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

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 Tóth (2001), and disproves a conjecture of Erdös and Révész (1984) and of Tóth (2001).

18 pages. Version 2 improves exposition, and adopted the journal template