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

For a one-dimensional simple symmetric random walk \((S_n)\) $(S_n)$ , an edge x (between points \(x-1\) $x-1$ 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 Tóth and Werner (Comb Probab Comput 6:359–369, 1997), and Ding and Shen (Ann Probab 46:2545–2561, 2018), disproves a conjecture mentioned in Remark 1 on page 368 of Tóth and Werner (1997).