How fast does a random walk cover a torus?

Abstract

We present high statistics simulation data for the average time $\langle T_{\rm cover}(L)\rangle$ that a random walk needs to cover completely a 2-dimensional torus of size $L\times L$. They confirm the mathematical prediction that $\langle T_{\rm cover}(L)\rangle \sim (L \ln L)^2$ for large $L$, but the prefactor {\it seems} to deviate significantly from the supposedly exact result $4/\pi$ derived by A. Dembo {\it et al.}, Ann. Math. {\bf 160}, 433 (2004), if the most straightforward extrapolation is used. On the other hand, we find that this scaling does hold for the time $ T_{\rm N(t)=1}(L)$ at which the average number of yet unvisited sites is 1, as also predicted previously. This might suggest (wrongly) that $\langle T_{\rm cover}(L)\rangle$ and $T_{\rm N(t)=1}(L)$ scale differently, although the distribution of rescaled cover times becomes sharp in the limit $L\to\infty$. But our results can be reconciled with those of Dembo {\it et al.} by a very slow and {\it non-monotonic} convergence of $\langle T_{\rm cover}(L)\rangle/(L \ln L)^2$, as had been indeed proven by Belius {\it et al.} [Prob. Theory \& Related Fields {\bf 167}, 1 (2014)] for Brownian walks, and was conjectured by them to hold also for lattice walks.