Session S04 - Random Walks and Related Topics

Thursday, July 15, 13:40 UTC-3

Exceptional points of random walks in planar domains

Marek Biskup

UCLA, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

I will consider exceptional sets of associated with the local time of the simple random walk in finite subsets of the square lattice. These sets approximate a nice bounded continuum planar domain in the scaling limit; the walk moves as the ordinary simple random walk inside the domain and, whenever it exits, it returns via a uniformly-chosen boundary edge in the next step. For the walk run up to a positive multiple of the cover time, I will show that the sets of suitably defined thick and thin points as well as the set of avoided (a.k.a. late) points are asymptotically 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” with non-trivial corrections to the limit law arising in the conversion to the actual time.

Joint work with Yoshihiro Abe (Chiba University, Japan) and Sangchul Lee (UCLA, United States).

View abstract PDF