Maximum and minimum of local times for two-dimensional random walk

2014/10/21 by Yoshihiro Abe, Abe, Yoshihiro
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1410.5601

1 figure

arxiv created 2014/10/21 · arxiv updated 2014/10/22

Abstract

We obtain the leading orders of the maximum and the minimum of local times for the simple random walk on the two-dimensional torus at time proportional to the cover time. We also estimate the number of points with large (or small) values of the local times. These are analogues of estimates on the two-dimensional Gaussian free fields by Bolthausen, Deuschel, and Giacomin [Ann. Probab., 29 (2001)] and Daviaud [Ann. Probab., 34 (2006)], but we have different exponents from the case of the Gaussian free field.

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