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

Abe, Yoshihiro

Electron. Commun. Probab. 20, Paper No. 22, 14 p. (2015).

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.

MSC:

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

Keywords:

local times; two-dimensional random walks; extrema; Gaussian free fields

Citations:

Zbl 1034.82018; Zbl 1104.60062