Extremes of local times for simple random walks on symmetric trees

Abstract

We consider local times of the simple random walk on the $b$-ary tree of depth $n$ and study a point process which encodes the location of the vertex with the maximal local time and the properly centered maximum over leaves of each subtree of depth $r_n$ rooted at the $(n-r_n)$ level, where $(r_n{)}_{n\ge 1}$ satisfies ${\lim }_{n\to \infty }{r}_n=\infty$ and ${\lim }_{n\to \infty }{r}_n/n\in [0,1)$. We show that the point process weakly converges to a Cox process with intensity measure $\alpha Z_{\infty }(dx)\otimes e^{-2\sqrt{\log b}y}dy$, where $\alpha >0$ is a constant and $Z_{\infty }$ is a random measure on $[0,1]$ which has the same law as the limit of a critical random multiplicative cascade measure up to a scale factor. As a corollary, we establish convergence in law of the maximum of local times over leaves to a randomly shifted Gumbel distribution.

48 pages, Update: several minor corrections, new proof for Lemma 2.1

References in corpus (3)

• Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field
• Freezing and decorated Poisson point processes
• Heat kernel estimates and related inequalities on metric graphs

Cited by in corpus (2)

• Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field
• Barrier estimates for a critical Galton--Watson process and the cover time of the binary tree