A Limit in Law for the Cover Time of Wired Planar Domains

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 Gravity Measure associated with the continuous Gaussian free field on the same domain. This resolves a well known open problem 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. Joint work with Santiago Saglietti (Catholic University, Santiago Chile),