Proof. We refer the reader to Proposition 2.1 in [11]. ◻

Proof. This follows directly from the fact that compatibility is checked in pairs. That is, \(\Gamma\in\mathscr{E}\) if and only if \(\gamma_1\neq\gamma_2\in\Gamma\Rightarrow\gamma_1\sim\gamma_2.\) ◻

Proof. Let \(\{A_1,...,A_n\}\) be a finite collection of non-empty disjoint sets. Then, \[\begin{align} \left(\sqcup_{i=1}^nA_i\right)^c&=\cap_{i=1}^nA_i^c=A_i^c\setminus\left(\sqcup_{j\neq i}A_j\right), \end{align}\] for all \(i\in[n]\). Hence, \[\label{eq:32set-relation} \begin{align} \left(\sqcup_{i=1}^nA_i\right)\times\left[\left(\sqcup_{i=1}^nA_i\right)^c\right]&=\sqcup_{i=1}^n\left[A_i\times\left(A_i^c\setminus\sqcup_{j\neq i}A_j\right)\right]\\ &=\left[\sqcup_{i=1}^n(A_i\times A_i^c)\right]\setminus\left[\sqcup_{i=1}^n\sqcup_{j\neq i}A_i\times A_j\right]. \end{align}\tag{21}\] Let \(A_\gamma=\mathrm{I}_-(\iota_\gamma(\Gamma))\), where \(\gamma\in\partial\Gamma\). Combining Equations 18 and 21 , we obtain \[\begin{align} H_h(\Gamma)=\sum_{\substack{x\in\mathrm{I}_-(\Gamma)\\y\in\mathrm{I}_-(\Gamma)^c}}2J_{xy}=\sum_{\gamma\in\partial\Gamma}\sum_{\substack{x\in\mathrm{I}_-(\iota_\gamma(\Gamma))\\y\in\mathrm{I}_-(\iota_\gamma(\Gamma))^c}}2J_{xy}-\sum_{\substack{\substack{\gamma,\gamma'\in\partial\Gamma\\\gamma\neq\gamma'}}}\sum_{\substack{x\in\mathrm{I}_-(\iota_\gamma(\Gamma))\\y\in\mathrm{I}_-(\iota_{\gamma'}(\Gamma))}}2J_{xy}, \end{align}\]The lemma has thus been proved. ◻

Proof. Given any finite set \(A\), we write \[\mathrm{P}_2(A)\mathrel{\vcenter{:}}=\{e\in\mathrm{P}(A):\lvert e\rvert=2\}.\]That is, we denote by \(\mathrm{P}_2(A)\) the collection of all subsets of \(A\) that contain exactly two elements. Let \(\{x_e:e\in \mathrm{P}_2(A)\}\subset\mathbb{R}\), then (recall Notation 1)\[\begin{align} \label{eq:32mayer-trick} \prod_{e\in\mathrm{P}_2(A)}(1+x_e)=\sum_{P\in\mathbbold{P}(A)}\prod_{Q\in P}\left[\sum_{G\in\mathbbold{G}(Q)}\prod_{e\in E(G)}x_e\right]. \end{align}\tag{31}\] Let \(\Lambda\Subset\mathbb{Z}\). Then, \[\begin{align} Z^+_{\Lambda,\beta,h}=\sum_{\sigma^+\in\Omega^+_\Lambda}e^{-\beta H_h^+(\sigma^+)}=1+\sum_{\overline{\Gamma}\in\mathscr{E}_\Lambda}e^{-\beta H_h(\overline{\Gamma})}=1+\sum_{\Gamma'_1\in\mathscr{P}_\Lambda}\sum_{\substack{\overline{\Gamma}_1\in\mathscr{E}\\\partial\overline{\Gamma}_1=\Gamma'_1}}e^{-\beta H_h(\overline{\Gamma}_1)}. \end{align}\] By Lemma 3 and Mayer’s trick (Equation 31 ), we have \[e^{-\beta H_h(\overline{\Gamma})}=\prod_{\gamma\in\partial\overline{\Gamma}}e^{-\beta H_{h,\gamma}(\overline{\Gamma})}\sum_{P\in\mathbbold{P}(\partial\overline{\Gamma})}\prod_{Q\in P}\sum_{G\in\mathbbold{G}(Q)}\prod_{e\in E(G)}\left(e^{\beta\Phi_e(\overline{\Gamma})}-1\right),\] for every \(\Gamma\in\mathscr{E}\). Hence,\[\begin{align} \sum_{\substack{\overline{\Gamma}_1\in\mathscr{E}\\\partial\overline{\Gamma}_1=\Gamma'_1}}e^{-\beta H_h(\overline{\Gamma}_1)}=\sum_{\substack{\overline{\Gamma}_1\in\mathscr{E}\\\partial\overline{\Gamma}_1=\Gamma'_1}}\prod_{\gamma_1\in\Gamma'_1}e^{-\beta H_{h,\gamma_1}(\overline{\Gamma}_1)}\sum_{X_1\in\mathbbold{P}(\Gamma'_1)}\prod_{\Gamma_1\in X_1}\sum_{G_1\in\mathbbold{G}(\Gamma_1)}\prod_{e_1\in E(G_1)}\left(e^{\beta\Phi_{e_1}(\overline{\Gamma}_1)}-1\right). \end{align}\]

Let us comment on the chosen notation. We write \(X_1\) in the hope that the reader identifies it with the same notation used earlier, where a graphical description of compatible collections of polymers is provided.

Back to the decomposition algorithm: note that, for each \(X_1\in\mathbbold{P}(\Gamma'_1)\), there is a natural bijection between \(\{\overline{\Gamma}_1\in\mathscr{E}:\partial\overline{\Gamma}_1=\Gamma'_1\}\) and \(\times_{\Gamma_1\in X_1}\{\overline{\Gamma}(\Gamma_1)\in\mathscr{E}:\partial(\overline{\Gamma}(\Gamma_1))=\Gamma_1\}\); the canonical identification being \[(\overline{\Gamma}(\Gamma_1):\Gamma_1\in X_1)\mapsto\overline{\Gamma}_1=\sqcup_{\Gamma_1\in X_1}\overline{\Gamma}(\Gamma_1),\] whose inverse is given by \[\overline{\Gamma}_1\mapsto\left(\iota_{\Gamma_1}(\overline{\Gamma}_1):\Gamma_1\in X_1\right).\] This bijection leaves the values of \(H_{h,\gamma_1}(\overline{\Gamma}_1)\) and \(\Phi_{e_1}(\overline{\Gamma}_1)\) invariant. That is, if \(\gamma_1\in\Gamma_1\), then \[\begin{align} H_{h,\gamma_1}(\overline{\Gamma}_1)=H_{h,\gamma_1}(\overline{\Gamma}(\Gamma_1)). \end{align}\]Similarly, if \(e_1\in\mathrm{P}_2(\Gamma_1)\), then \[\begin{align} \Phi_{e_1}(\overline{\Gamma}_1)=\Phi_{e_1}(\overline{\Gamma}(\Gamma_1)). \end{align}\] In this case, we arrive at \[\begin{align} \label{eq:32iteration-polymer-factorization} \sum_{\substack{\overline{\Gamma}_1\in\mathscr{E}\\\partial\overline{\Gamma}_1=\Gamma'_1}}e^{-\beta H_h(\overline{\Gamma}_1)}=\sum_{X_1\in\mathbbold{P}(\Gamma'_1)}\prod_{\Gamma_1\in X_1}Z_{\beta,h}(\Gamma_1)=\sum_{X_1\in\mathbbold{P}(\Gamma'_1)}\prod_{\Gamma_1\in X_1}z^+_{\beta,h}(\Gamma_1 )Z^*_{\beta,h}(\Gamma_1). \end{align}\tag{32}\] On the other hand, for each \(\gamma_1\in\Gamma_1\), we have \[\begin{align} Z^*_{\beta,h}(\gamma_1)&=\sum_{\substack{\overline{\Gamma}_1\in\mathscr{E}\\\partial\overline{\Gamma}_1=\{\gamma_1\}}}e^{-\beta H^*_{h,\gamma_1}(\overline{\Gamma}_1)}=1+\sum_{\substack{\Gamma'_2\in\mathscr{P}\\\Gamma'_2\in\mathscr{I}(\gamma_1)}}\sum_{\substack{\overline{\Gamma}_2\in\mathscr{E}\\\partial\overline{\Gamma}_2=\Gamma'_2}}e^{-\beta H_{h}(\overline{\Gamma}_2)}. \end{align}\]Since \(\mathscr{I}(\gamma_1)\) might be empty, the sum can be \(0\). This happens if, for example, \(\mathrm{diam}(\gamma_1)\leq 2M\). If not, we proceed with the decomposition by applying Equation 32 . We have thus provided an algorithm that writes the initial partition function in terms of a gas of polymers. ◻

Proof. Let \(\overline{\Gamma}\in\mathscr{E}\) such that \(\partial\overline{\Gamma}=\Gamma\). We begin by establishing a uniform (on \(\overline{\Gamma}\)) bound on \[\label{eq:32g-overline-Gamma-definition} g_\beta(\overline{\Gamma})\mathrel{\vcenter{:}}=\sum_{G\in\mathbbold{G}(\Gamma)}\prod_{e\in E(G)}\left(e^{\beta \Phi_e(\overline{\Gamma})}-1\right).\tag{39}\] Since \(\Phi_e(\overline{\Gamma})\leq \Phi(e)\), for any \(e\in\mathrm{P}_2(\Gamma)\), and every connected graph \(G\in\mathbbold{G}(\Gamma)\) has at least one generating tree \(T\in\mathbbold{T}(\Gamma)\), we have \[g_\beta(\overline{\Gamma})\leq\sum_{T\in\mathbbold{T}(\Gamma)}\prod_{e\in E(T)}\left(e^{\beta\Phi(e)}-1\right)\sum_{\substack{G\in\mathbbold{G}(\Gamma)\\E(T)\subset E(G)}}\prod_{e\in E(G)\setminus E(T)}\left(e^{\beta \Phi(e)}-1\right).\]The non-negativity of \(\Phi(e)\) implies that\[\sum_{\substack{G\in\mathbbold{G}(\Gamma)\\E(T)\subset E(G)}}\prod_{e\in E(G)\setminus E(T)}(e^{\beta\Phi(e)}-1)\leq\sum_{\substack{E\subset\mathrm{P}_2(\Gamma)\\E\cap E(T)=\emptyset}}\prod_{e\in E}(e^{\beta\Phi(e)}-1)=\prod_{e\in\mathrm{P_2}(\Gamma)\setminus E(T)}e^{\beta\Phi(e)}.\]On the other hand, since \(x\geq 0\Rightarrow e^x-1\leq xe^x\), we have \[\prod_{e\in E(T)}\left(e^{\beta\Phi(e)}-1\right)\leq \left(\prod_{e\in E(T)}\beta\Phi(e)\right)\exp\left(\sum_{e\in E(T)}\beta\Phi(e)\right).\]Hence \[g_\beta(\overline{\Gamma})\leq \sum_{T\in\mathbbold{T}(\Gamma)}\prod_{e\in E(T)}\beta\Phi(e)\prod_{e\in P_2(\Gamma)}e^{\beta\Phi(e)}.\]By Lemma 3 and Hypothesis 1, \[\begin{align} \sum_{e\in\mathrm{P}_2(\Gamma)}\Phi(e)=-H(\Gamma)+\sum_{\gamma\in\Gamma}H(\gamma)\leq\frac{1}{8}\sum_{\gamma\in\Gamma}H(\gamma). \end{align}\]Since \(\inf_{\gamma\in\mathscr{C}}H(\gamma)\geq 2\), \(\lvert E(T)\rvert=\lvert \Gamma\rvert-1\) and \(x\geq 0\Rightarrow x\leq e^x\), we have \[\beta\geq1\Rightarrow\prod_{e\in E(T)}\beta\leq\prod_{\gamma\in\Gamma}4(\beta/4)\leq \prod_{\gamma\in\Gamma}4e^{\frac{1}{4}\beta}\leq \prod_{\gamma\in\Gamma}e^{\log4+\frac{1}{8}\beta H(\gamma)}.\] Thus, we obtain \[\label{eq:32g-beta-bound-upper-activity} g_\beta(\overline{\Gamma})\leq \left(\prod_{\gamma\in\Gamma}e^{\log 4+\frac{1}{4}\beta H(\gamma)}\right)\sum_{T\in\mathbbold{T}(\Gamma)}\prod_{e\in E(T)}\Phi(e).\tag{40}\] Let \(\gamma\in\Gamma\). By Hypothesis 1, we have \[\label{eq:32hyp-M-to-choice-upper-activity} -H_\gamma(\overline{\Gamma})\leq -H^*_\gamma(\overline{\Gamma})-(7/8)H(\gamma).\tag{41}\] Combining Equations 40 and 41 with Equation 27 , we obtain\[z^+_\beta(\Gamma)\leq \left(\prod_{\gamma\in\Gamma}e^{\log 4-\frac{5}{8}\beta H(\gamma)}\right)\sum_{T\in\mathbbold{T}(\Gamma)}\prod_{e\in E(T)}\Phi(e).\]Choosing \(\beta\geq 8\log2\), we arrive at Equation 36 .

By Equation 18 , we have \[\label{eq:32field-activity-dealing-with-the-term} H^*_{h,\gamma}(\overline{\Gamma})-H_{h,\gamma}(\overline{\Gamma})=H^*_\gamma(\overline{\Gamma})-H_\gamma(\overline{\Gamma})+\sum_{x\in\mathrm{I}_-(\iota^*_\gamma(\overline{\Gamma}))}2h_x-\sum_{x\in\mathrm{I}_-(\iota_\gamma(\overline{\Gamma}))}2h_x\leq H^*_\gamma(\overline{\Gamma})-H_\gamma(\overline{\Gamma})+\sum_{x\in\mathrm{I}_-(\gamma)}2\lvert h_x\rvert.\tag{42}\] Combining Equation 42 with Equation 36 , we obtain Equation 37 . This concludes the proof of the lemma. ◻

Proof. Let \(\varphi\in\mathrm{Inj}(m,n)\) be an injection, we need to define a canonical permutation \(\tilde{\varphi}\mathrel{\vcenter{:}}=\tilde{\varphi}(\varphi)\in \mathrm{Sym}(n)\) such that \((\tilde{\varphi}\circ\varphi)(i)=i\), for all \(i\in [m]\). If \(j\in\mathrm{Im}(\varphi)\), we define \(\tilde{\varphi}(j)=\varphi^{-1}(j)\in[m]\). If \(j\in [n]\setminus\mathrm{Im}(\varphi)\), we simply extend \(\tilde{\varphi}\) such that if \(j,k\in[n]\setminus\mathrm{Im}(\varphi)\) and \(j<k\), then \(\tilde{\varphi}(j)<\tilde{\varphi}(k)\). We have thus constructed a canonical injection \(\mathrm{Inj}(m,n)\ni\varphi\rightarrow\tilde{\varphi}\in \mathrm{Sym}(n)\).

Denoting the usual left-action of \(\mathrm{Sym}(n)\) on \(\mathscr{V}^n\) by \(F_{\cdot}\), the permutation \(\tilde{\varphi}\) induces a bijection \(F_{\tilde{\varphi}}:\mathscr{V}^n\rightarrow\mathscr{V}^n\) given by\[(F_{\tilde{\varphi}}\mathbf{v})_i=\vartheta_{\tilde{\varphi}(i)}.\]Introducing the variable change \(\mathbf{v}'=F_{\tilde{\varphi}^{-1}}\mathbf{v}\), we obtain \[\overline{\mathrm{W}}_{\beta,n}(\mathbf{w})=\sum_{\varphi\in\mathrm{Inj}(m,n)}\frac{1}{n!}\sum_{T\in\mathbbold{T}_n}\sum_{\mathbf{v}'\in\mathscr{V}^n}\mathbbold{1}_{\mathbf{v}'|_{[m]}=\mathbf{w}}w_\beta (T,F_{\tilde{\varphi}}\mathbf{v}').\]Analogously, consider the left-action \(F^\star_{\cdot}\) of \(\mathrm{Sym}(n)\) on \(\mathbbold{T}_n\) given by \[E(F^\star_{\tilde{\varphi}}T)\mathrel{\vcenter{:}}=\left\{\{\tilde{\varphi}(j),\tilde{\varphi}(k)\}:\{j,k\}\in E(T)\right\}.\] In this case, we have \(w_\beta(T,F_{\tilde{\varphi}}\mathbf{v}')=w_\beta(F^\star_{\tilde{\varphi}}T,\mathbf{v'})\).

