda_M\setminus\Lambda_{d(x)/10}(x)}=\hat{\sigma}|_{ \Lambda_M\setminus\Lambda_{d(x)/10}(x)}=\xi, \quad \sigma|_{ \Lambda_M\setminus\Lambda_{2^t}(x)}=\hat{\sigma}|_{ \Lambda_M\setminus\Lambda_{2^t}(x)}.\] Let \(\mathsf{Plus}(\sigma)\) denote the set of plus spins in \(\sigma\). Then \(\mathsf{Plus}(\hat{\sigma})=\mathsf{Plus}(\sigma)\setminus\mathcal{C}_x.\) Recall \(\mathcal{P}_0\) from Definition 5. Let \(\mathsf{CP}\) denote the connected component of \(\mathcal{P}_0\) in \(\mathsf{Plus}(\sigma).\) Then the connected component of \(\mathcal{P}_0\) in \(\mathsf{Plus}(\hat{\sigma})=\mathsf{Plus}(\sigma)\setminus\mathcal{C}_x\) is a subset of \(\mathsf{CP}\setminus \mathcal{C}_x.\)

Recall that \(\mathcal{X}^+\) is the event that there exists a plus circuit in \(\Lambda_{M/2}\setminus\Lambda_{M/4}\) that connects to \(\mathcal{P}_0\). Since \(\mathcal{X}^+\) is an increasing event, flipping the sign of \(\mathcal{C}_x\) can affect \(X^+\) only if \(\mathcal{X}^+\) holds when \(s_x=1\) but fails when \(s_x=-1\). Therefore, \(\mathsf{Plus}(\sigma)\) contains a plus circuit in \(\Lambda_{M/2}\setminus\Lambda_{M/4}\) connected to \(\mathcal{P}_0,\) hence \(\mathsf{CP}\) contains a plus circuit in \(\Lambda_{M/2}\setminus\Lambda_{M/4}\). But we also require that \(\mathsf{Plus}(\hat{\sigma})\) does not contain such circuit, so there are two possible cases:

• At least one point \(v\in\mathsf{CP}\setminus\mathcal{C}_x\) is not connected to \(\mathcal{P}_0\). Assume \(\pi=[v=v_0,v_1,\cdots,v_L]\) is a path in \(\mathsf{CP}\) from \(v_0\) to \(\mathcal{P}_0\). (If multiple such paths exist, select the first one according to a pre-fixed order.) Therefore we know \(\pi\not\subset \mathsf{CP}\setminus \mathcal{C}_x.\) Thus \(\pi\cap\Lambda_{2^t}(x)\neq \emptyset.\) This motivates us to define the following entrance time and exit time through \(\Lambda_{2^t}(x)\): \[l_2=\min\{1\leq l\leq L: v_l\in\partial\Lambda_{2^t}(x)\}, andl_1=\max\{1\leq l<l_2: v_l\in\partial\Lambda_{d(x)/10}(x)\};\] \[l_3=\max\{1\leq l\leq L: v_l\in\partial\Lambda_{2^t}(x)\}, andl_4=\min\{l_3\leq l\leq L: v_l\in\partial\Lambda_{d(x)/10}(x)\}.\] Here we recall that \(2^t\leq d(x)/10\). From the above definitions, we see that \(1\leq l_1 < l_2 < l_3 < l_4 \leq L\). We further define \[\pi_1=\{v_{l_1},v_{l_1+1},\cdots,v_{l_2}\}, \quad \pi_2=\{v_{l_3}, v_{l_3+1},\cdots,v_{l_4}\},\] which correspond to the ‘first’ and ‘last’ plus crossings of \(\pi\) in the annulus \(\Lambda_{d(x)/10}(x)\setminus\Lambda_{2^t}{(x)}.\) We claim that \(\pi_1\) and \(\pi_2\) are not connected through plus spins in \(\mathsf{CP}\cap\big(\Lambda_{d(x)/10}(x)\setminus\Lambda_{2^t}{(x)}\big) .\) Otherwise, this yields a path from \(v\) to \(\mathcal{P}_0\) that avoids \(\Lambda_{2^t}(x)\). Hence, \(v\) is connected to \(\mathcal{P}_0\) in \(\mathsf{CP}\setminus \mathcal{C}_x\), contradicting the definition of \(v\). See Figure 1 for an illustration.

• Every point in \(\mathsf{CP}\setminus\mathcal{C}_x\) is connected to \(\mathcal{P}_0,\) then \(\mathcal{X}^+\) fails only if \(\mathsf{CP}\setminus\mathcal{C}_x\) contains no plus circuit in \(\Lambda_{M/2}\setminus\Lambda_{M/4}\). Let \(\gamma\) denote the plus circuit in \(\mathsf{CP}\cap (\Lambda_{M/2}\setminus\Lambda_{M/4}).\) (If multiple such circuits exist, select the first one according to a pre-fixed order.) Assume \(\gamma=[v_0,v_1,v_2,\cdots,v_m=v_0]\) such that \(v_{i-1}\sim v_i\) for \(i=1,2,\cdots,m.\) Since \(v_0\) could be chosen arbitrarily in the circuit, we could assume that \(v_0\not\in\Lambda_{d(x)/10}(x).\) The rest is similar to the first case.

Let \(\mathtt L_t\) be the event that \(\{\mathcal{C}_x\cap\partial\Lambda_{2^t}(x)=\emptyset\}\cap \{ \mathcal{C}_x\cap\partial\Lambda_{2^{t-1}}(x)\neq\emptyset\}\). We conclude from the above discussion that \[\{\mathcal{C}_x\text{ is pivotal to } \mathcal{X}^+\}\subset\bigcup_{t=1}^{\lceil\log_2(d(x)/10)\rceil}\mathtt L_t \cap \mathtt{Con_2}(2^t, d(x)/10).\label{eq:decomposition-of-pivotal32cle32for32plus32circuit}\tag{181}\] Let \(\mathscr{B}_t\) denote the collection of boundary conditions on \(\partial\Lambda_{2^t}(x)\), i.e., \(\mathscr{B}_t=\{1,-1\}^{\partial\Lambda_{2^t}(x)}\). Here, notice that \(\mathtt L_t\) is measurable with respect to the configurations on \(\Lambda_{2^t}\). For \(\nu\in \mathscr{B}_t\), let \[\Psi_t(\nu):=\mu^{\xi}_{\Lambda_{d(x)/10}(x)}\Big(\sigma|_{\partial\Lambda_{2^t}(x)}=\nu\, \Big).\] Applying the DMP and Lemma 28, we obtain that \[\begin{align} &\sum_{t=1}^{\lceil\log_2(d(x)/10)\rceil}\bar\mu^{\xi}_{\Lambda_{d(x)/10}(x)}(\mathtt L_t\cap \mathtt{Con_2}(2^t,d(x)/10))\nonumber\\ {=}& \sum_{t=1}^{\lceil\log_2(d(x)/10)\rceil}\sum_{\nu\in \mathscr{B}_t}\Psi_t(\nu)\bar\mu_{\Lambda_{2^t}(z)}^{\nu}(\mathtt L_t)\times\mu_{\Lambda_{2^t,d(x)/10}(z)}^{\nu,\xi}\big(\mathtt{Con_2}(2^t,d(x)/10)\big)\nonumber~~~\text{(by DMP)}\\ {\le} &\sum_{t=1}^{\lceil\log_2(d(x)/10)\rceil}\left[\sum_{\nu\in \mathscr{B}_t}\Psi_t(\nu)\times\bar\mu_{\Lambda_{2^t}(z)}^{\nu}(\mathtt L_t)\right]\times C_1(\frac{d(x)}{2^t})^{-\frac{1}{4}}\nonumber~~~(\text{by Lemma \ref{lem:32BK32for32Ising32weak32four32arm}}), \end{align}\] where \(\mu_{\Lambda_{2^t,d(x)/10}(z)}^{\nu,\xi}(\mathtt{Con_2}(2^t,d(x)/10)):=1\) if \(2^t\ge d(x)/10\). Applying the total law of probability and the change from \(\bar\mu^{\xi}_{\Lambda_{d(x)/10}(x)}\) to \(\phi^{\xi}_{\Lambda_{d(x)/10}(x)}\), we obtain that \[\begin{align} &\sum_{t=1}^{\lceil\log_2(d(x)/10)\rceil}\left[\sum_{\nu\in \mathscr{B}_t}\Psi_t(\nu)\times\bar\mu_{\Lambda_{2^t}(z)}^{\nu}(\mathtt L_t)\right]\times C_1(\frac{d(x)}{2^t})^{-\frac{1}{4}}\nonumber\\ =~&\sum_{t=1}^{\lceil\log_2(d(x)/10)\rceil}\phi_{\Lambda_{d(x)/10}(x)}^{\xi}(\mathtt L_t)\times C_1(\frac{d(x)}{2^t})^{-\frac{1}{4}}\le C_2\sum_{t=1}^{\lceil\log_2(d(x)/10)\rceil}2^{-\frac{t}{8}}\times (\frac{d(x)}{2^t})^{-\frac{1}{4}}\le C_3 d(x)^{-\frac{1}{8}}.\nonumber \end{align}\] Thus, we complete the proof of the Lemma 29 for \(\mathcal{W}=\mathcal{X}^+.\)

