On the trace of a discrete-time simple random walk on \(\mathbb {Z}^d\) for \(d\ge 2\) , we consider the evolution of favorite sites, i.e., sites that achieve the maximal local time at a certain time. For \(d=2\) , we show that almost surely three favorite sites occur simultaneously infinitely often and eventually there is no simultaneous occurrence of four favorite sites. For \(d\ge 3\) , we derive sharp asymptotics of the number of favorite sites. This answers an open question of Erdős and Révész (1984), which was brought up again by Dembo (2005).