Since \(F^\star_{\tilde{\varphi}}\) is a bijection for any \(\tilde{\varphi}\in\mathrm{Sym}(n)\), introducing the variable change \(T'=F^\star_{\tilde{\varphi}}T\), we have \[\label{eq:32W-bar-n-definition} \overline{\mathrm{W}}_{\beta,n}(\mathbf{w})=\sum_{\varphi\in\mathrm{Inj}(m,n)}\frac{1}{n!}\sum_{T'\in\mathbbold{T}_n}\sum_{\mathbf{v'\in\mathscr{V}^n}}\mathbbold{1}_{\mathbf{v}'|_{[m]}=\mathbf{w}}w_\beta(T',\mathbf{v}').\tag{48}\] A simple counting argument and the relabeling of mute indices gives us the first part of Equation 47 . The second equality follows from the observation that \(\mathscr{V}_n\) is invariant under permutations of indices. The lemmas has thus been proved. ◻

Proof. Recalling Equation 51 , we have \[\begin{align} \mathrm{S}_{\beta,t}(\varOmega)\leq&\prod_{p=1}^{t-1}\sum_{n_p=1}^\infty\frac{1}{n_p!}\prod_{k_p=1}^{n_p}\left(\sum_{l_{p-1}=1}^{n_{p-1}}\sum_{\vartheta_{p,k_p}}\ell_\beta(\vartheta_{p,k_p},\vartheta_{p-1,l_{p-1}})\right)\\&\times \sum_{n_t=1}^\infty\frac{1}{n_t!}\prod_{k_t=1}^{n_t}\sum_{l_{t-1}=1}^{n_{t-1}}\sum_{\vartheta_{t,k_t}}\ell_{\beta/2}(\vartheta_{t,k_t},\vartheta_{t-1,l_{t-1}})\eqqcolon\bar{\mathrm{S}}_{\beta,t}(\varOmega). \end{align}\] We prove the validity of Equation 53 by showing that there exists \(\beta_0>0\) such that if \(\beta\geq \beta_0\), then\[\begin{align} \label{eq:32S-bar-recurrence} \bar{\mathrm{S}}_{\beta,t+1}(\varOmega)\leq 2^{-1}\bar{\mathrm{S}}_{\beta,t}(\varOmega), \end{align}\tag{54}\] for all \(t\in\mathbb{N}\) and all \(\varOmega\in\mathscr{V}\). In this case, \[\begin{align} \frac{\overline{\mathrm{W}}_\beta(\varOmega)}{v_\beta(\varOmega)}\leq 1+\sum_{t=1}^\infty \bar{\mathrm{S}}_{\beta,t}(\varOmega)\leq 1+\sum_{t=1}^\infty2^{-(t-1)}\bar{\mathrm{S}}_{\beta,1}(\varOmega)\leq 2(1+\bar{\mathrm{S}}_{\beta,1}(\varOmega)). \end{align}\]By the definition of \(\bar{\mathrm{S}}_{\beta,1}(\varOmega)\) and Equation 52 , we have \[\begin{align} 1+\bar{\mathrm{S}}_{\beta,1}(\varOmega)&=1+\sum_{n=1}^\infty\frac{1}{n!}\prod_{k=1}^n\sum_{\vartheta_k}\ell_{\beta/2}(\vartheta_k,\varOmega)=\sum_{n=0}^\infty\frac{1}{n!}(\mathcal{L}_{\beta/2}(\varOmega))^n=\exp\mathcal{L}_{\beta/2}(\varOmega). \end{align}\]By Hypothesis 5, if \(\beta\geq\beta_0\) is sufficiently large, then, for every \(\varOmega\in\mathscr{V}\), we have \[\mathcal{L}_{\beta/2}(\varOmega)\leq\frac{1}{2}v'_{\beta/2}(\varOmega)\leq\frac{1}{2} v'_\beta(\varOmega),\]where the second inequality is a consequence of Hypothesis 4. In this case, \(\beta_0>0\) must be chosen so that \[2v_\beta(\varOmega)\exp(2^{-1} v'_\beta(\varOmega))\leq\eta v_{\beta/2}(\varOmega)\iff 4\eta^{-2}\leq\exp(v'_\beta(\varOmega))\]holds for \(\beta\geq\beta_0\) and all \(\varOmega\in\mathscr{V}\). This can certainly be done by Hypothesis 4.

Let us now prove that if \(\beta_0\) is sufficiently large, then \(\bar{\mathrm{S}}_{\beta,t+1}(\varOmega)\leq 2^{-1}\bar{\mathrm{S}}_{\beta,t}(\varOmega)\) for all \(t\geq 1\) and all \(\varOmega\in\mathscr{V}\). By the definition of \(\mathcal{L}_{\beta/2}(\cdot)\), we have \[\sum_{n_{t+1}=1}^{\infty}\frac{1}{n_{t+1}!}\prod_{k_{t+1}=1}^{n_{t+1}}\sum_{l_t=1}^{n_t}\sum_{\vartheta_{t+1,k_{t+1}}}\ell_{\beta/2}(\vartheta_{t+1,k_{t+1}},\vartheta_{t,l_t})= e^{\sum_{l_t=1}^{n_t}\mathcal{L}_{\beta/2}(\vartheta_{t,l_t})}-1.\] Recalling that \(x>0\Rightarrow(e^x-1<xe^x)\wedge(x<e^x)\), we arrive at \[\label{eq:32generation-eating-operation} \sum_{n_{t+1}=1}^{\infty}\frac{1}{n_{t+1}!}\prod_{k_{t+1}=1}^{n_{t+1}}\sum_{l_t=1}^{n_t}\sum_{\vartheta_{t+1,k_{t+1}}}\ell_{\beta/2}(\vartheta_{t+1,k_{t+1}},\vartheta_{t,l_t})\leq \frac{1}{2}\prod_{l_t=1}^{n_t}e^{3\mathcal{L}_{\beta/2}(\vartheta_{t,l_t})}.\tag{55}\] By Hypotheses 4 and 5, there exists \(\beta_0>0\) such that if \(\beta\geq\beta_0\), then \[\begin{align} e^{3\mathcal{L}_{\beta/2}(\vartheta_1)}\ell_\beta(\vartheta_1,\vartheta_2)\leq\ell_{\beta/2}(\vartheta_1,\vartheta_2), \end{align}\]for all \(\vartheta_1,\vartheta_2\in\mathscr{V}\). Combining this inequality with Equation 55 , we obtain Equation 54 . Thus, the lemma has been proved. ◻

Proof. By Equation 49 , we have \[\overline{\mathrm{W}}_\beta(\mathbf{w})=\sum_{n=2}^\infty\frac{1}{(n-2)!}\sum_{T\in\mathbbold{T}_n}\sum_{\mathbf{v}\in\mathscr{V}^n}\mathbbold{1}_{\mathbf{v}|_{[2]}=\mathbf{w}}w_\beta(T,\mathbf{v}).\]Introducing the application \(\Psi_{n,2}\), we obtain \[\overline{\mathrm{W}}_\beta(\mathbf{w})=\sum_{n=2}^\infty\frac{1}{(n-2)!}\sum_{T\in\mathbbold{T}_n}\sum_{\underline{T}\in\underline{\mathbbold{T}}_{n,2}}\mathbbold{1}_{\Psi_{n,2}(T)=\underline{T}}\sum_{\mathbf{v}\in\mathscr{V}^n}\mathbbold{1}_{\mathbf{v}|_{[2]}=\mathbf{w}}w_\beta(T,\mathbf{v}).\]Inverting the order of the sums and recalling that \(\Psi_{n,2}(T)=T[2]\), we have \[\overline{\mathrm{W}}_\beta(\mathbf{w})=\sum_{\underline{T}\in\underline{\mathbbold{T}}_2}\sum_{n=2}^\infty\frac{1}{(n-2)!}\sum_{T\in\mathbbold{T}_n}\mathbbold{1}_{T[2]=\underline{T}}\sum_{\mathbf{v}\in\mathscr{V}^n}\mathbbold{1}_{\mathbf{v}|_{[2]}=\mathbf{w}}w_\beta(T,\mathbf{v}).\]We note that \(\underline{\mathbbold{T}}_2\) is in one-to-one correspondence with pairs \((\underline{n},\underline{\varphi})\), where \(\underline{n}\geq 2\) is a natural number and \(\underline{\varphi}\) is an injection from \([\underline{n}]\) to \(\mathbb{N}\) such that \(\underline{\varphi}(1)=1\) and \(\underline{\varphi}(\underline{n})=2\). This bijection is given by

1. \(V(\underline{T})=\underline{\varphi}([\underline{n}])\),

2. \(E(\underline{T})=\{\{\underline{\varphi}(i),\underline{\varphi}(i+1)\}:i\in [\underline{n}-1]\}\).

We also note that if \(T[2]=\underline{T}\), then \(\textrm{Im}(\underline{\varphi})\subset [n]\). That is, if \(\underline{N}\mathrel{\vcenter{:}}= \max\textrm{Im}(\underline{\varphi})\leq n\).

For each \(\underline{T}\in\underline{\mathbbold{T}}_2\), provided \(n\geq\underline{N}\), the set \(\Psi_{n,2}^{-1}(\underline{T})\) is in a one-to-one correspondence with

1. an ordered partition \(\mathbf{P}=(P_1,...,P_{\underline{n}})\) of \([n]\) such that \(i\in[\underline{n}]\Rightarrow\underline{\varphi}(i)\in P_i\);

2. a vector of trees \(\mathbf{T}=(T_1,...,T_{\underline{n}})\), such that \(T_i\in\mathbbold{T}(P_i)\).

The correspondence is given by \[E(T)=E(\underline{T})\sqcup\bigsqcup_{i=1}^{\underline{n}}E(T_i).\]In this case, if \(T\in\Psi_{n,2}^{-1}(\underline{T})\), then there exists \((\mathbf{P},\mathbf{T})\) such that \[\label{eq:32weight-usual-cluster-2} w_\beta(T,\mathbf{v})=\prod_{i=1}^{\underline{n}}w_\beta(T_i,\mathbf{v}|_{P_i})\prod_{j=1}^{\underline{n}-1}e(\vartheta_{\underline{\varphi}(j)},\vartheta_{\underline{\varphi}(j+1)}).\tag{57}\] By Equation 57 , we have \[\overline{\mathrm{W}}_\beta(\mathbf{w})=\sum_{\underline{n}=2}^\infty\sum_{\underline{\varphi}}\sum_{n=\underline{N}}^\infty\frac{1}{(n-2)!}\sum_{\mathbf{P}}\sum_{\mathbf{T}}\sum_{\mathbf{v}\in\mathscr{V}^n}\mathbbold{1}_{\mathbf{v}|_{[2]}=\mathbf{w}}\prod_{i=1}^{\underline{n}}w_\beta(T_i,\mathbf{v}|_{P_i})\prod_{j=1}^{\underline{n}-1}e(\vartheta_{\underline{\varphi}(j)},\vartheta_{\underline{\varphi}(j+1)}),\]and an analogous formula for \(\mathrm{W}_\beta(\mathbf{w})\).

It is useful to separate the sum term indexed by \(\underline{n}=2\). Note that in this case there is a single injection \(\underline{\varphi}\) and \(\underline{N}=2\). Furthermore, the term involving edge functions simplifies to \(e(\varOmega_1,\varOmega_2)\). Since the indices are mute, the matter becomes counting the set of partitions \(\mathbf{P}=(P_1,P_2)\) of \([n]\) such that \(1\in P_1\) and \(2\in P_2\). One readily finds that the term indexed by \(\underline{n}=2\) is equal to \[\overline{\mathrm{W}}_\beta(\varOmega_1)e(\varOmega_1,\varOmega_2)\overline{\mathrm{W}}_\beta(\varOmega_2).\]Since \(\mathscr{V}_{n_1+n_2}\subset\mathscr{V}_{n_1}\times\mathscr{V}_{n_2}\), where equality is not necessary, in the case of \(\mathrm{W}\), the same argument yields an upper bound: the term indexed by \(\underline{n}=2\) is bounded above by \[\mathrm{W}_\beta(\varOmega_1)e(\varOmega_1,\varOmega_2)\mathrm{W}_\beta(\varOmega_2).\]

Henceforth, we suppose that \(\underline{n}\geq 3\). Let us begin by eliminating the sum over \(\underline{\varphi}\) and transform it into a sum over \(\underline{N}\geq\underline{n}\). For each \(\underline{\varphi}\), this will be done by relabeling \(i\in[n]\) (not \(\underline{n}\) or \(\underline{N}\)) so that for \(i\in[\underline{n}]\), \(\underline{\varphi}(i)=i\). This operation will introduce a combinatorial factor which we compute in the following. Given \(\underline{N}\), one must simply count the amount of injections from \([\underline{n}]\) to \(\mathbb{N}\) such that \(\underline{\varphi}(1)=1\), \(\underline{\varphi}(\underline{n})=2\) and \(\max\textrm{Im}(\underline{\varphi})=\underline{N}\). There are \((\underline{n}-2)\) possible indices such that \(\underline{\varphi}(i)=\underline{N}\), in which case one must simply distribute \((\underline{N}-3)\) values over the remaining \((\underline{n}-3)\) indices. That is, one obtains the following combinatorial factor \[(\underline{n}-2)\times(\underline{N}-3)...(\underline{N}-\underline{n}+1)=(\underline{n}-2)\frac{(\underline{N}-3)!}{(\underline{N}-\underline{n})!}.\] The sum over \(\underline{n}\geq3\) becomes \[\sum_{\underline{n}=3}^\infty\sum_{\underline{N}=\underline{n}}^\infty\sum_{n=\underline{N}}^\infty\frac{(\underline{N}-3)!(\underline{n}-2)}{(\underline{N}-\underline{n})!(n-2)!}\sum_{\mathbf{P}}\sum_{\mathbf{T}}\sum_{\mathbf{v}\in\mathscr{V}^n}\mathbbold{1}_{\vartheta_1=\varOmega_1}\mathbbold{1}_{\vartheta_{\underline{n}}=\varOmega_2}\prod_{i=1}^{\underline{n}}w_\beta(T_i,\mathbf{v}|_{P_i})\prod_{j=1}^{\underline{n}-1}e(\vartheta_{j},\vartheta_{j+1}).\]We note that the ordered partition now satisfies \(i\in P_i\), for \(i\in [\underline{n}]\).