Now we explain how to deal with the other cases when \(\mathcal{W}\in\{\mathcal{X}^-,\mathcal{Y}^+,\mathcal{Y}^-\}.\) To this end, we claim that 180 actually holds for a large class of events \(\mathcal{W}\). Let \(\mathscr P_0\) denote the set of events that two disjoint subsets of \(\partial\Lambda_M\cup\Lambda_{M/2}\cup\Lambda_{M/4}\) are connected by plus spins, i.e., \[\mathscr P_0=\big\{\{A{\leftrightarrow} B\}:A,B\subset\bar{\partial}\Lambda_M=\partial\Lambda_M\cup\partial\Lambda_{M/2}\cup\partial\Lambda_{M/4}, A\cap B=\emptyset \big\}.\] Moreover, define \[\mathscr P=\{\bigcap_{i=1}^n\mathcal{A}_i:n<\infty, \mathcal{A}_i\in\mathscr P_0for 1\leq i\leq n\}\] to be the finite intersection of events in \(\mathscr P_0.\)

Note that all the events in \(\mathscr P_0\) are increasing, and therefore so are events in \(\mathscr P.\) For any \(\mathcal{W}=\bigcap_{i=1}^n \mathcal{A}_n\in\mathscr P,\) where \(\mathcal{A}_i=\{A_i{\leftrightarrow} B_i\}\), if \(\mathcal{C}_x\) is pivotal to \(\mathcal{W}\), then \(\mathcal{W}\) holds with \(s_x=1\) and fails with \(s_x=0.\) Let \(\mathcal{A}_j\) be the first one (according to some fixed order on \(\mathscr P_0\)) among events \(\mathcal{A}_1\) to \(\mathcal{A}_n\) such that \(\mathcal{C}_x\) is pivotal to \(\mathcal{A}_j\) and \(\gamma\) is the first path (according to some fixed order on paths) from \(A_j\) to \(B_j\) such that all the spins on \(\gamma\) are plus if \(s_x=1\). We could follow the arguments (and notations) above to show that the event \(\mathsf{Con}_2(2^t,d(x)/10)\) occurs and thus the desired bound holds.

Now recall that \[(\mathcal{X}^-)^c=\{\partial\Lambda_{M/2}{\leftrightarrow}\partial\Lambda_{M/4}\},\quad \mathcal{Y}^+=\bigcap_{U\in\mathcal{U}}\{U{\leftrightarrow}\mathcal{P}_0\},\quad (\mathcal{Y}^-)^c= \{\mathcal{P}_1{\leftrightarrow}\mathcal{P}_0\} ,\] therefore, \((\mathcal{X}^-)^c,\mathcal{Y}^+,(\mathcal{Y}^-)^c\in\mathscr P.\) Noting that by definition \(\mathcal{C}_x\) being pivotal to \(\mathcal{W}\) is equivalent to \(\mathcal{C}_x\) being pivotal to \(\mathcal{W}^c\) completes the proof. ◻

Proof of Proposition 35 given Lemma 30. We just prove ?? since ?? is essentially the same. Recall the definition of \(\Xi_I'\) from Corollary 4 and recall that \(R=\mathbf{1}_{\mathcal{R}}\). We get from Lemma 27 that \[\begin{align} \label{eq:proof-of-Etau-3} \Big|\langle \sigma^I\cdot R\rangle_{\Lambda_{M}}^{\xi}\Big|=&\Big|\langle \sigma^I\cdot R\cdot \mathbf{1}_{\mathtt{NoEx}}\rangle_{\Lambda_{M}}\Big|\le \langle R\cdot \mathbf{1}_{\mathtt{NoEx}}\rangle_{\Lambda_{M}}\nonumber \\ \leq & \sum_{\bar\eta_I\in\Xi_I'}\bar\mu^{\xi}_{\Lambda_{M}}(\bar\eta_I)\cdot \mu^{\xi,\bar\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(\mathcal{R}). \end{align}\tag{185}\]

Combining Lemma 30 and Corollary 4 with 185 , we have that \[\begin{align} \Big|\langle \sigma^I\cdot R\rangle_{\Lambda_{M}}^{\xi}\Big|&\leq C^{|I|}\mu^{\xi}_{\Lambda_{M}}(\mathcal{R})\sum_{\eta_I\in\Xi_I'}\bar\mu^{\xi}_{\Lambda_{M}}(\eta_I)\le C^{|I|}\langle R\rangle_{\Lambda_{M}}^{\xi}\cdot F(\bar\partial\Lambda_{M},I),\label{eq:byproduct} \end{align}\tag{186}\] This completes the proof by recalling the definition that \(\langle \sigma^I\mid \mathcal{R}\rangle_{\Lambda_{M}}^{\xi}=\frac{\langle \sigma^I\cdot R\rangle_{\Lambda_{M}}^{\xi}}{\langle R\rangle_{\Lambda_{M}}^{\xi}}.\) ◻

Proof. For any \(x\in I\), let \(\mathcal{A}_x\) denote the event that there exists a plus circuit in the annulus \(\Lambda_{\frac{d_x}{8}}(x)\setminus\Lambda_{\frac{d_x}{9}}(x)\) and let \(\mathcal{A}=\cap_{x\in I }\mathcal{A}_x.\) By FKG and RSW, we have that \[\mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta _I}(\mathcal{A})\geq \prod_{x\in I} \mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta _I}(\mathcal{A}_x)\ge C_1^{|I|}.\label{eq:32plus32circuit32event32probability32O}\tag{187}\] On the event \(\mathcal{A}\), we can explore the innermost plus circuit in each annulus \(\Lambda_{\frac{d_x}{8}}(x)\setminus\Lambda_{\frac{d_x}{9}}(x)\) and denote this collection of circuits by \(\gamma=\{\gamma_x\}_{x\in I}.\) Let \(\mathcal{O}_\gamma\) denote the event that the exploration outcome is \(\gamma\) and \(\mathscr O\) denote all the possible exploration outcomes \(\gamma\). Then we have that \[\label{eq:32plus32circuit32event32O32in32each32box32decomposition} \begin{align} \mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta _I}(\mathcal{A})&=\sum_{\gamma\in\mathscr O} \mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta _I}(\mathcal{O}_\gamma),\\ and \mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta _I}(\mathcal{V}_i\cap\mathcal{A}) &=\sum_{\gamma\in\mathscr O}\mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta_I}(\mathcal{V}_i\cap\mathcal{O}_\gamma). \end{align}\tag{188}\] Let \(\Gamma=\cup_{x\in I}\Gamma_x\) where \(\Gamma_x\) is the interior of \(\gamma_x\). Note that on the event \(\mathcal{O}_\gamma,\) all the spins on \(\gamma\) are positive. In addition, \(\mathcal{V}_i\) depends only on the connectivity of plus spins and thus, conditioned on \(\{\sigma_v=1,\forall v\in\cup_{x\in I}\gamma_x\},\) it does not depend on the configuration in \(\Gamma\setminus(\cup_{x\in I}\gamma_x)\). Then we get that \[\mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta _I}(\mathcal{V}_i\mid \mathcal{O}_\gamma)=\mu_{\Lambda_M\setminus\Gamma}^{\xi,+}(\mathcal{V}_i)= \mu_{\Lambda_M}^{\xi}(\mathcal{V}_i\mid \sigma_v=1,\forall v\in\cup_{x\in I}\gamma_x)\label{eq:32plus32circuit32boundary32reduction32in32O}\tag{189}\] where the equality comes from the fact that \(\mu_{\Lambda_M\setminus\Gamma}^{\xi,+}\) and \(\mu_{\Lambda_M}^{\xi}(\cdot\mid \sigma_v=1,\forall v\in\cup_{x\in I}\gamma_x)\) have the same law on \(\Lambda_M\setminus\Gamma.\) FKG and RSW yield that \[\mu_{\Lambda_M}^{\xi}(\mathcal{V}_i\mid \{\sigma_v=1,\forall v\in\cup_{x\in I}\gamma_x\}\stackrel{\mathrm{FKG}}{\geq}\mu_{\Lambda_M}^{\xi}(\mathcal{V}_i)\stackrel{\mathrm{RSW}}{\geq} C_2.\] Plugging into 189 , we get that \[\mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta _I}(\mathcal{V}_i\mid \mathcal{O}_\gamma)\ge C_2.\label{eq:32Vi32prob32conditioned32on32plus32circuit32O}\tag{190}\] Combining 187 , 188 and 190 , we obtain that \[\begin{align} \mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta _I}(\mathcal{V}_i)\geq ~&\mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta _I}(\mathcal{V}_i\cap\mathcal{A}) \stackrel{\eqref{eq:32plus32circuit32event32O32in32each32box32decomposition}}{=}\sum_{\gamma\in\mathscr O}\mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta_I}(\mathcal{V}_i\cap\mathcal{O}_\gamma)\\=~&\sum_{\gamma\in\mathscr O}\mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta_I}(\mathcal{O}_\gamma)\mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta_I}(\mathcal{V}_i\mid \mathcal{O}_\gamma)\\ \stackrel{\eqref{eq:32Vi32prob32conditioned32on32plus32circuit32O}}{\ge} &C_2\sum_{\gamma\in\mathscr O}\mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta_I}(\mathcal{O}_\gamma)\stackrel{\eqref{eq:32plus32circuit32event32O32in32each32box32decomposition}}{=} C_2\mu_{\Lambda_M\setminus\mathrm{Ball}(I)}^{\xi,\eta _I}(\mathcal{A})\stackrel{\eqref{eq:32plus32circuit32event32probability32O}}{\ge} C_2\cdot C_1^{|I|}.\qedhere \end{align}\] ◻

