Interlacement limit of a stopped random walk trace on a torus

We first introduce some notation used in this paper. In Section 1, we denoted the discrete torus $\mathbb{T}_{N}=[-N/2,N/2)^{d}\cap\mathbb{Z}^{d}$, $d\geq 3$ and the canonical projection map $\varphi_{N}:\mathbb{Z}^{d}\to\mathbb{T}_{N}$. From here on, we will omit the $N$-dependence and write $\varphi$ and $\tau_{\mathbf{x}}$ instead.

We write vertices and subsets of the torus in bold, i.e. $\mathbf{x}\in\mathbb{T}_{N}$ and $\mathbf{K}\subset\mathbb{T}_{N}$. In order to simplify notation, in the rest of the paper we abbreviate $\mathbf{K}=\tau_{\mathbf{x}}\varphi(K)$.

Let $(Y_{t})_{t\geq 0}$ be a discrete-time lazy simple random walk on $\mathbb{Z}^{d}$, that is,

We denote the corresponding lazy random walk on $\mathbb{T}_{N}$ by $(\mathbf{Y}_{t})_{t\geq 0}=(\varphi(Y_{t}))_{t\geq 0}$. Let $\mathbf{P}_{x^{\prime}}$ denote the distribution of the lazy random walk on $\mathbb{Z}^{d}$ started from $x^{\prime}\in\mathbb{Z}^{d}$, and write $\mathbf{P}_{\mathbf{x}}$ for the distribution of the lazy random walk on $\mathbb{T}_{N}$ started from $\mathbf{x}=\varphi(x^{\prime})\in\mathbb{T}_{N}$. We write $p_{t}(x^{\prime},y^{\prime})=\mathbf{P}_{x^{\prime}}[Y_{t}=y^{\prime}]$ for the $t$-step transition probability. Further notation we will use:

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  $L=mN$, where $L^{2}\sim AN^{d}$ as $N\to\infty$ for some constant $A\in(0,\infty)$

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  $D=(-m,m)^{d}$, rescaled box, indicates which copy of the torus the walk is in

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  $n=\lfloor N^{\delta}\rfloor$ for some $2<\delta<d$, long enough for the mixing property on the torus, but short compared to $L^{2}$

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  $\mathbf{x_{0}}\in\mathbf{K}$ is a fixed point of $\mathbf{K}$

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  we write points in the original lattice $\mathbb{Z}^{d}$ with a prime, such as $y^{\prime}$, and decompose a point $y^{\prime}$ as $yN+\mathbf{y}$ with $y$ in another lattice isomorphic to $\mathbb{Z}^{d}$ and $\mathbf{y}=\varphi(y^{\prime})\in\mathbb{T}_{N}$

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  $T=\inf\{t\geq 0:Y_{t}\not\in(-L,L)^{d}\}$, the first exit time from $(-L,L)^{d}$

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  $S=\inf\{\ell\geq 0:Y_{n\ell}\not\in(-L,L)^{d}\}$, so that the first multiple of $n$ when the rescaled point $Y_{n\ell}/N$ is not in $(-m,m)^{d}$ equals $S\cdot n$

We omit the dependence on $d$ and $N$ from some notation above for simplicity.

In this section, we collect some known results required for the proof of Theorem 1.1. We will rely heavily on the Local Central Limit Theorem (LCLT) [9, Chapter 2], with error term, and the Martingale maximal inequality [9, Eqn. (12.12) of Corollary 12.2.7]. We will also use Equation (6.31) in [9], that relates $\mathrm{Cap}(K)$ to the probability that a random walk started from the boundary of a large ball with radius $n$ hits the set $K$ before exiting the ball. In estimating some error terms in our arguments, sometimes we will use the Gaussian upper and lower bounds [5]. We also need to derive a lemma related to the mixing property on the torus [10, Theorem 5.6] to show that the starting positions of different stretches are not far from uniform on the torus; see Lemma 2.1.

We recall the LCLT from [9, Chapter 2]. The following is a specialisation of [9, Theorem 2.3.11] to lazy simple random walk. The covariance matrix $\Gamma$ and the square root $J^{*}(x)$ of the associated quadratic form are given by

where $I$ is the $(d\times d)$-unit matrix.

