%0 Journal Article
%A Grassberger, Peter
%T How fast does a random walk cover a torus?
%J Physical review / E
%V 96
%N 1
%@ 2470-0045
%C Woodbury, NY
%I Inst.
%M FZJ-2017-07803
%P 012115
%D 2017
%X 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.
%F PUB:(DE-HGF)16
%9 Journal Article
%$ pmid:29347167
%U <Go to ISI:>//WOS:000405194200005
%R 10.1103/PhysRevE.96.012115
%U https://juser.fz-juelich.de/record/840250