Proof. Since \(\mathcal{I}\cap\mathcal{Y}^-\in\mathscr F_0,\) we can define \(\hat{\mathscr C_0}\subset\mathscr C_0\) to be the collection of \(A\) such that \(\{\mathcal{C}_0=A\}\subset\mathcal{I}\cap\mathcal{Y}^-\). Then we have that \[\mu^{\xi}_{\Lambda_{M}}(\{\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\}\cap\mathcal{I}\cap\mathcal{Y}^-)=\sum_{A\in \hat{\mathscr{C}_0}}\mu^{\xi}_{\Lambda_{M}}(\mathcal{C}_0=A) \cdot \mu^{\xi}_{\Lambda_{M}}(\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\mid \mathcal{C}_0=A).\label{eq:32event32E32localize32132with32interface}\tag{191}\] Recall the discussions before 189 . We want to apply a similar argument to the event \(\{\mathcal{C}_0=A\}.\) Note that all the spins on \(\partial_{\mathrm{ext}}\mathcal{C}_0\cap \Lambda_M=\partial_{\mathrm{ext}}A\cap \Lambda_M\) are minus and \(\mathcal{C}_1\) is the plus spin cluster, thus conditioned on \(\{\sigma_v=-1,\forall v\in\partial_{\mathrm{ext}}A\cap \Lambda_M\},\) the event \(\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\) does not depend on the configuration in \(A\setminus\partial A\). Therefore, for the same reason as 189 , we get that \[\mu^{\xi}_{\Lambda_{M}}(\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\mid \mathcal{C}_0=A)=\mu^{\xi}_{\Lambda_{M}}(\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\mid \sigma_v=-1,\forall v\in\partial_{\mathrm{ext}}A\cap \Lambda_M).\label{eq:32event32E32localize32232with32interface}\tag{192}\] Now, both the events \(\{\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\}\) and \(\{\sigma_v=-1,\forall v\in\partial_{\mathrm{ext}}A\cap \Lambda_M\}\) are decreasing, we can use FKG and RSW to obtain that \[\mu^{\xi}_{\Lambda_{M}}(\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\mid \sigma_v=-1,\forall v\in\partial_{\mathrm{ext}}A\cap \Lambda_M)\stackrel{\mathrm{FKG}}{\geq} \mu^{\xi}_{\Lambda_{M}}(\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset)\geq \mu^{\xi}_{\Lambda_{M}}(\mathcal{A}^-)\stackrel{\mathrm{RSW}}{\ge} C_1,\nonumber\] where \(\mathcal{A}^-\) denotes the event that there exists a minus circuit in \(\Lambda_{M}\setminus\Lambda_{0.95M}\), which implies \(\{\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\}\). ◻

Proof. Let \(\mathcal{A}'=\{\mathcal{C}_1\cap \Lambda_{0.95M}=\emptyset\}\). Note that the events \(\mathcal{A}'\) and \(\mathcal{Y}^-\) are decided by \(\mathcal{C}_1,\) i.e. \(\mathcal{A}',\mathcal{Y}^-\in \mathscr F_1\) (recall the definition of \(\mathscr F_1\) from Definition 9). Let \(\hat{\mathscr C}_1\) denote the set of \(A\) such that \(\{\mathcal{C}_1=A\}\subset \mathcal{A}'\cap \mathcal{Y}^-,\) then we have that \[\begin{align} \label{eq:bound-plus-crossing-given-Y-1} \mu^{\xi}_{\Lambda_M}(\mathcal{V}_1\cap \mathcal{Y}\cap \mathcal{A}') =\sum_{A\in\hat{\mathscr C_1}}\mu^{\xi}_{\Lambda_M}(\{\mathcal{C}_1=A\})\mu^{\xi}_{\Lambda_M}(\mathcal{V}_1\cap \mathcal{Y}^+\mid \mathcal{C}_1=A). \end{align}\tag{193}\] Similarly, we also have that \[\label{eq:bound-plus-crossing-given-Y-5} \mu_{\Lambda_M}^{\xi}(\mathcal{A}'\cap\mathcal{Y})=\sum_{A\in\hat{\mathscr C_1}}\mu^{\xi}_{\Lambda_M}(\{\mathcal{C}_1=A\})\mu^{\xi}_{\Lambda_M}(\mathcal{Y}^+\mid \{\mathcal{C}_1=A\}).\tag{194}\] Recall that all the spins on \(\partial_{\mathrm{ext}}\mathcal{C}_1\cap\Lambda_M\) are minus. By DMP and FKG, we have that \[\label{eq:bound-plus-crossing-given-Y-2} \mu^{\xi}_{\Lambda_M}(\mathcal{V}_1\cap \mathcal{Y}^+\mid \mathcal{C}_1=A)\stackrel{\mathrm{DMP}}{=}\mu^{\xi,-}_{\Lambda_M\setminus A}(\mathcal{V}_1\cap \mathcal{Y}^+)\stackrel{\mathrm{FKG}}{\geq} \mu^{\xi,-}_{\Lambda_M\setminus A}(\mathcal{V}_1)\mu^{\xi,-}_{\Lambda_M\setminus A}(\mathcal{Y}^+).\tag{195}\] Here, \((\xi,-)\) denotes the boundary condition which is \(\xi\) on \(\partial\Lambda_M\) and which is minus on \(\partial_{\mathrm{ext}}\mathcal{C}_1\cap\Lambda_M.\)

Note that \(\mathcal{V}_1\) is measurable with respect to the configuration inside \(\Lambda_{M/2}\), while every \(A\in\hat{\mathscr C}_1\) is contained in \(\Lambda_M\setminus\Lambda_{0.95M}\). Hence, by the mixing property of Lemma 38, \[\label{eq:bound-plus-crossing-given-Y-3} \mu^{\xi,-}_{\Lambda_M\setminus A}(\mathcal{V}_1)\geq C_1\mu^{\xi}_{\Lambda_M}(\mathcal{V}_1).\tag{196}\] Plugging 196 into 195 and then into 193 , we obtain that \[\begin{align} \mu^{\xi}_{\Lambda_M}(\mathcal{V}_1\cap \mathcal{Y}\cap \mathcal{A}')&~\geq~ \mu^{\xi,-}_{\Lambda_M\setminus A}(\mathcal{V}_1)\sum_{A\in\hat{\mathscr C_1}}\mu^{\xi}_{\Lambda_M}(\{\mathcal{C}_1=A\})\mu^{\xi,-}_{\Lambda_M\setminus A}(\mathcal{Y}^+)\nonumber\\&~\geq~C_1\mu^{\xi}_{\Lambda_M}(\mathcal{V}_1)\sum_{A\in\hat{\mathscr C_1}}\mu^{\xi}_{\Lambda_M}(\{\mathcal{C}_1=A\})\mu^{\xi,-}_{\Lambda_M\setminus A}(\mathcal{Y}^+).\label{eq:32V132Y32separation} \end{align}\tag{197}\] For the same reason as in 189 , we get that \(\mu^{\xi,-}_{\Lambda_M\setminus A}(\mathcal{Y}^+)=\mu^{\xi}_{\Lambda_M}(\mathcal{Y}^+\mid \{\mathcal{C}_1=A\}).\) Plugging into 197 and combining with 194 , we get that \[\begin{align} \mu^{\xi}_{\Lambda_M}(\mathcal{V}_1\cap \mathcal{Y}\cap \mathcal{A}') &~\ge C_1\mu^{\xi}_{\Lambda_M}(\mathcal{V}_1)\sum_{A\in\hat{\mathscr C_1}}\mu^{\xi}_{\Lambda_M}(\{\mathcal{C}_1=A\})\mu^{\xi}_{\Lambda_M}(\mathcal{Y}^+\mid \{\mathcal{C}_1=A\})\nonumber\\ &~=C_1\mu^{\xi}_{\Lambda_M}(\mathcal{V}_1)\mu_{\Lambda}^{\xi}(\mathcal{A}'\cap\mathcal{Y})\geq C_2\mu^{\xi}_{\Lambda_M}(\mathcal{V}_1)\mu_{\Lambda}^{\xi}(\mathcal{Y}),\label{eq:bound-plus-crossing-given-Y-4} \end{align}\tag{198}\] where the last inequality comes from choosing \(\mathcal{I}=\mathcal{Y}^+\) in Lemma 32 and recalling that \(\mathcal{A}'=\{\mathcal{C}_1\cap \Lambda_{0.95M}=\emptyset\}\). By RSW, we get that \(\mu^{\xi}_{\Lambda_M}(\mathcal{V}_1)\geq C_3\). Plugging into 198 and noting that \(\mu^{\xi}_{\Lambda_M}(\mathcal{V}_1\cap \mathcal{Y})\geq \mu^{\xi}_{\Lambda_M}(\mathcal{V}_1\cap \mathcal{Y}\cap \mathcal{A}')\) completes the proof of Lemma 33. ◻

