Multifractal spectrum of branching random walks on free groups

TL;DR

This paper performs a multifractal analysis of the limit set of a branching random walk on free groups, revealing the Hausdorff dimensions of subfractals and phase transitions related to the escape rate of trajectories.

Contribution

It extends previous work by calculating the Hausdorff dimensions of subfractals and identifying a phase transition in the escape rate for the limit set of the branching random walk.

Findings

Determined Hausdorff dimensions of subfractals (r) within the limit set.

Identified a unique (r) where the Hausdorff dimension of (r) equals that of the entire limit set.

Discovered a phase transition at the spectral radius R, with escape rate (r) > 0 for r < R and (R) = 0.

Abstract

A symmetric branching random walk (BRW) on a free group $\mathbb{F}$ is transient if and only if the mean offspring number $r$ does not exceed $R$, the reciprocal of the spectral radius of the underlying random walk. In this regime, the limit set $\Lambda_r$ -- consisting of all ends of $\mathbb{F}$ to which the BRW's particle trajectories converge -- is a proper random subset of the boundary $\partial \mathbb{F}$. Hueter and Lalley (2000) determined the Hausdorff dimension of $\Lambda_r$ and proved that $\dim_{\mathrm{H}} \Lambda_r \le (1/2)\dim_{\mathrm{H}} \partial \mathbb{F}$, with equality possible only when $r = R$. In this paper, we further extend this study by conducting a multifractal analysis of the limit set $\Lambda_r$. We obtain the Hausdorff dimensions of the subfractals $\Lambda_r(\alpha) \subset \Lambda_r$, which consist of all ends of $\mathbb{F}$ approached by particle trajectories escaping at rate $\alpha \in [0,1]$. Notably, there exists a unique $\alpha(r) \in [0,1]$ such that \[ \dim_{\mathrm{H}} \Lambda_r = \dim_{\mathrm{H}} \Lambda_r(\alpha(r)). \] Moreover, an interesting phase transition occurs: $\alpha(r) > 0$ for $r < R$ while $\alpha(R) = 0$.