Multifractal spectrum of branching random walks on free groups

Abstract

Consider a symmetric branching random walk on a free group $\mathbb{F}$ in the transient regime $1<r\le R$, where $r$ is the mean offspring number and $R$ is the reciprocal of the spectral radius of the underlying random walk. 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 ${\dim }_{\mathrm{H}}{\Lambda }_r\le \frac{1}{2}{\dim }_{\mathrm{H}}\partial \mathbb{F}$ with equality possible only when $r=R$. We further extend this study by conducting a multifractal analysis of the limit set ${\Lambda }_r$. We obtain the Hausdorff dimensions of the sub-fractals ${\Lambda }_r(\alpha )\subset {\Lambda }_r$ which consist of all ends of $\mathbb{F}$ approached by particle trajectories escaping at the rate $\alpha \in [0,1]$. Notably, there exists a unique $\alpha (r)\in [0,1]$ such that \begin{equation} \dim_{\mathrm{H}} Λ_r = \dim_{\mathrm{H}} Λ_r( α(r) ). \end{equation} Moreover, the maximizing speed exhibits a phase transition: $\alpha (r)>0$ for $1<r<R$, whereas $\alpha (R)=0$.

53 pages, 1 figure. Fixed several mistakes and filled some gaps. Comments are welcome