How fast does a random walk cover a torus?

How fast does a random walk cover a torus? 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)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 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)10.1007/s00440-015-0689-6] for Brownian walks, and was conjectured by them to hold also for lattice walks. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review E American Physical Society (APS)

How fast does a random walk cover a torus?

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How fast does a random walk cover a torus?

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)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 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)10.1007/s00440-015-0689-6] for Brownian walks, and was conjectured by them to hold also for lattice walks.
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Publisher
The American Physical Society
Copyright
Copyright © ©2017 American Physical Society
ISSN
1539-3755
eISSN
550-2376
D.O.I.
10.1103/PhysRevE.96.012115
Publisher site
See Article on Publisher Site

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)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 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)10.1007/s00440-015-0689-6] for Brownian walks, and was conjectured by them to hold also for lattice walks.

Journal

Physical Review EAmerican Physical Society (APS)

Published: Jul 10, 2017

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