Exceptional points of random walks in planar domains
I will consider random walks in finite subsets of the square lattice that approximate nice bounded continuum planar domains in the scaling limit. The walk moves as the simple random walk inside the domain and, whenever it exits, it returns back via a uniformly-chosen boundary edge in the next step. Running the walk up to a positive multiple of the cover time, I will show that the scaling limits of various exceptional sets of the local time — specifically, the sets of suitably defined thick and thin points as well as the set of avoided (a.k.a. late) points — are distributed according to versions of the Liouville Quantum Gravity in the underlying continuum domain. The results are cleanest when the walk is parametrized by the local time spent at the “boundary vertex” with non-trivial corrections to the limit law arising in the conversion to the actual time. Based on joint papers with Yoshihiro Abe and Sangchul Lee.