The next step is to relabel the ordered partitions. Consider a vector \(\mathbf{n}=(n_1,...,n_{\underline{n}})\) such that \(n_i\geq 1\) and \(n_1+...+n_{\underline{n}}=n\). Since \(i\in P_i\), there are \[\frac{(n-\underline{n})!}{\prod_{i=1}^{\underline{n}}(n_i-1)!}\]ordered partitions \(\mathbf{P}\) such that \(\lvert P_i\rvert=n_i\). In this case, relabeling them in terms of \(\mathbf{n}\) and recalling Equation 49 , the sum becomes \[\sum_{\underline{n}=3}^\infty\sum_{\underline{N}=\underline{n}}^\infty\sum_{n=\underline{N}}^\infty C(\underline{n},\underline{N},n)\sum_{\mathbf{n}}\sum_{\mathbf{v}\in\mathscr{V}^{\underline{n}}}\mathbbold{1}_{\vartheta_1=\varOmega_1}\mathbbold{1}_{\vartheta_{\underline{n}}=\varOmega_2}\prod_{i=1}^{\underline{n}}\overline{\mathrm{W}}_{\beta,n_i}(\vartheta_i)\prod_{j=1}^{\underline{n}-1}e(\vartheta_j,\vartheta_{j+1}),\]where \[C(\underline{n},\underline{N},n)\mathrel{\vcenter{:}}= \frac{(\underline{n}-2)(\underline{N}-3)!(n-\underline{n})!}{(\underline{N}-\underline{n})!(n-2)!}=\binom{\underline{N}-3}{\underline{n}-3}\binom{n-2}{\underline{n}-2}^{-1}.\]Once again, in the case of \(\mathrm{W}_\beta(\cdot)\), the analogous formula is an upper bound.

The final step consists of showing that, for all \(\mathbf{v}\in\mathscr{V}^{\underline{n}}\), \[\sum_{\underline{N}=\underline{n}}^\infty\sum_{n=\underline{N}}^\infty C(\underline{n},\underline{N},n)\sum_{\mathbf{n}}\prod_{i=1}^{\underline{n}}\overline{\mathrm{W}}_{\beta,n_i}(\vartheta_i)=\prod_{i=1}^{\underline{n}}\sum_{n_i=1}^\infty\overline{\mathrm{W}}_{\beta, n_i}(\vartheta_i)=\prod_{i=1}^{\underline{n}}\overline{\mathrm{W}}_\beta(\vartheta_i),\]where we recall Equation 49 . Noting that \[\sum_{\underline{N}=\underline{n}}^\infty\sum_{n=\underline{N}}^\infty\sum_{\mathbf{n}}=\sum_{n_1=...=n_{\underline{n}}=1}^\infty\sum_{n=\underline{n}}^\infty\sum_{\underline{N}=\underline{n}}^n\mathbbold{1}_{n=n_1+...+n_{\underline{n}}},\]it is sufficient to show that \[\label{eq:32computation-mute-indices} \sum_{n=\underline{n}}^\infty\sum_{\underline{N}=\underline{n}}^n\mathbbold{1}_{n=n_1+...+n_{\underline{n}}}C(\underline{n},\underline{N},n)=1.\tag{58}\] We recall that if \(r,s\geq 0\), then \[\sum_{t=0}^r\binom{s+t}{t}=\binom{r+s+1}{r+1}.\]Hence, \[\sum_{\underline{N}=\underline{n}}^n\binom{\underline{N}-3}{\underline{n}-3}=\sum_{\underline{N}=0}^{n-\underline{n}}\binom{\underline{n}-3+\underline{N}}{\underline{N}}=\binom{n-2}{\underline{n}-2}.\]We have thus shown that Equation 58 holds. The Lemma has thus been proved. ◻

Proof. First, employing an idea very similar to that in the proof of Proposition 3.7 in [25], we obtain a bound for \(\mathrm{E}'_{\beta,n}(\varOmega_1,\varOmega_2)\). Then, we conclude by Equation 61 . We provide the algorithm operated on \(\mathrm{E}'_{\beta,n}(\varOmega_1,\varOmega_2)\) below. Its crucial feature is the fact that no vertex is visited more than twice.

There is a state \(s\) that takes values in \(\{L,R\}\), which indicates in which direction the pointer will move next: \(L\) is for left and \(R\) is for right. The starting position of the pointer is \(n_0=\lfloor n/2\rfloor+1\) (recall that there are \(n+2\) vertices). If \(n\) is odd, the starting state is \(s_0=L\). If \(n\) is even, the starting state is \(s_0=R\).

After setting up the starting position, we run the following monotone algorithm. More precisely, we run an algorithm that can only increase the value of the estimated sum.

In the first step, we check whether the pointer is over an extremal vertex. If it is, then the algorithm ends. Otherwise, we proceed to the second step.

In the second step, we update the weight of the vertex indicated by the pointer. More precisely, its current weight, which is equal to \(v_\beta(\cdot)v'_\beta(\cdot)^{\xi}\), where \(\xi\in\{0,1,2\}\), is updated to \(2v_{\beta/2}(\cdot)\). This is possible due to Hypothesis 4 and \(x^2\leq 2e^x\), if \(x>0\).

In the third step, the pointer moves in the direction indicated by the state, then it updates (\(L\rightarrow R\) or \(R\rightarrow L\)).

In the fourth step, if \(s=L\) (\(s=R\)), then we remove the vertex to the left (right) of the pointer employing Hypothesis 6. More precisely, if \(\beta\geq \beta_0(\eta)\), then \[\mathcal{R}^2_{\beta/2}(\vartheta_1,\vartheta_2)\leq (\eta/4)[v'_{\beta/2}(\vartheta_1)+v'_{\beta/2}(\vartheta_2)]e(\vartheta_1,\vartheta_2)\leq (\eta/4)[v'_\beta(\vartheta_1)+v'_\beta(\vartheta_2)]e(\vartheta_1,\vartheta_2),\]where the second inequality follows from Hypothesis 4. In this way, a single sum is split into two: in the first, the weight of the vertex indicated by the pointer has been multiplied by \(v'_\beta(\cdot)\); in the second, the vertex to the direction indicated by the state of the pointer has been multiplied by \(v'_\beta(\cdot)\). We move to the first step.

For a detailed graphical description of the algorithm, see Figure 10. By the end, we obtain \[\begin{align} \mathrm{E}'_{\beta,n}(\varOmega_1,\varOmega_2)&\leq 2^n2^n(\eta/4)^n\max\left\{v'_\beta(\varOmega_1)^{\xi_1}v'_\beta(\varOmega_2)^{\xi_2}:\xi_1,\xi_2\in\{0,1,2\}\right\}e(\Omega_1,\Omega_2)\\ &=\eta^nv'_\beta(\varOmega_1)^2e(\varOmega_1,\varOmega_2)v'_\beta(\varOmega_2)^2, \end{align}\]where the equality is a consequence of Hypothesis 4, as long as \(\beta\geq\beta_0\) is sufficiently large. The first term \(2^n\) comes from the number of sums generated as the algorithm runs. The second term \(2^n\) comes from the second step. Recalling Equation 61 , this concludes the proof of the lemma. ◻

Proof. The proof is a direct application of Lemmas 8 and 10. We have\[\begin{align} \mathrm{W}_\beta(\varOmega_1,\varOmega_2)&\leq v_{\beta/2}(\varOmega_1)e(\varOmega_1,\varOmega_2)v_{\beta/2}(\varOmega_2)+\sum_{n=1}^\infty v_{\beta/2}(\varOmega_1)\mathrm{E}'_{\beta/2,n}(\varOmega_1,\varOmega_2)v_{\beta/2}(\varOmega_2). \end{align}\]By Hypothesis 4, we can bound the first term by \[\begin{align} v_{\beta/2}(\varOmega_1)e(\varOmega_1,\varOmega_2)v_{\beta/2}(\varOmega_2)\leq 2^{-1}v_{\beta/4}(\varOmega_1)e(\varOmega_1,\varOmega_2)v_{\beta/4}(\varOmega_2). \end{align}\]Let us now bound the second term. Choose \(\eta=9^{-1}\) in Lemma 10. Recalling that \[v'_{\beta/2}(\vartheta)^2v_{\beta/2}(\vartheta)\leq 2!v_{\beta/4}(\vartheta)=2v_{\beta/4}(\vartheta),\] we obtain \[\sum_{n=1}^\infty v_{\beta/2}(\varOmega_1)\mathrm{E}'_{\beta/2,n}(\varOmega_1,\varOmega_2)v_{\beta/2}(\varOmega_2)\leq 2^{-1}v_{\beta/4}(\varOmega_1)e(\varOmega_1,\varOmega_2)v_{\beta/4}(\varOmega_2).\]The corollary has thus been proved. ◻

Proof. On the one hand, since \(T\) is a tree, we have \(\lvert E(T)\rvert=n-1\). On the other hand, we have \[\begin{align} \lvert E(T)\rvert&=\frac{1}{2}\sum_{v\in [n]}\delta_T(v)=\frac{1}{2}\sum_{v\in [m]}\delta_T(v)+\frac{1}{2}\sum_{v=m+1}^n\delta_T(v)\geq \frac{1}{2}m+\frac{3}{2}(n-m)=\frac{1}{2}(3n-2m). \end{align}\]We thus obtain \(n\leq 2m-2\). ◻

Proof. Since the combinatorial part of the argument is identical to that of the proof of Lemma 9, we just describe the argument in terms of operations of trees.

In the first step, we contract every tree to ensure that the set of leaves is a subset of \([m]\). Then, we relabel them to ensure that their vertex set is given by \([n]\), for some \(n\geq m\). That is, recalling Notation 1, we obtain \[\label{eq:32m-points-abstract-sum-step-one} \overline{\mathrm{W}}_\beta(\mathbf{w})=\sum_{n=m}^\infty\frac{1}{(n-m)!}\sum_{T\in\overline{\mathbbold{T}}_{n,m}}\sum_{\mathbf{v}\in\mathscr{V}^n}\mathbbold{1}_{\mathbf{v}|_{[m]}=\mathbf{w}}\prod_{i=1}^n\overline{\mathrm{W}}_{\beta}(\vartheta_i)\prod_{\{j,k\}\in E(T)}e(\vartheta_j,\vartheta_k).\tag{65}\]

In the second step, we contract lines to guaranty that if a vertex is not labeled by an element of \([m]\), then it has degree at least \(3\). Let \(\Psi^*_m:\overline{\mathbbold{T}}_m\rightarrow\mathbbold{T}^*_m\) be the application defined by the composition of the following two maps. First, let \(T\in\overline{\mathbbold{T}}_m\), define \[V(T^*)\mathrel{\vcenter{:}}=[m]\sqcup\{i\in V(T): (i>m)\wedge (\delta_T(i)\geq 3)\}.\]Let \(i,j\in V(T^*)\), we denote the (unordered) path in \(T\) connecting \(i\) and \(j\) by \(\mathfrak{p}_T(i,j)\). In this case, we also define \[E(T^*)=\{\{i,j\}\in\mathrm{P}_2(V(T^*)):V(\mathfrak{p}_T(i,j))\cap V(T^*)=\{i,j\}\}.\]Since \(T^*\mathrel{\vcenter{:}}= (V(T^*),E(T^*))\in\overline{\mathbbold{T}}^*_m\), we have just constructed an application \(\overline{\Psi}^*_m:\overline{\mathbbold{T}}_m\rightarrow\overline{\mathbbold{T}}^*_m\).

Let \([m]\subset B\) such that \(\lvert B\rvert\leq 2m-2\). We denote by \(\varphi_B:B\rightarrow[\lvert B\rvert]\) the canonical bijection satisfying \[b_1<b_2\in B\Rightarrow\varphi_B(b_1)<\varphi_B(b_2).\]The family \(\{\varphi_B:([m]\subset B\in\mathrm{P}(\mathbb{N}))\wedge(\lvert B\rvert\leq 2m-2)\}\) induces a canonical relabeling application \(\overline{\varphi}_m:\overline{\mathbbold{T}}^*_m\rightarrow\mathbbold{T}^*_m\). Finally, we define \(\Psi^*_m\mathrel{\vcenter{:}}=\overline{\varphi}_m\circ\overline{\Psi}^*_m\). For a visual description, see Figure 12.

Proceeding as in the proof of Lemma 9, we transform Equation 65 into Equation 64 . Once again, we note that in the case of the global compatibility condition, we obtain upper bounds. ◻

Proof. The cases \(m=1\) and \(m=2\) have already been dealt with, and hence we assume that \(m\geq 3\). The starting point is Equation 66 . Our strategy is to show that if \(m\leq n\leq n+1\leq 2m-2\), then \[\label{eq:32iterative-step-m-point-bound} \mathrm{W}^*_{\beta,n+1}(\mathbf{w})\leq \frac{1}{2}\mathrm{W}^*_{\beta/2,n}(\mathbf{w}).\tag{71}\] This is indeed sufficient: assume that Equation 71 holds. An easy induction argument proves that if \(0\leq k\leq m-2\), then \(\mathrm{W}^*_{\beta,m+k}(\mathbf{w})\leq 2^{-k}\mathrm{W}^*_{2^{-k}\beta,m}(\mathbf{w})\). Therefore, \[\label{eq:32consequence-of-iterative-step-m-point-bound} \mathrm{W}_\beta(\mathbf{w})\leq\sum_{n=m}^{2m-2}\mathrm{W}^*_{\beta/4,n}(\mathbf{w})= \sum_{k=0}^{m-2}\mathrm{W}^*_{\beta/4,m+k}(\mathbf{w})\leq\sum_{k=0}^{m-2}\frac{1}{2^k}\mathrm{W}^*_{2^{-2-k}\beta,m}(\mathbf{w}).\tag{72}\] Since \(\mathbbold{T}^*_{m,m}=\mathbbold{T}_m\), we have \[\mathrm{W}^*_{2^{-2-k}\beta,m}(\mathbf{w})=\sum_{T\in\mathbbold{T}_m}\prod_{i=1}^mv_{2^{-2-k}\beta}(\varOmega_i)\prod_{\{j,k\}\in E(T)}e(\varOmega_j,\varOmega_k).\]Thus, if \(0\leq k<m-2\), then, by Hypothesis 4, we may suppose that \[\label{eq:32W39-same-index-decreasing-beta-bound} \mathrm{W}^*_{2^{-2-k}\beta,m}(\mathbf{w})\leq\frac{1}{2}\sum_{T\in\mathbbold{T}_m}\prod_{i=1}^mv_{2^{-m}\beta}(\varOmega_i)\prod_{\{j,k\}\in E(T)}e(\varOmega_j,\varOmega_k)=\frac{1}{2}\mathrm{W}^*_{2^{-m}\beta,m}(\mathbf{w}).\tag{73}\] Inserting Equation 73 into Equation 72 , we obtain \[\mathrm{W}_\beta(\mathbf{w})\leq\left(\sum_{k=0}^{m-3}\frac{1}{2^{k+1}}+\frac{1}{2^{m-2}}\right)\mathrm{W}^*_{2^{-m}\beta,m}(\mathbf{w})=\sum_{T\in\mathbbold{T}_m}\prod_{i=1}^mv_{2^{-m}\beta}(\varOmega_i)\prod_{\{j,k\}\in E(T)}e(\varOmega_j,\varOmega_k).\]To finish the proof, let us show that Equation 71 holds.

By Hypothesis 7, there exists \(\beta_0\) such that if \(\beta\geq\beta_0\), then \[\mathcal{R}^k_\beta(\mathbf{v})\leq (4\lvert\mathbbold{T}_{2m-2}\rvert)^{-1}\sum_{\varphi\in\mathrm{Sym}(k)}v'_\beta(\vartheta_{\varphi(k)})\prod_{i=1}^{k-1}e(\vartheta_{\varphi(i)},\vartheta_{\varphi(i+1)}),\]for \(3\leq k\leq 2m-3\) and all \(\mathbf{v}\in\mathscr{V}_k\). Hence, recalling Equation 69 , we obtain \[\mathrm{W}^*_{\beta,n+1}(\mathbf{w})=\sum_{T^*\in\mathbbold{T}^*_{n+1,m}}\mathrm{W}_\beta(\mathbf{w},T^*)\leq \sum_{T^*\in\mathbbold{T}^*_{n+1,m}}(2\lvert\mathbbold{T}_{2m-2}\rvert)^{-1}\mathrm{W}^*_{\beta/2,n}(\mathbf{w})\leq\frac{1}{2}\mathrm{W}^*_{\beta/2,n}(\mathbf{w}),\]where the final inequality holds since \(T^*\in\mathbbold{T}^*_{n+1,m}\) implies \(\delta_{T^*}(n+1)\in [3,2m-3]\) and \(m\leq n\leq 2m-2\) implies \(\lvert\mathbbold{T}^*_{n,m}\rvert\leq\lvert\mathbbold{T}_{2m-2}\rvert\). The theorem has thus been proved. ◻

Proof. By definition, we have \[\mathcal{L}_\beta(\gamma_1)=\sum_{\gamma\in\mathscr{C}}e^{-\beta H(\gamma)}\Phi(\gamma,\gamma_1)\mathbbold{1}_{\gamma \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_1}=\sum_{[\gamma]\in\bar{\mathscr{C}}}e^{-\beta H([\gamma])}\sum_{\gamma\in[\gamma]}\Phi(\gamma,\gamma_1)\mathbbold{1}_{\gamma \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_1},\]where we introduce the equivalence classes of contours under integer translations.

