January 01, 1970
We obtain sharp asymptotic probabilities for upward deviations of the maximum local time of discrete- and continuous-time simple random walks on \(\mathbb{Z}^d\), \(d\ge 3\). For downward deviations, we prove the sharp continuous-time asymptotics and the discrete-time upper bound. Together with the loop-pruning paper [1], which proves the matching discrete-time lower bound via a loop-pruning construction, this yields the sharp downward-deviation asymptotics in discrete time as well. We also derive Gumbel-type consequences at the logarithmic scale.
The maximum local time of simple random walk is a classical object in the study of random walks. In dimensions \(d \ge 3\), the walk is transient, so each fixed site is visited only finitely many times almost surely, while the trajectory up to time \(n\) still explores many sites. The competition among these visited sites leads to an extremal quantity of logarithmic order. The classical work of Erdős and Taylor identified this logarithmic scale, and subsequent work of Révész studied the maximum local time more directly in the transient setting; see [2], [3]. For the upper-tail mechanism relevant to the present paper, an especially useful predecessor is the work of Csáki, Földes, Révész, Rosen and Shi on frequently visited sets, whose two-point estimate for joint local times will play an important role in our analysis; see [4].
A broader adjacent literature concerns thick points, frequent points, and related extreme local-time phenomena in two dimensions. On the Brownian side, Dembo, Peres, Rosen and Zeitouni developed the thick-point picture for planar Brownian motion and, as a consequence, resolved the Erdős–Taylor conjecture for planar random walk; see [5]. On the random-walk side, Bass and Rosen obtained asymptotics for frequent points in two dimensions; see [6]. More recently, Jego studied thick points of random walk through their connection with the Gaussian free field; see [7]. We also mention Rosen’s recent tightness result for thick points in two dimensions; see [8].
While these works clarify the typical scale of the maximum local time and, in related settings, the structure of thick or frequent points, they leave open a natural large-deviation question. The present paper, together with the loop-pruning paper [1], answers this question for simple random walks in transient dimensions \(d\ge 3\). We remark that intimately related but different extremal problems for cover times were studied in [9]–[11].
We work throughout with the discrete-time simple random walk \((S_n)_{n \ge 0}\) on \(\mathbb{Z}^d\), \(d \ge 3\) as well as the corresponding continuous-time walk obtained by attaching i.i.d.\(\mathrm{Exp}(1)\) holding times to the discrete-time walk, which we denote by \((Y_t)_{t \ge 0}\). We write \(\mathbb{P}\) for the law governing both walks. For \(x \in \mathbb{Z}^d\), let \[\xi(n,x) := \sum_{k=0}^n \mathbf{1}_{\{S_k=x\}}, \qquad \ell(t,x) := \int_0^t \mathbf{1}_{\{Y_s=x\}}\,ds\] be the local times of the discrete-time and continuous-time walks, respectively, and set \[\xi^*(n) := \max_{x \in \mathbb{Z}^d} \xi(n,x), \qquad \ell^*(t) := \max_{x \in \mathbb{Z}^d} \ell(t,x).\] Thus \(\xi^*(n)\) and \(\ell^*(t)\) record the local time at a most frequently visited site. Let \[\gamma = \gamma_d := \mathbb{P}\bigl(S_n \neq 0 \text{ for all } n \ge 1\bigr), \qquad \alpha = \alpha_d := -\frac{1}{\log(1-\gamma)}.\] It is classical that \[\lim_{t \to \infty} \frac{\ell^*(t)}{\log t} = \gamma^{-1}, \qquad \lim_{n \to \infty} \frac{\xi^*(n)}{\log n} = \alpha\] almost surely; see, for instance, Theorem 13 in [2]. Thus the natural scales of the maximum local time are \(\gamma^{-1}\log t\) in continuous time and \(\alpha \log n\) in discrete time.
Theorem 1 (Upward deviations). For any \(\beta > 1\) and \(u \in \mathbb{R}\), as \(t \to \infty\) or \(n \to \infty\), we have \[\begin{align} \mathbb{P}\bigl(\ell^*(t) > \beta \cdot \gamma^{-1}\log t + u\bigr) &= (1+o(1)) \gamma e^{-\gamma u} t^{-(\beta-1)}, \label{eq:conti95UD}\\ \mathbb{P}\bigl(\xi^*(n) > \beta \cdot \alpha \log n + u\bigr) &= (1+o(1)) \gamma n (1-\gamma)^{\lfloor \beta \cdot \alpha \log n + u \rfloor}. \label{eq:discrete95UD} \end{align}\] {#eq: sublabel=eq:eq:conti95UD,eq:eq:discrete95UD}
Note that \[n^{-\beta}(1-\gamma)^u \le (1-\gamma)^{\lfloor \beta \cdot \alpha \log n + u \rfloor} < n^{-\beta}(1-\gamma)^{u-1}.\] Thus, setting \[c_{\beta,n,u} := n^\beta (1-\gamma)^{-u}(1-\gamma)^{\lfloor \beta \cdot \alpha \log n + u \rfloor} \in [1,(1-\gamma)^{-1}),\] we may rewrite ?? as \[\mathbb{P}\bigl(\xi^*(n) > \beta \cdot \alpha \log n + u\bigr) = (1+o(1)) c_{\beta,n,u}\gamma (1-\gamma)^u n^{-(\beta-1)}.\]
In this paper, the downward problem is fully resolved for the continuous-time walk. For the discrete-time walk, we prove the corresponding upper bound, while the matching lower bound, which is much more involved due to the discrete-time nature of the walk, is proved in the loop-pruning paper [1] using a novel construction called the “loop-pruned random walk”.
Theorem 2 (Downward deviations). For any \(\beta \in (0,1]\) and \(u \in \mathbb{R}\), as \(t \to \infty\) or \(n \to \infty\), we have \[\begin{align} \mathbb{P}\bigl(\ell^*(t) \le \beta \cdot \gamma^{-1}\log t + u\bigr) &= \exp\bigl\{-(1+o(1)) \gamma e^{-\gamma u} t^{1-\beta}\bigr\}, \label{eq:conti95DD}\\ \mathbb{P}\bigl(\xi^*(n) \le \beta \cdot \alpha \log n + u\bigr) &\le \exp\bigl\{-(1+o(1)) c_{\beta,n,u}\gamma (1-\gamma)^u n^{1-\beta}\bigr\}. \label{eq:discrete95DD} \end{align}\] {#eq: sublabel=eq:eq:conti95DD,eq:eq:discrete95DD}
Remark 3. We formulate the results for simple random walk in order to keep the exposition transparent. We expect that the arguments can be adapted to broader classes of transient finite-range random walks.
Remark 4 (Gumbel fluctuations). When \(\beta = 1\), the preceding results yield Gumbel-type fluctuations around the logarithmic centering:
In the continuous-time case, \(\ell^*(t) - \gamma^{-1}\log t\) converges in distribution to a Gumbel law with mode \(\gamma^{-1}\log \gamma\) and scale parameter \(\gamma^{-1}\).
In the discrete-time case, the present paper gives the downward upper bound. Combining this with the matching lower bound from the subsequent paper [1] yields the corresponding lattice, or discretized, Gumbel-type asymptotics, with the arithmetic oscillation encoded by \(c_{1,n,u}\).
One may compare these results with Jego’s exit-time result in dimension \(d\ge3\), where for the continuous-time walk stopped at the first exit time \(\tau_N\) of \(V_N\), the centred maximum \(\sup_{x\in V_N}\ell_{\tau_N}^x-2g\log N\) converges to a randomly shifted Gumbel law [7]. The same mesoscopic block decomposition underlying our proofs should likewise yield large-deviation estimates for the maximal local time up to exit times, by combining the deterministic-time one-block estimates with standard exit-time bounds.
One may also compare these with the Gumbel fluctuation of the cover time of tori in dimension \(d \ge 3\); see (1.2) of [9].
Remark 5 (Moderate deviations). The estimates above can also be used in the moderate-deviation regime where the threshold approaches the typical logarithmic scale. If \(a_t\) is positive, \(a_t\to\infty\), and \(a_t=o(\log t)\), then \[\begin{align} \mathbb{P}\bigl(\ell^*(t)>\gamma^{-1}\log t+a_t\bigr) &=(1+o(1))\gamma e^{-\gamma a_t},\\ \mathbb{P}\bigl(\ell^*(t)\le \gamma^{-1}\log t-a_t\bigr) &=\exp\{-(1+o(1))\gamma e^{\gamma a_t}\}. \end{align}\] Similarly, if \(a_n\) is positive, \(a_n\to\infty\), and \(a_n=o(\log n)\), then the discrete-time upward tail satisfies \[\begin{align} \mathbb{P}\bigl(\xi^*(n)>\alpha\log n+a_n\bigr) &=(1+o(1))\gamma n(1-\gamma)^{\lfloor \alpha\log n+a_n\rfloor}. \end{align}\] For the discrete-time downward tail, the argument in the present paper gives the corresponding upper bound \[\begin{align} \mathbb{P}\bigl(\xi^*(n)\le \alpha\log n-a_n\bigr) &\le \exp\{-(1+o(1))\gamma n(1-\gamma)^{\lfloor \alpha\log n-a_n\rfloor}\}. \end{align}\] The matching lower bound in this moderate-deviation regime requires the discrete-time loop-pruning construction developed in the loop-pruning paper [1].
These results exhibit a clear asymmetry between the two tails. Upward deviations are caused by the appearance of a single unusually thick site, and their probabilities are polynomial, up to the arithmetic correction in discrete time. Downward deviations, by contrast, are collective: one must prevent the walk from creating any site whose local time exceeds a subcritical threshold, and the resulting cost is stretched exponential, with exponent \(1-\beta\).
We now briefly comment on the proof for downward deviations. The upper bounds are obtained by decomposing the trajectory into many time intervals and requiring that no interval produce a local time above the target scale. For the continuous-time lower bound, we force the walk, once a site approaches the threshold, to spend only very short future holding times there, thereby preventing any further substantial accumulation at that site. To make this strategy affordable, we further decompose the path into segments and distinguish between local accumulation, where a single segment already creates a near-thick site, and global accumulation, where several separated segments contribute to the same site. The latter mechanism is much more costly, since it requires returns across long time gaps. Under a suitable conditioning that rules out local accumulation on slightly shortened segments, one shows that, with high probability, each site is visited by only boundedly many segments, which yields control of both the number of dangerous sites and the number of their later returns.
The rest of the paper is organized as follows. Section 2 collects the standard heat-kernel, hitting, and local-time estimates used throughout. Section 3 proves the upward-deviation asymptotics for both discrete-time and continuous-time walks. Section 4 treats the downward regime: it establishes the upper bounds in both settings and the continuous-time lower bound, while the discrete-time lower bound is proved in the loop-pruning paper [1].
XL is supported by the National Key R&D Program of China (No. 2021YFA1002700). YZ is supported by the China Postdoctoral Science Foundation (Nos. 2023M743721 and 2025T180850).
We collect here several standard estimates for transient simple random walk on \(\mathbb{Z}^d\), \(d\ge 3\), that will be used throughout the proofs.
We begin with the standard heat-kernel bound.
Lemma 1 (Theorem 2.3.9 in [12]). Uniformly for \(n\in \mathbb{N}^+\), \(t\in [1,\infty)\) and \(x\in \mathbb{Z}^d\), \[\begin{align} \label{eq:lclt} {\mathbb{P}}(S_n=x)\lesssim n^{-d/2};\qquad {\mathbb{P}}(Y_t=x)\lesssim t^{-d/2}. \end{align}\qquad{(1)}\]
The next two simple consequences will be used repeatedly.
Corollary 1. Uniformly for \(n\in\mathbb{N}^+\) and \(t,r\ge 1\), \[\begin{align} \label{eq:uniform95in95range} {\mathbb{P}}(\left\lvert S_n \right\rvert<r)\lesssim \Big(\frac{r}{n^{1/2}}\Big)^d;\qquad {\mathbb{P}}(\left\lvert Y_t \right\rvert<r)\lesssim \Big(\frac{r}{t^{1/2}}\Big)^d. \end{align}\qquad{(2)}\]
Corollary 2. Uniformly for \(x\in \mathbb{Z}^d\) and \(m\in \mathbb{N}^+\), \[\begin{align} \label{eq:late95hit} {\mathbb{P}}\big(x\in S[m,\infty)\big)\lesssim m^{-(d/2-1)}. \end{align}\qquad{(3)}\]
We also recall the classical asymptotic behavior of the hitting probability.
