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

We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by N ) versions D N ⊆ Z 2 of bounded open domains D ⊆ 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 θ -multiple of the cover time of D N , the sets of suitably defined λ -thick (i.e., heavily visited) and λ -thin (i.e., lightly visited) points are, as N → ∞ , distributed according to the Liouville Quantum Gravity Z λ D with parameter λ -times the critical value. For θ < 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 θ D . 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.