LA Probability Forum

Let X be a simple random walk in $\mathbb{Z}_n^d$ with $d\geq 3$ and let $t_{\rm{cov}}$ be the expected time it takes for X to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set $\mathcal{L}_\alpha$ of points that have not been visited by time $\alpha t_{\rm{cov}}$ and prove that it exhibits a phase transition: there exists $\alpha_*$ so that for all $\alpha>\alpha_*$ and all $\epsilon>0$ there exists a coupling between $\mathcal{L}_\alpha$ and two i.i.d. Bernoulli sets $\mathcal{B}^{\pm}$ on the torus with parameters $n^{-(a\pm\epsilon)d}$ with the property that $\mathcal{B}^-\subseteq \mathcal{L}_\alpha\subseteq \mathcal{B}^+$ with probability tending to 1 as $n\to\infty$. When $\alpha\leq \alpha_*$, we prove that there is no such coupling.