Uniformity of the late points of random walk on Z_n^d for 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 $\text{U}(t)$ consist of those $x\in {\mathbb{Z}}_n^d$ which have not been visited by $X$ by time $t$. Let $\text{tcov}$ 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 $0 < \alpha_0(d) \leq \alpha_1(d) < 1$andatime$t_* = \tcov(1+o(1))$as$n \to \infty$suchthatthefollowingistrue\mathrm{.}For$\alpha > \alpha_1(d)(resp.\\alpha < \alpha_0(d)$),thetotalvariationdistancebetweenthelawof$\U(\alpha t_*)$andthelawofi\mathrm{.}i\mathrm{.}d\mathrm{.}Bernoullirandomvariablesindexedby$\Z_n^d$withsuccessprobability$n^{-\alpha d}$tendsto$0(resp.\1$)as$n \to \infty$\mathrm{.}Let$\tau_\alpha$bethefirsttime$t$that$|\U(t)| = n^{d-\alpha d}$. We also show that the total variation distance between the law of $\text{U}({\tau }_{\alpha })$ 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 $n\to \mathrm{\infty }$.