Lemma 2 (Lemma 17.8 in [13]). There exists a constant \(C_d>0\) such that \[\begin{align} \label{eq:hit95prob} {\mathbb{P}}\big(x\in S[0,\infty)\big)=\frac{C_d+o(1)}{\left\lvert x \right\rvert^{d-2}}, \qquad \text{as }\left\lvert x \right\rvert\to\infty. \end{align}\qquad{(4)}\]
We next record the one-point and two-point local-time tails.
Lemma 3 (Eq. (3.1) in [2]). For any \(m\in \mathbb{N}\) and \(s\ge 0\), \[\begin{align} \label{eq:local95time950} {\mathbb{P}}\big(\xi(\infty,0)> m\big)=(1-\gamma)^m; \qquad {\mathbb{P}}\big(\ell(\infty,0)> s\big)=e^{-\gamma s}. \end{align}\qquad{(5)}\] In particular, for \(\beta>0\), \(n\in\mathbb{N}\) and \(t>0\), \[\begin{align} \label{eq:local95time95039} \begin{aligned} &{\mathbb{P}}\big(\xi(\infty,0)>\beta\alpha\log n + u\big) =c_{\beta,n,u}(1-\gamma)^u n^{-\beta};\\ &{\mathbb{P}}\big(\ell(\infty,0)>\beta\gamma^{-1}\log t+u\big)=t^{-\beta}e^{-\gamma u}. \end{aligned} \end{align}\qquad{(6)}\]
As a key input for the upward deviation analysis, we will also need a two-point estimate.
Proposition 6 (Eq. (4.1) in [4]). Let \(t_y:={\mathbb{P}}\big(y\in S[0,\infty)\big),\; y\in\mathbb{Z}^d\setminus\{0\}.\) Then \(\xi(\infty,0)+\xi(\infty,y)\) has geometric distribution on \(\mathbb{N}^+\) with success probability \(\gamma/(1+t_y)\). Namely, for all \(u\in\mathbb{N}\), \[\begin{align} \label{eq:CFRRZ1} {\mathbb{P}}\big(\xi(\infty,0)+\xi(\infty,y)>u\big) =\Big(1-\frac{\gamma}{1+t_y}\Big)^u. \end{align}\qquad{(7)}\] Consequently, there exists \(\delta\in(0,1)\) such that for all \(u\in\mathbb{N}\), \[\begin{align} \label{eq:two95point95discrete} \sup_{y\in \mathbb{Z}^d\setminus\{0\}} {\mathbb{P}}\big(\xi(\infty,0)+\xi(\infty,y)>2u\big) \le (1-\gamma)^{(1+\delta)u} ={\mathbb{P}}\big(\xi(\infty,0)>u\big)^{1+\delta}. \end{align}\qquad{(8)}\]
Corollary 3. For any \(y\in \mathbb{Z}^d\setminus\{0\}\), the sum \(\ell(\infty,0)+\ell(\infty,y)\) has exponential distribution with mean \((1+t_y)/\gamma\). Namely, for all \(s\ge 0\), \[\begin{align} \label{eq:two95point95continuous} {\mathbb{P}}\big(\ell(\infty,0)+\ell(\infty,y)>s\big) =\exp\Big(-\frac{\gamma}{1+t_y}s\Big). \end{align}\qquad{(9)}\]
This follows from Proposition 6, since \(\xi(\infty,0)+\xi(\infty,y)\) is geometric with parameter \(\gamma/(1+t_y)\), and conditionally on this value the local time is a sum of i.i.d.\(\mathrm{Exp}(1)\) variables.
We begin with the discrete-time simple random walk. For \(x\in \mathbb{Z}^d\) and integers \(0\le n_1 \le n_2\), define the local time accumulated between times \(n_1\) and \(n_2\) as \[\xi([n_1,n_2],x) := \xi(n_2,x) - \xi(n_1 - 1,x),\] with \(\xi(-1,x):=0\).
To observe an upward deviation, it suffices for a single site to accumulate unusually large local time. This motivates a reformulation of the event in terms of a counting variable. Let \[m_n:=\lfloor \beta\alpha\log n + u\rfloor,\quad \mathcal{N} = \mathcal{N}(n,\beta) := \#\big\{x\in\mathbb{Z}^d:\xi(n,x)> m_n\big\}.\] Since \(\xi^*(n)\) is integer-valued, \(\{\xi^*(n)>\beta\alpha\log n+u\}=\{\xi^*(n)>m_n\}\). Then we have \[\left\{ \xi^*(n) > m_n \right\} = \left\{ \mathcal{N} \geq 1 \right\}.\]
To estimate \({\mathbb{P}}(\mathcal{N}\ge1)\), we use the first and second moment methods. The key input is Proposition 6, which implies that joint thick-point events at two distinct sites are negligible at the scale of the first moment. As a result, the first two moments of \(\mathcal{N}\) are asymptotically equivalent.
Lemma 4. As \(n \to \infty\), \[\begin{align} \mathbb{E}[\mathcal{N}] &= \big(1 + o(1)\big)\, c_{\beta,n,u} \gamma\,(1-\gamma)^u n^{1 - \beta}; \label{eq:first95moment}\\ \mathbb{E}[\mathcal{N}^2] &= \big(1 + o(1)\big)\, c_{\beta,n,u} \gamma\,(1-\gamma)^u n^{1 - \beta}.\label{eq:second95moment} \end{align}\] {#eq: sublabel=eq:eq:first95moment,eq:eq:second95moment}
Proof of Theorem 1 assuming Lemma 4. This follows directly from Lemma 4 via the first and second moment methods. ◻
We now turn to the proof of Lemma 4. For this, we use the representation \[\begin{align} \label{eq:represent} \mathcal{N}=\sum_{j=0}^n \mathbf{1}\big\{\xi([j,n],S_j)=m_n+1\big\}. \end{align}\tag{1}\] Indeed, each site \(x\) with \(\xi(n,x)>m_n\) contributes exactly once, namely at the time of its \((m_n+1)\)-st last visit before time \(n\).
This representation reduces the moment estimates for \(\mathcal{N}\) to estimates on the event \(\big\{\xi([j,n],S_j)=m_n+1\big\}.\) The next estimate will be used to justify replacing, away from the terminal time \(n\), the event \(\big\{\xi([j,n],S_j)=m_n+1\big\}\) by its infinite-horizon counterpart \(\big\{\xi([j,\infty),S_j)=m_n+1\big\},\) up to a negligible error.
Lemma 5. For any \(\beta>1\) and \(\varepsilon>0\), and any integer sequence \((k_n)\) satisfying \(k_n=\beta\alpha\log n+O(1)\) as \(n\to\infty\), one has \[\begin{align} \label{eq:accumulate95early} \sup_{j\ge \varepsilon n} {\mathbb{P}}\big(\xi(\infty,0)\ge k_n,\;\xi(j,0)<k_n\big) =o(n^{-\beta}). \end{align}\qquad{(10)}\]
Proof. Since \(\xi(j,0)\) is nondecreasing in \(j\), it suffices to prove that \[{\mathbb{P}}\big(\xi(\infty,0) \ge k_n,\;\xi(\lfloor \varepsilon n\rfloor,0)< k_n\big) =o(n^{-\beta}).\] Let \(T_0:=0\), and for \(j\ge 1\) define the \(j\)-th return time to the origin by \[T_j:=\inf\{k>T_{j-1}:S_k=0\}.\] Then \[\big\{\xi(\infty,0)\ge k_n,\;\xi(\lfloor\varepsilon n\rfloor,0)<k_n\big\} \subseteq \{T_{k_n}<\infty,\;T_{k_n}>\varepsilon n\}.\]
On the event \(\{T_{k_n}>\varepsilon n\}\), at least one of the \(k_n\) gaps \(T_j-T_{j-1}\), \(1\le j\le k_n\), must be at least \(L_n:=\frac{\varepsilon n}{k_n}\). Hence \[\{T_{k_n}<\infty,\;T_{k_n}>\varepsilon n\} \subseteq \bigcup_{j=1}^{k_n}\{T_{k_n}<\infty,\;T_j-T_{j-1}\ge L_n\}.\] Therefore, \[\label{eq:Aj95new} \text{LHS of \eqref{eq:accumulate95early}} \le \sum_{j=1}^{k_n}{\mathbb{P}}\big(T_{k_n}<\infty,\;T_j-T_{j-1}\ge L_n\big).\tag{2}\]
By the strong Markov property, \[\begin{align} &{\mathbb{P}}\big(T_{k_n}<\infty,\;T_j-T_{j-1}\ge L_n\big) = {\mathbb{P}}(T_{j-1}<\infty)\, {\mathbb{P}}\big(T_1\ge L_n,\;T_{k_n-j}<\infty\big)\\ \le\;& {\mathbb{P}}(T_{j-1}<\infty)\, {\mathbb{P}}(L_n\le T_1<\infty)\, {\mathbb{P}}(T_{k_n-j}<\infty)\\ =\;&{\mathbb{P}}\big(\xi(\infty,0)>j-1\big)\, {\mathbb{P}}\big(0\in S[\lfloor L_n\rfloor,\infty)\big)\, {\mathbb{P}}\big(\xi(\infty,0)>k_n-j\big) \overset{\eqref{eq:local95time950}}{\underset{\eqref{eq:late95hit}}{\lesssim}} n^{-\beta} \Big(\frac{n}{\log n}\Big)^{-(d/2 - 1)}. \end{align}\] Substituting this into 2 , we obtain \[\text{LHS of \eqref{eq:accumulate95early}} \lesssim (\log n)\,n^{-\beta}\Big(\frac{n}{\log n}\Big)^{-(d/2-1)} =o(n^{-\beta}).\qedhere\] ◻
Proof of ?? in Lemma 4. By 1 and the Markov property, \[\begin{align} {\mathbb{E}}[\mathcal{N}] &=\sum_{j=0}^n {\mathbb{P}}\big(\xi([j,n],S_j)=m_n+1\big)\\ &=\sum_{j=0}^n {\mathbb{P}}\big(\xi(n-j,0)=m_n+1\big) =\sum_{j=0}^n {\mathbb{P}}\big(\xi(j,0)=m_n+1\big). \end{align}\]
Fix \(\varepsilon\in(0,1)\) and decompose \[{\mathbb{E}}[\mathcal{N}] = \sum_{0\le j<\varepsilon n}{\mathbb{P}}\big(\xi(j,0)=m_n+1\big) + \sum_{\varepsilon n\le j\le n}{\mathbb{P}}\big(\xi(j,0)=m_n+1\big) =:I_1+I_2.\]
For \(I_1\), we simply use the bound \[I_1 \le \varepsilon n\,{\mathbb{P}}\big(\xi(\infty,0)> m_n\big) \overset{\eqref{eq:local95time950}}{=}\varepsilon c_{\beta,n,u}(1-\gamma)^u n^{1-\beta}.\]
We now turn to \(I_2\). Uniformly for \(j\ge \varepsilon n\), \[\begin{align} &\big|{\mathbb{P}}\big(\xi(j,0)=m_n+1\big)-{\mathbb{P}}\big(\xi(\infty,0)=m_n+1\big)\big|\\ \le\,& {\mathbb{P}}\big(\xi(\infty,0)> m_n,\;\xi(j,0)\le m_n\big) +{\mathbb{P}}\big(\xi(\infty,0)> m_n+1,\;\xi(j,0)\le m_n+1\big) \overset{\eqref{eq:accumulate95early}}{=}o(n^{-\beta}). \end{align}\] Therefore, \[\begin{align} {\mathbb{P}}\big(\xi(j,0)=m_n+1\big) ={\mathbb{P}}\big(\xi(\infty,0)=m_n+1\big)+o(n^{-\beta}) \overset{\eqref{eq:local95time950}}{=} c_{\beta,n,u}\gamma(1-\gamma)^u n^{-\beta}+o(n^{-\beta}) \end{align}\] uniformly for \(j\ge \varepsilon n\). Summing over such \(j\), we obtain \[I_2 = (1-\varepsilon) c_{\beta,n,u}\gamma(1-\gamma)^u n^{1-\beta} +o(n^{1-\beta}).\]
Combining the estimates for \(I_1\) and \(I_2\), we get \[\begin{align} (1-\varepsilon)\gamma c_{\beta,n,u}(1-\gamma)^u n^{1-\beta}&+o(n^{1-\beta}) \le {\mathbb{E}}[\mathcal{N}]\\ &\le \bigl((1-\varepsilon)\gamma+\varepsilon\bigr)c_{\beta,n,u}(1-\gamma)^un^{1-\beta} +o(n^{1-\beta}). \end{align}\] Since \(\varepsilon>0\) is arbitrary, this proves ?? . ◻
Proof of ?? in Lemma 4. Recall from 1 that \[\mathcal{N}=\sum_{j=0}^n X_j, \qquad X_j:=\mathbf{1}\{\xi([j,n],S_j)=m_n+1\}.\] Hence \[{\mathbb{E}}[\mathcal{N}^2]-{\mathbb{E}}[\mathcal{N}] =2\sum_{0\le i<j\le n}{\mathbb{E}}[X_iX_j].\] Moreover, if \(i<j\) and \(S_i=S_j\), then \(X_iX_j=0\), since \(\xi([i,n],S_i)\ge \xi([j,n],S_j)+1.\) Therefore, it suffices to show that \[\begin{align} \label{eq:different95ij} \sum_{0\le i<j\le n}{\mathbb{E}}\big[X_iX_j\mathbf{1}_{\{S_i\ne S_j\}}\big] =o(n^{1-\beta}). \end{align}\tag{3}\]
Let \(\delta\in(0,1)\) be the constant from ?? , and fix \(\varepsilon\) small enough such that \[\begin{align} \label{eq:eps95small} (1+\delta)\varepsilon \beta /2+\varepsilon<\delta\beta\qquad\text{and}\qquad \varepsilon<\beta-1. \end{align}\tag{4}\] We first consider the short-time regime \(j-i\le n^\varepsilon\).
