A limit law for the most favorite point of simple random walk on a regular tree

TL;DR

This paper investigates the asymptotic distribution of the maximum local time at leaf vertices for a continuous-time random walk on a finite regular tree, revealing convergence to a randomly-shifted Gumbel law as the tree depth grows.

Contribution

It introduces a limit law for the favorite points of a random walk on a regular tree, connecting local times to a derivative-martingale structure and extending understanding of extremal behaviors.

Findings

Maximal local time converges to a randomly-shifted Gumbel distribution.

The shift is characterized by a derivative-martingale related to local times.

Results hold as the tree depth tends to infinity.

Abstract

We consider a continuous-time random walk on a regular tree of finite depth and study its favorite points among the leaf vertices. For the walk started from a leaf vertex and stopped upon hitting the root we prove that, in the limit as as the depth of the tree tends to infinity, the suitably scaled and centered maximal time spent at any leaf converges to a randomly-shifted Gumbel law. The random shift is characterized using a derivative-martingale like object associated with square-root local-time process on the tree.