Branching Random Walks on relatively hyperbolic groups

TL;DR

This paper studies the behavior of branching random walks on relatively hyperbolic groups, showing that the growth rate of the trace and the Hausdorff dimension of the limit set are directly related to the Green function's growth rate.

Contribution

It establishes a precise connection between the trace growth rate, the Hausdorff dimension of the limit set, and the Green function for branching random walks on relatively hyperbolic groups.

Findings

Trace growth rate equals the Green function growth rate.

Hausdorff dimension of the limit set is proportional to the Green function growth rate.

Results hold for a range of offspring mean values up to the Green function's radius of convergence.

Abstract

Let $\Gamma$ be a non-elementary relatively hyperbolic group with a finite generating set. Consider a finitely supported admissible and symmetric probability measure $\mu$ on $\Gamma$ and a probability measure $\nu$ on $\mathbb{N}$ with mean $r$. Let $\mathrm{BRW}(\Gamma,\nu,\mu)$ be the branching random walk on $\Gamma$ with offspring distribution $\nu$ and base motion given by the random walk with step distribution $\mu$. It is known that for $1 < r \leq R$ with $R$ the radius of convergence for the Green function of the random walk, the population of $\mathrm{BRW}(\Gamma,\nu,\mu)$ survives forever, but eventually vacates every finite subset of $\Gamma$. We prove that in this regime, the growth rate of the trace of the branching random walk is equal to the growth rate $\omega_\Gamma(r)$ of the Green function of the underlying random walk. We also prove that the Hausdorff dimension of the limit set $\Lambda(r)$, which is the random subset of the Bowditch boundary consisting of all accumulation points of the trace of $\mathrm{BRW}(\Gamma,\nu,\mu)$, is equal to a constant times $\omega_\Gamma(r)$.