For \(0\le i<j\le n\), define \[U_{i,j}:=\xi([i,j-1],S_i), \qquad V_{i,j}:=\xi([j,n],S_i)+\xi([j,n],S_j).\] If \(X_i=X_j=1\) and \(S_i\ne S_j\), then \[U_{i,j}+V_{i,j} =\xi([i,n],S_i)+\xi([j,n],S_j) =2(m_n+1).\] Set \(K:=\Big\lfloor\frac{1}{\varepsilon}\Big\rfloor+1.\) It follows that \[\begin{align} &\{X_i=X_j=1,\;S_i\ne S_j\}\\ \subseteq\,& \bigcup_{k=1}^{K} \Bigl\{ U_{i,j}\in[(k-1)\varepsilon (m_n+1),\;k\varepsilon (m_n+1)),\; V_{i,j}\ge (2-k\varepsilon)(m_n+1),\; S_i\ne S_j \Bigr\}. \end{align}\] Therefore, \[\begin{align} &{\mathbb{E}}\big[X_iX_j\mathbf{1}_{\{S_i\ne S_j\}}\big]\\ \le\,& \sum_{k=1}^{K} {\mathbb{P}}\Bigl( U_{i,j}\in[(k-1)\varepsilon (m_n+1),\;k\varepsilon (m_n+1)),\; V_{i,j}\ge (2-k\varepsilon)(m_n+1),\; S_i\ne S_j \Bigr)\\ \le\,& \sum_{k=1}^{K}{\mathbb{P}}\big(\xi(\infty,0)\ge (k-1)\varepsilon (m_n+1)\big) \sup_{y\in\mathbb{Z}^d\setminus\{0\}} {\mathbb{P}}\big(\xi(\infty,0)+\xi(\infty,y)\ge (2-k\varepsilon)(m_n+1)\big)\\ \le\,& \sum_{k=1}^{K} (1-\gamma)^{(k-1)\varepsilon (m_n+1)+(1+\delta)(1-k\varepsilon/2)(m_n+1)}\\ \le\,& K(1-\gamma)^{(1+\delta)(1-\varepsilon/2)m_n}\le C n^{-(1+\delta)\beta +(1+\delta)\varepsilon \beta /2}, \end{align}\] for some constant \(C=C(\varepsilon,\beta)>0\).
Summing over all pairs \((i,j)\) with \(j-i\le n^\varepsilon\), we get \[\begin{align} \label{eq:sum95small} \sum_{\substack{0\le i<j\le n:\\ j-i\le n^\varepsilon}} {\mathbb{E}}\big[X_iX_j\mathbf{1}_{\{S_i\ne S_j\}}\big] &\le C\,n^{1+\varepsilon}n^{-(1+\delta)\beta+(1+\delta)\varepsilon \beta /2} \overset{\eqref{eq:eps95small}}{=}o(n^{1-\beta}). \end{align}\tag{5}\]
Thus the short-time contribution is \(o(n^{1-\beta})\). It remains to estimate the sum over pairs with \(j-i>n^\varepsilon\). For this regime, we further localize the gap \(j-i\). More precisely, for \(\eta\in[\varepsilon,1]\), we consider pairs satisfying \(n^\eta\le j-i<n^{\frac{10}{9}\eta},\) and on each such block we distinguish between the two spatial regimes \[0<|S_i-S_j|<n^{\eta/8} \qquad\text{and}\qquad |S_i-S_j|\ge n^{\eta/8}.\]
We begin with the near-displacement case.
Lemma 6. For any \(\eta>0\), \[\begin{align} \label{eq:sum95eta} \sum_{\substack{0\le i<j\le n:\\ j-i\in[n^\eta,n^{\frac{10}{9}\eta})}} {\mathbb{E}}\bigl[X_iX_j\mathbf{1}_{\{0<|S_i-S_j|<n^{\eta/8}\}}\bigr] =o(n^{1-\beta}). \end{align}\qquad{(11)}\]
Proof. Fix \(0\le i<j\le n\) with \(j-i\in[n^\eta,n^{\frac{10}{9}\eta})\). By the Markov property, \[\begin{align} &{\mathbb{E}}\bigl[X_iX_j\mathbf{1}_{\{0<|S_i-S_j|<n^{\eta/8}\}}\bigr] \le {\mathbb{E}}\bigl[\mathbf{1}_{\{0<|S_i-S_j|<n^{\eta/8}\}}X_j\bigr]\\ \le\,& {\mathbb{P}}\bigl(0<|S_{j-i}|<n^{\eta/8}\bigr)\, {\mathbb{P}}\bigl(\xi(\infty,0)> m_n\bigr)\overset{\eqref{eq:uniform95in95range}}{\lesssim}\Bigl(\frac{n^{\eta/8}}{(j-i)^{1/2}}\Bigr)^d n^{-\beta}\le n^{-\frac{9}{8}\eta-\beta}. \end{align}\] Therefore, \[\sum_{\substack{0\le i<j\le n:\\ j-i\in[n^\eta,n^{\frac{10}{9}\eta})}} {\mathbb{E}}\bigl[X_iX_j\mathbf{1}_{\{0<|S_i-S_j|<n^{\eta/8}\}}\bigr] \lesssim n^{1+\frac{10}{9}\eta}\,n^{-\frac{9}{8}\eta-\beta} =o(n^{1-\beta}).\qedhere\] ◻
Let \[K=K(\varepsilon):=\max\Bigl\{k\ge 0:\Bigl(\frac{10}{9}\Bigr)^k\varepsilon\le 1\Bigr\}, \qquad \eta_k:=\Bigl(\frac{10}{9}\Bigr)^k\varepsilon.\] Applying Lemma 6 on each shell, we obtain \[\begin{align} \label{eq:small95dis} \sum_{k=0}^K \sum_{\substack{0\le i<j\le n:\\ j-i\in[n^{\eta_k},\,n^{\frac{10}{9}\eta_k})}} {\mathbb{E}}\bigl[X_iX_j\mathbf{1}_{\{0<|S_i-S_j|<n^{\eta_k/8}\}}\bigr] =o(n^{1-\beta}). \end{align}\tag{6}\]
We next consider the far-displacement case. By ?? , uniformly for \(y\in\mathbb{Z}^d\) with \(|y|\ge n^{\eta/8}\), \[t_y\lesssim |y|^{-(d-2)}\le n^{-\eta/8}.\] Hence, by ?? , uniformly for \(u\in[0,2(m_n+1)]\) and \(|y|\ge n^{\eta/8}\), \[\begin{align} \label{eq:large95dis} {\mathbb{P}}\bigl(\xi(\infty,0)+\xi(\infty,y)\ge u\bigr) = \bigl(1-\gamma/(1+t_y)\bigr)^{u-1} = (1-\gamma)^{u-1}\Bigl(1+O\Bigl(\frac{\log n}{n^{\eta/8}}\Bigr)\Bigr). \end{align}\tag{7}\]
Repeating the argument from the short-time regime, but using 7 in place of ?? , we obtain \[{\mathbb{E}}\bigl[X_iX_j\mathbf{1}_{\{|S_i-S_j|\ge n^{\eta/8}\}}\bigr] \le C n^{-(2\beta-\varepsilon)}\] for some constant \(C=C(\varepsilon,\beta)>0\), uniformly for all \(0\le i<j\le n\) satisfying \(j-i\in[n^\eta,n^{\frac{10}{9}\eta}).\) Therefore, \[\begin{align} \label{eq:big95dis} \sum_{k=0}^K \sum_{\substack{0\le i<j\le n:\\ j-i\in[n^{\eta_k},\,n^{\frac{10}{9}\eta_k})}} {\mathbb{E}}\bigl[X_iX_j\mathbf{1}_{\{|S_i-S_j|\ge n^{\eta_k/8}\}}\bigr] \le (K+1)n^2\cdot C n^{-(2\beta-\varepsilon)}\overset{\eqref{eq:eps95small}}{=} o(n^{1-\beta}). \end{align}\tag{8}\]
Combining 5 , 6 , and 8 , we obtain 3 . This completes the proof of ?? . ◻
We now turn to the continuous-time walk. Couple the discrete-time and continuous-time simple random walks \((S_n)_{n\ge0}\) and \((Y_t)_{t\ge0}\) by \[Y_t=S_{N_t},\] where \((N_t)_{t\ge0}\) is an independent rate-\(1\) Poisson process. Thus \((S_n)\) is the discrete skeleton of \((Y_t)\). For \(n\in\mathbb{N}\), let \[N^{-1}(n):=\inf\{t\ge0:N_t=n\}\] be the \(n\)-th jump time of \(Y\). We define the local time sampled at jump times by \[\begin{align} \label{def:tilde95ell} \widetilde{\ell}(n,x):=\ell(N^{-1}(n+1),x), \qquad \widetilde{\ell}^{*}(n):=\max_{x\in\mathbb{Z}^d}\widetilde{\ell}(n,x)=\ell^{*}(N^{-1}(n+1)). \end{align}\tag{9}\] For integers \(1\le n_1\le n_2\), write \[\widetilde{\ell}([n_1,n_2],x):=\widetilde{\ell}(n_2,x)-\widetilde{\ell}(n_1-1,x).\]
Fix \(\alpha\in(1/2,1)\). Standard moderate deviations for the Poisson process imply that there exists \(C=C(\alpha)>0\) such that for all \(t\ge1\), \[{\mathbb{P}}\Bigl(\bigl|N_t-t\bigr|>t^\alpha\Bigr)\le e^{-Ct^{2\alpha-1}}.\] As a consequence, \[\begin{align} \label{eq:moderate} {\mathbb{P}}\Bigl( \widetilde{\ell}^{*}(\lfloor t-t^\alpha\rfloor) \le \ell^{*}(t) \le \widetilde{\ell}^{*}(\lfloor t+t^\alpha\rfloor) \Bigr) \ge 1-e^{-Ct^{2\alpha-1}}. \end{align}\tag{10}\]
Therefore, to prove the upward deviation asymptotics for \(\ell^{*}(t)\), it suffices to show the following.
