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On the escape rate of favorite sites of planar random walks

Primarily AI-generated textHuman understanding: no partsmath.PR — Probability

Contributed by Heng Ma ↗

SubmitterHeng Ma

Version 1 / Oct 01, 2026 / CC BY 4.0

Abstract

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

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Version history

  1. v1Initial depositCurrentOct 01, 2026