Sesión Estadística, Probabilidad y Ciencias de DatosTightness of the Cover Time of Wired Planar Domains
Santiago Saglietti
Pontificia Universidad Católica de Chile, Chile - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
We consider a continuous time simple random walk (CTSRW) on a finite subset of the square lattice with wired boundary conditions. This CTSRW transitions at edge rate 1 on the graph obtained from the (lattice) closure of this subset by contracting its boundary into one vertex. We study the cover time of such a walk, namely the time it takes for the walk to visit all vertices in the graph. Taking a sequence of subsets obtained as scaled lattice versions of a nice planar domain, we show that the square root of the cover time, when normalized by the size of the corresponding subset, is tight around an explicit function f(N), where N denotes the scale parameter. Our proof is based on a delicate comparison of the local time field of the walk with the extremal landscape of the discrete Gaussian Free Field on the same subset combined with sharp asymptotics for the coupon collector problem. Joint work with Oren Louidor.