How fast does a random walk cover a torus?

Peter Grassberger

Index: 10.1103/PhysRevE.96.012115

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Abstract

We present high statistics simulation data for the average time 〈Tcover(L)〉 that a random walk needs to cover completely a two-dimensional torus of size L×L. They confirm the mathematical prediction that 〈Tcover(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)], if the most straightforward extrapolation is used. On the other hand, we find that this scaling does hold for the time TN(t)=1(L) at which the average number of yet unvisited sites is 1, as also predicted previously. This might suggest (wrongly) that 〈Tcover(L)〉 and TN(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 〈Tcover(L)〉/(LlnL)2, as had been indeed proven by Belius et al. [Probab. Theory Relat. Fields 167, 461 (2017)] for Brownian walks, and was conjectured by them to hold also for lattice walks.