Exceptional points of two-dimensional random walks at multiples of the cover time

Abstract

We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by N) versions $$D_N\subseteq {\mathbb {Z}}^2$$ of bounded open domains $$D\subseteq {\mathbb {R}}^2$$ . Upon exit from $$D_N$$ , the walk lands on a “boundary vertex” and then reenters $$D_N$$ through a random boundary edge in the next step. In the parametrization by the local time at the “boundary vertex” we prove that, at times corresponding to a $$\theta $$ -multiple of the cover time of $$D_N$$ , the sets of suitably defined $$\lambda $$ -thick (i.e., heavily visited) and $$\lambda $$ -thin (i.e., lightly visited) points are, as $$N\rightarrow \infty $$ , distributed according to the Liouville Quantum Gravity $$Z^D_\lambda $$ with parameter $$\lambda $$ -times the critical value. For $$\theta <1$$ , also the set of avoided vertices (a.k.a. late points) and the set where the local time is of order unity are distributed according to $$Z^D_{\sqrt{\theta }}$$ . The local structure of the exceptional sets is described as well, and is that of a pinned Discrete Gaussian Free Field for the thick and thin points and that of random-interlacement occupation-time field for the avoided points. The results demonstrate universality of the Gaussian Free Field for these extremal problems.