The escape rate of favorite sites of simple random walk and Brownian motion

Abstract

Consider a simple symmetric random walk on the integer lattice $\ZB$. For each n, let $V(n)$ denote a favorite site (or most visited site) of the random walk in the first n steps. A somewhat surprising theorem of Bass and Griffin [Z. Wahrsch. Verw. Gebiete 70 (1985) 417--436] says that V is almost surely transient, thus disproving a previous conjecture of Erdős and Révész [Mathematical Structures--Computational Mathematics--Mathematical Modeling 2 (1984) 152--157]. More precisely, Bass and Griffin proved that almost surely, $\liminf_{n\to \infty} {|V(n)| \over n^{1/2}(\log n)^{-\gamma}}$ equals $0$ if $\gamma<:1$, and is infinity if $\gamma>11$ (eleven). The present paper studies the rate of escape of $V(n)$. We show that almost surely, the "lim\,inf'' expression in question is 0 if $\gamma\leq 1$, and is infinity otherwise. The corresponding problem for Brownian motion is also studied.