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 $N\to\infty$, 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.