Proof of 183 given Lemma 34.. Recall that \(\mathcal{R}=\mathcal{Y}\cap \{X=0\}.\) \[\begin{align} ~&\mu_{\Lambda_M}^{\xi,\eta_I}(\mathcal{R})=\mu_{\Lambda_M}^{\xi,\eta_I}(\{X=0\}\cap\mathcal{Y})\leq \mu_{\Lambda_M}^{\xi,\eta_I}(\mathcal{Y}) \stackrel{(*)}{\leq} c_{13}^{|I|}\mu_{\Lambda_M}^{\xi}(\mathcal{Y})\stackrel{(**)}{\leq} c_{12}\cdot c_{13}^{|I|}\mu_{\Lambda_M}^{\xi}(\mathcal{Y}\cap \mathcal{V}_1). \end{align}\] Here, (*) uses Lemma 34 and (**) uses Lemma 33. Combining with the fact that \(\mathcal{V}_1\subset\{X=0\}\), we get that \[\mu_{\Lambda_M}^{\xi,\eta_I}(\mathcal{R})\leq c_{12}\cdot c_{13}^{|I|}\mu_{\Lambda_M}^{\xi}(\mathcal{Y}\cap \{X=0\})\leq c_{9}^{|I|}\mu_{\Lambda_M}^{\xi}(\mathcal{R})\qedhere\] by choosing \(c_9\) relying on \(c_{12}\) and \(c_{13}\). ◻

Proof of Lemma 34. With the notations in Definition 10, it suffices to show that for \(1\le p\le k\) \[\label{eq:32two32rare32event32probability32with32interface32and32ball-induction32step} \mu_{\Lambda_M}^{\xi,\eta_I^p}(\mathcal{Y})\le c_9\mu_{\Lambda_M}^{\xi,\eta_I^{p-1}}(\mathcal{Y}).\tag{200}\]

For simplicity, from now on we fix \(1\le p\le k\) and write \[x=x_p, B=\Lambda_{d_{x}/10}(x), \eta=\eta_I^{p-1}, \eta^B=\eta_I^p, B^{s}=\Lambda_{sd_{x}/10}(x) for s=8,9,10,\] until the end of this proof.

Now for the measures \(\mu_{\Lambda_M}^{\xi,\eta^B}\) and \(\mu_{\Lambda_M}^{\xi,\eta},\) we explore the configurations outside \(B^{10}.\) That is, let \(\hat{\Sigma}=\{-1,1\}^{(\Lambda_M\setminus B^{10})\cup\partial B^{10}}\) be the configuration set, then we have \[\begin{align} \label{eq:explore-outside-10B} \mu_{\Lambda_M}^{\xi,\eta^B}(\mathcal{Y}) =\sum_{\hat{\xi}\in\hat{\Sigma}}\mu_{\Lambda_M}^{\xi,\eta^B}(\hat{\xi})\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\mathcal{Y}) =\sum_{\hat{\xi}\in\hat{\Sigma}}\mu_{\Lambda_M}^{\xi,\eta^B}(\hat{\xi})\mu_{ B^{10}}^{\hat{\xi},\eta^B}( \mathcal{Y}) =\sum_{\hat{\xi}\in\hat{\Sigma}}\mu_{\Lambda_M}^{\xi,\eta^B}(\hat{\xi})\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\hat{\mathcal{Y}}), \end{align}\tag{201}\] and a similar expansion holds if we replace \(\eta^B\) by \(\eta\). Here \(\hat{\mathcal{Y}}\) is defined in the following sense: let \(\hat{\mathcal{P}}_0\) denote the “push forward" of \(\mathcal{P}_0\) on \(\partial B^{10}\) given the information \(\hat{\xi}\in\hat{\Sigma}\), that is, the set of points on \(\partial{B^{10}}\) that is connected to \(\mathcal{P}_0\) in \(\hat{\xi}\) via plus spins, we can define \(\hat{\mathcal{P}}_1, \hat{U}\) similarly. Then we can define the events of \(\hat{\mathcal{Y}}^+\) and \(\hat{\mathcal{Y}}^-\) to be the push forward of \(\mathcal{Y}^+\) and \(\mathcal{Y}^-,\) i.e. \(\hat{\mathcal{Y}}^+\) is the event that every component of \(\hat{\mathcal{U}}\) is connected to \(\hat{\mathcal{P}}_0\) and \(\hat{\mathcal{Y}}^-\) is the event that \(\hat{\mathcal{P}}_1\) is not connected to \(\hat{\mathcal{P}}_0\). Finally, we define \(\hat{\mathcal{Y}}=\hat{\mathcal{Y}}^+\cap \hat{\mathcal{Y}}^-.\)

Now we define two events \[\begin{align} &\mathcal{A}_1:=\{the connected component of \hat{\mathcal{P}}_1does not intersect withB^9\},\\ &\mathcal{A}_2:=\{there exsits a plus spin circuit in B^9\setminus B^8\}, \end{align}\] and claim that there exists a constant \(C>0\) such that the following inequalities hold for any \(\xi\in \Sigma_M^c\) and \(\hat{\xi}\in\hat{\Sigma}\): \[\begin{align} &\mu_{\Lambda_M}^{\xi,\eta}(\hat{\xi})\geq C\mu_{\Lambda_M}^{\xi,\eta^B}(\hat{\xi});\tag{202}\\ & \mu_{B^{10}}^{\hat{\xi},\eta^B}(\mathcal{A}_1\mid\hat{\mathcal{Y}})\geq C;\tag{203}\\ &\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\mathcal{A}_2\mid \mathcal{A}_1\cap \hat{\mathcal{Y}})\geq C;\tag{204}\\ &\mu_{ B^{10}}^{\hat{\xi}}(\mathcal{A}_2\cap \hat{\mathcal{Y}})\geq C\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\mathcal{A}_2\cap \hat{\mathcal{Y}})\tag{205}. \end{align}\]

Now we quickly explain why these inequalities hold. 203 follows directly from Lemma 32. Combining 198 with RSW, we get that \[\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\mathcal{A}_2\mid \mathcal{A}_1\cap \hat{\mathcal{Y}})\stackrel{\eqref{eq:bound-plus-crossing-given-Y-4}}{\ge}\frac{C_1\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\mathcal{A}_2)\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\hat{\mathcal{ Y}})}{\mu_{ B^{10}}^{\hat{\xi},\eta^B}( \mathcal{A}_1\cap \hat{\mathcal{Y}})}\stackrel{RSW}{\ge}\frac{C_2\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\hat{\mathcal{ Y}})}{\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\mathcal{A}_1\cap \hat{\mathcal{Y}})}\ge C_2.\]Thus we finish the proof of 204 . Both 202 and 205 come from the mixing property of Lemma 38. 202 claims that the configuration inside \(B\) does not influence the configuration outside \(B^{10}\) too much. To prove 205 , it suffices to show that the event \(\mathcal{A}_2\cap \hat{\mathcal{Y}}\) relies only on the configuration in \((B^8)^c\). We prove this by contradiction. Assume that \(\sigma_1,\sigma_2\) are two configurations that coincide on \((B^8)^c\), such that \(\sigma_1\in \mathcal{A}_2\cap\hat{\mathcal{Y}}\) but \(\sigma_2\) does not. Then \(\sigma_1\) contains a plus circuit \(\gamma\) in \(B^9\setminus B^8\), so does \(\sigma_2.\) Now, given the existence of \(\gamma,\) whether \(\hat{\mathcal{P}_0}\) is connected to \(\hat{\mathcal{P}_1}\) or \(\hat{\mathcal{U}}\) is decided outside \(\gamma,\) on which region \(\sigma_1\) coincides with \(\sigma_2.\) Hence we arrive at the contradiction since \(\sigma_2\notin \mathcal{A}_2\cap\hat{\mathcal{Y}}\).