Let $\bar{p}_{t}(x^{\prime})$ denote the estimate of $p_{t}(x^{\prime})$ that one obtains by the LCLT, for lazy simple random walk. We have

The lazy simple random walk $(Y_{t})_{t\geq 0}$ in $\mathbb{Z}^{d}$ is aperiodic, irreducible with mean zero, finite second moment, and finite exponential moments. All joint third moments of the components of $Y_{1}$ vanish.

The Martingale maximal inequality in [9, Eqn. (12.12) of Corollary 12.2.7] is stated as follows. Let $(Y^{(i)}_{t})_{t\geq 0}$ denote the $i$-th coordinate of $(Y_{t})_{t\geq 0}$ ($1\leq i\leq d$). The standard deviation $\sigma$ of $Y^{(i)}_{1}$ is given by $\sigma^{2}=(2d)^{-1}$. For all $t\geq 1$ and all $r>0$ we have

Now we state the result of [9, Eqn. (6.31)]. Recall that $B_{r}(o)$ is the discrete ball centred at $o$ with radius $r$. Let for any subset $B\subset\mathbb{Z}^{d}$

Let $\partial B_{r}(o)=\{y^{\prime}\in\mathbb{Z}^{d}\setminus B_{r}(o):\text{$\exists x^{\prime}\in B_{r}(o)$ such that $|x^{\prime}-y^{\prime}|=1$}\}$. For a given finite set $K\subseteq\mathbb{Z}^{d}$, let $H_{K}$ denote the hitting time

Then we have

Here $\mathrm{Cap}(K)$ is the capacity of $K$; see [9, Section 6.5], which states the analogous statement for the simple random walk. Since we consider the lazy random walk, this introduces a factor of $1/2$.

In estimating some error terms in our arguments, sometimes we will use the Gaussian upper and lower bounds [5]: there exist constants $C=C(d)$ and $c=c(d)$ such that

Recall that the norm $|\cdot|$ refers to the Euclidean norm.

Regarding mixing times, recall that lazy simple random walk on the $N$-torus mixes in time $N^{2}$ [10, Theorem 5.6]. With this in mind we derive the following simple lemma.

Recall that $2<\delta<d$ and $n=\lfloor N^{\delta}\rfloor$.

In this section we state some propositions to be used in Section 3 to prove Theorem 1.1. The propositions will be proved in Section 4.

The strategy of the proof is to consider stretches of length $n$ of the walk, and estimate the small probability in each stretch that $\mathbf{K}$ is hit by the projection. What makes this strategy work is that we can estimate, conditionally on the starting and end-points of a stretch, the probability that $\mathbf{K}$ is hit, and this event asymptotically decouples from the number of stretches. The number of stretches will be the random variable $S$. Since $nS\approx T$, and $T$ is $\Theta(N^{d})$ in probability, we have that $S$ is $\Theta(N^{d}/n)$. In Lemma 2.2 below we show a somewhat weaker estimate for $S$ (which suffices for our needs).

The main part of the proof will be to show that during a fixed stretch, $\mathbf{K}$ is not hit with probability

Heuristically, conditionally on $S$ this results in the probability

and we will conclude by showing that $(n/N^{d})S$ converges in distribution to a constant multiple of the Brownian exit time $\sigma_{1}$.

The factor $n/N^{d}$ in (6) arises as the expected time spent by the projected walk at a fixed point of the torus during a given stretch. The capacity term arises as we pass from expected time to hitting probability.

