Uniformity of the late points of random walk on $${\mathbb {Z}}_{n}^{d}$$ for $$d \ge 3$$d≥3

Suppose that X is a simple random walk on \({\mathbb {Z}}_n^d\) for \(d \ge 3\) and, for each t, we let \({\mathcal {U}}(t)\) consist of those \(x \in {\mathbb {Z}}_n^d\) which have not been visited by X by time t. Let \(t_{\mathrm {cov}}\) be the expected amount of time that it takes for X to visit every site of \({\mathbb {Z}}_n^d\). We show that there exists \(0 < \alpha _0(d) \le \alpha _1(d) < 1\) and a time \(t_* = t_{\mathrm {cov}}(1+o(1))\) as \(n \rightarrow \infty \) such that the following is true. For \(\alpha > \alpha _1(d)\) (resp. \(\alpha < \alpha _0(d)\)), the total variation distance between the law of \({\mathcal {U}}(\alpha t_*)\) and the law of i.i.d. Bernoulli random variables indexed by \({\mathbb {Z}}_n^d\) with success probability \(n^{-\alpha d}\) tends to 0 (resp. 1) as \(n \rightarrow \infty \). Let \(\tau _\alpha \) be the first time t that \(|{\mathcal {U}}(t)| = n^{d-\alpha d}\). We also show that the total variation distance between the law of \({\mathcal {U}}(\tau _\alpha )\) and the law of a uniformly chosen set from \({\mathbb {Z}}_n^d\) with size \(n^{d-\alpha d}\) tends to 0 (resp. 1) for \(\alpha > \alpha _1(d)\) (resp. \(\alpha < \alpha _0(d)\)) as \(n \rightarrow \infty \).