Uniformity of the late points of random walk on Z_n^d for d >= 3

TL;DR

This paper investigates the distribution of unvisited points in a high-dimensional random walk on a finite grid, revealing a phase transition in uniformity and distribution of late points as the walk approaches cover time.

Contribution

It establishes the asymptotic behavior of unvisited sites, showing a phase transition in their distribution from uniform to independent Bernoulli variables near the cover time.

Findings

For large n, the distribution of unvisited sites converges to Bernoulli variables with specific success probabilities.

A phase transition occurs at certain thresholds, changing the distribution of late points from uniform to independent.

The results quantify the uniformity of late points in high-dimensional random walks near the cover time.

Abstract

Suppose that $X$ is a simple random walk on $\Z_n^d$ for $d \geq 3$ and, for each $t$, we let $\U(t)$ consist of those $x \in \Z_n^d$ which have not been visited by $X$ by time $t$. Let $\tcov$ be the expected amount of time that it takes for $X$ to visit every site of $\Z_n^d$. We show that there exists $0 < \alpha_0(d) \leq \alpha_1(d) < 1$ and a time $t_* = \tcov(1+o(1))$ as $n \to \infty$ such that the following is true. For $\alpha > \alpha_1(d)$ (resp.\ $\alpha < \alpha_0(d)$), the total variation distance between the law of $\U(\alpha t_*)$ and the law of i.i.d.\ Bernoulli random variables indexed by $\Z_n^d$ with success probability~$n^{-\alpha d}$ tends to~$0$ (resp.\ $1$) as $n \to \infty$. Let $\tau_\alpha$ be the first time $t$ that $|\U(t)| = n^{d-\alpha d}$. We also show that the total variation distance between the law of $\U(\tau_\alpha)$ and the law of a uniformly chosen set from $\Z_n^d$ with size $n^{d-\alpha d}$ tends to $0$ (resp.\ $1$) for $\alpha > \alpha_1(d)$ (resp.\ $\alpha < \alpha_0(d)$) as $n \to \infty$.