On the escape rate of favorite sites of planar random walks

Abstract — v1

For planar simple random walk, the favorite sites at time $n$ are the sites whose local time at time $n$ is maximal. We prove that, almost surely, for every $\gamma>1/2$ and every $c>0$, all favorite sites lie outside the ball centered at the origin with radius $c\sqrt n/(\log n)^\gamma$ for all sufficiently large $n$. At the critical exponent $\gamma=1/2$, almost surely, for every $c>0$, the entire favorite sites lies within distance $c\sqrt{n/\log n}$ of the origin infinitely often.