Multifractal spectrum of branching random walks on free groups

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 ${\mathrm{\Lambda }}_r$ – consisting of all ends of $\mathbb{F}$ to which the BRW’s particle trajectories converge – is a proper random subset of the boundary $\mathrm{\partial }\mathbb{F}$. Hueter and Lalley (2000) determined the Hausdorff dimension of ${\mathrm{\Lambda }}_r$ and proved that ${\dim }_{\mathrm{H}}{\mathrm{\Lambda }}_r\le (1\mathrm{/}2){\dim }_{\mathrm{H}}\mathrm{\partial }\mathbb{F}$, with equality possible only when $r=R$. In this paper, we further extend this study by conducting a multifractal analysis of the limit set ${\mathrm{\Lambda }}_r$. We obtain the Hausdorff dimensions of the subfractals ${\mathrm{\Lambda }}_r(\alpha )\subset {\mathrm{\Lambda }}_r$, 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 $${\dim }_{\mathrm{H}}{\mathrm{\Lambda }}_r={\dim }_{\mathrm{H}}{\mathrm{\Lambda }}_r(\alpha (r))\mathrm{.}$$ Moreover, an interesting phase transition occurs: $\alpha (r)>0$ for $r<R$ while $\alpha (R)=0$.