Marek Biskup

Exceptional points of planar random walks

I will discuss exceptional sets of 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 enters a ``boundary vertex’’ and then returns back to the domain 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 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 conclusions are cleanest when the walk is parametrized by the local time spent at the “boundary vertex”; non-trivial corrections to the limit law arise in the conversion to the actual time. Based on joint papers with Yoshihiro Abe and Sangchul Lee as well as papers with Oren Louidor on the exceptional points of the discrete Gaussian Free Field.