The Escape Rate of Favorite Edges of Simple Random Walk

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 ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}\frac{\mathrm{\mid }E(n)\mathrm{\mid }}{\sqrt{n}\cdot (\log n{)}^{-\gamma }}$ equals 0 if $\gamma \le 1$, and is infinity otherwise. We also obtain a law of the iterated logarithm for $E(n)$.