For the above approach to work, we need a small probability event on which the number of stretches or end-points of stretches are not sufficiently well-behaved. First, we will need to restrict to realizations where $(\sqrt{\log\log n})^{-1}(N^{d}/n)\leq S\leq\log N(N^{d}/n)$, which occurs with high probability as $N\to\infty$ (see Lemma 2.2 below). Second, suppose that the $\ell$-th stretch starts at the point $y^{\prime}_{\ell-1}$ and ends at the point $y^{\prime}_{\ell}$, that is, $y^{\prime}_{\ell-1}$ and $y^{\prime}_{\ell}$ are realizations of $Y_{(\ell-1)n}$ and $Y_{\ell n}$. In order to have a good estimate of the probability that $\mathbf{K}$ is hit during this stretch, we will need to impose a few conditions on $y^{\prime}_{\ell-1}$ and $y^{\prime}_{\ell}$. One of these is that the displacement $|y^{\prime}_{\ell}-y^{\prime}_{\ell-1}|$ is not too large: we will require that for all stretches, it is at most $f(n)\sqrt{n}$, for a function to be chosen later that increases to infinity with $N$. We will be able to choose $f(n)$ of the form $f(n)=C_{1}\sqrt{\log N}$ in such a way that this restriction holds for all stretches with high probability. A third condition we need to impose, that will also hold with high probability, is that $y^{\prime}_{\ell-1}$ is at least a certain distance $N^{\zeta}$ from $\varphi^{-1}(\mathbf{K})$ for a parameter $0<\zeta<1$ (this will only be required for $\ell\geq 1$, and is not needed for the first stretch starting with $y^{\prime}_{0}=o$). The reason we need this is to be able to appeal to (4) to extract the $\mathrm{Cap}(\mathbf{K})$ contribution, when we know that $\mathbf{K}$ is hit from a long distance (we will take $r=N^{\zeta}$ in (4)). The larger the value of $\zeta$, the better error bound we get on the approach to $\mathrm{Cap}(\mathbf{K})$. On the other hand, $\zeta$ should not be too close to $1$, because we want the separation of $y^{\prime}_{\ell-1}$ from $\mathbf{K}$ to occur with high enough probability.

The set $\mathcal{G}_{\zeta,C_{1}}$ defined below represents realizations of $S$ and the sequence $Y_{n},Y_{2n},\dots,Y_{Sn}$ satisfying the above restrictions. Proposition 2.1 below implies that these restrictions hold with high probability. First, we will need $2<\delta<d$ to satisfy the inequality

This can be satisfied if $d\geq 3$ and $\delta$ is sufficiently close to $d$, say $\delta=\frac{7}{8}d$. Since the left-hand side of the left-hand inequality in (7) equals $(\delta/d)(d-2)$, we can subsequently choose $\zeta$ such that we also have

With the parameter $\zeta$ fixed satisfying the above, we now define:

where $f(n)=C_{1}\,\sqrt{\log N}$ and recall that we write $y^{\prime}_{\ell}=y_{\ell}N+\mathbf{y}_{\ell}$ and $y^{\prime}_{\ell-1}=y_{\ell-1}N+\mathbf{y}_{\ell-1}$, and we define $y^{\prime}_{0}=o$. The time $\tau$ is corresponding to a particular value of the exit time $S$, so $y_{\ell}\in D$ for $1\leq\ell<\tau$ and $y_{\tau}\notin D$. The parameter $C_{1}$ will be chosen in the course of the proof. See Figure 1 for a visual illustration of the sequence $y^{\prime}_{0},y^{\prime}_{1},\dots$ in the definition of $\mathcal{G}_{\zeta,C_{1}}$.

The next lemma shows that the restriction made on the time-parameter $\tau$ in the definition of $\mathcal{G}_{\zeta,C_{1}}$ holds with high probability for $S$.

The starting point for the proof of Theorem 1.1 is the following proposition that decomposes the probability we are interested in into terms involving single stretches of duration $n$.

Central to the proof of Theorem 1.1 is the following proposition, that estimates the probability of hitting a copy of $\mathbf{K}$ during a “good stretch” where the displacement $|y^{\prime}_{\ell}-y^{\prime}_{\ell-1}|$ is almost of order $\sqrt{n}$. This will not hold for all stretches with high probability, but the fraction of stretches for which it fails will vanish asymptotically.

In addition to the above proposition (that we prove in Section 4.2), we will need a weaker version for the remaining “bad stretches” that have less restriction on the distance $|y^{\prime}_{\ell}-y^{\prime}_{\ell-1}|$. This will be needed to estimate error terms arising from the “bad stretches”, and it will also be useful to demonstrate some of our proof ideas for other error terms arising later in the paper. It will be proved in Section 4.1.

Our final proposition is needed to estimate the number of stretches that are “bad”.