Proposition 7. Set \(b_n:=\beta\gamma^{-1}\log n + u\). As \(n\to\infty\), \[\begin{align} \label{eq:upward95ell42} {\mathbb{P}}\bigl(\widetilde{\ell}^{*}(n)>b_n\bigr) = (1+o(1))\,\gamma e^{-\gamma u}\,n^{-(\beta-1)}. \end{align}\qquad{(12)}\]
To this end, define \[\mathcal{N}'=\mathcal{N}'(n,\beta):=\#\{x\in\mathbb{Z}^d:\widetilde{\ell}(n,x)>b_n\}.\] Then \(\{\widetilde{\ell}^{*}(n)>b_n\}=\{\mathcal{N}'\ge1\}\). Moreover, \(\mathcal{N}'\) admits the representation \[\mathcal{N}' = \sum_{j=0}^n \mathbf{1}\Bigl\{ \widetilde{\ell}([j,n],S_j)>b_n,\; \widetilde{\ell}([j+1,n],S_j)\le b_n \Bigr\}.\] Indeed, each site counted by \(\mathcal{N}'\) contributes exactly once, namely at the last jump time at which the accumulated jump-time local time at that site crosses the level \(b_n\).
The proof of ?? now proceeds in the same way as in the discrete-time case, with \(\widetilde{\ell}\) in place of \(\xi\), and with Corollary 3 replacing the corresponding discrete two-point estimate. Since the argument is entirely parallel, we omit the details.
In this section, we prove the downward deviation estimate stated in Theorem 2. Throughout, we assume that \(\beta \in (0,1]\).
The upper bound is straightforward in both the continuous-time and discrete-time settings. We give the argument for the continuous-time walk, since the discrete-time case is completely analogous.
Fix \(\beta'\in(0,\beta)\) and partition the path \(Y{[0,t]}\) into sections of equal length \(t^{\beta'}\): \[Y[0,t]\supset\bigcup_{j=1}^{\lfloor t^{1-\beta'}\rfloor}Y\big[(j-1)t^{\beta'},jt^{\beta'}\big].\] Then the event \(\{\ell^*(t)\le \beta\cdot \gamma^{-1}\log t\}\) implies that the maximum local time in each section must be no more than \(\beta \cdot \gamma^{-1} \log t=\frac{\beta}{\beta'}\cdot \gamma^{-1} \log t^{\beta'}\). Hence, \[\begin{align} &{\mathbb{P}}\big(\ell^*(t)\le \beta\cdot \gamma^{-1}\log t+u\big) \le{\mathbb{P}}\Big\{\ell^*\big(t^{\beta'}\big)\le \beta\cdot \gamma^{-1}\log t+u\Big\}^{\lfloor t^{1-\beta'}\rfloor}\\ \overset{\eqref{eq:conti95UD}}{=}&\Big\{1-\big(1+o(1)\big)\gamma\cdot e^{-\gamma u}\big(t^{\beta'}\big)^{-(\beta/\beta'-1)}\Big\}^{\lfloor t^{1-\beta'}\rfloor} =\exp\big\{-\big(1+o(1)\big)\gamma e^{-\gamma u} t^{1-\beta}\big\}. \end{align}\] This yields the desired upper bound.
In this subsection, we prove the lower bound in Theorem 2 for the continuous-time simple random walk. For simplicity, we restrict ourselves to the case \(u=0\), since the proof for general \(u\in\mathbb{R}\) is entirely analogous.
We now fix \(\kappa\in(1-\beta/2,1)\). It is enough to prove the following jump-time estimate: \[\begin{align} \mathbb{P}\Bigl( \widetilde{\ell}^{*}\!\bigl(\lfloor n+n^\kappa\rfloor\bigr) \le \beta \gamma^{-1}\log n \Bigr) \ge \exp\Bigl\{-(1+o(1))\gamma n^{1-\beta}\Bigr\}, \qquad n\to\infty . \label{eq:conti-lower-jumptime} \end{align}\tag{11}\]
By the moderate deviation estimate 10 , with probability at least \(1-\exp\{-Ct^{2\kappa-1}\}\), one has \[\widetilde{\ell}^{*}\!\bigl(\lfloor t-t^{\kappa}\rfloor\bigr) \le \ell^{*}(t) \le \widetilde{\ell}^{*}\!\bigl(\lfloor t+t^{\kappa}\rfloor\bigr),\] and therefore, \[\begin{align} &\mathbb{P}\Bigl(\ell^{*}(t)\le \beta\gamma^{-1}\log t\Bigr)\\ \ge\,& \mathbb{P}\Bigl( \widetilde{\ell}^{*}\!\bigl(\lfloor t+t^\kappa\rfloor\bigr) \le \beta\gamma^{-1}\log t \Bigr) - \mathbb{P}\Bigl( \ell^{*}(t)>\widetilde{\ell}^{*}\!\bigl(\lfloor t+t^\kappa\rfloor\bigr) \Bigr) \\ \overset{\eqref{eq:conti-lower-jumptime}}{\ge}& \exp\Bigl\{-(1+o(1))\gamma t^{1-\beta}\Bigr\} - \exp\{-Ct^{2\kappa-1}\} = \exp\Bigl\{-(1+o(1))\gamma t^{1-\beta}\Bigr\}, \end{align}\] since \(2\kappa-1>1-\beta\). Thus 11 implies the desired continuous-time lower bound.
It therefore suffices to prove that for every \(\delta>0\),
\[\begin{align} \mathbb{P}\Bigl( \widetilde{\ell}^{*}\!\bigl(\lfloor n+n^\kappa\rfloor\bigr) \le \beta \gamma^{-1}\log n \Bigr) \ge \exp\Bigl\{-(1+\delta)\gamma n^{1-\beta}\Bigr\} \label{eq:conti-lower-jumptime-delta} \end{align}\tag{12}\] for all sufficiently large \(n\). Set \(\widehat n:=\lfloor n+n^\kappa\rfloor\).
We now describe a forcing strategy for the event \(\{\widetilde{\ell}^{*}(\widehat n)\le \beta\gamma^{-1}\log n\}.\) Fix a small \(\eta>0\), to be chosen later. Set \[\Lambda_n:=\beta\gamma^{-1}\log n-\eta . \label{eq:Lambda95n}\tag{13}\] The idea is to monitor those sites whose local time first reaches the preliminary level \(\Lambda_n\), and then force all future holding times at such sites to be very short. More precisely, once a site reaches level \(\Lambda_n\), we require the walk to leave it within time \(\eta/2\); if the walk later returns to that site, then the next holding time is required to be at most \(\eta/4\), then at most \(\eta/8\), and so on. Since \(\sum_{k=1}^{\infty}\frac{\eta}{2^{k}}=\eta,\) this guarantees that after the local time first reaches \(\Lambda_n\), the total additional local time accumulated at that site is less than \(\eta\), and hence the final local time remains below \(\beta\gamma^{-1}\log n\).
We now formalize this construction. Define stopping times \((\tau_j)_{j\ge1}\) by \[\begin{align} \tau_1:=\,&\inf\big\{t\ge0:\ell(t,Y_t)>\Lambda_n\big\},\qquad and for \(j\ge2\), \\ \tau_j:=\,& \inf\Bigl\{ t\ge0:\ell(t,Y_t)>\Lambda_n,\, Y_t\notin\{Y_{\tau_1},\dots,Y_{\tau_{j-1}}\} \Bigr\}, \end{align}\] with the convention \(\inf\varnothing=\infty\). Thus \(\tau_j\) is the first time at which a new site reaches the level \(\Lambda_n\). Let \[\mathcal{N}:=\sup\big\{j\ge1:\tau_j<N^{-1}(\widehat n)\big\},\] with the convention \(\sup\varnothing=0\). Hence \(\mathcal{N}\) is the number of distinct sites that reach level \(\Lambda_n\) before the \(\widehat n\)-th jump time. For each \(1\le j\le \mathcal{N}\), define the holding time immediately after \(\tau_j\) by \[h_j^{0}:=\inf\big\{s\ge0:Y_{\tau_j+s}\neq Y_{\tau_j}\big\}. \label{eq:hj0}\tag{14}\] Recursively, for \(k\ge1\), let \[\tau_j^{k}:=\inf\big\{t\ge \tau_j^{k-1}+h_j^{k-1}:Y_t=Y_{\tau_j}\big\}, \qquad h_j^{k}:=\inf\big\{s\ge0:Y_{\tau_j^{k}+s}\neq Y_{\tau_j}\big\}, \label{eq:taujk-hjk}\tag{15}\] where \(\tau_j^{k}\) is the \(k\)-th return time to \(Y_{\tau_j}\) after \(\tau_j\), and \(h_j^{k}\) is the corresponding holding time. We also write \[\mathcal{M}_j:=\sup\big\{k\ge1:\tau_j^{k}<N^{-1}(\widehat n)\big\},\] so that \(\mathcal{M}_j\) is the number of returns to \(Y_{\tau_j}\) before time \(N^{-1}(\widehat n)\).
We now introduce the forcing event \[B:= \Bigl\{ h_j^{k}<\eta/2^{k+1} \;\text{for all }1\le j\le \mathcal{N},\;0\le k\le \mathcal{M}_j \Bigr\}. \label{eq:forcing95event95B}\tag{16}\]
By construction, on the event \(B\), every site that ever reaches the level \(\Lambda_n\) can accumulate at most an additional amount \(\sum_{k=0}^{\infty}\eta/2^{k+1}=\eta\) of local time afterwards. Hence no site can exceed the level \(\Lambda_n+\eta=\beta\gamma^{-1}\log n\), and therefore \[B\subset \Bigl\{ \widetilde{\ell}^{*}(\widehat n)\le \beta\gamma^{-1}\log n \Bigr\}.\] Consequently, \[\mathbb{P}\Bigl( \widetilde{\ell}^{*}(\widehat n)\le \beta\gamma^{-1}\log n \Bigr)\ge \mathbb{P}(B). \label{eq:lower95bound95by95B}\tag{17}\]
To estimate \(\mathbb{P}(B)\), we isolate two ingredients. The first gives the conditional probability of the forcing event \(B\) once the number of dangerous sites and the numbers of later returns to them are fixed. The second shows that these quantities are typically small, with the correct exponential cost.
Proposition 8. We have \[\begin{align} \label{eq:condi95prob} \mathbb{P}\Bigl( B \,\Big|\, \mathcal{N},\;(\mathcal{M}_j)_{1\le j\le \mathcal{N}} \Bigr) = \prod_{j=1}^{\mathcal{N}}\prod_{k=0}^{\mathcal{M}_j} \bigl(1-e^{-\eta/2^{k+1}}\bigr). \end{align}\qquad{(13)}\]
Proposition 9. For every \(\delta>0\), one can choose \(\eta>0\) sufficiently small so that there exist constants \(b<1-\beta\) and \(c>0\) satisfying \[\begin{align} \label{eq:NM95bound} \mathbb{P}\Bigl( \mathcal{N}\le n^b,\; \mathcal{M}_j\le c\log n \;\text{for all }1\le j\le \mathcal{N} \Bigr) \ge \exp\Bigl\{-(1+\delta/2+o(1))\gamma n^{1-\beta}\Bigr\}. \end{align}\qquad{(14)}\]
Remark 10. The exponent \(b\) in Proposition 9 is allowed to be negative. In that case, \(\mathcal{N}_n\le n^b\) is equivalent to \(\mathcal{N}_n=0\).