Let us estimate the sum over an equivalence class \([\gamma]\). We have \[\sum_{\substack{\gamma\in[\gamma]}}\Phi(\gamma,\gamma_1)\mathbbold{1}_{\gamma \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_1}=\sum_{\substack{x\in\mathrm{I}_-(\gamma_1)\\y\in\mathrm{I}_-(\gamma_1)^c}}4J_{xy}\sum_{\gamma\in[\gamma]}\mathbbold{1}_{y\in\mathrm{I}_-(\gamma)}\mathbbold{1}_{\gamma \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_1}.\] Since \(\mathbbold{1}_{\gamma_1 \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma}\leq 1\) and at most \(\lvert\mathrm{I}_-(\gamma)\rvert\leq \mathrm{diam}(\gamma)\) elements \(\gamma\in[\gamma]\) satisfy \(y\in\mathrm{I}_-(\gamma)\), we obtain \[\sum_{\substack{\gamma\in[\gamma]}}\Phi(\gamma,\gamma_1)\mathbbold{1}_{\gamma \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_1}\leq 2\mathrm{diam}(\gamma)H(\gamma_1).\] Finally, by Hypotheses 2 and 3, we obtain \[\begin{align} \mathcal{L}_\beta(\gamma)\leq 2\sum_{[\gamma_1]\in\overline{\mathscr{C}}}e^{-(\beta-c_0)H([\gamma_1])} H(\gamma)\leq 2e^{-c_2\beta/2} H(\gamma), \end{align}\] where the last inequality follows by choosing \(\beta_0\geq 2c_0\) (recall Hypothesis 3). The lemma has thus been proved. ◻

Proof. This is a direct consequence of Lemmas 8 and 13. ◻

Proof. Let \(x,y,z\in\mathbb{R}\), then \[\begin{align} \label{eq:32reverse-holder} \lvert x-y\rvert^{\alpha}&\leq (\lvert x-z\rvert+\lvert z-x\rvert)^\alpha\leq 2^{\alpha-1}\lvert x-z\rvert^\alpha+2^{\alpha-1}\lvert z-y\rvert^\alpha, \end{align}\tag{82}\] where the second inequality is a reverse Hölder inequality. For each permutation \(\varphi\in \mathrm{Sym}(k)\), define \[\begin{align} E_\varphi=\{i\leq j\in [k]\Rightarrow\lvert a_{\varphi(i)}-b_{\varphi(i)}\rvert\geq\lvert a_{\varphi(j)}-b_{\varphi(j)}\rvert\}\subset\bigtimes_{i=1}^k(A_i\times B). \end{align}\] Let \((a_1,b_1,...,a_k,b_k)\in E_\varphi\) and \(i\in[k-1]\), then, by Equation 82 , \[\lvert a_{\varphi(i)}-a_{\varphi(i+1)}\rvert^\alpha\leq \left(2\lvert a_{\varphi(i)}-b_{\varphi(i)}\rvert+\mathrm{diam}(B)\right)^\alpha\leq 2^{2\alpha-1}\lvert a_{\varphi(i)}-b_{\varphi(i)}\rvert^\alpha+2^{\alpha-1}[1+\mathrm{diam}(B)]^\alpha.\]On the other hand, since \(x,y\geq2\Rightarrow x+y\leq xy\), we obtain \[\lvert a_{\varphi(i)}-a_{\varphi(i+1)}\rvert^\alpha\leq 4^{\alpha}(1+\mathrm{diam}(B))^\alpha\lvert a_{\varphi(i)}-b_{\varphi(i)}\rvert^\alpha.\]Hence, \[\begin{align} \prod_{i=1}^{k-1}\lvert a_{\varphi(i)}-a_{\varphi(i+1)}\rvert^\alpha\leq 4^{\alpha k}(1+\mathrm{diam}(B))^{\alpha k}\prod_{i=1}^{k-1}\lvert a_{\varphi(i)}-b_{\varphi(i)}\rvert^\alpha. \end{align}\] In this case, for any \((a_1,b_1,...,a_k,b_k)\in E_\varphi\), we have \[\begin{align} \frac{\prod_{i=1}^{k-1}\lvert a_{\varphi(i)}-a_{\varphi(i+1)}\rvert^{\alpha}}{\prod_{i=1}^k\lvert a_i-b_i\rvert^{\alpha}}\leq \frac{\left(4^{\alpha}(1+\mathrm{diam}(B))^{\alpha}\right)^{k}}{\lvert a_{\varphi(k)}-b_{\varphi(k)}\rvert^\alpha}. \end{align}\]On the other hand, since \(1\leq \sum_{\varphi\in S_k} \mathbbold{1}_{E_\varphi}\), we arrive at \[\begin{align} \prod_{i=1}^k\Phi(A_i,B)\leq \left(16^{\alpha}\lvert B\rvert(1+\mathrm{diam}(B))^\alpha\right)^{k}\sum_{\varphi\in \mathrm{Sym}(k)}\sum_{\substack{a_i\in A_i\\b\in B\\i\in[k]}}\frac{1}{\lvert a_{\varphi(k)}-b\rvert^{\alpha}}\prod_{j=1}^{k-1}\frac{1}{\lvert a_{\varphi(j)}-a_{\varphi(j+1)}\lvert^\alpha}. \end{align}\]Finally, since \(\mathrm{dist}(A_{\varphi(k)},B)\leq\lvert a_{\varphi(k)}-b\rvert\) and \[\begin{align} \sum_{\substack{a_i\in A_i\\i\in[k]}}\prod_{j=1}^{k-1}\frac{1}{\lvert a_{\varphi(j)}-a_{\varphi(j+1)}\lvert^\alpha}\leq \prod_{j=1}^{k-1}\sum_{\substack{a_{\varphi(j)}\in A_{\varphi(j)}\\a_{\varphi(j+1)}\in A_{\varphi(j+1)}}}\frac{1}{\lvert a_{\varphi(j)}-a_{\varphi(j+1)}\rvert^{\alpha}}, \end{align}\]we arrive at Equation 81 . The lemma has thus been proved. ◻

Proof. Introducing equivalence classes as in the proof of Lemma 13, we have \[\mathcal{R}^k_ {\beta}(\gamma_1,...,\gamma_k)=\sum_{[\gamma]\in\overline{\mathscr{C}}}e^{-\beta H([\gamma])}\sum_{\gamma\in [\gamma]}\prod_{i=1}^k\Phi(\gamma_i,\gamma)\mathbbold{1}_{\gamma_i \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma}.\] On the other hand, by Lemma 14, we obtain \[\prod_{i=1}^k\Phi(\gamma_i,\gamma)\mathbbold{1}_{\gamma_i \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma}\leq f_k([\gamma])\sum_{\varphi\in \mathrm{Sym}(k)}\frac{\mathbbold{1}_{\gamma_{\varphi(k)} \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma}}{\mathrm{dist}\left(\mathrm{I}_-(\gamma_{\varphi(k)}),\mathrm{I}_-(\gamma)\right)^\alpha}\prod_{j=1}^{k-1}\Phi\left(\gamma_{\varphi(j)},\gamma_{\varphi(j+1)}\right),\] where, by Hypothesis 2, \[\begin{align} f_k([\gamma])&=16^{\alpha k}\mathrm{diam}([\gamma])^{2\alpha k}\leq 16^{\alpha k}e^{2\alpha c_0kH([\gamma])} \end{align}\]For any \(l\in[k]\) and \(n\in\mathbb{N}\), there are at most \(2\lvert \gamma_l\rvert\) elements of \([\gamma]\) that satisfy \(\mathrm{dist}(\mathrm{I}_-(\gamma_l),\mathrm{I}_-(\gamma))=n\). Hence, \[\sum_{\gamma\in[\gamma]}\frac{\mathbbold{1}_{\gamma_l \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma}}{\mathrm{dist}(\mathrm{I}_-(\gamma_l),\mathrm{I}_-(\gamma))^\alpha}\leq 2\lvert\gamma_l\rvert\sum_{n=1}^\infty\frac{1}{n^\alpha}\leq\frac{\alpha}{\alpha-1}H(\gamma_l).\]Finally, we arrive at \[\begin{align} \mathcal{R}_{\beta}^k(\gamma_1,...,\gamma_k)&\leq\frac{16^{\alpha k}\alpha}{\alpha-1}\sum_{\varphi\in \mathrm{Sym}(k)}\prod_{j=1}^{k-1}\Phi(\gamma_{\varphi(j)},\gamma_{\varphi(j+1)})\sum_{[\gamma]\in\overline{\mathscr{C}}}e^{-(\beta-2\alpha c_0k)H([\gamma])} H\left(\gamma_{\varphi(k)}\right)\\ &\leq \frac{16^{\alpha k}\alpha}{\alpha-1}e^{-\frac{1}{2}c_2\beta}\sum_{\varphi\in \mathrm{Sym}(k)} H\left(\gamma_{\varphi(k)}\right)\prod_{j=1}^{k-1}\Phi\left(\gamma_{\varphi(j)},\gamma_{\varphi(j+1)}\right), \end{align}\]where the last inequality is a consequence of Hypothesis 3 and \(\beta\geq 4\alpha c_0k\). The corollary has thus been proved. ◻

Proof. This is a direct consequence of Corollaries 2 and 4. ◻

Proof. This is a direct consequence of Theorem 4 and Corollary 4. ◻

Proof. We have \[\begin{align} \sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{\Gamma_1\not\sim\Gamma}\bar{z}_\beta(\Gamma)&=\sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{\left\{\substack{\exists\gamma_1\in\Gamma_1,\gamma\in\Gamma\\\textrm{s.t. }\gamma_1\not\sim\gamma}\right\}}\bar{z}_\beta(\Gamma)+\sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{\left\{\substack{\forall\gamma_1\in\Gamma_1,\gamma\in\Gamma\\\Rightarrow\gamma_1\sim\gamma}\right\}}\mathbbold{1}_{\Gamma_1\not\sim\Gamma}\bar{z}_\beta(\Gamma). \end{align}\] Let us begin by dealing with the first sum. Recalling Equation 79 , Corollary 3 and introducing equivalence classes of contours under integer translations, we obtain \[\mathrm{S}_0\mathrel{\vcenter{:}}=\sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{\left\{\substack{\exists\gamma_1\in\Gamma_1,\gamma\in\Gamma\\\textrm{s.t. }\gamma_1\not\sim\gamma}\right\}}\bar{z}_\beta(\Gamma)\leq\sum_{\gamma_1\in\Gamma_1}\sum_{\gamma\in\mathscr{C}}\mathbbold{1}_{\gamma\not\sim\gamma_1}\sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{\gamma\in\Gamma}\bar{z}_\beta(\Gamma)\leq \sum_{\gamma_1\in\Gamma_1}\sum_{[\gamma]\in\overline{\mathscr{C}}}e^{-\frac{1}{4}\beta H([\gamma])}\sum_{\gamma\in[\gamma]}\mathbbold{1}_{\gamma\not\sim\gamma_1}.\]By Definition 1, we have \[\sum_{\gamma\in[\gamma]}\mathbbold{1}_{\gamma\not\sim\gamma_1}\leq\sum_{x^*_1\in\gamma_1}\sum_{\gamma\in[\gamma]}\mathbbold{1}_{\mathrm{dist}(x^*_1,\gamma)\leq M\mathrm{diam}(\gamma)^{3/2}}\leq 3M\lvert\gamma_1\rvert\mathrm{diam}([\gamma])^{3/2}\leq 2M\mathrm{diam}([\gamma])^{3/2}H(\gamma_1).\]Combining Hypotheses 2 and 3, choosing \(\beta\geq 12c_0+4\log(2M)\), we obtain\[\mathrm{S}_0\leq e^{-c_2\beta/8}\sum_{\gamma_1\in\Gamma_1} H(\gamma_1).\]For the second sum, we again divide it into two parts. If \(\Gamma_1,\Gamma\in\mathscr{P}\) are such that \(\Gamma_1\cap\Gamma=\emptyset\), \(\Gamma_1\sqcup\Gamma\in\mathscr{E}\) and \(\Gamma_1\not\sim\Gamma\), then one of the following assertions holds:

1. there exist \(\gamma\in\Gamma\) and \(\gamma_1,\gamma'_1\in\Gamma_1\) such that \(\gamma_1\trianglelefteq\gamma\) and \(\gamma_1' \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma\);

2. or, there exists \(\gamma,\gamma'\in\Gamma\) and \(\gamma_1\in\Gamma_1\) such that \(\gamma\trianglelefteq\gamma_1\) and \(\gamma' \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_1\).