We are now ready to prove 200 : \[\begin{align} ~~&\mu_{\Lambda_M}^{\xi,\eta_I^{p-1}}(\mathcal{Y}) \stackrel{\eqref{eq:explore-outside-10B}}{=}\sum_{\hat{\xi}\in\hat{\Xi}}\mu_{\Lambda_M}^{\xi,\eta}(\hat{\xi})\mu_{ B^{10}}^{\hat{\xi},\eta}(\hat{\mathcal{Y}}) \stackrel{\eqref{proof32of32177-0}}{\geq} C\sum_{\hat{\xi}\in\hat{\Xi}}\mu_{\Lambda_M}^{\xi,\eta^B}(\hat{\xi})\mu_{ B^{10}}^{\hat{\xi},\eta}(\mathcal{A}_2\cap\hat{\mathcal{Y}})\\ \stackrel{\eqref{proof32of32177-3}}{\geq} &C^2 \sum_{\hat{\xi}\in\hat{\Xi}}\mu_{\Lambda_M}^{\xi,\eta^B}(\hat{\xi})\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\mathcal{A}_2\cap\hat{\mathcal{Y}}) \geq C^2 \sum_{\hat{\xi}\in\hat{\Xi}}\mu_{\Lambda_M}^{\xi,\eta^B}(\hat{\xi})\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\mathcal{A}_1\cap\mathcal{A}_2\cap\hat{\mathcal{Y}})\\ \stackrel{\eqref{proof32of32177-2}}{\geq} &C^3 \sum_{\hat{\xi}\in\hat{\Xi}}\mu_{\Lambda_M}^{\xi,\eta^B}(\hat{\xi})\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\mathcal{A}_1\cap\hat{\mathcal{Y}}) \stackrel{\eqref{proof32of32177-1}}{\geq} C^4 \sum_{\hat{\xi}\in\hat{\Xi}}\mu_{\Lambda_M}^{\xi,\eta^B}(\hat{\xi})\mu_{ B^{10}}^{\hat{\xi},\eta^B}(\hat{\mathcal{Y}}) \stackrel{\eqref{eq:explore-outside-10B}}{=}C^4\mu_{\Lambda_M}^{\xi,\eta_I^p}(\mathcal{Y}). \end{align}\] Choosing \(c_{13}\geq C^{-4}\) completes the proof of Lemma 34. ◻

Proof. The proof of 208 is rather direct. Note that \[\begin{align} \sum_{\gamma}\mu^{\xi}_{\Lambda_{M}}(\gamma) \geq\mu_{\Lambda_M}^\xi(there exists a plus-minus circuit in \Lambda_{M/2}\setminus\Lambda_{M/4}) \end{align}\] since the summation is taken over all possible choices of \(\gamma.\) Moreover, a sufficient condition for the existence of such \(\gamma\) is that there exists a plus circuit in \(\Lambda_{M/2}\setminus\Lambda_{3M/8}\) and a minus circuit in \(\Lambda_{3M/8}\setminus\Lambda_{M/4}\), which occurs with positive probability by the RSW inequality of two dimensional critical Ising model.

Then we focus on proving 209 via induction in \(k\). For \(k=1,\) notice that \[\begin{align} ~&\sum_{\gamma}\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(\gamma,\tau(\gamma)=1)\\ =~&\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(there exists a plus-minus circuit in \Lambda_{M/2}\setminus\Lambda_{M/4})\\ \leq ~& 1-\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(\partial\Lambda_{M/2}\leftrightarrow\partial\Lambda_{M/4}) \leq 1-c_{10}^{|I|}, \end{align}\] where the last inequality uses Lemma 31.

For the induction step, we first define \(\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(\gamma',\tau(\gamma')=k,\gamma,\tau(\gamma)=k+1)\) to be the probability that \(\gamma'\) is the \(k\)-th circuit and \(\gamma\) is the \((k+1)\)-th circuit. Note that \[\begin{align} \sum_{\gamma}\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(\gamma,\tau(\gamma)=k+1) =\sum_{\gamma',\gamma}\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(\gamma',\tau(\gamma')=k,\gamma,\tau(\gamma)=k+1).\label{eq:32interface32exploration32probability32induction32step} \end{align}\tag{210}\] Let \(\Gamma'\) denote the region enclosed by \(\gamma'\) and let \(\mathcal{A}(\Gamma')\) denote the event that there exists a plus-minus circuit in \(\Lambda_{M/2}\setminus(\Gamma'\cup\mathrm{Ball}(I))\), then we get from DMP that \[\begin{align} &\sum_{\gamma',\gamma}\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(\gamma',\tau(\gamma')=k,\gamma,\tau(\gamma)=k+1)\nonumber\\=~&\sum_{\gamma'}\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(\gamma',\tau(\gamma')=k)\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{(\mathrm{Ball}(I)\cup\Gamma')}}\big(\mathcal{A}(\Gamma')\big)\nonumber\\ \leq ~&\sum_{\gamma'}\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(\gamma',\tau(\gamma')=k)\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}\big(\mathcal{A}(\Gamma')\big).\label{eq:32interface32exploration32probability32induction32step321} \end{align}\tag{211}\] Let \(\mathcal{B}\) denote the event that there exists a plus spin crossing from \(\partial\Lambda_{M/2}\) to \(\partial\Lambda_{M/4}\). Note that \(\mathcal{A}(\Gamma')\subset\mathcal{A}(\Lambda_{M/4})\) and by planar duality \(\mathcal{A}(\Lambda_{M/4})\subset\mathcal{B}^c\). We get from Lemma 31 that \[\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}\big(\mathcal{A}(\Gamma')\big)\le \mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}\big(\mathcal{B}^c\big)\le 1-c_{10}^{|I|}.\nonumber\]Combined with 211 , it yields that \[\begin{align} &\sum_{\gamma',\gamma}\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(\gamma',\tau(\gamma')=k,\gamma,\tau(\gamma)=k+1)\nonumber\\\le~&\sum_{\gamma'}\mu^{\xi,\eta_I}_{\Lambda_{M}\setminus{\mathrm{Ball}(I)}}(\gamma',\tau(\gamma')=k)(1-c_{10}^{|I|}) \leq (1-c_{10}^{|I|})^{k+1}. \end{align}\] Plugging into 210 and choosing \(c_{14}\) depending on \(c_{10}\) completes the proof. ◻

Proof of 184 . Fix a circuit \(\gamma_0\subset\Lambda_{M/2}\setminus\Lambda_{M/4}\) and let \(\Gamma_0\) be the region enclosed by \(\gamma_0\).

Combining Lemmas 36 and 35 with 206 , we obtain that \[\begin{align} \mu_{\Lambda_M}^{\xi}(\mathcal{X}\cap\mathcal{Y})&\ge \sum_{\gamma}\mu_{\Lambda_M}^{\xi}(\gamma)\cdot \mu_{\Lambda_M\setminus\Gamma}^{\xi,+}(\mathcal{Y}\cap\{\gamma\leftrightarrow\mathcal{P}_0\})~~~\text{(by \eqref{eq:32two32rare32event32exploring32the32interface})}\nonumber\\&\ge \sum_{\gamma}\mu_{\Lambda_M}^{\xi}(\gamma)\cdot \frac{1}{c_{15}}\mu_{\Lambda_M\setminus\Gamma_0}^{\xi,+}(\mathcal{Y}\cap\{\gamma_0\leftrightarrow\mathcal{P}_0\})~~~\text{(by Lemma~\ref{lem:32two32rare32event32probability32with32different32interface32explored})}\nonumber\\&\ge \frac{c_{14}}{c_{15}}\cdot \mu_{\Lambda_M\setminus\Gamma_0}^{\xi,+}(\mathcal{Y}\cap\{\gamma_0\leftrightarrow\mathcal{P}_0\})~~~\text{(by Lemma~\ref{lem:32interface32exploration32probability32with32number})}.\label{eq:32two32rare32event32exploring32the32interface32lower32bound321} \end{align}\tag{214}\] Combining Lemmas 37, 36 and 34 with 207 , we get that \[\begin{align} \mu_{\Lambda_M}^{\xi,\eta_I}(\mathcal{X}\cap\mathcal{Y})&\le \sum_{k=0}^{\infty} \sum_{\gamma}\mu_{\Lambda_M}^{\xi,\eta_I}(\gamma,\tau(\gamma)=k)\cdot \mu_{\Lambda_M\setminus\Gamma}^{\xi,\eta_I,+}(\mathcal{Y}\cap\{\gamma\leftrightarrow\mathcal{P}_0\})~~~\text{(by \eqref{eq:32two32rare32event32exploring32the32interface32with32number})}\nonumber\\&\le \sum_{k=0}^{\infty} \sum_{\gamma}\mu_{\Lambda_M}^{\xi,\eta_I}(\gamma,\tau(\gamma)=k)\cdot c_{16}^{|I|}\mu_{\Lambda_M\setminus\Gamma}^{\xi,+}(\mathcal{Y}\cap\{\gamma\leftrightarrow\mathcal{P}_0\})~~~\text{(by Lemma~\ref{lem:32two32rare32event32probability32with32interface32and32ball-X611}))}\nonumber\\&\le\sum_{k=0}^{\infty} \sum_{\gamma}\mu_{\Lambda_M}^{\xi,\eta_I}(\gamma,\tau(\gamma)=k)\cdot c_{15} c_{16}^{|I|}\mu_{\Lambda_M\setminus\Gamma_0}^{\xi,+}(\mathcal{Y}\cap\{\gamma_0\leftrightarrow\mathcal{P}_0\})~~~\text{(by Lemma~\ref{lem:32two32rare32event32probability32with32different32interface32explored})}\nonumber\\&\le c_{15} c_{16}^{|I|}\cdot \sum_{k=0}^{\infty}(1-c_{14}^{|I|})^k\cdot \mu_{\Lambda_M\setminus\Gamma_0}^{\xi,+}(\mathcal{Y}\cap\{\gamma_0\leftrightarrow\mathcal{P}_0\})~~~\text{(by Lemma~\ref{lem:32two32rare32event32probability32with32interface32and32ball})}\nonumber\\&= c_{15} (\frac{c_{16}}{c_{14}})^{|I|}\cdot \mu_{\Lambda_M\setminus\Gamma_0}^{\xi,+}(\mathcal{Y}\cap\{\gamma_0\leftrightarrow\mathcal{P}_0\}).\label{eq:32two32rare32event32exploring32the32interface32lower32bound322} \end{align}\tag{215}\] Combining 214 and 215 , we finish the proof of 184 . ◻

