Favorite sites of one-dimensional asymmetric simple random walk

TL;DR

This paper investigates the long-term behavior of favorite sites in one-dimensional asymmetric simple random walks, proving their infinite recurrence and analyzing their growth rate.

Contribution

It establishes that favorite sites occur infinitely often and provides the asymptotic growth rate of their number in such random walks.

Findings

Favorite sites occur infinitely often almost surely.

The asymptotic growth rate of favorite sites is characterized.

The study advances understanding of site visitation patterns in asymmetric random walks.

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.