That is, \(\Gamma\in A_1(\Gamma_1)\cup A_2(\Gamma_1)\), where \[A_1(\Gamma_1)\mathrel{\vcenter{:}}=\{\Gamma\in\mathscr{P}:\exists\gamma\in\Gamma,\exists\gamma_1,\gamma_1'\in\Gamma_1\textrm{ s.t. }(\gamma_1\trianglelefteq\gamma)\wedge(\gamma'_1 \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma)\},\] and \[A_2(\Gamma_1)\mathrel{\vcenter{:}}=\{\Gamma\in\mathscr{P}:\exists\gamma,\gamma'\in\Gamma,\exists\gamma_1\in\Gamma_1\textrm{ s.t. }(\gamma\trianglelefteq\gamma_1)\wedge(\gamma' \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_1)\}.\] We have, by Corollary 3, \[\mathrm{S}_1\mathrel{\vcenter{:}}=\sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{\Gamma\in A_1(\Gamma_1)}\bar{z}_\beta(\Gamma)\leq\sum_{\gamma_1\in\Gamma_1}\sum_{\gamma\in\mathscr{C}}\mathbbold{1}_{\gamma_1\trianglelefteq\gamma}\sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{\gamma\in\Gamma}\bar{z}_\beta(\Gamma)\leq\sum_{\gamma_1\in\Gamma_1}\sum_{\gamma\in\mathscr{C}}\mathbbold{1}_{\gamma_1\trianglelefteq\gamma}e^{-\frac{1}{4}\beta H(\gamma)}.\] Introducing equivalence classes of contours under integer translations, combining Hypotheses 2 and 3 and choosing \(\beta\geq8c_0\), we obtain \[\begin{align} \mathrm{S}_1\leq \sum_{\gamma_1\in\Gamma_1}\sum_{[\gamma]\in\overline{\mathscr{C}}}e^{-\frac{1}{4}\beta H([\gamma])}\sum_{\gamma\in[\gamma]}\mathbbold{1}_{\gamma_1\trianglelefteq\gamma}\leq\sum_{\gamma_1\in\Gamma_1}\sum_{[\gamma]\in\overline{\mathscr{C}}}\mathrm{diam}([\gamma])e^{-\frac{1}{4}\beta H([\gamma])}\leq e^{-c_2\beta/8}\sum_{\gamma_1\in\Gamma_1} H(\gamma_1). \end{align}\] On the other hand, we have \[\mathrm{S}_2\coloneq\sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{\Gamma\in A_2(\Gamma_1)}\bar{z}_\beta(\Gamma)\leq \sum_{\gamma_1\in\Gamma_1}\sum_{\gamma,\gamma'\in\mathscr{C}}\mathbbold{1}_{\gamma\trianglelefteq\gamma_1}\mathbbold{1}_{\gamma' \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_1}\mathbbold{1}_{\gamma \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma'}\sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{\gamma,\gamma'\in\Gamma}\bar{z}_\beta(\Gamma).\]By Corollary 5, we obtain \[\mathrm{S}_2\leq \sum_{\gamma_1\in\Gamma_1}\sum_{\gamma,\gamma'\in\mathscr{C}}\mathbbold{1}_{\gamma\trianglelefteq\gamma_1}\mathbbold{1}_{\gamma' \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_1}\mathbbold{1}_{\gamma \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma'}\Phi(\gamma,\gamma')e^{-\frac{1}{8}\beta[H(\gamma)+H(\gamma')]}.\]Introducing equivalence classes of contours under integer translations and bounding \(\Phi(\gamma,\gamma')\) under the constraint \(\mathbbold{1}_{\gamma\trianglelefteq\gamma_1}\mathbbold{1}_{\gamma' \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_1}\), we obtain \[\mathrm{S}_2\leq\sum_{\gamma_1\in\Gamma_1}\sum_{[\gamma],[\gamma']\in\overline{\mathscr{C}}}e^{-\frac{1}{8}\beta[H([\gamma])+H([\gamma'])]}\sum_{\substack{\gamma\in[\gamma]\\\gamma'\in[\gamma']}}\sum_{\substack{x\in\mathrm{I}_-(\gamma_1)\\y\in\mathrm{I}_-(\gamma_1)^c}}\frac{4\mathbbold{1}_{x\in\mathrm{I}_-(\gamma)}\mathbbold{1}_{y\in\mathrm{I}_-(\gamma')}}{\lvert x-y\rvert^\alpha}.\]By Hypotheses 2 and 3, choosing \(\beta\geq 16c_0\), we obtain\[\mathrm{S}_2\leq \sum_{\gamma_1\in\Gamma_1} 2H(\gamma_1)\left(\sum_{[\gamma]\in\overline{\mathscr{C}}}\mathrm{diam}([\gamma])e^{-\frac{1}{8}\beta H([\gamma])}\right)^2\leq e^{-c_2\beta/8}\sum_{\gamma_1\in\Gamma_1} 2H(\gamma_1).\]Combining the bounds for \(\mathrm{S}_i\), \(i=0,1,2\), we obtain Equation 83 and Lemma 15 has thus been proved. ◻

Proof. This is a direct consequence of Lemmas 8 and 15. ◻

Proof of Theorem 1. First, let us deal with the matter of convergence in the absence of an external field. By Equation 76 , it is sufficient to apply the machinery developed in Section 4. More precisely, with Lemma 15, we transform the sum over trees of polymers into a sum over trees of contours. Once again, we apply the machinery developed in Section 4; for all the details, see Section 5.2. That is, \[\begin{align} \lvert \Lambda\rvert P_\beta(\Lambda)\leq\sum_{\Gamma\in\mathscr{P}_\Lambda}\overline{\mathrm{W}}_\beta(\Gamma)\leq\sum_{\Gamma\in\mathscr{P}_\Lambda}\bar{z}_{\beta/2}(\Gamma)\leq\sum_{\gamma\in\mathscr{C}_\Lambda}\mathrm{W}_{\beta/4}(\gamma)\leq \sum_{\gamma\in\mathscr{C}_\Lambda}e^{-\frac{1}{8}\beta H(\gamma)}. \end{align}\] The first inequality is a consequence of Equation 76 . The second inequality is a consequence of Corollary 7 (\(\eta=1\)). The third inequality is a consequence of Equation 79 . Finally, the fourth inequality is a consequence of Corollary 3.

Let \([\gamma]\) be an equivalence class of contours associated to the action of \(\mathbb{Z}\) by translation. There are at most \(\lvert\Lambda\rvert\) elements of \([\gamma]\) in \(\mathscr{C}_\Lambda\). Hence, \[\begin{align} P_\beta(\Lambda)\leq \sum_{[\gamma]\in\overline{\mathscr{C}}}e^{-\frac{1}{8}\beta H([\gamma])}\leq e^{-\frac{1}{8}c_2\beta}, \end{align}\]where the second inequality is a consequence of Hypothesis 3 (recall Equation 26 ). The first part of Theorem 1 has thus been proved.

For the second part of Theorem 1, it is a direct application of Lemma 6 and \(\lvert e^z\rvert\leq e^{\lvert z\rvert}\). Take \(r_0\mathrel{\vcenter{:}}= (8\lvert A\rvert)^{-1}\). Since \(H(\gamma)\geq 2\), for all \(\gamma\in\mathscr{C}\), we have \[\beta E_{\lvert h\rvert}(\gamma)+\log2\leq 2\beta \lvert h\rvert \lvert A\rvert+\log2\leq \frac{1}{4}\beta H(\gamma),\]where the last inequality holds if \(\beta\geq 4\log2\). That is, if \(h\in\mathbb{D}_{A,r_0}\), then \[\lvert\bar{z}_{\beta,h}(\Gamma)\rvert\leq \bar{z}_{\beta/2}(\Gamma),\] for all \(\Gamma\in\mathscr{P}\). We thus conclude the proof of Theorem 1. ◻

Proof. Let \(\mathbf{x}\) and \(\bar{\mathbf{x}}\) satisfy the hypotheses of the lemma. We shall construct an injection from \(\mathrm{Sym}(l)\) to \(\mathrm{Sym}(k)\), denoted by \(\varphi\mapsto\tilde{\varphi}\), such that \(\bar{x}_{\varphi(l)}=x_{\tilde{\varphi}(k)}\) and \[\label{eq:32C-x-x-C-bar-x-bar-x} \prod_{i=1}^{l-1}C(\bar{x}_{\varphi(i)},\bar{x}_{\varphi(i+1)})=\prod_{j=1}^{k-1}C(x_{\tilde{\varphi}(j)},x_{\tilde{\varphi}(j+1)}).\tag{84}\] Since \(C\) is positive, the existence of such an injection proves the lemma.

For \(i\in[l]\), define \(K_i\mathrel{\vcenter{:}}=\{j\in[k]:x_j=\bar{x}_i\}\).

Denote by \(\phi_i:[\lvert K_i\rvert]\rightarrow K_i\) the bijection that orders \(K_i\); that is, \(j<j'\implies \phi_i(j)<\phi_i(j')\).

Let \(\varphi\in\mathrm{Sym}(l)\). It is convenient to write \[k[\varphi](i)\mathrel{\vcenter{:}}=\sum_{p=1}^i\lvert K_{\varphi(p)}\rvert,\] with the convention \(k[\varphi](0)=0\). Note that \(k[\varphi](l)=k\) and that \(i\mapsto k[\varphi](i)\) is strictly increasing. In order to define \(\tilde{\varphi}\), we proceed in blocks: if \(j\in[k]\), then there exists a unique \(i(j)\in[l]\) such that \(k[\varphi](i(j)-1)<j\leq k[\varphi](i(j))\). Define \[\tilde{\varphi}(j)\mathrel{\vcenter{:}}= \phi_{\varphi(i(j))}\left(j-k[\varphi](i(j))\right).\]While this may seem rather involved, the graphical description is rather straightforward, see Figure 14.

Let us first show that \(\varphi\mapsto\tilde{\varphi}\) is an injection. If \(\varphi_1\neq\varphi_2\in\mathrm{Sym}(l)\), then there exists a smallest \(i\in[l]\) such that \(\varphi_1(i)\neq\varphi_2(i)\). Thus, if \(j=k[\varphi_1](i-1)+1=k[\varphi_2](i-1)+1\), then \(\tilde{\varphi}_1(j)\in K_{\varphi_1(i)}\) and \(\tilde{\varphi}_2(j)\in K_{\varphi_2(i)}\). Since \(K_{\varphi_1(i)}\cap K_{\varphi_2(i)}=\emptyset\), we conclude that \(\varphi\mapsto\tilde{\varphi}\) is an injection.

Finally, let us now show that Equation 84 holds. If \(i\in[l]\), then \[\prod_{j=k[\varphi](i-1)+1}^{k[\varphi](i)-1}C(x_{\tilde{\varphi}(j)},x_{\tilde{\varphi}(j+1)})=\prod_{j=k[\varphi](i-1)+1}^{k[\varphi](i)-1}C(\bar{x}_{\varphi(i)},\bar{x}_{\varphi(i)})=1.\]Therefore, we obtain \[\prod_{j=1}^{k-1}C(x_{\tilde{\varphi}(j)},x_{\tilde{\varphi}(j+1)})=\prod_{i=1}^{l-1}C(x_{\tilde{\varphi}(k[\varphi](i))},x_{\tilde{\varphi}(k[\varphi](i)+1)})=\prod_{i=1}^{l-1}C(\bar{x}_{\varphi(i)},\bar{x}_{\varphi(i+1)}).\]The reader may readily check that \(\bar{x}_{\varphi(l)}=x_{\tilde{\varphi}(k)}\). The lemma has thus been proved. ◻

Proof. Let \(\bar{\mathbf{x}}=(\bar{x}_1,...,\bar{x}_l)\) such that \(i\neq j\Rightarrow\bar{x}_i\neq\bar{x}_j\) and \(\{x_i:i\in[k]\}=\{\bar{x}_i:i\in[l]\}\). Then \[\begin{align} \prod_{i=1}^kC(x_i,y)=\prod_{j=1}^lC(\bar{x}_j,y)^{k_j}, \end{align}\]where \(k_j=\lvert\{i\in[k]:x_i=\bar{x}_j\}\rvert\). Hence, \[\prod_{i=1}^kC(x_i,y)\leq\prod_{j=1}^lC(\bar{x}_j,y)=\prod_{j=1}^l\frac{\mathbbold{1}_{\bar{x}_j\neq y}}{\lvert \bar{x}_j-y\rvert^\alpha}+\sum_{i=1}^l\mathbbold{1}_{\bar{x}_i=y}\prod_{j\neq i}\frac{\mathbbold{1}_{\bar{x}_j\neq y }}{\lvert y-\bar{x}_j\rvert^\alpha}.\]By Lemma 14, we bound the first term by \[\prod_{j=1}^l\frac{\mathbbold{1}_{\bar{x}_j\neq y}}{\lvert \bar{x}_j-y\rvert^\alpha}\leq 16^{\alpha l}\prod_{i=1}^l\mathbbold{1}_{\bar{x}_i\neq y}\sum_{\varphi\in \mathrm{Sym}(l)}C(\bar{x}_{\varphi(l)},y)\prod_{j=1}^{l-1}C(\bar{x}_{\varphi(j)},\bar{x}_{\varphi(j+1)}).\]Let us now bound the second term. For \(i\in[l]\), define \(\mathbf{z}^i=(\bar{x}_1,...,\bar{x}_{i-1},\bar{x}_{i+1},...,\bar{x}_l)\). Thus, the same Lemma 14 implies that \[\mathbbold{1}_{\bar{x}_i=y}\prod_{j=1}^{l-1}\frac{\mathbbold{1}_{z^i_j\neq y}}{\lvert z^i_j-y\rvert^{\alpha}}\leq 16^{\alpha(l-1)}\mathbbold{1}_{\bar{x}_i=y}\prod_{j\neq i}\mathbbold{1}_{\bar{x}_j\neq y}\sum_{\varphi\in\mathrm{Sym}(l-1)}\prod_{j=1}^{l-2}C(z^{i}_{\varphi(j)},z^i_{\varphi(j+1)})\cdot C(z^i_{\varphi(l-1)},\bar{x}_i).\]There is a canonical bijection \(\mathrm{Sym}(l-1)\ni\varphi\leftrightarrow\tilde{\varphi}\in\mathrm{Sym}(l)\) given by \[\tilde{\varphi}(j)\mathrel{\vcenter{:}}=\begin{cases} i, &\textrm{if }j=l;\\ \varphi(j), &\textrm{if }\varphi(j)<i;\\ \varphi(j)+1, &\textrm{if }\varphi(j)\geq i. \end{cases}\]Recalling that \(\mathbbold{1}_{\bar{x}_i=y}=\mathbbold{1}_{\bar{x}_i=y}C(\bar{x}_i,y)\), the right-hand side can be written as\[16^{\alpha (l-1)}\mathbbold{1}_{\bar{x}_i=y}\prod_{j\neq i}\mathbbold{1}_{\bar{x}_i\neq\bar{x}_j}\sum_{i=1}^l\sum_{\substack{\varphi\in\mathrm{Sym}(l)\\\varphi(l)=i}}C(\bar{x}_{\varphi(l)},y)\prod_{j=1}^{l-1}C(\bar{x}_{\varphi(j)},\bar{x}_{\varphi(j+1)}).\]Finally, since \[\prod_{i=1}^l\mathbbold{1}_{\bar{x}_i\neq y}+\sum_{i=1}^l\mathbbold{1}_{\bar{x}_i=y}\prod_{j\neq i}\mathbbold{1}_{\bar{x}_i\neq \bar{x}_j}\leq 1,\]we obtain that\[\prod_{i=1}^kC(x_i,y)\leq 16^{\alpha l}\sum_{\varphi\in\mathrm{Sym}(l)}C(\bar{x}_{\varphi(l)},y)\prod_{i=1}^{l-1} C(\bar{x}_{\varphi(i)},\bar{x}_{\varphi(i+1)}).\]By Lemma 16, the proof is finished. ◻

Proof. Equation 88 follows from a direct iteration of Equation 87 . ◻

Proof. By Remark 1, there are two cases to consider. First, if there exist \(\gamma_1\in\Gamma_1\) and \(\gamma_2\in\Gamma_2\) such that \(\gamma_1\not\sim\gamma_2\), then \[\begin{align} \mathrm{dist}(\mathrm{I}_-(\gamma_1),\mathrm{I}_-(\gamma_2))&\leq 1+\mathrm{dist}(\gamma_1,\gamma_2)\leq 1+M\min\{\mathrm{diam}(\gamma_1),\mathrm{diam(\gamma_2)}\}^{3/2}\\&\leq 2M\min\{\mathrm{diam}(\gamma_1),\mathrm{diam(\gamma_2)}\}^{3/2}. \end{align}\]That is, \((\gamma_1,\gamma_2)\in\mathscr{A}\). If \(\Gamma_1\not\sim\Gamma_2\) and \((\gamma_1,\gamma_2)\in\Gamma_1\times\Gamma_2\Rightarrow \gamma_1\sim\gamma_2\), then there exist \(\gamma_1\in\Gamma_1\) and \(\gamma_2\in\Gamma_2\) such that \(\mathrm{I}_-(\gamma_1)\cap\mathrm{I}_-(\gamma_2)\neq\emptyset\). That is, \((\gamma_1,\gamma_2)\in\mathscr{A}\). The lemma has thus been proved. ◻

Proof. The first bound follows directly from Hypothesis 2.

Let us prove the second one. In general, we have \[\label{eq:32trivial-phi-bound} \Phi(\gamma_1,\gamma_2)\leq\frac{\mathrm{diam}(\gamma_1)\mathrm{diam}(\gamma_2)}{\mathrm{dist}(\gamma_1,\gamma_2)^\alpha}.\tag{90}\] On the other hand, by the triangular inequality, we have \[\lvert x-y\rvert\leq \mathrm{diam}(\gamma_1)+\mathrm{dist}(\gamma_1,\gamma_2)+\mathrm{diam}(\gamma_2).\]Dividing and multiplying by \(2\) and using \(x,y\geq 2\Rightarrow x+y\leq xy\), we arrive at \[\label{eq:32crude-x-y-bound} \lvert x-y\rvert^\alpha\leq 2^{2\alpha}\mathrm{dist}(\gamma_1,\gamma_2)^\alpha\mathrm{diam}(\gamma_1)^\alpha\mathrm{diam}(\gamma_2)^\alpha.\tag{91}\] Combining Hypothesis 2 with Equations 90 and 91 , we obtain the second bound.

