How fast does a random walk cover a torus?

Abstract

We present high statistics simulation data for the average time 〈T_{cover}(L)〉 that a random walk needs to cover completely a two-dimensional torus of size L×L. They confirm the mathematical prediction that 〈T_{cover}(L)〉∼(LlnL)^{2} for large L, but the prefactor seems to deviate significantly from the supposedly exact result 4/π derived by Dembo et al. [Ann. Math. 160, 433 (2004)ANMAAH0003-486X10.4007/annals.2004.160.433], if the most straightforward extrapolation is used. On the other hand, we find that this scaling does hold for the time T_{N(t)=1}(L) at which the average number of yet unvisited sites is 1, as also predicted previously. This might suggest (wrongly) that 〈T_{cover}(L)〉 and T_{N(t)=1}(L) scale differently, although the distribution of rescaled cover times becomes sharp in the limit L→∞. But our results can be reconciled with those of Dembo et al. by a very slow and nonmonotonic convergence of 〈T_{cover}(L)〉/(LlnL)^{2}, as had been indeed proven by Belius et al. [Probab. Theory Relat. Fields 167, 461 (2017)10.1007/s00440-015-0689-6] for Brownian walks, and was conjectured by them to hold also for lattice walks.