  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
$d\to\infty$.