Finally, let us prove the third bound. Let \((\gamma_1,\gamma_2)\in\mathscr{A}\), \(x_1\in\mathrm{I}_-(\gamma_1)\) and \(x_2\in\mathrm{I}_-(\gamma_2)\). Then \[\lvert x-y\rvert\leq \mathrm{diam}(\gamma_1)+\mathrm{dist}(\mathrm{I}_-(\gamma_1),\mathrm{I}_-(\gamma_2))+\mathrm{diam}(\gamma_2).\]By the definition of \(\mathcal{A}\), \[\mathrm{dist}(\mathrm{I}_-(\gamma_1),\mathrm{I}_-(\gamma_2))\leq 2M\mathrm{diam}(\gamma_1)^{3/4}\mathrm{diam}(\gamma_2)^{3/4}.\] A clever application of \(x,y\geq 2\Rightarrow x+y\leq xy\) gives us \[M\mathrm{diam}(\gamma_1)^{3/4}\mathrm{diam}(\gamma_2)^{3/4}+\mathrm{diam}(\gamma_1)\leq 2M\mathrm{diam}(\gamma_1)\mathrm{diam}(\gamma_2)^{3/4}.\] Thus, \[\lvert x-y\rvert\leq M\mathrm{diam}(\gamma_1)^{3/4}\mathrm{diam}(\gamma_2)^{3/4}(2\mathrm{diam}(\gamma_1)^{1/4}+2\mathrm{diam}(\gamma_2)^{1/4}).\]Finally, combining another application of \(x,y\geq 2\Rightarrow x+y\leq xy\) with Hypothesis 2, we obtain the third bound. The lemma has thus been proved. ◻

Proof. The corollary follows from Lemmas 19 and 20 (item 3): \[\begin{align} \mathbbold{1}_{\varGamma_1\not\sim\varGamma_2}&\leq \sum_{\substack{\gamma_1\in\varGamma_1\\\gamma_2\in\varGamma_2}}\mathbbold{1}_{\mathscr{A}}(\gamma_1,\gamma_2)\leq\sum_{\substack{\gamma_1\in\varGamma_1\\\gamma_2\in\varGamma_2}}\sum_{\substack{x_1\in\mathrm{I}_-(\gamma_1)\\x_2\in\mathrm{I}_-(\gamma_2)}}\mathbbold{1}_{\mathscr{A}}(\gamma_1,\gamma_2)\mathbbold{1}_{x_1\in\mathrm{I}_-(\gamma_1)}\mathbbold{1}_{x_2\in\mathrm{I}_-(\gamma_2)}\\ &\leq (4M)^\alpha\sum_{\substack{\gamma_1\in\varGamma_1\\\gamma_2\in\varGamma_2}}e^{\alpha c_0 H(\gamma_1)}e^{\alpha c_0H(\gamma_2)}\sum_{\substack{x_1\in\gamma_1\\x_2\in\gamma_2}}C(x_1,x_2). \end{align}\] ◻

Proof. By Corollary 7 and Equations 102 and 106 , we have \[\overline{\mathrm{E}}_{\beta,n}(\varGamma_1,\varGamma_2)\leq\sum_{J\subset [n]}\overline{\mathrm{E}}_{\beta,J}(\varGamma_1,\varGamma_2),\]where \(\Upsilon=(\gamma_{0,1},\gamma_{1,0},...,\gamma_{n,n+1},\gamma_{n+1,n})\in\varGamma_1\times \mathscr{C}^{2n}\times\varGamma_2\), \(\mathbf{\Gamma}=(\Gamma_1,...,\Gamma_n)\in\mathscr{P}^n\) and \[\overline{\mathrm{E}}_{\beta,J}(\varGamma_1,\varGamma_2)\mathrel{\vcenter{:}}=\sum_{\Upsilon}\tilde{\mathbbold{1}}_J(\Upsilon)\tilde{\mathbbold{1}}_\mathscr{A}(\Upsilon)\sum_{\mathbf{\Gamma}}\bar{z}_{\beta/2} (\Gamma_k)\prod_{k=1}^n\mathbbold{1}_{\gamma_{k,k-1},\gamma_{k,k+1}\in\Gamma_k}.\]By Corollaries 3 and 5, we obtain \[\overline{\mathrm{E}}_{\beta, J}(\varGamma_1,\varGamma_2)\leq\sum_{\Upsilon}\overline{\mathbbold{1}}_J(\Upsilon)\tilde{\mathbbold{1}}_{\mathscr{A}}(\Upsilon)\prod_{i=1}^ne^{-\frac{1}{16}\beta[H(\gamma_{i,i-1})+H(\gamma_{i,i+1})]}\prod_{j\in J}\Phi(\gamma_{j,j-1},\gamma_{j,j+1}).\]We have thus bounded \(\overline{\mathrm{E}}_{\beta,n}\) in terms of contours.

Let us transform this bound in terms of contours into a bound in terms of lattice sites. Since \(\gamma\in\mathscr{C}\Rightarrow\lvert\mathrm{I}_-(\gamma)\rvert\geq 1\), we have \[\label{eq:32contours-to-points} 1\leq \sum_{\mathbf{x}}\prod_{k=0}^n\mathbbold{1}_{x_{k,k+1}\in\mathrm{I}_-(\gamma_{k,k+1})}\mathbbold{1}_{x_{k+1,k}\in\mathrm{I}_-(\gamma_{k+1,k})},\tag{108}\] where \(\mathbf{x}=(x_{0,1},x_{1,0},...,x_{n,n+1},x_{n+1,n})\in\mathbb{Z}^{2n+2}\).

By Lemma 20, if \(j\in J\), \(x_{j,j-1}\in\mathrm{I}_-(\gamma_{j,j-1})\) and \(x_{j,j+1}\in\mathrm{I}_-(\gamma_{j,j+1})\), then \[\label{eq:32in-J-contours-to-points} \Phi(\gamma_{j,j-1},\gamma_{j,j+1})\leq 4^{\alpha}e^{2\alpha c_0\left[H(\gamma_{j,j-1})+H(\gamma_{j,j+1})\right]}C(x_{j,j-1},x_{j,j+1}).\tag{109}\] If \(j\in [n]\setminus J\), \(x_{j,j-1}\in\mathrm{I}_-(\gamma_{j,j-1})\) and \(x_{j,j+1}\in\mathrm{I}_-(\gamma_{j,j+1})\), then \[\label{eq:32out-J-contours-to-points} 1\leq e^{\alpha c_0[H(\gamma_{j,j-1})+H(\gamma_{j,j+1})]}C(x_{j,j-1},x_{j,j+1}).\tag{110}\] If \((\gamma_{i,i+1},\gamma_{i+1,i})\in\mathscr{A}\), \(x_{i,i+1}\in\mathrm{I}_-(\gamma_{i,i+1})\) and \(x_{i+1,i}\in\mathrm{I}_-(\gamma_{i+1,i})\), then \[\label{eq:32in-scrA-contours-to-points} 1\leq (4M)^\alpha e^{\alpha c_0[H(\gamma_{i,i+1})+H(\gamma_{i+1,i})]}C(x_{i,i+1},x_{i+1,i}).\tag{111}\]

Condensing notation as in Figure 16, combining Equations 109 , 110 and 111 , we obtain \[\overline{\mathrm{E}}_{\beta,J}(\varGamma_1,\varGamma_2)\leq (8M)^{2\alpha n}\sum_{\mathbf{x}}\sum_{\Upsilon}\overline{\mathbbold{1}}_J(\Upsilon,\mathbf{x})C(\mathbf{x})\epsilon_\alpha(\Upsilon)\epsilon_\beta(\Upsilon).\] Let us now sum over \(\Upsilon'=(\gamma_{1,0},\gamma_{1,2},...,\gamma_{n,n-1},\gamma_{n,n+1})\in\mathscr{C}^{2n}\). As often done with estimates with contours, let us introduce equivalence classes of contours under integer translations. Noting that \(x_{i,j}\) belongs to the minus interior of \(\lvert\mathrm{I}_-(\gamma_{i,j})\rvert\leq e^{\alpha c_0H(\gamma_{i,j})}\) elements of \([\gamma_{i,j}]\), by Hypothesis 3, we obtain \[\label{eq:32summing-over-Upsilon-prime} \sum_{\Upsilon'}\overline{\mathbbold{1}}_J(\Upsilon,\mathbf{x})\epsilon_\beta(\Upsilon)\leq \mathbbold{1}_{x_{0,1}\in\mathrm{I}_-(\gamma_{0,1})}\mathbbold{1}_{x_{n+1,n}\in\mathrm{I}_-(\gamma_{n+1,n})}e^{-n\frac{c_2}{8}(\beta-64\alpha c_0)}.\tag{112}\] We must now sum over \(\mathbf{x}'=(x_{1,0},x_{1,2}...,x_{n,n-1},x_{n,n+1})\in\mathbb{Z}^{2n}\). Iterating Lemma 17, we obtain \[\label{eq:32summing-over-x-prime} \sum_{\mathbf{x}'}C(\mathbf{x})\leq 2^{8\alpha n}(1+2\zeta(\alpha))^{2n}C(x_{0,1},x_{n+1,n}).\tag{113}\] Choosing \(\beta\geq 128\alpha c_0+(32\alpha/c_2)\log[2^7M(1+2\zeta(\alpha))]\), and combining Equations 92 , 112 and 113 , we obtain \[\overline{\mathrm{E}}_{\beta,J}(\varGamma_1,\varGamma_2)\leq e^{-nc_2\beta/16}\tilde{\mathrm{E}}_\alpha(\varGamma_1,\varGamma_2).\]Since \(\lvert \mathrm{P}([n])\rvert=2^n\), if \(\beta\geq (16/c_2)\log 2\), then \[\label{eq:32summing-over-n-E-bar-n} \sum_{n=1}^\infty\overline{\mathrm{E}}_{\beta,n}(\varGamma_1,\varGamma_2)\leq \tilde{\mathrm{E}}_\alpha(\varGamma_1,\varGamma_2).\tag{114}\] Combining Equations 92 and@eq:eq:32summing-over-n-E-bar-n , we obtain Equation 107 . The lemma has thus been proved. ◻

Proof. If \(A\subset\mathrm{I}_-(\gamma)\), then \(\mathrm{diam}(A)+1\leq \mathrm{diam}(\gamma)\). Therefore, recalling Hypothesis 2, we have\[\sum_{\gamma\in\mathscr{C}}\mathbbold{1}_{A\subset\mathrm{I}_-(\gamma)}e^{-\beta H(\gamma)}\leq \sum_{n=\lceil 1+\log_2(1+\mathrm{diam}(A))\rceil}2^{c_1n}2^{-c_0\beta n}\leq [2(1+\mathrm{diam}(A))]^{-\frac{c_0\beta}{2}},\]where the second inequality holds if \(\beta\geq 2(1+c_1)\). The lemma has thus been proved. ◻

Proof. By Corollary 7, we have \[\overline{\mathrm{W}}_\beta(\{A\})\leq\sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{A\subset\mathrm{I}_-(\Gamma)}\overline{\mathrm{W}}_\beta(\Gamma)\leq \sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{A\subset\mathrm{I}_-(\Gamma)}\bar{z}_{\beta/2}(\Gamma).\] By the definition of polymer, we may write \[\sum_{\Gamma\in\mathscr{P}}\mathbbold{1}_{A\subset\mathrm{I}_-(\Gamma)}\bar{z}_{\beta/2}(\Gamma)=\sum_{P\in\mathbbold{P}(A)}\sum_{\Upsilon}\tilde{\mathbbold{1}}_P(\{\gamma_a\})\sum_{\Gamma\in\mathscr{P}}\bar{z}_{\beta/2}(\Gamma)\prod_{a\in A}\mathbbold{1}_{a\in\mathrm{I}_-(\gamma_a)}\mathbbold{1}_{\gamma_a\in\Gamma},\]where \(\Upsilon=(\gamma_a:a\in A)\in\mathscr{C}^A\) and \[\tilde{\mathbbold{1}}_P(\Upsilon)\mathrel{\vcenter{:}}=\prod_{Q\in P}\prod_{q_1,q_2\in Q}\mathbbold{1}_{\gamma_{q_1}=\gamma_{q_2}}\prod_{\substack{R,S\in P\\R\neq S}}\prod_{\substack{r\in R\\s\in S}}\mathbbold{1}_{\gamma_r \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_s}.\]Note that if \(\tilde{\mathbbold{1}}_P(\Upsilon)=1\), then there are just \(\lvert P\rvert\) diverse contours. Thus, we write\[\label{eq:32Upsilon-partition} \Upsilon_P=(\gamma_Q:Q\in P)\in\mathscr{C}_P\mathrel{\vcenter{:}}=\{\Upsilon\in\mathscr{C}^P:(Q\neq R\in P)\Rightarrow\gamma_Q \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_R\}.\tag{117}\]

By Corollaries 3, 5 and 6, we obtain \[\overline{\mathrm{W}}_\beta(\{A\})\leq\sum_{P\in\mathbbold{P}(A)}\sum_{T\in\mathbbold{T}(P)}\sum_{\Upsilon_P}\prod_{Q\in P}\mathbbold{1}_{Q\subset\mathrm{I}_-(\gamma_Q)}e^{-\frac{\beta H(\gamma_Q)}{2^{\lvert P\rvert+2}}}\prod_{\{R,S\}\in E(T)}\Phi(\gamma_R,\gamma_S).\] Consider \(\{R,S\}\in E(T)\). Since \(R\subset \mathrm{I}_-(\gamma_R)\) and \(S\subset\mathrm{I}_-(\gamma_S)\), there exist \(x_S\in\mathrm{I}_-(\gamma_S)\) and \(x_R\in\mathrm{I}_-(\gamma_R)\) such that \(\lvert x_R-x_S\rvert=\mathrm{diam}(R\sqcup S)\). By Lemma 20, we obtain \[\Phi(\gamma_R,\gamma_S)\leq 2^{2\alpha}\mathrm{diam}(\gamma_1)^{1+\alpha}\mathrm{diam}(\gamma_2)^{1+\alpha}\mathrm{diam}(R\sqcup S)^{-\alpha}.\]Fix \(Q\in P\). By Hypothesis 2 and Lemma 22, if \(\beta\geq 2^{\lvert A\rvert+2}c_0(1+\alpha)\lvert A\rvert\), then \[\sum_{\gamma_Q\in\mathscr{C}}\mathbbold{1}_{Q\subset\mathrm{I}_-(\gamma_i)}\mathrm{diam}(\gamma_Q)^{(1+\alpha)\delta_T(Q)}e^{-\frac{\beta H(\gamma_Q)}{2^{\lvert P\rvert+2}}}\leq [2(1+\mathrm{diam}(Q))]^{-\frac{c_0\beta}{2^{\lvert P\rvert+4}}}.\] Combining all these estimates, we obtain \[\label{eq:32W-bar-A-in-term-of-trees} \overline{\mathrm{W}}_\beta(\{A\})\leq 2^{3\alpha\lvert A\rvert}\sum_{P\in\mathbbold{P}(A)}\sum_{T\in\mathbbold{T}(P)}\prod_{Q\in P}[2(1+\mathrm{diam}(Q))]^{-\frac{c_0\beta}{2^{\lvert P\rvert+4}}}\prod_{\{R,S\}\in E(T)}\mathrm{diam}(R\sqcup S)^{-\alpha}.\tag{118}\]