Proof of Lemma 36. Recall Definition 9 and recall that \(\mathcal{Y}=\mathcal{Y}^-\cap \mathcal{Y}^+.\) Choosing \(\mathcal{I}=\mathcal{Y}^+\cap\{\gamma_2\leftrightarrow\mathcal{P}_0\}\) in Lemma 32, we have that \[\mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_2}(\mathcal{C}_1\cap \Lambda_{0.95M}=\emptyset\mid \mathcal{Y}\cap\{\gamma_2\leftrightarrow\mathcal{P}_0\})\ge C_1.\label{eq:32P95132cluster32trapped32lower32bound32with32single32condition}\tag{216}\] Next, we want to explore the cluster \(\mathcal{C}_1\) in \(\Lambda_M\). Note that by exploring \(\mathcal{C}_1\), we already see whether the event \(\mathcal{Y}^-\cap \{\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\}\) occurs. Thus we define \(\mathfrak{C}_1\) to be the collection of \(A\) such that \(\{\mathcal{C}_1=A\}\) is possible and \(\{\mathcal{C}_1=A\}\) implies \(\mathcal{Y}^-\cap \{\mathcal{C}_1\cap \Lambda_{0.95M}=\emptyset\}\). Then we have \[\begin{align} &~\mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_1}( \mathcal{Y}\cap\{\gamma_1\leftrightarrow\mathcal{P}_0\}\cap\{\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\})\nonumber\\=&\sum_{A\in\mathfrak{C}_1} \mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_1}(\mathcal{C}_1=A)\times\mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_1\cup A)}(\mathcal{Y}^+\cap\{\gamma_1\leftrightarrow\mathcal{P}_0\}),\tag{217}\\ &~\mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_2}( \mathcal{Y}\cap\{\gamma_2\leftrightarrow\mathcal{P}_0\}\cap\{\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\})\nonumber\\=&\sum_{A\in\mathfrak{C}_1} \mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_2}(\mathcal{C}_1=A)\times\mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_2\cup A)}(\mathcal{Y}^+\cap\{\gamma_2\leftrightarrow\mathcal{P}_0\}).\tag{218} \end{align}\] Here, for any region \(\Gamma\), we use \(\mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma\cup A)}\) to denote the Gibbs measure on \(\Lambda_{M}\setminus(\Gamma\cup A)\) with plus boundary condition on \(\partial_{\mathrm{ext}}\Gamma\cap{\partial(\Lambda_M\setminus(\Gamma\cup A))}\) and minus boundary condition on \(\partial_{\mathrm{ext}} A\cap{\partial(\Lambda_M\setminus(\Gamma\cup A))}\).

Note that \(\Gamma_1\subset\Lambda_{M/2}\) and \(\Gamma_2\subset\Lambda_{M/2}\). By the mixing property Lemma 38, we get that for any \(A\subset\Lambda_M\setminus\Lambda_{0.95M}\) that \[\mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_1}(\mathcal{C}_1=A)\ge C_2\mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_2}(\mathcal{C}_1=A).\label{eq:32mixing32property32in32two32rare32events32given32different32region}\tag{219}\] Let \(\mathcal{A}\) denote the event that there exists a plus circuit in \(\Lambda_{0.8M}\setminus\Lambda_{3M/4}\) which is connected to \(\gamma_1\). Then by RSW, we get that \[\mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_1\cup A)}(\mathcal{A})\ge C_3.\label{eq:32plus32circuit32existence32in32interface32comparison}\tag{220}\] Moreover, we decompose the event \(\mathcal{A}\) into the disjoint union of \(\mathcal{A}_{\hat{\gamma}}, \hat{\gamma}\in\mathscr S,\) where \(\mathcal{A}_{\hat{\gamma}}\) is the event that \(\hat{\gamma}\) is the innermost plus circuit connected to \(\gamma_1\) in \(\Lambda_{0.8M}\setminus\Lambda_{3/4M}\) and \(\mathscr S\) is the set of all possible \(\hat{\gamma}.\) Noting that \(\gamma_2\subset\Lambda_{M/2}\subset\hat{\Gamma}\), where \(\hat{\Gamma}\) is the interior of \(\hat{\gamma},\) it follows from DMP and CBC that \[\begin{align} & \mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_1\cup A)}(\mathcal{Y}^+\cap\{\gamma_1\leftrightarrow\mathcal{P}_0\}\mid\mathcal{A}_{\hat{\gamma}}) \stackrel{\mathrm{DMP}}{=}\mu^{\xi,+,-}_{\Lambda_{M}\setminus(\hat{\Gamma}\cup A)}(\mathcal{Y}^+\cap\{\hat{\gamma}\leftrightarrow\mathcal{P}_0\})\nonumber\\ \stackrel{\mathrm{CBC}}{\geq} &\mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_2\cup A)}(\mathcal{Y}^+\cap\{\hat{\gamma}\leftrightarrow\mathcal{P}_0\}) \geq \mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_2\cup A)}(\mathcal{Y}^+\cap\{\gamma_2\leftrightarrow\mathcal{P}_0\}). \label{eq:32plus32boundary32condition32connectivity32comparison32inequality} \end{align}\tag{221}\]

Combining 220 with 221 , we get that \[\begin{align} &\mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_1\cup A)}(\mathcal{Y}^+\cap\{\gamma_1\leftrightarrow\mathcal{P}_0\})\nonumber\\ \ge ~&\sum_{\hat{\gamma}\in\mathscr S}\mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_1\cup A)}(\mathcal{Y}^+\cap\{\gamma_1\leftrightarrow\mathcal{P}_0\}\mid\mathcal{A}_{\hat{\gamma}})\cdot \mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_1\cup A)}(\mathcal{A}_{\hat{\gamma}})\nonumber\\\ge~ & \mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_2\cup A)}(\mathcal{Y}^+\cap\{\gamma_2\leftrightarrow\mathcal{P}_0\})\sum_{\hat{\gamma}\in\mathscr S}\mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_1\cup A)}(\mathcal{A}_{\hat{\gamma}})\nonumber\\ =~& \mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_2\cup A)}(\mathcal{Y}^+\cap\{\gamma_2\leftrightarrow\mathcal{P}_0\})\mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_1\cup A)}(\mathcal{A})\nonumber\\ \geq ~& C_3\mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_2\cup A)}(\mathcal{Y}^+\cap\{\gamma_2\leftrightarrow\mathcal{P}_0\}).\label{eq:32plus32boundary32condition32connectivity32comparison32inequality322} \end{align}\tag{222}\] Plugging 219 and 222 into 217 and 218 , we get that \[\begin{align} &\mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_1}( \mathcal{Y}\cap\{\gamma_1\leftrightarrow\mathcal{P}_0\}\cap\{\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\})\nonumber\\ \ge~& C_2\sum_{A\in\mathfrak{C}_1} \mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_2}(\mathcal{C}_1=A)\times C_3 \mu^{\xi,+,-}_{\Lambda_{M}\setminus(\Gamma_2\cup A)}(\mathcal{Y}^+\cap\{\gamma_1\leftrightarrow\mathcal{P}_0\})\nonumber \\=~& C_2C_3 \mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_2}( \mathcal{Y}\cap\{\gamma_2\leftrightarrow\mathcal{P}_0\}\cap\{\mathcal{C}_1\cap\Lambda_{0.95M}=\emptyset\}).\label{eq:32two32rare32event32different32region32comparison32conditioned32on32trapped} \end{align}\tag{223}\] Combining 223 with 216 , we get that \[\begin{align} &\mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_1}(\mathcal{Y}\cap\{\gamma_1\leftrightarrow\mathcal{P}_0\})\ge \mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_1}( \mathcal{Y}\cap\{\gamma_1\leftrightarrow\mathcal{P}_0\}\cap\{\mathcal{P}_1\not\leftrightarrow \Lambda_{0.95M}\})\nonumber\\ \ge& C_2C_3\mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_2}( \mathcal{Y}\cap\{\gamma_2\leftrightarrow\mathcal{P}_0\}\cap\{\mathcal{P}_1\not\leftrightarrow \Lambda_{0.95M}\})\ge C_1C_2C_3 \mu^{\xi,+}_{\Lambda_{M}\setminus\Gamma_2}( \mathcal{Y}\cap\{\gamma_2\leftrightarrow\mathcal{P}_0\}). \end{align}\]Thus we finish the proof of Lemma 36. ◻