Proof of 12 assuming Propositions 8 and 9. From ?? and ?? , we have \[\begin{align} {\mathbb{P}}(B)&\ge {\mathbb{P}}\big(B,\,\mathcal{N}\le n^b,\,\mathcal{M}_j\le c\log n\;\forall\,1\le j\le \mathcal{N}\big)\\ &\ge \Big[\prod_{j=1}^{\lfloor n^b \rfloor}\prod_{k=0}^{c\log n}\big(1-\exp\{-\eta/2^{k+1}\}\big)\Big]\exp\big\{-\big(1+\delta/2+o(1)\big)\,\gamma n^{1-\beta}\big\}\\ &\ge \exp\big\{-\big(1+\delta/2+o(1)\big)\,\gamma n^{1-\beta}\big\}, \end{align}\] which (along with 17 ) implies 12 . ◻
We begin by introducing some notation. For \(x\in \mathbb{Z}^d\) and \(j\ge 1\), let \(T^x_j\) denote the \(j\)-th hitting time of \(x\): \[\begin{align} T^x_1&:=\inf\big\{n\ge 0: S_n=x\big\},\qquad T^x_j:=\inf\big\{n > T^x_{j-1}:S_n=x\big\}\quad\text{for }j\ge 2, \end{align}\] with the convention \(\inf\emptyset=\infty\). Define the holding time at \(x\) after the \(j\)-th hitting as \[\begin{align} \sigma(j,x):=\mathbf{1}\{T^x_j<\infty\}\cdot\big[N^{-1}(T^x_j+1)-N^{-1}(T^x_j)\big]. \end{align}\]
Next, we focus on the points \(\big(Y_{\tau_j}:1\le j\le \mathcal{N}\big)\). For each such point, we isolate the special holding time during which the total local time reaches the threshold \(\beta \cdot \gamma^{-1} \log n\). Specifically, for each \(1 \le j \le \mathcal{N}\), define \[\mathcal{R}_j:=\inf\Big\{k\in\mathbb{N}^+: \sum_{i=1}^k\,\sigma(i,Y_{\tau_j})\ge \Lambda_n\Big\}=\xi(\widehat n,Y_{\tau_j})-\mathcal{M}_j.\] Then, immediately following the \(\mathcal{R}_j\)-th visit to \(Y_{\tau_j}\), the addition of the holding time \(\sigma(\mathcal{R}_j, Y_{\tau_j})\) causes the local time at \(Y_{\tau_j}\) to exceed the threshold \(\Lambda_n=\beta \gamma^{-1} \log n - \eta\). We split this holding time into two parts, representing the contributions before and after this threshold: \[\sigma(\mathcal{R}_j,Y_{\tau_j})=\sigma_1(Y_{\tau_j})+\sigma_2(Y_{\tau_j}),\] with \[\begin{align} \sigma_1(Y_{\tau_j}):=\Lambda_n-\sum_{i=1}^{\mathcal{R}_j-1}\sigma(i,Y_{\tau_j}),\qquad \sigma_2(Y_{\tau_j}):=\sum_{i=1}^{\mathcal{R}_j}\sigma(i,Y_{\tau_j})-\Lambda_n. \end{align}\]
Lemma 7. Let \((\gamma_i:i\ge 1)\) be i.i.d.\(\mathrm{Exp}(1)\) random variables. Conditionally on \(\big\{\xi(\widehat n,x):x\in\mathbb{Z}^d\big\}\), \(\mathcal{N}\), \(\big(Y_{\tau_j}:1\le j\le \mathcal{N}\big)\) and \(\big(\mathcal{M}_j:1\le j\le \mathcal{N}\big)\), the following hold:
The holding times \(\Big\{\sigma(i,x),\,\sigma_1(x)\mathbf{1}_{x\in \{Y_{\tau_j}\}},\, \sigma_2(x)\mathbf{1}_{x\in \{Y_{\tau_j}\}}:1\le i\le \xi(\widehat n,x)\Big\}\) are independent across different \(x\in\mathbb{Z}^d\);
For any \(x\in \mathbb{Z}^d\setminus\{Y_{\tau_j}\}\), \[\Big(\sigma(i,x):1\le i\le \xi(\widehat n,x)\Big)\overset{d}{=} \Big(\big(\gamma_i:1\le i\le \xi(\widehat n,x)\big)\,\Big|\,\sum_{i=1}^{\xi(\widehat n,x)}\gamma_i<\Lambda_n\Big);\]
For any \(x\in \{Y_{\tau_j}\}\), \[\begin{align} &\Big(\sigma_1(x),\,\sigma(i,x):1\le i\le \mathcal{R}_j-1\Big)\overset{d}{=} \Big(\big(\gamma_i:1\le i\le \mathcal{R}_j\big)\,\Big|\,\sum_{i=1}^{\mathcal{R}_j}\gamma_i=\Lambda_n\Big),\\ &\Big(\sigma_2(x),\sigma(i,x):\mathcal{R}_j+1\le i\le \xi(\widehat n,x)\Big)\overset{d}{=}\big(\gamma_i:1\le i\le \xi(\widehat n,x)-\mathcal{R}_j\big), \end{align}\] and these two collections of holding times are mutually independent.
Proof. Conditionally on \(\{\xi(\widehat n,x): x \in \mathbb{Z}^d\}\), we have:
the holding times \(\big\{\sigma(j,x):x\in \mathbb{Z}^d,\,1\le j\le \xi(\widehat n,x)\big\}\) are i.i.d. \(\text{Exp}(1)\) random variables;
for any \(N\in \mathbb{N}\), distinct points \((y_j : 1 \le j \le N) \subset \mathbb{Z}^d\) and non-negative integers \((m_j : 1 \le j \le N)\), the event \[\Big\{\mathcal{N}=N,\,(Y_{\tau_j})=(y_j),\,(\mathcal{M}_j)=(m_j)\Big\}\] equals the event that
for any \(x\in \mathbb{Z}^d\setminus\{y_j\}\), \(\sum_{i=1}^{\xi(\widehat n,x)}\sigma(i,x)<\Lambda_n,\)
for any \(x\in\{y_j\}\), \(\sum_{i=1}^{R_j-1}\sigma(i,x)\le\Lambda_n<\sum_{i=1}^{R_j}\sigma(i,x),\) where \(R_j:=\xi(\widehat n,x)-m_j\).
The conclusions follow directly from the properties of i.i.d.\(\text{Exp}(1)\) variables under conditioning. ◻
Proof of Proposition 8. Recall the definitions of \(h^k_j\) in 14 and 15 . We observe that for \(1\le j\le \mathcal{N}\), \[\begin{align} h^0_j=\sigma_2(Y_{\tau_j})\quad\text{and}\quad h^k_j=\sigma(\mathcal{R}_j+k,Y_{\tau_j})\quad\text{for }k\ge 1. \end{align}\] Therefore, by Lemma 7, conditionally on \(\big\{\xi(\widehat n,x): x \in \mathbb{Z}^d\big\}\), \(\mathcal{N}\), \(\big(Y_{\tau_j}: 1 \le j \le \mathcal{N}\big)\), and \(\big(\mathcal{M}_j : 1 \le j \le \mathcal{N}\big)\), the collection \(\big(h^k_j : 1 \le j \le \mathcal{N},\;0 \le k \le \mathcal{M}_j\big)\) consists of i.i.d.\(\text{Exp}(1)\) random variables. This immediately yields ?? . ◻
Recall the partition of the path \(Y[0, t]\) from Section 4.1. In a discrete-time analogue, we partition the path \(S[0, n]\) into segments \(S(I_j)\) (see 18 below), where each \(I_j\) represents a discrete time interval.
For any point \(x \in \mathbb{Z}^d\), the local time at \(x\) can exceed the threshold \(\beta\cdot\gamma^{-1}\log n\) via one of two mechanisms:
Local accumulation: The local time at \(x\) accumulated during a single interval \(I_j\) already exceeds the threshold.
Global accumulation: The local time at \(x\) remains below the threshold in each individual \(I_j\), but the cumulative sum across multiple intervals pushes it above the threshold.
Between these two, global accumulation is significantly more costly, as it requires the point \(x\) to be visited across several disjoint intervals.
As in the estimate from Section 4.1, the probability that no point exceeds the threshold via local accumulation matches the lower bound. We therefore use this estimate to bound the cost of our strategy in handling the local case. On the other hand, the number of points that exceed the threshold through global accumulation is small. Consequently, the cost of controlling their subsequent holding times is also small, and can in principle be bounded by \(\exp\left\{ - o(1)\, \gamma n^{1 - \beta} \right\}\).
The remaining issue is to separate the costs of ruling out local accumulation and ruling out global accumulation. The key observation is that, after conditioning on (a suitable modification of) the event that local accumulation does not occur, the strong Markov property implies that, with high probability, each point \(x\in S[0,\widehat n]\) is visited by at most a constant number \(M=M(\beta_1,\beta_2)\) of segments \(S(I_j)\). In other words, the conditioning effectively constrains only finitely many intervals at each site.
Since the probability that no local threshold is exceeded on any fixed finite collection of intervals is \(1+o(1)\), it follows that, up to a \(1+o(1)\) factor, the event of no local accumulation and the event that our forcing strategy succeeds against global accumulation are asymptotically decoupled. This is the mechanism behind Proposition 9.
We now implement this idea. Fix constants \[\beta_1\in(0,\beta), \qquad \beta_2\in\Bigl(\frac{2}{d}\beta_1,\beta_1\Bigr).\] We partition the path \(S[0,\widehat n]\) into segments of equal discrete length \(\lfloor n^{\beta_1}\rfloor\). Set \(K:=\left\lfloor \frac{\widehat n}{\lfloor n^{\beta_1}\rfloor}\right\rfloor+1.\) For \(j=1,\dots,K-1\), define \[I_j:=\bigl[(j-1)\lfloor n^{\beta_1}\rfloor,\; j\lfloor n^{\beta_1}\rfloor\bigr),\] and for the final interval set \[I_K:=\bigl[(K-1)\lfloor n^{\beta_1}\rfloor,\; \widehat n\bigr].\] We write the corresponding path segments as \[\begin{align} \label{eq:partition} S(I_j)= \begin{cases} S\bigl[(j-1)\lfloor n^{\beta_1}\rfloor,\; j\lfloor n^{\beta_1}\rfloor\bigr), & 1\le j\le K-1,\\[3pt] S\bigl[(K-1)\lfloor n^{\beta_1}\rfloor,\; \widehat n\bigr], & j=K. \end{cases} \end{align}\tag{18}\] Thus, \(S[0,\widehat n]=\cup_{j=1}^K S(I_j)\). Next, we shorten each segment by removing its initial \(\lfloor n^{\beta_2}\rfloor\) steps. For \(j=1,\dots,K-1\), define \[I_j':=\bigl[(j-1)\lfloor n^{\beta_1}\rfloor+\lfloor n^{\beta_2}\rfloor,\; j\lfloor n^{\beta_1}\rfloor\bigr),\] and for \(j=K\), set \[I_K':=\bigl[(K-1)\lfloor n^{\beta_1}\rfloor+\lfloor n^{\beta_2}\rfloor,\; \widehat n\bigr].\] Recall that for integers \(1\le n_1\le n_2\) and \(x\in\mathbb{Z}^d\), \[\widetilde{\ell}([n_1,n_2],x):=\widetilde{\ell}(n_2,x)-\widetilde{\ell}(n_1-1,x).\] We also define the maximum local time over an interval by \[\widetilde{\ell}^{*}([n_1,n_2]) :=\max_{x\in\mathbb{Z}^d}\widetilde{\ell}([n_1,n_2],x), \qquad \widetilde{\ell}^{*}([n_1,n_2)) :=\widetilde{\ell}^{*}([n_1,n_2-1]).\] To define the no-local-accumulation event, fix \(\delta>0\), and choose \(\eta>0\) sufficiently small so that \[\begin{align} \label{eq:ensure} e^{2\gamma\eta}-1<\delta/2. \end{align}\tag{19}\] We then set \[\begin{align} \label{def:A} A:=\bigcap_{j=1}^K A_j, \qquad A_j:=\left\{ \widetilde{\ell}^{*}(I_j') < \beta\gamma^{-1}\log n-\eta \right\}. \end{align}\tag{20}\] We first estimate the probability of a single block. Let \[m_n:=\lfloor n^{\beta_1}-n^{\beta_2}\rfloor .\] By ?? , for all sufficiently large \(n\), uniformly in \(1\le j\le K\), \[\begin{align} \mathbb{P}(A_j^{\rm c}) &\le \mathbb{P}\left\{ \widetilde{\ell}^{*}(m_n) > \frac{\beta}{\beta_1}\gamma^{-1}\log m_n-2\eta \right\}\\ &= (1+o(1))\,\gamma e^{2\gamma\eta}\, (n^{\beta_1})^{-(\beta/\beta_1-1)}. \end{align}\] In view of 19 , this gives \[\begin{align} \label{eq:single-Aj} \mathbb{P}(A_j) \ge 1-(1+\delta/2+o(1))\,\gamma\, (n^{\beta_1})^{-(\beta/\beta_1-1)}. \end{align}\tag{21}\] Since the events \(A_1,\ldots,A_K\) depend on disjoint collections of increments and holding times, they are independent. Hence \[\begin{align} \label{eq:pa} \begin{aligned} \mathbb{P}(A) =\prod_{j=1}^K\mathbb{P}(A_j)&\ge \left( 1-(1+\delta/2+o(1))\,\gamma\,(n^{\beta_1})^{-(\beta/\beta_1-1)} \right)^K \\ &= \exp\left\{-(1+\delta/2+o(1))\,\gamma\,n^{1-\beta}\right\}. \end{aligned} \end{align}\tag{22}\] We denote the corresponding conditional measure by \[\mathbb{Q}=\mathbb{Q}(n):=\mathbb{P}(\,\cdot\,|\,A).\] It remains to prove the following two estimates under the conditional measure \(\mathbb{Q}\).
