Maximum and minimum of local times for two-dimensional random walk. (English) Zbl 1321.60151

Summary: 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 E. Bolthausen et al. [Ann. Probab. 29, No. 4, 1670–1692 (2001; Zbl 1034.82018)] and O. Daviaud [Ann. Probab. 34, No. 3, 962–986 (2006; Zbl 1104.60062)], but we have different exponents from the case of the Gaussian free field.


60J55 Local time and additive functionals
60G70 Extreme value theory; extremal stochastic processes
60G50 Sums of independent random variables; random walks
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60G60 Random fields
60G15 Gaussian processes
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