A limit law for the cover time of the two-dimensional discrete torus

Abstract

We determine the limiting distribution of the cover time of simple random walk on the two-dimensional discrete torus. For the continuous-time walk with total jump rate one on $(\mathbb Z/N\mathbb Z)^2$, let $T_N$ denote its cover time. We prove that $\frac{T_N}{(2/\pi)N^2\log N}-2\log N+\log\log N \Longrightarrow G+\log(\kappa Z)$, where $G$ is a standard Gumbel random variable, $Z$ is the total mass of the critical Gaussian multiplicative chaos associated with the zero-average Gaussian free field on the unit torus, $G$ and $Z$ are independent, and $\kappa>0$ is deterministic. This answers the limit-law question suggested by Aldous and recorded by Dembo, Peres, Rosen and Zeitouni. The proof identifies the random fluctuations in the number of small, well-separated unvisited components at a deterministic time before coverage, and then estimates the time needed to visit the remaining components.