Proof of Lemma 37. The proof is very similar to that of Lemma 34. So we just provide a sketch and point out the differences here. With the notations in Definition 10 and the proof of Lemma 34, it suffices to show that \[\mu_{\Lambda\setminus\Gamma}^{\xi,\eta_I^p,+}(\mathcal{Y}\cap\{\gamma\leftrightarrow\mathcal{P}_0\})\leq c_9\mu_{\Lambda\setminus\Gamma}^{\xi,\eta_I^{p-1},+}(\mathcal{Y}\cap\{\gamma\leftrightarrow\mathcal{P}_0\}).\] As in the proof of Lemma 34, we want to explore the configuration \(\hat{\xi}\) outside \(B^{10}\). Recall the definitions of the push forward sets and events \(\hat{\mathcal{P}_0},\hat{\mathcal{P}_1},\hat{\mathcal{U}}\) and \(\hat{\mathcal{Y}}\) from Lemma 34. We further define the push forward of \(\{\gamma\leftrightarrow\mathcal{P}_0\},\) to be \[\hat{\mathcal{T}}=\{\gamma\leftrightarrow\hat{\mathcal{P}_0} in B^{10}\}\cup\{\gamma\leftrightarrow\hat{\mathcal{P}_0} in \hat{\xi}\}.\] As in the proof of Lemma 34, we only have to show that for some universal constant \(C>0\), the following inequalities hold: \[\begin{align} &\mu_{\Lambda_M\setminus\Gamma}^{\xi,+}(\hat{\xi})\geq C\mu_{\Lambda_M\setminus\Gamma}^{\xi,\eta^B,+}(\hat{\xi});\tag{224}\\ & \mu_{B^{10}\setminus\Gamma}^{\hat{\xi},\eta^B,+}(\mathcal{A}_1'\mid\hat{\mathcal{Y}}\cap\hat{\mathcal{T}})\geq C;\tag{225}\\ &\mu_{ B^{10}\setminus\Gamma}^{\hat{\xi},\eta^B,+}(\mathcal{A}_2'\mid \mathcal{A}_1'\cap \hat{\mathcal{Y}}\cap\hat{\mathcal{T}})\geq C;\tag{226}\\ &\mu_{ B^{10}\setminus\Gamma}^{\hat{\xi},+}(\mathcal{A}_2'\cap \hat{\mathcal{Y}}\cap\hat{\mathcal{T}})\geq C\mu_{ B^{10}\setminus\Gamma}^{\hat{\xi},\eta^B,+}(\mathcal{A}_2'\cap \hat{\mathcal{Y}}\cap \hat{\mathcal{T}})\tag{227}, \end{align}\] where \(\mathcal{A}_1',\mathcal{A}_2'\) are the events \(\mathcal{A}_1,\mathcal{A}_2\) (recall the definitions from the proof of Lemma 34) restricted in \(B^{10}\setminus\Gamma,\) i.e., \[\begin{align} \mathcal{A}_1':=\{&the connected component of \hat{\mathcal{P}}_1does not intersect withB^9in B^{10}\setminus\Gamma\},\\ \mathcal{A}_2':=\{&there existsS\subset B^9\setminus (B^8\cup\Gamma) such that \sigma_v=1, \forall v\in S\\ & and any path from \partial B^9 to B^8must intersect withS \}. \end{align}\]

The reasons why 224 to 226 hold are the same as those for 202 to 204 . Next, we show 227 : similar to 205 , it still comes from the mixing property of Lemma 38 and the fact that whether \(\mathcal{A}_2'\cap \hat{\mathcal{Y}}\cap \hat{\mathcal{T}}\) holds only relies on the configuration outside \(B^8\). The only difference is that we now have a new event \(\hat{\mathcal{T}},\) but note that \(\gamma\) is a circuit in \(\Lambda_{M/2}\setminus\Lambda_{M/4},\) therefore either \(\gamma\) lies outside \((B^8)^c\), or it intersects the set \(S\) from \(\mathcal{A}_2'.\) In both cases, the connecting event does not depend on the configurations inside \(B^8.\) ◻

Proof of Lemma 38. For the proof, we use the disagreement Ising model. We first prove that for any configuration \(\eta\) on \(\Lambda_K\), we have \[\frac{\mu_{\Lambda_N}^{\xi_1}(\sigma|_{\Lambda_K}=\eta)}{\mu_{\Lambda_N}^{\xi_2}(\sigma|_{\Lambda_K}=\eta)}\le 1+C (\frac{N}{K})^{-\frac{1}{8}}.\label{eq:32spatial32mixing32step320}\tag{230}\] Now we define the joint partition function for \(\bar\mu_{\Lambda_N\setminus\Lambda_K}^{\xi_1,\eta}\) and \(\bar\mu_{\Lambda_N}^{\xi_2}\) as \[\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)=\bar{\mathcal{Z}}_{\Lambda_N\setminus\Lambda_K}^{\xi_1,\eta}\cdot \bar{\mathcal{Z}}^{\xi_2}_{\Lambda_N}=\sum_{\bar\sigma\in\Xi(\Lambda_N\setminus\Lambda_K)}\sum_{\bar\sigma'\in\Xi(\Lambda_N)}\exp\left[-\beta_c\left(\bar H^{\xi_1,\eta}(\bar\sigma)+H^{\xi_2}(\bar\sigma')\right)\right],\] where \(\Xi(\cdot)\) denotes the configuration space for the extended Ising model; similarly we could define \(\bar{\mathcal{Z}}(\xi_2,\eta;\xi_1).\) We also define the joint partition function for \(\bar\mu_{\Lambda_N}^{\xi_1}\) and \(\bar\mu_{\Lambda_N}^{\xi_2}\) as \[\bar{\mathcal{Z}}(\xi_1;\xi_2)=\bar{\mathcal{Z}}_{\Lambda_N}^{\xi_1}\cdot \bar{\mathcal{Z}}^{\xi_2}_{\Lambda_N}=\sum_{\bar\sigma\in\Xi(\Lambda_N)}\sum_{\bar\sigma'\in\Xi(\Lambda_N)}\exp\left[-\beta_c\left(\bar H^{\xi_1}(\bar\sigma)+H^{\xi_2}(\bar\sigma')\right)\right],\]

Note that \[\mu_{\Lambda_N}^{\xi_1}(\sigma|_{\Lambda_K}=\eta)=\frac{\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)}{\bar{\mathcal{Z}}(\xi_1;\xi_2)},\quad \mu_{\Lambda_N}^{\xi_2}(\sigma|_{\Lambda_K}=\eta)=\frac{\bar{\mathcal{Z}}(\xi_2,\eta;\xi_1)}{\bar{\mathcal{Z}}(\xi_2;\xi_1)}.\] In addition, we have by symmetry \[\bar{\mathcal{Z}}(\xi_1;\xi_2)=\bar{\mathcal{Z}}(\xi_2;\xi_1).\]

As a consequence, \[\frac{\mu_{\Lambda_N}^{\xi_1}(\sigma|_{\Lambda_K}=\eta)}{\mu_{\Lambda_N}^{\xi_2}(\sigma|_{\Lambda_K}=\eta)}=\frac{\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)}{\bar{\mathcal{Z}}(\xi_2,\eta;\xi_1)}.\]

