20/05/2026, 17:00 — 18:00 Europe/Lisbon —
Online
Santiago Saglietti, Pontificia Universidad Católica de Chile
A Limit in Law for the Cover Time and Last Visited Vertex of Wired Planar Domains
We derive a scaling limit in law as N tends to infinity for the cover time by a simple random walk of the subgraph of the square lattice obtained by discretizing an N-scale blow up of a planar domain and adding a wired boundary. The limiting distribution is that of a Gumbel random variable shifted by an independent (random) quantity which is equal to the full mass of a variant of the critical Liouville Quantum Gravity Measure on the same domain. We also derive a limit in law for the rescaled location of the last visited vertex by the walk. Here the limit turns out to be precisely the (expected) critical Liouville Measure, normalized by its total mass. Both limits hold jointly with the limiting joint law explicitly described. These results resolve well-known open problems in the field, in the case of wired boundary conditions. The proof is based on comparison with the extremal landscape of the discrete Gaussian Free Field and, in particular, with that of the discrete Gaussian Free Field conditioned to have zero average. Joint work with Oren Louidor (Technion).
Senha do zoom: 359751
