The Escape Rate of Favorite Edges of Simple Random Walk

Abstract

Consider a simple symmetric random walk on the integer lattice $\mathbb{Z}$. Let $E(n)$ denote a favorite edge of the random walk at time $n$. In this paper, we study the escape rate of $E(n)$, and show that almost surely $\liminf_{n\to\infty}\frac{|E(n)|}{\sqrt{n}\cdot(\log n)^{-\gamma}}$ equals 0 if $\gamma\le 1$, and is infinity otherwise. We also obtain a law of the iterated logarithm for $E(n)$.