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The Escape Rate of Favorite Edges of Simple Random Walk
Chen-Xu Hao · 2023-03-23 · via math.PR updates on arXiv.org

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 $\liminf_{n\to\infty}\frac{|E(n)|}{\sqrt{n}\cdot(\log n)^{-γ}}$ equals 0 if $γ\le 1$, and is infinity otherwise. We also obtain a law of the iterated logarithm for $E(n)$.

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