How uniformly a random walker covers a finite lattice

Harald Freund and Peter Grassberger

Physica A: Statistical Mechanics and its Applications, 1993, vol. 192, issue 3, 465-470

Abstract: We study the distribution of the number of visits a random walker makes at a given site on a finite lattice with N sites, during a very long walk which visits each site a large number of times. For regular hypercubic lattices in all dimensions we find normal central limit behavior, but with anomalously large variance in ⩽2 dimensions. In 2 dimensions, the ratio of the variance over the average number of visits increases logarithmically with N, while it increases ≈N in one dimension. We confront this with the case of self-repelling (also called “true self-avoiding”) walks. There, the variance remains bounded for all times, and increases logarithmically with N in 2 dimensions.

Date: 1993
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:192:y:1993:i:3:p:465-470

DOI: 10.1016/0378-4371(93)90048-9

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