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

Abstract

Let $X$ be a simple random walk on $\mathbb {Z}_{n}^{d}$ with $d\geq 3$ and let $t_{\mathrm {cov}}$ be the expected cover time. We consider the set $\mathcal {U}_{\alpha }$ of points of $\mathbb {Z}_{n}^{d}$ that have not been visited by the walk by time $\alpha t_{\mathrm {cov}}$ for $\alpha \in (0,1)$. It was shown in [6] 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 [6] 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 [5] we greatly simplify the proof of [6] and find a constant $\alpha _{1}(d)$ which converges to $3/4$ as $d\to \infty $. We prove analogous results for the high points of the Gaussian free field.