A Limit in Law for the Cover Time and Last Visited Vertex of Wired Planar Domains

TL;DR

This paper establishes the limiting distribution laws for the cover time and last visited vertex of a random walk on scaled planar domains with wired boundaries, linking them to Liouville Quantum Gravity measures.

Contribution

It provides the first rigorous derivation of the joint limiting distribution for cover time and last visited vertex in wired planar domains, connecting random walk behavior to Liouville Quantum Gravity.

Findings

Limit law for cover time as a Gumbel distribution shifted by a quantum gravity measure

Limit law for last visited vertex as the normalized Liouville measure

Joint distribution explicitly characterized and proven for wired boundary conditions

Abstract

We derive a scaling limit in law for the cover time of a simple random walk on a lattice version of a scaled-up planar domain with wired boundary conditions. The limiting distribution is that of a Gumbel Random Variable shifted randomly by an independent 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 scaled location of the vertex visited last by the walk. Here the limit turns out to be precisely the 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 a version there-of obtained by conditioning the average value of the field to be zero.