Chen--Stein method for the uncovered set of random walk on $\mathbb{Z}_{n}^{d}$ for $d \geq 3$

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 $\mathcal{U}_\alpha$ 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 $d\to\infty$.