Phase transition for the late points of random walk

Abstract

Let $X$ be a random walk on the torus of side length $N$ in dimension $d\ge 3$ with uniform starting point, and $t_{\text{cov}}$ be the expected value of its cover time, which is the first time that $X$ has visited every vertex of the torus at least once. For $\alpha >0$, the set ${\mathcal{L}}^{\alpha }$ of $\alpha$-late points consists of those points not visited by $X$ at time $\alpha t_{\text{cov}}$. We prove the existence of a value ${\alpha }_{\ast }\in (\frac{1}{2},1)$ across which ${\mathcal{L}}^{\alpha }$ trivialises as follows: for all $\alpha >{\alpha }_{\ast }$ and $\epsilon \ge N^{-c}$ there exists a coupling of ${\mathcal{L}}^{\alpha }$ and two occupation sets ${\mathcal{B}}^{{\alpha }_{\pm }}$ of i.i.d. Bernoulli fields having the same density as ${\mathcal{L}}^{\alpha \pm \epsilon }$, which is asymptotic to $N^{-(\alpha \pm \epsilon )d}$, with the property that the inclusion ${\mathcal{B}}^{{\alpha }_{+}}\subseteq {\mathcal{L}}^{\alpha }\subseteq {\mathcal{B}}^{{\alpha }_{-}}$ holds with high probability as $N\to \infty$. On the contrary, when $\alpha \le {\alpha }_{\ast }$ there is no such coupling. Corresponding results also hold for the vacant set of random interlacements at high intensities. The transition at ${\alpha }_{\ast }$ corresponds to the (dis-)appearance of `double-points' (i.e. neighboring pairs of points) in ${\mathcal{L}}^{\alpha }$. We further describe the law of ${\mathcal{L}}^{\alpha }$ for $\alpha >\frac{1}{2}$ by adding independent patterns to ${\mathcal{B}}^{{\alpha }_{\pm }}$. In dimensions $d\ge 4$ these are exactly all two-point sets. When $d=3$ one must also include all connected three-point sets, but no other.

68 pages, 2 figures