Now, let \[\begin{align} \mathcal{A}:=\{(\bar\sigma,\bar\sigma')\in\Xi(\Lambda_N\setminus\Lambda_K)\otimes\Xi(\Lambda_N):& there exists an agreement *-circuit\\ &separating \partial\Lambda_Nand \Lambda_K \}. \end{align}\] Then, by planar duality, we get that \[\begin{align} \mathcal{A}^c=\{(\bar\sigma,\bar\sigma')\in\bar\Sigma(\Lambda_N\setminus\Lambda_K)\otimes\bar\Sigma(\Lambda_N):& there exists an pre-disagreement path \\ &connecting \partial\Lambda_Nand \Lambda_K \}. \end{align}\] Now we define \[\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)[\mathcal{A}]:=\sum_{(\bar\sigma,\bar\sigma')\in\mathcal{A}}\exp\left[-\beta_c\left(\bar H^{\xi_1,\eta}(\bar\sigma)+H^{\xi_2}(\bar\sigma')\right)\right],\] and similarly \(\bar{\mathcal{Z}}(\xi_2,\eta;\xi_1)[\mathcal{A}].\) By the swapping identity of [19], we have \[\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)[\mathcal{A}]=\bar{\mathcal{Z}}(\xi_2,\eta;\xi_1)[\mathcal{A}].\] Moreover, we get from CBC that \[\begin{align} &\left|\frac{\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)[\mathcal{A}]}{\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)}-1\right|=\left|\bar{\mu}^{\xi_1,\eta}_{{\Lambda_N\setminus\Lambda_K}}\otimes \bar{\mu}^{\xi_2}_{{\Lambda_N}}(\mathcal{A})-1\right|=\bar{\mu}^{\xi_1,\eta}_{{\Lambda_N\setminus\Lambda_K}}\otimes \bar{\mu}^{\xi_2}_{{\Lambda_N}}(\mathcal{A}^c)\le \bar{\mu}^{+/-,+/-}_{{\Lambda_N\setminus\Lambda_K}}(\mathcal{A}^c). \end{align}\]Combined with [22] and RSW, we get that \[\begin{align} \left|\frac{\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)[\mathcal{A}]}{\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)}-1\right|\le\bar{\mu}^{+/-,+/-}_{{\Lambda_N\setminus\Lambda_K}}(\mathcal{A}^c)= 1-\left(\frac{1-\phi_{\Lambda_N\setminus\Lambda_K}^{w,w}(\partial\Lambda_N\leftrightarrow\partial\Lambda_K)}{1+\phi_{\Lambda_N\setminus\Lambda_K}^{w,w}(\partial\Lambda_N\leftrightarrow\partial\Lambda_K)}\right)^2 \leq C\left(\frac{K}{N}\right)^{\frac{1}{8}}.\label{eq:32spatial32mixing32step321} \end{align}\tag{231}\] Applying 231 twice, we get that \[\begin{align} \frac{\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)}{\bar{\mathcal{Z}}(\xi_2,\eta;\xi_1)}=\frac{\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)}{\bar{\mathcal{Z}}(\xi_1,\eta;\xi_2)[\mathcal{A}]}\cdot \frac{\bar{\mathcal{Z}}(\xi_2,\eta;\xi_1)[\mathcal{A}]}{\bar{\mathcal{Z}}(\xi_2,\eta;\xi_1)}\le \Big(1+C\left(\frac{K}{N}\right)^{\frac{1}{8}}\Big)^2.\nonumber \end{align}\]Thus we finish the proof of 230 . To finish the proof of 228 , we get from 230 again that \[\frac{\mu_{\Lambda_N}^{\xi_1}(\sigma|_{\Lambda_K}=\eta)}{\mu_{\Lambda_N}^{\xi_2}(\sigma|_{\Lambda_K}=\eta)}\ge \frac{1}{1+C (\frac{N}{K})^{-\frac{1}{8}}}\ge 1-C (\frac{N}{K})^{-\frac{1}{8}}.\label{eq:32spatial32mixing32step322}\tag{232}\] Combining 230 and 232 gives the desired result. ◻

Proof. Let \(0 < \rho \ll 1\), \(r > 0\) and \(x \in \delta \mathbb{Z}^2\). Let \(\Lambda_{\delta} \subset \mathbb{Z}^2\) be a square centered at \(x\). For \(R > 0\), let us set \(\Lambda_{R,\delta}(x) := \Lambda_{R}(x) \cap \delta \mathbb{Z}^2\), where \(\Lambda_{R}(x)\) is the Euclidean square of side-length \(2R\) centered at \(x\). For \(0 < R_1 < R_2\), we say that there is an arm of sign \(+\), respectively \(-\), in the annulus \(\Lambda_{R_{2},\delta}(x) \setminus \Lambda_{R_{1},\delta}(x)\) if there is a nearest neighbor path joining \(\partial \Lambda_{R_{1},\delta}(x)\) to \(\partial \Lambda_{R_{2},\delta}(x)\) along which all vertices have sign \(+1\), respectively \(-1\). For \(0 < R_1 < R_2\) and \(x \in \delta \mathbb{Z}^2\), we denote by \(A_2(x;R_1,R_2)\) the event that there exist two arms of opposite sign crossing \(\Lambda_{R_{2},\delta}(x) \setminus \Lambda_{R_{1},\delta}(x)\). It is then easy to see that \[\mu_{\delta, \Lambda_{\delta}}^{\pm}[{\operatorname{dist}}(x,\gamma_{\delta}) \leq r] \leq \mu_{\delta, \Lambda_{\delta}}^{\pm}[A_2(x;r,{\operatorname{dist}}(x,\partial \Lambda_{\delta})].\] By [56], there exist \(\delta_{0} > 0\) and an absolute constant \(C>0\) such that \[\mu_{\delta, \Lambda_{\delta}}^{\pm}[A_2(x;r,{\operatorname{dist}}(x,\partial \Lambda_{\delta})] \leq C\bigg( \frac{r}{{\operatorname{dist}}(x,\partial \Lambda_{\delta})}\bigg)^{\frac{5}{8}-\rho},\] which concludes the proof of the lemma. ◻

Proof. Let \(0 < \rho \ll 1\), \(k \in \mathbb{N}\), \((x_1,\dots,x_k) \in D_{\delta}^k\) and \((r_1,\dots,r_k) \in (0,\infty)^k\). Using the same notations as in the proof of Lemma 39, we have that \[\label{ineq-visit-interface-two-arm-event} \mu_{\delta}^{\pm}\bigg[ \bigcap_{j=1}^{k} \{ {\operatorname{dist}}(x_j, \gamma_{\delta}) \leq r_j \}\bigg] \leq \mu_{\delta}^{\pm}\bigg[ \bigcap_{j=1}^{k} A_{2}(x_j;r_j,d_{\delta}(x_j)) \bigg].\tag{233}\] We now would like to use the Markov property of the Ising model to decouple the events \[\{A_{2}(x_j;r_j,d_{\delta}(x_j))\}_{1 \leq j \leq k}.\] However, the outer squares \(\Lambda_{d_{\delta}(x_j),\delta}(x)\) may overlap each other. We therefore consider smaller outer squares to obtain from 233 that \[\mu_{\delta}^{\pm}\bigg[ \bigcap_{j=1}^{k} \{ {\operatorname{dist}}(x_j, \gamma_{\delta}) \leq r_j \}\bigg] \leq \mu_{\delta}^{\pm}\bigg[ \bigcap_{j=1}^{k} A_{2}(x_j;r_j,d_{\delta}(x_j) \wedge L(x_j)) \bigg],\] where \(L(x_j)\) is defined as in the statement of the lemma. Using now the Markov property of the Ising model in the squares \((\Lambda_{\tilde{L}(x_j),\delta}(x))_j\), we obtain that \[\mu_{\delta}^{\pm}\bigg[ \bigcap_{j=1}^{k} \{ {\operatorname{dist}}(x_j, \gamma_{\delta}) \leq r_j \}\bigg] \leq \mu_{\delta}^{\pm} \bigg[ \mu_{\Lambda_{\tilde{L}(x_j),\delta}(x)}^{\sigma_{\partial \Lambda_{\tilde{L}(x_j),\delta}(x)}}\big[ A_{2}(x_j;r_j,\frac{1}{2}d_{\delta}(x_j) \wedge \frac{1}{4}L(x_j)) \big] \bigg]\] where we have set \(\tilde{L}(x_j) = d_{\delta}(x_j) \wedge L(x_j)\). [56] guarantees that when \(\beta =\beta_c\), changing the boundary conditions does not change by more than a multiplicative constant the probability of events depending on edges that are at macroscopic distance from the boundary. This implies that there exists a constant \(c>0\) such that \({\mathbb{P}}_{\delta}^{\pm}\)-almost surely, for any \(x_j\), \[\label{mixing-2-arm-event} \mu_{\Lambda_{\tilde{L}(x_j),\delta}(x)}^{\sigma_{\partial \Lambda_{\tilde{L}(x_j),\delta}(x)}}\big[ A_{2}(x_j;r_j,\frac{1}{2}d_{\delta}(x_j) \wedge \frac{1}{4}L(x_j)) \big] \leq c\mu_{\Lambda_{\tilde{L}(x_j),\delta}(x)}^{\pm}\big[ A_{2}(x_j;r_j,\frac{1}{2}d_{\delta}(x_j) \wedge \frac{1}{4}L(x_j)) \big]\tag{234}\] where under \({\mathbb{P}}_{\Lambda_{\tilde{L}(x_j),\delta}(x)}^{\pm}\), the Ising model has boundary conditions \(-1\) on the bottom side of \(\partial \Lambda_{\tilde{L}(x_j),\delta}(x)\) and \(+1\) boundary conditions elsewhere. Lemma 39 provides an upper bound on the right-hand side of 234 , which combined with 233 , yields that there exist \(\delta_0>0\) (independent of \(k\) and \((x_1,\dots,x_k)\)) and an absolute constant \(C>0\) such that \[\begin{align} \mu_{\delta}^{\pm}\bigg[ \bigcap_{j=1}^{k} \{ {\operatorname{dist}}(x_j, \gamma_{\delta})\leq r_j \}\bigg] &\leq c^k \prod_{j=1}^{k} \mu_{\Lambda_{\tilde{L}(x_j),\delta}(x)}^{\pm}\big[ A_{2}(x_j;r_j,\frac{1}{2}d_{\delta}(x_j) \wedge \frac{1}{4}L(x_j)) \big] \\ & \leq C^k c^k \bigg( \prod_{j=1}^{k} \frac{r_j}{\frac{1}{2}d_{\delta}(x_j) \wedge \frac{1}{4}L(x_j))} \bigg)^{\frac{5}{8}-\rho}. \end{align}\] This directly implies the lemma. ◻