Extremal process of the local time of simple random walk on a regular tree

TL;DR

This paper investigates the extremal behavior of local times of simple random walks on regular trees, revealing convergence to a decorated Poisson process with connections to branching structures.

Contribution

It establishes the limiting extremal process of local times on regular trees, linking it to decorated Poisson processes and branching random walks.

Findings

Extremal process converges to a decorated Poisson point process.

Decorations match those of the tree-indexed Markov chain.

Proof uses a Lindeberg-type swap and isomorphism theorem.

Abstract

We study a continuous-time simple random walk on a regular rooted tree of depth $n$ in two settings: either the walk is started from a leaf vertex and run until the tree root is first hit or it is started from the root and run until it has spent a prescribed amount of time there. In both cases we show that the extremal process associated with centered square-root local time on the leaves tends, as $n\to\infty$, to a decorated Poisson point process with a random intensity measure. While the intensity measure is specific to the local-time problem at hand, the decorations are exactly those for the tree-indexed Markov chain (a.k.a. Branching Random Walk or Gaussian Free Field) with normal step distribution. The proof demonstrates the latter by way of a Lindeberg-type swap of the decorations of the two processes which itself relies on a well-known isomorphism theorem.