Proposition 11. There exists a constant \(c=c(\beta_1,\beta_2)>0\) such that \[\begin{align} \label{eq:o1} \mathbb{Q}\bigl(\mathcal{M}_j\le c\log n \text{ for all }1\le j\le \mathcal{N}\bigr)=1-o(1). \end{align}\qquad{(15)}\]
Proposition 12. There exists a constant \(b=b(\beta_1,\beta_2)<1-\beta\) such that \[\begin{align} \label{eq:o2} \mathbb{Q}\bigl(\mathcal{N}\le n^b\bigr)=1-o(1). \end{align}\qquad{(16)}\]
As explained in the main idea, a key step is to control how many segments can visit a given point. We begin by establishing a quantitative version of this statement. For \(1\le n_1\le n_2\le K\), write \[I_{[n_1,n_2]}:=I_{n_1}\cup\cdots\cup I_{n_2}, \qquad I'_{[n_1,n_2]}:=I'_{n_1}\cup\cdots\cup I'_{n_2}.\]
Lemma 8. Uniformly for \(1\le i<j\le K\) with \(j-i\ge2\), \(k\in I_i\), and \(m\in I_j\), we have \[\begin{align} \mathbb{Q}\bigl(S_k=S_m\,\big|\,S(I_{[1,i]}),\, S(I_{[j,K]})\bigr) &\lesssim \bigl((j-i-1)n^{\beta_2}\bigr)^{-d/2}. \label{eq:SlSm} \end{align}\qquad{(17)}\] Consequently, uniformly for \(1\le i\le K-2\) and \(k\in I_i\), \[\begin{align} \label{eq:432hit} \mathbb{Q}\bigl(S_k\in S(I_{[i+2,K]})\,\big|\,S(I_{[1,i]})\bigr) \lesssim n^{-\{(d/2)\beta_2-\beta_1\}}. \end{align}\qquad{(18)}\]
Proof. Fix \(k\) and \(m\). Conditionally on \(S(I_{[1,i]})\), \(S(I_{[j,K]})\), and \(S(I'_{[i+1,j-1]})\), the vector \[\bigl(S_{i\lfloor n^{\beta_1}\rfloor}-S_k\bigr) +\bigl(S_m-S_{(j-1)\lfloor n^{\beta_1}\rfloor}\bigr) +\sum_{r=i+1}^{j-1}\bigl(S_{(r+1)\lfloor n^{\beta_1}\rfloor}-S_{r\lfloor n^{\beta_1}\rfloor+\lfloor n^{\beta_2}\rfloor}\bigr)\] is determined by the conditioning, and hence equals some fixed vector \(x\in\mathbb{Z}^d\). Therefore, the event \(\{S_k=S_m\}\) is equivalent to \[\left\{ \sum_{r=i+1}^{j-1}\bigl(S_{r\lfloor n^{\beta_1}\rfloor+\lfloor n^{\beta_2}\rfloor}-S_{r\lfloor n^{\beta_1}\rfloor}\bigr) =-x \right\}.\] Under the conditioning, the sum above has the same law as the position of a \(d\)-dimensional simple random walk after \((j-i-1)\lfloor n^{\beta_2}\rfloor\) steps. By ?? , there exists a universal constant \(c>0\) such that \[\mathbb{Q}\bigl( S_k=S_m\,\big|\,S(I_{[1,i]}),\,S(I_{[j,K]}),\,S(I'_{[i+1,j-1]}) \bigr) \le \frac{c}{\bigl((j-i-1)n^{\beta_2}\bigr)^{d/2}}.\] This proves ?? . The bound ?? then follows from the union bound. ◻
For \(1\le i\le K\), let \(\mathcal{G}_i:=\sigma\bigl(S(I_{[1,i]})\bigr).\) The next stopping-time version follows immediately from ?? , by applying that estimate on each event \(\{\tau=i\}\).
Corollary 4. Let \(\tau\in[1,K-2]\) be a stopping time with respect to the filtration \((\mathcal{G}_i)\). Then, for some universal constant \(c>0\), conditionally on \(\mathcal{G}_\tau\), for any \(k\in I_\tau\), \[\begin{align} \label{eq:stopping} \mathbb{Q}\bigl(S_k\in S(I_{[\tau+2,K]})\,\big|\,\mathcal{G}_\tau\bigr) \le c\,n^{-\{(d/2)\beta_2-\beta_1\}}. \end{align}\qquad{(19)}\]
We now estimate the maximal number of segments that can visit a single point. For \(x\in\mathbb{Z}^d\), define \[\mathcal{J}_x=\mathcal{J}_x(n):=\{1\le j\le K:x\in S(I_j)\}.\]
Proposition 13. There exists a constant \(M=M(\beta_1,\beta_2)\in\mathbb{N}^+\) such that \[\begin{align} \label{eq:Jx} \mathbb{Q}\Bigl(\sup_{x\in\mathbb{Z}^d}\#\mathcal{J}_x\le M\Bigr)=1-o(1). \end{align}\qquad{(20)}\]
Proof. We first observe that \[\begin{align} \label{eq:observe1} \Bigl\{\sup_{x\in\mathbb{Z}^d}\#\mathcal{J}_x>M\Bigr\} = \bigcup_{k=0}^{\widehat n} \bigl\{S_k\notin S[1,k-1],\;\#\mathcal{J}_{S_k}>M\bigr\}. \end{align}\tag{23}\] Fix \(1\le i\le K\) and \(k\in I_i\), and define stopping times \[\tau_0(k):=i, \qquad \tau_r(k):=\inf\{m\in[\tau_{r-1}(k)+2,K]:S_k\in S(I_m)\}, \qquad r\ge1.\] Then \[\begin{align} \label{eq:observe2} \bigl\{S_k\notin S[1,k-1],\;\#\mathcal{J}_{S_k}>M\bigr\} \subset \bigl\{\tau_{\lfloor M/2\rfloor-1}(k)<\infty\bigr\}. \end{align}\tag{24}\] We now apply ?? iteratively to the sequence of stopping times \(\tau_0(k),\dots,\tau_{\lfloor M/2\rfloor-2}(k)\). This yields the following bound, for some universal constant \(c>0\): \[\mathbb{Q}\bigl(\tau_{\lfloor M/2\rfloor-1}(k)<\infty\bigr) \le c\,n^{-(\lfloor M/2\rfloor-1)\{(d/2)\beta_2-\beta_1\}}.\] Since \((d/2)\beta_2-\beta_1>0\), we may choose \(M=M(\beta_1,\beta_2)\) sufficiently large so that \[(\lfloor M/2\rfloor-1)\{(d/2)\beta_2-\beta_1\}>1.\] Combining 23 and 24 and using the union bound, we obtain \[\mathbb{Q}\Bigl(\sup_{x\in\mathbb{Z}^d}\#\mathcal{J}_x>M\Bigr) \le \widehat n\cdot c\,n^{-(\lfloor M/2\rfloor-1)\{(d/2)\beta_2-\beta_1\}} =o(1).\qedhere\] ◻
We record the following simple observation, which will be used in the proof of Proposition 12. For each \(1\le j\le K\), set \[a_j:=(j-1)\lfloor n^{\beta_1}\rfloor,\qquad b_j:=j\lfloor n^{\beta_1}\rfloor\wedge \widehat n.\] Let \(\widehat Y^{(j)}\) be the \(j\)-th continuous-time block translated to start from time \(0\) and position \(0\): \[\widehat Y^{(j)} :=\bigl(Y_{N^{-1}(a_j)+t}-Y_{N^{-1}(a_j)}: 0\le t\le N^{-1}(b_j)-N^{-1}(a_j)\bigr).\] For each subset \(J\subset\{1,\dots,K\}\), define \[\begin{align} \label{def:QJ} \mathbb{Q}_J:= \mathbb{P}\Bigl(\,\cdot\,\Big| \bigcap_{j\in J}A_j\Bigr). \end{align}\tag{25}\] Thus, unlike \(\mathbb{Q}\), which conditions on all the events \(A_j\), the measure \(\mathbb{Q}_J\) conditions only on those \(A_j\) with \(j\in J\).
Lemma 9. For any event \(E\in \sigma\bigl(\widehat Y^{(j)}:j\in J\bigr),\) we have \[\begin{align} \label{eq:QJ} \mathbb{Q}(E)=\mathbb{Q}_J(E). \end{align}\qquad{(21)}\]
Proof. Proposition 13 provides a bound on the number of segments visiting any given point. We now further bound the maximal discrete local time within each segment. Together, these yield the desired bound on \(\mathcal{M}_j\). Denote by \[\xi^*(I_j):=\sup_{x\in{\mathbb{Z}}^d}\xi(I_j,x)=\sup_{x\in{\mathbb{Z}}^d}\sum_{r\in I_j}\mathbf{1}_{\{S_r=x\}}\] the maximal discrete local time accumulated during the segment \(I_j\). From ?? , for each \(j=1,\ldots,K\), we have \[{\mathbb{P}}\big(\xi^*(I_j)>(1+\beta_1)\cdot\gamma^{-1}\log n\big)\lesssim (n^{\beta_1})^{-(\frac{1+\beta_1}{\beta_1}-1)}=n^{-1}.\] Recalling the definitions of \(A\) and \(A_j\) in 20 , it follows that \[\begin{align} {\mathbb{Q}}\big(\xi^*(I_j)>(1+\beta_1)\cdot\gamma^{-1}\log n\big) \le{\mathbb{P}}\big(\xi^*(I_j)>(1+\beta_1)\cdot\gamma^{-1}\log n\big)/{\mathbb{P}}(A_j)\lesssim n^{-1}. \end{align}\] Hence, recalling that \(K=\left\lfloor \frac{\widehat n}{\lfloor n^{\beta_1}\rfloor}\right\rfloor+1\sim n^{1-\beta_1}\), we obtain \[\begin{align} \label{eq:upwards} {\mathbb{Q}}\big(\xi^*(I_j)\le (1+\beta_1)\cdot\gamma^{-1}\log n\;\forall\,j=1,\ldots,K\big)\ge 1-CKn^{-1}=1-o(1). \end{align}\tag{26}\] Now observe that \[\begin{align} &\big\{\sup_{x\in{\mathbb{Z}}^d}\#\mathcal{J}_x\le M\big\}\cap \big\{\xi^*(I_j)\le (1+\beta_1)\cdot\gamma^{-1}\log n\;\forall\,j=1,\ldots,K\big\}\\ &\subset\big\{\mathcal{M}_j\le M(1+\beta_1)\cdot\gamma^{-1}\log n\;\forall\,1\le j\le \mathcal{N}\big\}. \end{align}\] Combining this inclusion with the bounds from ?? and 26 , the desired result follows with \(c=M(1+\beta_1)\cdot \gamma^{-1}\). ◻
Proof. Fix \(M\in\mathbb{N}^+\) so that ?? holds. It suffices to prove that \[\mathbb{E}^{\mathbb{Q}}\Bigl( \mathcal{N}\cdot \mathbf{1}\Bigl\{\sup_{x\in\mathbb{Z}^d}\#\mathcal{J}_x\le M\Bigr\} \Bigr) =o(n^b) \qquad\text{for some } b<1-\beta.\] Step 1: Reduction to a blockwise estimate. We begin with the representation \[\mathcal{N} = \sum_{k=0}^{\widehat n} \mathbf{1}\bigl\{ S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n \bigr\}.\] It follows that \[\begin{align} \mathbb{E}^{\mathbb{Q}}\Bigl( \mathcal{N}\cdot \mathbf{1}\Bigl\{\sup_{x\in\mathbb{Z}^d}\#\mathcal{J}_x\le M\Bigr\} \Bigr)\le\,& \sum_{k=0}^{\widehat n} \mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\#\mathcal{J}_{S_k}\le M \Bigr) \\ =\,& \sum_{i=1}^{K}\sum_{k\in I_i} \mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\#\mathcal{J}_{S_k}\le M \Bigr). \end{align}\] Since \(K\sim n^{1-\beta_1}\), it remains to show that there exists \(u>0\) such that, uniformly for \(i\in[1,K]\), \[\begin{align} \label{eq:N95sum} \sum_{k\in I_i} \mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\#\mathcal{J}_{S_k}\le M \Bigr) = o\bigl(n^{-(\beta-\beta_1)-u}\bigr). \end{align}\tag{27}\] Step 2: Decomposition according to \(\mathcal{J}_{S_k}\). To estimate the left-hand side of 27 , we decompose according to the possible values of \(\mathcal{J}_{S_k}\): \[\begin{align} &\mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\#\mathcal{J}_{S_k}\le M \Bigr) \\ &\qquad = \sum_{\substack{J\subset\{i,\dots,K\}\\ \#J\le M}} \mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=J \Bigr). \end{align}\] We split the index sets \(J\) into the following three classes:
\(J=\{i\}\);
\(J=[i,j]\) for some \(j\in[i+1,i+M-1]\);
\(J\) is not an interval.
