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Three Favorite Sites Occurs Infinitely Often for One-dimensional Simple Random Walk


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 talk, 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).  Joint work with Jianfei Shen.


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