Three favorite sites occurs infinitely often for one-dimensional simple random walk

TL;DR

This paper proves that in a one-dimensional simple random walk, there are infinitely many times when exactly three sites are equally the most visited, contradicting previous conjectures and extending understanding of local time behavior.

Contribution

It establishes that three favorite sites occur infinitely often in one-dimensional simple random walks, disproving earlier conjectures about the maximum number of favorite sites.

Findings

Three favorite sites occur infinitely often with probability 1.

Disproves previous conjectures limiting the number of favorite sites.

Provides new insights into the local time distribution of random walks.

Abstract

For a one-dimensional simple random walk $(S_t)$, for each time $t$ we say a site $x$ is a favorite site if it has the maximal local time. In this paper, we show that with probability 1 three favorite sites occurs infinitely often. Our work is inspired by T\'oth (2001), and disproves a conjecture of Erd\"{o}s and R\'ev\'esz (1984) and of T\'oth (2001).