We shall prove later that, uniformly for \(i\in[1,K]\), the contributions of these three classes satisfy \[\begin{align} &\sum_{k\in I_i} \mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=\{i\} \Bigr)= O\bigl(n^{-(\beta-\beta_1)-(\beta_1-\beta_2)}\bigr);\tag{28} \\[4pt] &\sum_{j=i+1}^{(i+M-1)\wedge K}\sum_{k\in I_i} \mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=[i,j] \Bigr)\tag{29}\\ &\le n^{-(\beta-\beta_1)-\beta_1/4+o(1)}; \notag\\[4pt] &\sum_{k\in I_i} \mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\; \mathcal{J}_{S_k}\text{ is not an interval},\;\# \mathcal{J}_{S_k}\le M \Bigr)\tag{30}\\ &\le n^{-(\beta-\beta_1)-((d/2)\beta_2-\beta_1)+o(1)}. \notag \end{align}\] Since \(\beta_2<\beta_1\) and \(\beta_1<\frac{d}{2}\beta_2\), the bounds in 28 –30 imply 27 . ◻
We now prove the three case estimates 28 –30 . Their mechanisms are as follows:
For 28 , all the local time is accumulated within a single segment. This is directly suppressed under the conditioning on \(A\), up to the short initial part removed from the segment.
For 29 , the index set \(J\) is a single interval of segments. Thus, if the local time at \(S_k\) is not already created inside \(I_i\), then it must be built up by revisits coming from later segments; away from the right endpoint of \(I_i\), this forces a delayed return, which yields an additional decay.
For 30 , the set \(J\) is disconnected. Each gap between successive blocks of \(J\) contributes an additional decay factor, and this gain outweighs the combinatorial complexity of summing over all such \(J\).
For later use, for each \(i\in[1,K]\), we write \[I_i^-:=(i-1)\lfloor n^{\beta_1}\rfloor, \qquad I_i^+:=i\lfloor n^{\beta_1}\rfloor\wedge \widehat n\] for the left and right endpoints of the interval \(I_i\).
Proof of 28 . Fix \(i\in[1,K]\). On \(\Bigl\{ S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=\{i\} \Bigr\},\) the site \(S_k\) is visited only during \(I_i\), and hence \[\widetilde{\ell}(\widehat n,S_k)=\widetilde{\ell}([k,I_i^+],S_k), \qquad I_i^+:=i\lfloor n^{\beta_1}\rfloor\wedge \widehat n.\] If \(k\in I_i'\), then \(A_i\) implies \[\widetilde{\ell}([k,I_i^+],S_k)\le \widetilde{\ell}^*(I_i')<\Lambda_n,\] so the corresponding contribution is zero. Therefore \[\sum_{k\in I_i} \mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=\{i\} \Bigr) \le \sum_{k\in I_i\setminus I_i'} \mathbb{Q}\Bigl( \widetilde{\ell}([k,I_i^+],S_k)\ge \Lambda_n \Bigr).\] By the definition of \(\mathbb{Q}\), \[\mathbb{Q}\Bigl( \widetilde{\ell}([k,I_i^+],S_k)\ge \Lambda_n \Bigr) \le \frac{\mathbb{P}\bigl(\widetilde{\ell}(\infty,0)\ge \Lambda_n\bigr)}{\mathbb{P}(A_i)} \lesssim n^{-\beta},\] uniformly in \(k\in I_i\setminus I_i'\). Summing over \(k\in I_i\setminus I_i'\), and using \(\#(I_i\setminus I_i')\lesssim n^{\beta_2}\), we obtain 28 . ◻
Proof of 29 . We first record the delayed-return estimate needed in the proof.
Lemma 10. Uniformly for \(m\in[1,\lfloor n^{\beta_1}\rfloor]\), \[\mathbb{P}\Big( \widetilde{\ell}(\infty,0)\ge \Lambda_n,\; 0\in S[m,\infty) \Big) \le \begin{cases} O(n^{-\beta}), & 1\le m\le \lfloor n^{\beta_1/2}\rfloor,\\[3pt] n^{-(\beta+\beta_1/4)+o(1)}, & \lfloor n^{\beta_1/2}\rfloor < m\le \lfloor n^{\beta_1}\rfloor. \end{cases}\]
Fix \(i\in[1,K]\), \(j\in[i+1,(i+M-1)\wedge K]\), and set \(J=[i,j]\). Recalling the definition of \(\mathbb{Q}_J\) in 25 , \[\begin{align} &\mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=J \Bigr)\\ \le\,& \mathbb{Q}\Bigl( \widetilde{\ell}([k,I_j^+],S_k)\ge \Lambda_n,\;S_k\in S(I_{[i+1,j]}) \Bigr) \\ \overset{\eqref{eq:QJ}}{=}&\mathbb{Q}_J\Bigl( \widetilde{\ell}([k,I_j^+],S_k)\ge \Lambda_n,\;S_k\in S(I_{[i+1,j]}) \Bigr)\\ \le\,& \frac{ \mathbb{P}\Bigl( \widetilde{\ell}([k,I_j^+],S_k)\ge \Lambda_n,\;S_k\in S(I_{[i+1,j]}) \Bigr) }{ \prod_{\ell\in J}\mathbb{P}(A_\ell) }. \end{align}\] Since \(\#J\le M\) and \(\mathbb{P}(A_\ell)=1-o(1)\) uniformly in \(\ell\), the denominator is \(1-o(1)\). Therefore, applying the Markov property and Lemma 10, we obtain \[\begin{align} &\mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=J \Bigr)\\ &\qquad \le (1+o(1)) \mathbb{P}\Bigl( \widetilde{\ell}(\infty,0)\ge \Lambda_n,\; 0\in S[\lfloor n^{\beta_1}\rfloor-k+I_i^-,\infty) \Bigr)\\ &\qquad \le \lfloor n^{\beta_1/2}\rfloor\cdot O(n^{-\beta}) + \lfloor n^{\beta_1}\rfloor\cdot n^{-(\beta+\beta_1/4)+o(1)} = n^{-(\beta-\beta_1)-\beta_1/4+o(1)}. \end{align}\] Since there are at most \(M\) admissible values of \(j\in[i+1,(i+M-1)\wedge K]\), summing over such \(j\) does not change the order of magnitude. This proves 29 . ◻
To prove Lemma 10, we need the following auxiliary estimate.
Lemma 11. As \(n\to\infty\), \[\begin{align} \label{eq:general95early} \mathbb{P}\Big( \widetilde{\ell}(\infty,0)\ge \Lambda_n,\; \widetilde{\ell}\big(\lfloor n^{\beta_1/2}/2\rfloor,0\big)<\Lambda_n \Big) \le n^{-(\beta+\beta_1/4)+o(1)}. \end{align}\qquad{(22)}\]
Proof. Define the inverse local time at the origin by \[\widetilde{T}_u:=\inf\{m\ge0:\widetilde{\ell}(m,0)\ge u\}, \qquad u\ge0.\] Fix \(\varepsilon>0\), and for \(j=1,\dots,\lfloor 1/\varepsilon\rfloor\), define \[\widetilde{B}_j:= \Bigl\{ \varepsilon\lfloor n^{\beta_1/2}/2\rfloor \le \widetilde{T}_{j\varepsilon \Lambda_n}-\widetilde{T}_{(j-1)\varepsilon \Lambda_n} <\infty \Bigr\}.\] Then \[\Bigl\{ \widetilde{\ell}(\infty,0)\ge \Lambda_n,\; \widetilde{\ell}\big(\lfloor n^{\beta_1/2}/2\rfloor,0\big)<\Lambda_n \Bigr\} \subset \bigcup_{j=1}^{\lfloor 1/\varepsilon\rfloor}\widetilde{B}_j,\] and therefore \[\mathbb{P}\Bigl( \widetilde{\ell}(\infty,0)\ge \Lambda_n,\; \widetilde{\ell}\big(\lfloor n^{\beta_1/2}/2\rfloor,0\big)<\Lambda_n \Bigr) \le \sum_{j=1}^{\lfloor 1/\varepsilon\rfloor} \mathbb{P}(\widetilde{T}_{\Lambda_n}<\infty,\widetilde{B}_j).\] By the strong Markov property, for each \(j\), \[\begin{align} &\mathbb{P}(\widetilde{T}_{\Lambda_n}<\infty,\widetilde{B}_j)\\ \le\,& \mathbb{P}\bigl(\widetilde{\ell}(\infty,0)\ge (j-1)\varepsilon \Lambda_n\bigr)\, \mathbb{P}\bigl(0\in S[\varepsilon\lfloor n^{\beta_1/2}/2\rfloor,\infty)\bigr) \mathbb{P}\bigl(\widetilde{\ell}(\infty,0)\ge (1-j\varepsilon)\Lambda_n\bigr)\\ \le\,& Cn^{-(1-\varepsilon)\beta-(d/2-1)\beta_1/2} \le Cn^{-(1-\varepsilon)\beta-\beta_1/4}, \end{align}\] since \(d\ge3\). Summing over \(j\le \lfloor 1/\varepsilon\rfloor\) and then letting \(\varepsilon\downarrow0\), we obtain ?? . ◻
Proof of Lemma 10. For \(1\le m\le \lfloor n^{\beta_1/2}\rfloor\), we simply use \[\mathbb{P}\Big( \widetilde{\ell}(\infty,0)\ge \Lambda_n,\; 0\in S[m,\infty) \Big) \le \mathbb{P}\bigl(\widetilde{\ell}(\infty,0)\ge \Lambda_n\bigr) \lesssim n^{-\beta}.\] Now assume \(\lfloor n^{\beta_1/2}\rfloor < m\le \lfloor n^{\beta_1}\rfloor\). Splitting according to whether \(\Lambda_n\) has been accumulated by time \(\lfloor n^{\beta_1/2}/2\rfloor\), we get \[\begin{align} &\mathbb{P}\Big( \widetilde{\ell}(\infty,0)\ge \Lambda_n,\; 0\in S[m,\infty) \Big)\\ \le\,& \mathbb{P}\Big( \widetilde{\ell}(\infty,0)\ge \Lambda_n,\; \widetilde{\ell}\big(\lfloor n^{\beta_1/2}/2\rfloor,0\big)<\Lambda_n \Big) + \mathbb{P}\Big( \widetilde{\ell}\big(\lfloor n^{\beta_1/2}/2\rfloor,0\big)\ge \Lambda_n,\; 0\in S[m,\infty) \Big). \end{align}\] The first term is bounded by \(n^{-(\beta+\beta_1/4)+o(1)}\) by Lemma 11. For the second term, the Markov property gives \[\begin{align} &\mathbb{P}\Big( \widetilde{\ell}\big(\lfloor n^{\beta_1/2}/2\rfloor,0\big)\ge \Lambda_n,\; 0\in S[m,\infty) \Big)\\ \le\,& \mathbb{P}\bigl(\widetilde{\ell}(\infty,0)\ge \Lambda_n\bigr)\, \sup_{x\in\mathbb{Z}^d} \mathbb{P}\Bigl(x\in S\bigl[m-\lfloor n^{\beta_1/2}/2\rfloor,\infty\bigr)\Bigr)\\ \overset{\eqref{eq:late95hit}}{=}& n^{-(\beta+(d/2-1)\beta_1/2)+o(1)} \le n^{-(\beta+\beta_1/4)+o(1)}. \end{align}\] This proves the lemma. ◻
Proof of 30 . For \(i\in[1,K]\), let \[\mathbb{J}_i:=\bigl\{J\subset\{i,\ldots,K\}:\;\#J\le M,\;\inf J=i,\;J \text{ is not an interval}\bigr\}.\] We first record the uniform estimate corresponding to a fixed non-interval index set \(J\).
