TY  - JOUR
AU  - Grassberger, Peter
TI  - How fast does a random walk cover a torus?
JO  - Physical review / E
VL  - 96
IS  - 1
SN  - 2470-0045
CY  - Woodbury, NY
PB  - Inst.
M1  - FZJ-2017-07803
SP  - 012115
PY  - 2017
AB  - 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.
LB  - PUB:(DE-HGF)16
C6  - pmid:29347167
UR  - <Go to ISI:>//WOS:000405194200005
DO  - DOI:10.1103/PhysRevE.96.012115
UR  - https://juser.fz-juelich.de/record/840250
ER  -