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

TL;DR

This paper investigates the distribution and local structure of exceptional points in two-dimensional random walks at multiples of the cover time, revealing universality with Liouville Quantum Gravity and Gaussian Free Field.

Contribution

It establishes the distribution of thick, thin, and avoided points of 2D random walks at cover time multiples, linking them to Liouville Quantum Gravity and Gaussian Free Field.

Findings

Thick and thin points follow Liouville Quantum Gravity distributions.

Avoided points and late points are distributed according to scaled Liouville Quantum Gravity.

Local structures resemble pinned Discrete Gaussian Free Field and random-interlacement occupation fields.

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.