Multifractal spectrum of branching random walks on free groups

Abstract: Consider a transient symmetric branching random walk (BRW) on a free group $\mathbb{F}$ indexed by a Galton-Watson tree $\mathcal{T}$ without leaves. The limit set $\Lambda$ is defined as the random subset of $\partial \mathbb{F}$ (the boundary of $\mathbb{F}$) consisting of all ends in $\partial \mathbb{F}$ to which particle trajectories converge. Hueter--Lalley (2000) determined the Hausdorff dimension of the limit set, and found that $ \dim_{\mathrm{H}} \Lambda \leq \frac{1}{2} \dim_{\mathrm{H}} \partial \mathbb{F}$. In this paper, we conduct a multifractal analysis for the limit set $\Lambda$. We compute almost surely and simultaneously, the Hausdorff dimensions of the sets $\Lambda(\alpha) \subset \Lambda$ consisting of all ends in $\partial \mathbb{F}$ to which particle trajectories, with a rate of escape $\alpha$, converge. Moreover, for isotropic BRWs, we obtain the dimensions of the sets $\Lambda(\alpha,\beta) \subset \Lambda$ which consist of all ends in $\partial \mathbb{F}$ to which particle trajectories, with the average rates of escape having limit points $[\alpha,\beta]$, converge. Finally, analogous to results of Attia--Barral (2014), we obtain the Hausdorff dimensions of the level sets $E(\alpha,\beta)$ of infinite branches in $\partial \mathcal{T}$ along which the averages of the BRW have $[\alpha,\beta]$ as the set of accumulation points.