Journal Article

FZJ-2017-07803

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png How fast does a random walk cover a torus?

Grassberger, P. (Corresponding author)FZJ*

2017
Inst. Woodbury, NY

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Please use a persistent id in citations: http://hdl.handle.net/2128/15984  doi:10.1103/PhysRevE.96.012115

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.

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Contributing Institute(s):

1. Jülich Supercomputing Center (JSC)

Research Program(s):

1. 511 - Computational Science and Mathematical Methods (POF3-511) (POF3-511)

Appears in the scientific report 2017

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Database coverage:
; ; ; Current Contents - Physical, Chemical and Earth Sciences ; Ebsco Academic Search ; IF < 5 ; JCR ; SCOPUS ; Science Citation Index ; Science Citation Index Expanded ; Thomson Reuters Master Journal List ; Web of Science Core Collection

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