Thick points of random walk and the Gaussian free field. (English) Zbl 1441.60036

Summary: We consider the thick points of random walk, i.e. points where the local time is a fraction of the maximum. In two dimensions, we answer a question of [A. Dembo et al., Acta Math. 186, No. 2, 239–270 (2001; Zbl 1008.60063)] and compute the number of thick points of planar random walk, assuming that the increments are symmetric and have a finite moment of order two. The proof provides a streamlined argument based on the connection to the Gaussian free field and works in a very general setting including isoradial graphs. In higher dimensions, we study the scaling limit of the set of thick points. In particular, we show that the rescaled number of thick points converges to a nondegenerate random variable and that the centred maximum of the local times converges to a randomly shifted Gumbel distribution.


60G50 Sums of independent random variables; random walks
60G15 Gaussian processes
60J55 Local time and additive functionals


Zbl 1008.60063
Full Text: DOI arXiv Euclid


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