Three Favorite Edges Occurs Infinitely Often for One-Dimensional Simple Random Walk

TL;DR

This paper proves that in a one-dimensional simple symmetric random walk, three favorite edges occur infinitely often with probability 1, disproving a previous conjecture and extending understanding of edge local times.

Contribution

It establishes the almost sure infinite occurrence of three favorite edges in one-dimensional simple symmetric random walks, challenging prior conjectures.

Findings

Three favorite edges occur infinitely often with probability 1.

Disproves a previous conjecture about the number of favorite edges.

Extends theoretical understanding of local times in random walks.

Abstract

For a one-dimensional simple symmetric random walk $(S_n)$, an edge $x$ (between points $x-1$ and $x$) is called a favorite edge at time $n$ if its local time at $n$ achieves the maximum among all edges. In this paper, we show that with probability 1 three favorite edges occurs infinitely often. Our work is inspired by T\'{o}th and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369], and Ding and Shen [Ann. Probab. {\bf 46} (2018) 2545-2561], disproves a conjecture mentioned in Remark 1 on page 368 of T\'{o}th and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369].