Favorite sites of one-dimensional asymmetric simple random walk

TL;DR

This paper analyzes the favorite sites of a one-dimensional asymmetric simple random walk with bias $p>q$. It introduces local time and thick points, and employs stopping times $T_m^k$ and associated constructions to quantify how many sites reach maximal local time. The authors prove that for any $r\ge1$, there are infinitely many times when exactly $r$ favorite sites occur, and they derive sharp asymptotics for the growth of the number of favorite sites, showing $\limsup_{n\to\infty} \frac{\#\mathcal{K}(n)}{\log \log n} = -\frac{1}{\log(1-2q)}$. The second main result provides precise bounds for the growth rate of the thick-site process via events $D_n^\varepsilon$, $E_n^\varepsilon$ and a refined Borel–Cantelli argument, establishing a method to control the limsup with the constant $\theta=-1/\log\gamma$, where $\gamma=1-2q$. Overall, the work extends the understanding of favorite sites from symmetric to asymmetric 1D walks and connects local-time extremal behavior to transience and thick-point structure.

Abstract

In this paper, we study favorite sites of one-dimensional asymmetric simple random walks. We show that almost surely, for any fixed integer $r\geq 1$, ``$r$ favorite sites" occurs infinitely often. We also give the asymptotic growth rate of the number of favorite sites.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 2.5
• Lemma 3.1
• Lemma 3.2
• proof : Proof.