We will conclude by employing the canonical application \(T_P:\mathbbold{T}(P)\rightarrow\mathbbold{T}(A)\) introduced at the beginning of this subsection. Endow \(A\) with the usual ordering of \(\mathbb{Z}\). Let \(P\in\mathbbold{P}(A)\) and \(Q\in P\), then \[\prod_{q=1}^{\lvert Q\rvert-1}\lvert a_{Q,q}-a_{Q,q+1}\rvert\leq \left(1+\sum_{q=1}^{\lvert Q\rvert-1}\lvert a_{Q,q}-a_{Q,q+1}\rvert\right)^{\lvert Q\rvert}\]Hence, if \(\beta\geq 2^{\lvert A\rvert+4}\lvert A\rvert \alpha c_0^{-1}\), then\[(1+\mathrm{diam}(Q))^{-\frac{c_0\beta}{2^{\lvert P\vert+4}}}\leq \prod_{q=1}^{\lvert Q\rvert-1}\lvert a_{Q,q}-a_{Q,q+1}\rvert^{-\alpha}.\]Furthermore, since \(\lvert a_{R,1}-a_{S,1}\rvert\leq\mathrm{diam}(R\sqcup S)\), for every \(P\in\mathbbold{P}(A)\), we have \[\prod_{Q\in P}(1+\mathrm{diam}(Q))^{-\frac{c_0\beta}{2^{\lvert P\rvert+4}}}\prod_{\{R,S\}\in E(T)}\mathrm{diam}(R\sqcup S)^{-\alpha}\leq\prod_{\{a_1,a_2\}\in E(T_P)}\lvert a_1-a_2\rvert^{-\alpha}.\] Finally, since the application \(T_P:\mathbbold{T}(P)\rightarrow\mathbbold{T}(A)\) is injective, we obtain \[\overline{\mathrm{W}}_\beta(\{A\})\leq\sum_{T\in\mathbbold{T}(A)}\left[\sum_{P\in\mathbbold{P}(A)}2^{3\alpha\lvert A\rvert-2^{-\lvert A\rvert-4}c_0\beta}\right]\prod_{\{a_1,a_2\}\in E(T)}\lvert a_1-a_2\rvert^{-\alpha}.\]Choosing \(\beta\geq 2^{\lvert A\rvert+5}\left[3\alpha\lvert A\rvert+\log_2\lvert\mathbbold{P}(A)\rvert\right]c_0^{-1}\), we obtain Equation 116 . The lemma has thus been proved. ◻

Proof of Theorem 2. By Equation 99 , we have \[\lvert\langle\sigma_x;\sigma_y\rangle^+_{\Lambda,\beta}\rvert\leq\overline{\mathrm{W}}_\beta(\{x,y\})+\overline{\mathrm{W}}_\beta(\{x\},\{y\}),\]where we write \(\overline{\mathrm{W}}_\beta(A_1,...,A_n)\) instead of \(\overline{\mathrm{W}}_\beta(\{A_1,...,A_n\})\).

We have already dealt with \(\overline{\mathrm{W}}_\beta(\{x,y\})\) in Lemma 23. Let us bound the term \(\overline{\mathrm{W}}_\beta(\{x\},\{y\})\). By Equation 100 , we have \[\overline{\mathrm{W}}_\beta(\{x\},\{y\})=\sum_{\Gamma_1,\Gamma_2\in\mathscr{P}}\mathbbold{1}_{x\in\mathrm{I}_-(\Gamma_1)}\mathbbold{1}_{y\in\mathrm{I}_-(\Gamma_2)}\overline{\mathrm{W}}_\beta(\Gamma_1)\overline{\mathrm{E}}_\beta(\Gamma_1,\Gamma_2)\overline{\mathrm{W}}_\beta(\Gamma_2).\]Let us write \[\begin{align} &\mathbbold{1}(\mathbf{\Gamma},\Upsilon,\mathbf{x})\mathrel{\vcenter{:}}=\mathbbold{1}_{x\in\mathrm{I}_-(\Gamma_1)}\mathbbold{1}_{y\in\mathrm{I}_-(\Gamma_2)}\mathbbold{1}_{\gamma_1\in\Gamma_1}\mathbbold{1}_{\gamma_2\in\Gamma_2}\mathbbold{1}_{x_1\in\mathrm{I}_-(\gamma_1)}\mathbbold{1}_{x_2\in\gamma_2},\\ &\epsilon_\alpha(\Upsilon)\mathrel{\vcenter{:}}= e^{\alpha c_0 H(\gamma_1)}e^{\alpha c_0 H(\gamma_2)},\\ &\xi_\beta(\mathbf{\Gamma})\mathrel{\vcenter{:}}= \bar{z}_{\beta/2}(\Gamma_1)\bar{z}_{\beta/2}(\Gamma_2), \end{align}\]where \(\mathbf{\Gamma}=(\Gamma_1,\Gamma_2)\in\mathscr{P}^2\), \(\Upsilon=(\gamma_1,\gamma_2)\in\mathscr{C}^2\) and \(\mathbf{x}=(x_1,x_2)\in\mathbb{Z}^2\).

By Corollary 7 and Lemma 21, we obtain \[\overline{\mathrm{W}}_\beta(\{x\},\{y\})\leq (8M)^\alpha\sum_{\mathbf{\Gamma}}\sum_{\Upsilon}\sum_{\mathbf{x}}\mathbbold{1}(\mathbf{\Gamma},\Upsilon,\mathbf{x})\xi_\beta(\mathbf{\Gamma})\epsilon_\alpha(\Upsilon)C(x_1,x_2).\]Let us sum over the polymers. Writing \(\Upsilon'=(\gamma_1',\gamma_2')\in\mathscr{C}^2\), recalling Equation 103 , and introducing a sum over \(J\subset [2]\) (as in the proof of Lemma 21), we have \[\mathbbold{1}(\mathbf{\Gamma},\Upsilon,\mathbf{x})=\sum_J\sum_{\Upsilon'}\mathbbold{1}_J(\Upsilon,\Upsilon',\mathbf{x})\mathbbold{1}_{\gamma_1,\gamma'_1\in\Gamma_1}\mathbbold{1}_{\gamma_2,\gamma'_2\in\Gamma_2},\]where \[\mathbbold{1}_J(\Upsilon,\Upsilon',\mathbf{x})\mathrel{\vcenter{:}}=\mathbbold{1}_{x\in\mathrm{I}_-(\gamma'_1)}\mathbbold{1}_{y\in\mathrm{I}_-(\gamma'_2)}\mathbbold{1}_{x_1\in\mathrm{I}_-(\gamma_1)}\mathbbold{1}_{x_2\in\gamma_2}\prod_{j\in J}\mathbbold{1}_{\gamma_j \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma'_j}\prod_{j\in[2]\setminus J}\mathbbold{1}_{\gamma_j=\gamma'_j}.\]Thus, by Corollaries 3 and 5, we obtain \[\label{eq:32two-point-correlation-summed-over-polymers} \overline{\mathrm{W}}_\beta(\{x\},\{y\})\leq (8M)^\alpha\sum_J\sum_{\Upsilon,\Upsilon'}\sum_{\mathbf{x}}\mathbbold{1}_J(\Upsilon,\Upsilon',\mathbf{x})\epsilon_\alpha(\Upsilon)\epsilon_\beta(\Upsilon,\Upsilon')C(x_1,x_2)\prod_{j\in J}\Phi(\gamma_j,\gamma'_j),\tag{119}\] where \[\epsilon_\beta(\Upsilon,\Upsilon')\mathrel{\vcenter{:}}= e^{-\frac{1}{8}\beta H(\gamma_1)}e^{-\frac{1}{8}\beta H(\gamma'_1)}e^{-\frac{1}{8}\beta H(\gamma_2)}e^{-\frac{1}{8}\beta H(\gamma'_2)}.\]By Lemma 20, we have \[\label{eq:32two-point-correlation-contours-to-points} \mathbbold{1}_{J}(\Upsilon,\Upsilon',\mathbf{x})C(x_1,x_2)\leq 16^\alpha \epsilon'_\alpha(\Upsilon,\Upsilon')C(x,x_1)C(x_1,x_2)C(x_2,y)\tag{120}\] where \[\epsilon'_\alpha(\Upsilon,\Upsilon')\mathrel{\vcenter{:}}= e^{2\alpha c_0H(\gamma_1)}e^{2\alpha c_0H(\gamma'_1)}e^{2\alpha c_0H(\gamma_2)}e^{2\alpha c_0H(\gamma'_2)}.\]Combining Hypotheses 2 and 3 with Equations 119 and 120 , and choosing \(\beta\geq 48\alpha c_0\), we obtain \[\overline{\mathrm{W}}_\beta(\{x\},\{y\})\leq (2^7M)^\alpha e^{-\frac{1}{4}c_2\beta}\sum_J\sum_{\mathbf{x}}C(x,x_1)C(x_1,x_2)C(x_2,y).\] Finally, by Lemma 17 and \(\lvert \mathrm{P}([2])\rvert=4<2^{2\alpha}\), we obtain \[\label{eq:32two-point-correlation-W-bar-bound-x-y} \overline{\mathrm{W}}_\beta(\{x\},\{y\})\leq (2^{17}M)^\alpha(1+2\zeta(\alpha))^2e^{-\frac{1}{4}c_2\beta}C(x,y).\tag{121}\]

Choosing \(\beta\geq(8\alpha/c_2)\log[2^{17}M(1+2\zeta(\alpha))]\) and \(c_3=\min\{2^{-7}(\log2)c_0,2^{-3}c_2\}\), by Lemma 23 (applied to \(A=\{x,y\}\)) and Equation 121 , we obtain Equation 1 . The theorem has thus been proved. ◻

Proof. Recalling Notation 1, we write \(\Upsilon=(\Upsilon_1,...,\Upsilon_n)\) and \(\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n)\), where \(\Upsilon_i\in\mathscr{C}^{E_T(i)}\) and \(\mathbf{x}_i\in\mathbb{Z}^{E_T(i)}\). Then \[\prod_{\{p,q\}\in E(T)}\tilde{\mathrm{E}}_\alpha(\Gamma_p,\Gamma_q)=\sum_{\Upsilon}\sum_{\mathbf{x}}\epsilon_\alpha(\Upsilon)C(\mathbf{x})\mathbbold{1}(\mathbf{\Gamma},\Upsilon)\mathbbold{1}(\Upsilon,\mathbf{x}),\]where \[\begin{align} &\epsilon_\alpha(\Upsilon)\mathrel{\vcenter{:}}=\prod_{i=1}^n\prod_{e\in E_T(i)}e^{\alpha c_0 H(\gamma_{i,e})},&\mathbbold{1}(\mathbf{\Gamma},\Upsilon)\mathrel{\vcenter{:}}=\prod_{i=1}^n\prod_{e\in E_T(i)}\mathbbold{1}_{\gamma_{i,e}\in\Gamma_i}\eqqcolon\prod_{i=1}^n\mathbbold{1}(\Gamma_i,\Upsilon_i),\\ &C(\mathbf{x})\mathrel{\vcenter{:}}=\prod_{\{p,q\}=e\in E(T)}C(x_{p,e},x_{q,e}), &\mathbbold{1}(\Upsilon,\mathbf{x})\mathrel{\vcenter{:}}=\prod_{i=1}^n\prod_{e\in E_T(i)}\mathbbold{1}_{x_{i,e}\in\mathrm{I}_-(\gamma_{i,e})}\eqqcolon\prod_{i=1}^n\mathbbold{1}(\Upsilon_i,\mathbf{x}_i). \end{align}\]On the other hand, we have \[\prod_{i=1}^m\mathbbold{1}_{A_i\in\mathrm{I}_-(\Gamma_i)}=\sum_{\Upsilon'}\sum_{\mathbf{x}'}\mathbbold{1}(\mathbf{x'},\mathbf{A})\mathbbold{1}(\Upsilon',\mathbf{x'})\mathbbold{1}(\mathbf{\Gamma},\Upsilon'),\]where \(\Upsilon'=(\Upsilon'_1,...,\Upsilon'_m)\) and \(\Upsilon'_i=(\gamma_{i,a_i}:a_i\in A_i)\in\mathscr{C}^{A_i}\), \(\mathbf{x}'=(\mathbf{x}'_1,...,\mathbf{x}'_m)\) and \(\mathbf{x}'_i=(x_{i,a_i}:a_i\in A_i)\in\mathbb{Z}^{A_i}\), and \[\begin{align} &\mathbbold{1}(\mathbf{x}',\mathbf{A})\mathrel{\vcenter{:}}= \prod_{i=1}^m\prod_{a_i\in A_i}\mathbbold{1}_{x_{i,a_i}=a_i}\eqqcolon\prod_{i=1}^m\mathbbold{1}(\mathbf{x}'_i,A_i),\\ &\mathbbold{1}(\Upsilon',\mathbf{x}')\mathrel{\vcenter{:}}=\prod_{i=1}^m\prod_{a_i\in A_i}\mathbbold{1}_{x_{i,a_i}\in\mathrm{I}_-(\gamma_{i,a_i})}\eqqcolon\prod_{i=1}^m\mathbbold{1}(\Upsilon'_i,\mathbf{x}'_i),\\ &\mathbbold{1}(\mathbf{\Gamma},\Upsilon')\mathrel{\vcenter{:}}=\prod_{i=1}^m\prod_{a_i\in A_i}\mathbbold{1}_{\gamma_{i,a_i}\in\Gamma_i}. \end{align}\]

We must now write \(\mathbbold{1}(\Gamma_i,\Upsilon_i)\mathbbold{1}(\Gamma_i,\Upsilon'_i)\) in a way that allows us to transform polymers into contours. There are two cases to consider: \(i\in [m]\) and \(m<i\leq n\). We begin with \(i\in[m]\). Let \[\mathbbold{1}(\Gamma_i,\Upsilon_i)\mathbbold{1}(\Gamma_i,\Upsilon'_i)=\mathbbold{1}(\Gamma_i,\Upsilon_i,\Upsilon'_i)=\prod_{e\in E_T(i)}\mathbbold{1}_{\gamma_{i,e}\in\Gamma_i}\prod_{a_i\in A_i}\mathbbold{1}_{\gamma_{i,a_i}\in\Gamma_i}.\]Introducing partitions \(P_i\in\mathbbold{P}(A_i\sqcup E_T(i))\), we have \[\mathbbold{1}(\Gamma_i,\Upsilon_i,\Upsilon'_i)=\sum_{P_i}\mathbbold{1}_{P_i}(\Upsilon_i,\Upsilon'_i)\mathbbold{1}(\Gamma_i,\Upsilon_i,\Upsilon'_i),\]where\[\mathbbold{1}_{P_i}(\Upsilon_i,\Upsilon'_i)\mathrel{\vcenter{:}}=\prod_{Q_i\in P_i}\prod_{q_i,q'_i\in Q_i}\mathbbold{1}_{\gamma_{i,q_i}=\gamma_{i,q'_i}}\prod_{\substack{R_i,S_i\in P_i\\R_i\neq S_i}}\prod_{\substack{r_i\in R_i\\s_i\in S_i}}\mathbbold{1}_{\gamma_{i,r_i} \mathrel{ \ooalign{ \hss\raisebox{-1pt}{\sim}\hss\cr \hss\raisebox{3.5pt}{\scalebox{0.4}{\triangle}}\hss\cr\\ } }\gamma_{i,s_i}}.\] Note that if \(\mathbbold{1}_{P_i}(\Upsilon_i,\Upsilon'_i)=1\), then there are \(\lvert P_i\rvert\) distinct contours. In this case, we have \[\mathbbold{1}_{P_i}(\Upsilon_i,\Upsilon'_i)\mathbbold{1}(\Gamma_i,\Upsilon_i,\Upsilon'_i)=\sum_{\Upsilon_{i,P_i}}\mathbbold{1}(\Upsilon_i,\Upsilon'_i,\Upsilon_{i,P_i})\mathbbold{1}(\Gamma_i,\Upsilon_{i,P_i}),\]where \(\Upsilon_{i,P_i}\in\mathscr{C}_{P_i}\) (see Equation 117 ), and \[\begin{align} &\mathbbold{1}(\Upsilon_i,\Upsilon'_i,\Upsilon_{i,P_i})=\prod_{Q_i\in P_i}\prod_{e\in E_T(i)}\mathbbold{1}_{\gamma_{i,e}=\gamma_{i,Q_i}}\prod_{a_i\in Q_i\cap A_i}\mathbbold{1}_{\gamma_{i,a_i}=\gamma_{i,Q_i}},\\&\mathbbold{1}(\Gamma_i,\Upsilon_{i,P_i})=\prod_{Q_i\in P_i}\mathbbold{1}_{\gamma_{i,Q_i}\in\Gamma_i}. \end{align}\] Thus, by Equation 79 and Corollaries 3, 5 and 6, we obtain \[\sum_{\Gamma_i}\bar{z}_{\beta/2}(\Gamma_i)\mathbbold{1}(\Gamma_i,\Upsilon_{i,P_i})\leq\sum_{T_i\in\mathbbold{T}(P_i)}\prod_{Q_i\in P_i}e^{-\frac{\beta H(\gamma_{i,Q_i})}{2^{\lvert P_i\rvert+2}}}\prod_{\{R_i,S_i\}\in E(T_i)}\Phi(\gamma_{i,R_i},\gamma_{i,S_i}).\]The case \(m<i\leq n\) is a simplification: we may formally consider \(A_i\equiv \emptyset\) and proceed exactly as we have done for the case \(i\in[m]\).