Lemma 12. Uniformly for \(i\in[1,K]\), \(J\in\mathbb{J}_i\), and \(k\in I_i\), the following holds. If \(J\) admits the interval decomposition \[\begin{align} \label{eq:J95decomposition} J=[i,i_1]\cup[i_2,i_3]\cup\cdots\cup[i_{2m},i_{2m+1}], \end{align}\qquad{(23)}\] where \(i=i_0\le i_1< i_2\le i_3<\cdots<i_{2m}\le i_{2m+1}\), and \(i_{2r}-i_{2r-1}\ge 2\) for \(r=1,\dots,m\), then \[\begin{align} \label{eq:fixed95J} \mathbb{Q}\Bigl( \widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=J \Bigr) \lesssim n^{-(\beta+(d/2)\beta_2-\beta_1)+o(1)} \prod_{r=1}^{m}\frac{1}{(i_{2r}-i_{2r-1}-1)^{d/2}}. \end{align}\qquad{(24)}\]
Assuming Lemma 12, we obtain \[\begin{align} &\sum_{k\in I_i} \mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\; \mathcal{J}_{S_k}\text{ is not an interval} \Bigr)\\ \le\;& \sum_{J\in\mathbb{J}_i}\sum_{k\in I_i} \mathbb{Q}\Bigl( \widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=J \Bigr)\\ \lesssim\;& \lfloor n^{\beta_1}\rfloor\, n^{-(\beta+(d/2)\beta_2-\beta_1)+o(1)} \sum_{\substack{J\in\mathbb{J}_i:\\ J\text{ has decomposition }\eqref{eq:J95decomposition}}} \prod_{r=1}^{m}\frac{1}{(i_{2r}-i_{2r-1}-1)^{d/2}}. \end{align}\] We now need the following elementary summation estimate.
Lemma 13. For every fixed \(i\in[1,K]\), \[\begin{align} \label{eq:summation} \sum_{\substack{J\in\mathbb{J}_i:\\ J\text{ has decomposition }\eqref{eq:J95decomposition}}} \prod_{r=1}^{m}\frac{1}{(i_{2r}-i_{2r-1}-1)^{d/2}} <\infty. \end{align}\qquad{(25)}\]
Assuming Lemma 13, we conclude that \[\begin{align} &\sum_{k\in I_i} \mathbb{Q}\Bigl( S_k\notin S[0,k-1],\;\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\; \mathcal{J}_{S_k}\text{ is not an interval} \Bigr)\\ =\,& n^{-(\beta-\beta_1)-((d/2)\beta_2-\beta_1)+o(1)}. \end{align}\] This completes the proof. ◻
Proof of Lemma 12. For an index set \(J\) with decomposition ?? , define \[J_r:=\bigcup_{\ell\in[i_{2r},i_{2r+1}]} I_\ell, \qquad r=0,\dots,m,\] where \(i_0:=i\). We first record the following blockwise estimate.
Lemma 14. Uniformly for \(i\in[1,K]\), \(k\in I_i\), an index set \(J\) with decomposition ?? , and \(t_0,\dots,t_m\in\mathbb{R}_+\), we have \[\label{eq:Jr} \begin{align} &\mathbb{Q}\Bigl( \widetilde{\ell}(J_r,S_k)\ge t_r\;\forall\,r=0,\ldots,m,\;\mathcal{J}_{S_k}=J \Bigr)\\ \lesssim\,& \left[ \prod_{r=1}^{m} \frac{n^{\beta_1}}{\bigl((i_{2r}-i_{2r-1}-1)n^{\beta_2}\bigr)^{d/2}} \right] \exp\Bigl\{-\gamma\sum_{r=0}^{m}t_r\Bigr\}. \end{align}\qquad{(26)}\]
Assuming Lemma 14, we now complete the proof of Lemma 12. Choose \(\varepsilon>0\) so small that \(\lfloor 1/\varepsilon\rfloor\ge M\). Since \(J\) contains at most \(M\) blocks, that is, \(m+1\le M\), it follows that on the event \(\bigl\{\widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=J\bigr\},\) there exists \((q_0,\dots,q_m)\in\mathbb{N}_0^{m+1}\) such that \[\sum_{r=0}^{m}q_r=\lfloor 1/\varepsilon\rfloor-M\qquad\text{and}\qquad \widetilde{\ell}(J_r,S_k)\ge q_r\varepsilon\Lambda_n,\qquad r=0,\dots,m.\] Hence, by the union bound, \[\begin{align} \mathbb{Q}\Bigl( \widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=J \Bigr)\le \sum_{\substack{(q_0,\dots,q_m)\in\mathbb{N}_0^{m+1}:\\ \sum_{r=0}^{m}q_r=\lfloor 1/\varepsilon\rfloor-M}} \mathbb{Q}\Bigl( \widetilde{\ell}(J_r,S_k)\ge q_r\varepsilon\Lambda_n\;\forall r,\;\mathcal{J}_{S_k}=J \Bigr). \end{align}\] Applying Lemma 14, we obtain \[\begin{align} &\mathbb{Q}\Bigl( \widetilde{\ell}(\widehat n,S_k)\ge \Lambda_n,\;\mathcal{J}_{S_k}=J \Bigr)\\ \lesssim\,& n^{-(\beta-c\varepsilon)} \left[ \prod_{r=1}^{m} \frac{n^{\beta_1}}{\bigl((i_{2r}-i_{2r-1}-1)n^{\beta_2}\bigr)^{d/2}} \right]\\ &\cdot\#\Bigl\{(q_0,\dots,q_m)\in\mathbb{N}_0^{m+1}:\sum_{r=0}^{m}q_r=\lfloor 1/\varepsilon\rfloor-M\Bigr\}. \end{align}\] Since \(\#\Bigl\{(q_0,\dots,q_m)\in\mathbb{N}_0^{m+1}:\sum_{r=0}^{m}q_r=\lfloor 1/\varepsilon\rfloor-M\Bigr\} \le (1/\varepsilon)^{M+1},\) letting \(\varepsilon\downarrow0\) proves ?? . ◻
Proof of Lemma 14. Let \(J\in\mathbb{J}_i\) admit the same interval decomposition as in ?? . Then for any \(k\in I_i\), \[\begin{align} &\bigl\{\widetilde{\ell}(J_r,S_k)\ge t_r \;\forall\, r\in[0,m],\;\mathcal{J}_{S_k}=J\bigr\} \\ &\subset \bigcup_{(k_r)_{r=1}^m\in \prod_{r=1}^m I_{i_{2r}}} \bigl\{ \widetilde{\ell}([k_r,I_{i_{2r+1}}^{+}],S_{k_r})\ge t_r \;\forall\, r\in[0,m],\; S_{k_r}=S_k \;\forall\, r=1,\dots,m \bigr\}, \end{align}\] where \(k_0:=k\). Consequently, \[\begin{align} \label{eq:union} \begin{aligned} &\mathbb{Q}\bigl( \widetilde{\ell}(J_r,S_k)\ge t_r \;\forall\, r\in[0,m],\;\mathcal{J}_{S_k}=J \bigr) \\ \le\;& \sum_{(k_r)_{r=1}^m\in \prod_{r=1}^m I_{i_{2r}}} \mathbb{Q}\bigl( \widetilde{\ell}([k_r,I_{i_{2r+1}}^{+}],S_{k_r})\ge t_r \;\forall\, r\in[0,m],\; S_{k_r}=S_k \;\forall\, r=1,\dots,m \bigr). \end{aligned} \end{align}\tag{31}\] Applying ?? with the above \(i\) and \(k\), and iteratively for \(j=i_{2m}, i_{2(m-1)},\ldots, i_2\), we get \[\begin{align} &\mathbb{Q}\bigl( \widetilde{\ell}([k_r,I^+_{i_{2r+1}}],S_{k_r})\ge t_r \;\forall\, r\in[0,m],\; S_{k_r}=S_k \;\forall\, r=1,\dots,m \bigr)\\ \le\,& \frac{1}{\big((i_{2m}-i_{2m-1}-1)n^{\beta_2}\big)^{d/2}}\\ &\cdot \mathbb{Q}\bigl( \widetilde{\ell}([k_r,I^+_{i_{2r+1}}],S_{k_r})\ge t_r \;\forall\, r\in[0,m],\; S_{k_r}=S_k \;\forall\, r=1,\dots,m-1 \bigr)\\ \le\,&\cdots\le \left[ \prod_{r=1}^{m} \frac{1}{\bigl((i_{2r}-i_{2r-1}-1)n^{\beta_2}\bigr)^{d/2}} \right] \mathbb{Q}\bigl( \widetilde{\ell}([k_r,I^+_{i_{2r+1}}],S_{k_r})\ge t_r \;\forall\, r\in[0,m] \bigr). \end{align}\] Moreover, \[\begin{align} &\mathbb{Q}\bigl( \widetilde{\ell}([k_r,I^+_{i_{2r+1}}],S_{k_r})\ge t_r \;\forall\, r\in[0,m] \bigr)\\ \overset{\eqref{eq:QJ}}{=}&\mathbb{Q}_J\bigl( \widetilde{\ell}([k_r,I^+_{i_{2r+1}}],S_{k_r})\ge t_r \;\forall\, r\in[0,m] \bigr)\\ \le\,& \big(1+o(1)\big)\mathbb{P}\bigl( \widetilde{\ell}([k_r,I^+_{i_{2r+1}}],S_{k_r})\ge t_r \;\forall\, r\in[0,m] \bigr)\\ \le\,& \prod_{r=0}^m \mathbb{P}\bigl( \widetilde{\ell}(\infty,0)\ge t_r\big)=\exp\Bigl\{-\gamma\sum_{r=0}^{m}t_r\Bigr\}. \end{align}\] Combining the above bounds, using that \(\#\Bigl(\prod_{r=1}^m I_{i_{2r}}\Bigr)=(n^{\beta_1})^m,\) and substituting into 31 , we obtain ?? . ◻
Proof of Lemma 13. Let \(J\in\mathbb{J}_i\) admit the same interval decomposition as ?? . For each \(1\le r\le m\), define the gap length \(g_r:=i_{2r}-i_{2r-1}-1\ge1,\) and the block lengths \[s_0:=i_1-i+1, \qquad s_r:=i_{2r+1}-i_{2r}+1\ge1,\quad 1\le r\le m.\] Since \(\#J\le M\), we have \(\sum_{r=0}^{m}s_r\le M.\) Thus the tuple \((m,s_0,\dots,s_m)\) can take only finitely many values. For each fixed such tuple, summing over all possible gap sequences \((g_1,\dots,g_m)\) yields \[\sum_{g_1=1}^{\infty}\cdots\sum_{g_m=1}^{\infty} \prod_{r=1}^{m}\frac{1}{g_r^{d/2}} = \left(\sum_{g=1}^{\infty}\frac{1}{g^{d/2}}\right)^m <\infty,\] since \(d \ge 3\). This proves ?? . ◻