Multifractal spectrum of branching random walks on free groups

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 $\hat{I}{›}_r$ -- 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 $\hat{I}{›}_r$ and proved that ${\dim }_{\mathrm{H}}\hat{I}{›}_r\le (1/2){\dim }_{\mathrm{H}}\partial \mathbb{F}$, with equality possible only when $r=R$. In this paper, we further extend this study by conducting a multifractal analysis of the limit set $\hat{I}{›}_r$. We obtain the Hausdorff dimensions of the subfractals $\hat{I}{›}_r(\hat{I}\pm )\subset \hat{I}{›}_r$, which consist of all ends of $\mathbb{F}$ approached by particle trajectories escaping at rate $\hat{I}\pm \in [0,1]$. Notably, there exists a unique $\hat{I}\pm (r)\in [0,1]$ such that \[ \dim_{\mathrm{H}} Λ_r = \dim_{\mathrm{H}} Λ_r(α(r)). \] Moreover, an interesting phase transition occurs: $\hat{I}\pm (r)>0$ for $r<R$ while $\hat{I}\pm (R)=0$.

59 pages, 1 figure, comments are welcome