Chen--Stein Method for the Uncovered Set of Random Walk on $\mathbb Z_n^d$ for $d \ge 3$

TL;DR

This paper simplifies the proof of a previous result on the uncovered set of a random walk on high-dimensional tori, using the Chen--Stein method, and refines the asymptotic behavior of a key constant as the dimension grows.

Contribution

The authors provide a simplified proof of the asymptotic behavior of the uncovered set in high dimensions and improve the constant's limit from 1 to 3/4.

Findings

Simplified proof of the uncovered set distribution using Chen--Stein method.

Established the limit of the constant (d) as dimension increases to 3/4.

Applied concentration results for Markov chains to analyze random walk coverage.

Abstract

Let $X$ be a simple random walk on $\mathbb{Z}_n^d$ with $d\geq 3$ and let $t_{\rm{cov}}$ be the expected cover time. We consider the set of points $\mathcal{U}_\alpha$ of $\mathbb{Z}_n^d$ that have not been visited by the walk by time $\alpha t_{\rm{cov}}$ for $\alpha\in (0,1)$. It was shown in [MS17] that there exists $\alpha_1(d)\in (0,1)$ such that for all $\alpha>\alpha_1(d)$ the total variation distance between the law of the set $\mathcal{U}_\alpha$ and an i.i.d. sequence of Bernoulli random variables indexed by $\mathbb{Z}_n^d$ with success probability $n^{-\alpha d}$ tends to $0$ as $n \to \infty$. In [MS17] the constant $\alpha_1(d)$ converges to $1$ as $d\to\infty$. In this short note using the Chen--Stein method and a concentration result for Markov chains of Lezaud we greatly simplify the proof of [MS17] and find a constant $\alpha_1(d)$ which converges to $3/4$ as $d\to\infty$.