Random Walks
Two simple random walks in the torus. Points are colored according to occupation measure (left) and hitting time (right). Pictures created by Raissa D’Souza.
Description:
Denote by T(x,r) the occupation measure of a disc of radius r around x by planar Brownian motion run till time 1, and let T(r) be the maximum of T(x,r) over x in the plane. We prove in [1] that T(r) is a.s. asymptotic to 2 r2 log r2 as r tends to 0, thus solving a problem posed by Perkins and Taylor (1987). Furthermore, for any a<2, the Hausdorff dimension of the set of points x for which T(x,r) is asymptotic to a r2log r2, is almost surely 2a. As a consequence, we prove a conjecture about planar simple random walk due to Erdős and Taylor (1960): The number of visits to the most frequently visited lattice site in the first n steps of the walk, is asymptotic to (log n)2/pi. We also show that for a between 0 and 1/pi, the number of points visited more than a(log n)2 times in the first n steps, is approximately n{1a pi}.
Hitting time results:
References:

Thick points for planar Brownian motion and the ErdosTaylor conjecture on random walk. (A. Dembo, Y. Peres, J. Rosen and O. Zeitouni). Acta Math. 186 no. 2, (2001), 239–270.

Cover Times for Brownian Motion and Random Walks in Two Dimensions. (A. Dembo, Y. Peres, J. Rosen, and O. Zeitouni). Ann. Math., 160 (2004).

Late Points for Random Walks in Two Dimensions. (A. Dembo, Y. Peres, J. Rosen, O. Zeitouni). Annals of Probability. To appear.