We must now employ Lemma 20 to sum over contours. More precisely, we must bound \[\label{eq:32many-point-correlations-sum-to-eliminate-contours} F(\mathbf{x},\mathbf{x}',\mathbf{P},\mathbf{T})\mathrel{\vcenter{:}}= C(\mathbf{x})\sum_{\Upsilon}\sum_{\Upsilon'}\sum_{\Upsilon_{\mathbf{P}}}\mathbbold{1}(\Upsilon',\mathbf{x}')\mathbbold{1}(\Upsilon,\mathbf{x})\mathbbold{1}(\Upsilon,\Upsilon',\Upsilon_{\mathbf{P}})\epsilon_\beta(\Upsilon_\mathbf{P})\varPhi(\Upsilon_\mathbf{P},\mathbf{T}),\tag{123}\] where \(\Upsilon_{\mathbf{P}}=(\Upsilon_{1,P_1},...,\Upsilon_{n,P_n})\), \(\mathbf{T}=(T_1,...,T_n)\) and \[\begin{align} \mathbbold{1}(\Upsilon,\Upsilon',\Upsilon_{\mathbf{P}})&\mathrel{\vcenter{:}}=\prod_{i=1}^n\mathbbold{1}(\Upsilon_i,\Upsilon'_i,\Upsilon_{i,P_i}),\\ \epsilon_\beta(\Upsilon_\mathbf{P})&\mathrel{\vcenter{:}}=\prod_{i=1}^n\prod_{Q_i\in P_i}e^{-\frac{\beta H(\gamma_{i,Q_i})}{2^{\lvert P_i\rvert+2}}},\\ \varPhi(\Upsilon_{\mathbf{P}},\mathbf{T})&\mathrel{\vcenter{:}}=\prod_{i=1}^n\prod_{\{R_i,S_i\}\in E(T_i)}\Phi(\gamma_{i,R_i},\gamma_{i,S_i}). \end{align}\]

First, for each \(i\in[n]\) and each \(P_i\in\mathbbold{P}(A_i\sqcup E_T(i))\), we order every \(Q_i\in P_i\) canonically as \(Q_i=\{q_{Q_i,1},...,q_{Q_{i},\lvert Q_i\rvert}\}\) such that

1. if \(\lvert Q_i\cap E_T(i)\rvert\mathrel{\vcenter{:}}= n(Q_i,T)>0\), then \(Q_i\cap E_T(i)=\{y_{Q_i,q_i}:q_i\in[n(Q_i,T)]\}\). Furthermore, the edges are ordered in ascending order: if \(y_{Q_i,q_i}=\{i,j_{Q_i,q_i}\}\) and \(y_{Q_i,q'_i}=\{i,j_{Q_i,q'_i}\}\), then \(q_i<q'_i\leq q_i(T)\) if and only if \(j_{Q_i,q_i}<j_{Q_i,q'_i}\);

2. the set \(\{y_{Q_i,q_i}:n(Q_i,T)<q_i\leq \lvert Q_i\rvert\}\subset A_i\) is also ordered in ascending order: if \(n(Q_i,T)<q_i<q'_i\), then \(y_{Q_i,q_i}<y_{Q_i,q'_i}\).

For reasons that shall be clear shortly, arguing as in Lemma 20, we obtain the following "phantom" bound\[\label{eq:32phantom-one-inside-partition-bound} 1\leq e^{(\lvert Q_i\rvert-1)\alpha c_0H(\gamma_{i,Q_i})}\prod_{q_i=1}^{\lvert Q_i\rvert-1}C\left(x_{i,y_{Q_i,q_i}},x_{i,y_{Q_i,q_i+1}}\right).\tag{124}\] Let us now bound \(\varPhi(\Upsilon_\mathbf{P},\mathbf{T})\) employing the canonical ordering described above.

Let \(\{R_i,S_i\}\in E(T_i)\), we write \(R_i=\{y_{R_i,r_i}:r_i\in [\lvert R_i\rvert]\}\) and \(S_i=\{y_{S_i,s_i}:s_i\in[\lvert S_i]\}\), in accordance with the canonical ordering. By Lemma 20, we have \[\label{eq:32T-i-edge-bounds} \Phi(\gamma_{i,R_i},\gamma_{i,S_i})\leq 4^\alpha e^{2\alpha c_0H(\gamma_{i,R_i})}e^{2\alpha c_0H(\gamma_{i,S_i})}C\left(x_{i,y_{R_i,1}},x_{i, y_{S_i,1}}\right).\tag{125}\] Since \(\lvert Q_i\rvert+\delta_{T_i}(Q_i)\leq 2N\), combining Equation 124 and 125 , we have \[\label{eq:32C-varPhi-bound} C(\mathbf{x})\varPhi(\Upsilon_{\mathbf{P}},\mathbf{T})\leq 4^{N\alpha}\bar\epsilon_{\alpha}(\Upsilon_{\mathbf{P}})C(\mathbf{x},\mathbf{x}',\mathbf{P},\mathbf{T}),\tag{126}\] where \[\begin{align} &\bar\epsilon_{\alpha}(\Upsilon_{\mathbf{P}})\mathrel{\vcenter{:}}=\prod_{i=1}^n\prod_{Q_i\in P_i}e^{4N\alpha c_0H(\gamma_{i,Q_i})},\\ &\frac{C(\mathbf{x},\mathbf{x}',\mathbf{P},\mathbf{T})}{C(\mathbf{x})}=\prod_{i=1}^n\prod_{Q_i\in P_i}\prod_{q_i=1}^{\lvert Q_i\rvert-1}C(x_{i,y_{Q_i,q_i}},x_{i,y_{Q_i,q_i+1}})\prod_{\{R_i,S_i\}\in E(T)}C(x_{i,y_{R_i,1}},x_{i,y_{S_i,1}}). \end{align}\]This might seem very complicated, but concretely we have \[C(\mathbf{x},\mathbf{x}',\mathbf{P},\mathbf{T})=\prod_{\{t,u\}\in E(T(\mathbf{P},\mathbf{T}))}C(x_t,x_u),\]where \(T(\mathbf{P},\mathbf{T})\) is the tree with vertex set \[V(T(\mathbf{P},\mathbf{T}))= \bigsqcup_{i=1}^n\{i\}\times (E_T(i)\sqcup A_i)\] and with the following set of edges\[\begin{align} E(T(\mathbf{P},\mathbf{T}))=&\left\{\{(i,e),(j,e)\}:\{i,j\}=e\in E(T)\right\}\\ &\sqcup\{\{(i,y_{Q_i,q_i}),(i,y_{Q_i,q_i+1})\}:i\in [n], Q_i\in P_i, q_i\in[\lvert Q_i\rvert-1]\}\\ &\sqcup\{\{(i,y_{R_i,1}),(i,y_{S_i,1})\}:i\in [n], \{R_i,S_i\}\in E(T_i)\}. \end{align}\]Note that \[\label{eq:32cardinality-special-tree-many-point-correlation} \lvert V(T(\mathbf{P},\mathbf{T}))\rvert=2(n-1)+N\tag{127}\] and\[\label{eq:32leaves-special-tree-many-point-correlation} \mathrm{L}(T(\mathbf{P},\mathbf{T}))\subset\sqcup_{i=1}^m\{i\}\times A_i\tag{128}\]

We are now in a position to sum over contours and obtain a decent upper bound for Equation 123 . By Equation 126 , we have \[\frac{F(\mathbf{x},\mathbf{x}',\mathbf{P},\mathbf{T})}{C(\mathbf{x},\mathbf{x}',\mathbf{P},\mathbf{T})}\leq 4^{N\alpha}\sum_{\Upsilon}\sum_{\Upsilon'}\sum_{\Upsilon_{\mathbf{P}}}\mathbbold{1}(\Upsilon',\mathbf{x}')\mathbbold{1}(\Upsilon,\mathbf{x})\mathbbold{1}(\Upsilon,\Upsilon',\Upsilon_{\mathbf{P}})\epsilon_\beta(\Upsilon_\mathbf{P})\bar\epsilon_\alpha(\Upsilon_\mathbf{P})\eqqcolon 4^{N\alpha}F'(\mathbf{P},\mathbf{T}).\]In fact, \(F'(\mathbf{P},\mathbf{T})\) is rather easy to bound. First, note that \[\mathbbold{1}(\Upsilon',\mathbf{x}')\mathbbold{1}(\Upsilon,\mathbf{x})\mathbbold{1}(\Upsilon,\Upsilon',\Upsilon_\mathbf{P})=\mathbbold{1}(\Upsilon,\Upsilon',\Upsilon_\mathbf{P})\prod_{i=1}^n\prod_{Q_i\in P_i}\prod_{q_i=1}^{\lvert Q_i\rvert}\mathbbold{1}_{x_{i,y_{Q_i,q_i}}\in \mathrm{I}_-(\gamma_{i,Q_i})}.\]Thus, \[F'(\mathbf{P},\mathbf{T})\leq \prod_{i=1}^n\prod_{Q_i\in P_i}\sum_{\gamma_{i,Q_i}}\mathbbold{1}_{x_{i,y_{Q_i,1}}\in\mathrm{I}_-(\gamma_{i,Q_i})}e^{2N\alpha c_0H(\gamma_{i,Q_i})}e^{-\frac{\beta H(\gamma_{i,Q_i})}{2^{\lvert P_i\rvert+2}}}.\]Introducing equivalence classes of contours under integer translation (note that \(\lvert\mathrm{I}_-(\gamma_{i,Q_i})\rvert\) elements of \([\gamma_{i,Q_i}]\) satisfy \(x_{i,y_{Q_i,1}}\in\mathrm{I}_-(\gamma_{i,Q_i})\)), recalling Hypotheses 2 and 3, and, finally, choosing \(\beta\geq 2^{N+5}N\alpha c_0\), we obtain\[F'(\mathbf{P},\mathbf{T})\leq \prod_{i=1}^n\prod_{Q_i\in P_i}e^{-\frac{\beta c_2}{2^{\lvert P_i\rvert+3}}}=\prod_{i=1}^ne^{-\frac{\lvert P_i\rvert\beta c_2}{2^{\lvert P_i\rvert+3}}}.\]Having eliminated the sums over contours, we have \[\label{eq:32overline-W-P-T-intermediary-bound} \overline{\mathrm{W}}_\beta(P,T)\leq 4^{N\alpha}\sum_{\mathbf{P}}\prod_{i=1}^ne^{-\frac{\lvert P_i\rvert\beta c_2}{2^{\lvert P_i\rvert+3}}}\sum_{\mathbf{T}}\sum_{\mathbf{x}'}\sum_{\mathbf{x}}\mathbbold{1}(\mathbf{x}',\mathbf{A})C(\mathbf{x},\mathbf{x}',\mathbf{P},\mathbf{T}).\tag{129}\] The final step is simply to apply Lemma 18, which can be done because of Equation 128 . Since \(n\leq 2m-2\leq 2N\), recalling Equation 127 , we obtain \[\label{eq:32C-x-x-prime-P-T-bound} \sum_{\mathbf{x}}C(\mathbf{x},\mathbf{x}',\mathbf{P},\mathbf{T})\leq \left[\lvert\mathbbold{T}_{3N}\rvert\right]^{2N}2^{27\alpha N^2}(1+2\zeta(\alpha))^{3N}\sum_{T'\in\mathbbold{T}(A_P)}\prod_{\{t,u\}\in E(T')}C(x_t,x_u),\tag{130}\] where \(A_P\mathrel{\vcenter{:}}=\sqcup_{i=1}^m\{i\}\times A_i\). On the other hand, \[\label{eq:32crudest-bound-over-partitions-trees} \sum_{\mathbf{P}}\sum_{\mathbf{T}}\prod_{i=1}^ne^{-\frac{\lvert P_i\rvert\beta c_2}{2^{\lvert P_i\rvert+3}}}\leq \left[\lvert \mathbbold{T}_{3N}\rvert\right]^N\lvert \mathbbold{P}_{3N}\rvert e^{-\frac{\beta c_2}{2^{3N+3}}}.\tag{131}\] Finally, we have \[\label{eq:32many-point-correlations-indicator-to-sum-of-trees} \sum_{\mathbf{x}'}\mathbbold{1}(\mathbf{x}',\mathbf{A})\sum_{T'\in\mathbbold{T}(A_P)}\prod_{\{t,u\}\in E(T')}C(x_t,x_u)=\sum_{T'\in\mathbbold{T}(A)}\prod_{\{a_1,a_2\}\in E(T')}\lvert a_1-a_2\rvert^{-\alpha}.\tag{132}\]

Combining Equations 129 , 130 , 131 and 132 , we obtain Equation 122 . The lemma has thus been proved. ◻

Proof of Theorem 3. The proof of the theorem follows from a rather direct application of Lemma 24 into Equation 99 . We have \[\lvert\langle:\sigma_A:\rangle^+_\Lambda\rvert\leq \overline{\mathrm{W}}_\beta(\{A\})+\sum_{m=2}^{N}\sum_{\substack{P\in\mathbbold{P}(A)\\\lvert P\rvert =m}}\sum_{n=m}^{2m-2}\sum_{T\in\mathbbold{T}^*_{n,m}}\frac{1}{(n-m)!}\overline{\mathrm{W}}'_\beta(P,T)\eqqcolon\overline{\mathrm{W}}_\beta(\{A\})+F_\beta(A).\]Bounding the first term is a direct application of Lemma 23. On the other hand, we have the following trivial bounds: \(\lvert\mathbbold{T}^*_{n,m}\rvert\leq \lvert \mathbbold{T}_{2N}\rvert\), \(m\leq N\) and \(\lvert\mathbbold{P}(A)\rvert\leq\lvert \mathbbold{P}_{N}\rvert\). Thus, by Lemma 24, \[F_\beta(A)\leq N\lvert\mathbbold{P}_N\rvert\lvert \mathbbold{T}_{2N}\rvert F(N)e^{-\frac{\beta c_2}{2^{3N+3}}}\sum_{T'\in\mathbbold{T}(A)}\prod_{\{a_1,a_2\}\in E(T')}\lvert a_1-a_2\rvert^{-\alpha}.\]Choosing \(\beta\geq 2^{3N+4}c_2^{-1}\log[N\lvert\mathbbold{P}_N\rvert\lvert\mathbbold{T}_{2N}\rvert F(N)]\) and \(c_4=\min\{2^{-3N-4}c_2,2^{-N-5}c_0\log2\}\), we obtain Equation 2 . The theorem has thus been proved. ◻