Exceptional points of discrete-time random walks in planar domains

Abstract

Given a sequence of lattice approximations $D_N\subset {\mathbb{Z}}^2$ of a bounded continuum domain $D\subset {\mathbb{R}}^2$ with the vertices outside $D_N$ fused together into one boundary vertex ϱ, we consider discrete-time simple random walks on $D_N\cup \{\mathit{\varrho }\}$ run for a time proportional to the expected cover time and describe the scaling limit of the exceptional level sets of the thick, thin, light and avoided points. We show that these are distributed, up a spatially-dependent log-normal factor, as the zero-average Liouville Quantum Gravity measures in D. The limit law of the local time configuration at, and nearby, the exceptional points is determined as well. The results extend earlier work by the first two authors who analyzed the continuous-time problem in the parametrization by the local time at ϱ. A novel uniqueness result concerning divisible random measures and, in particular, Gaussian Multiplicative Chaos, is derived as part of the proofs.