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

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/π)N^2\log N}-2\log N+\log\log N \Lo