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

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 $\mathcal{U}_α$ of $\mathbb{Z}_n^d$ that have not been visited by the walk by time $αt_{\rm{cov}}$ for $α\in (0,1)$. It was shown in [MS17] that there exists $α_1(d)\in (0,1)$ such that for all $α>α_1(d)$ the total variation distance between the law of the set $\mathcal{U}_α$ and an i.i.d. sequence of Bernoulli random variables indexed by $\mathbb{Z}_n^d$ with success probability $n^{-αd}$ tends to $0$ as $n \to \infty$. In [MS17] the constant $α_1(d)$ 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 $α_1(d)$ which converges to $3/4$ as $d\to\infty$.

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