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Abstract:Consider a symmetric branching random walk on a free group $\mathbb{F}$ in the transient regime $1<r\leq R$, 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 $\dim_{\mathrm{H}} \Lambda_r \leq \frac{1}{2} \dim_{\mathrm{H}} \partial \mathbb{F}$ 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}} \Lambda_r = \dim_{\mathrm{H}} \Lambda_r( \alpha(r) ). \end{equation} Moreover, the maximizing speed exhibits a phase transition: $\alpha(r)>0$ for $1<r<R$, whereas $\alpha(R)=0$.

Submission history

From: Heng Ma [view email]
[v1] Mon, 2 Sep 2024 16:01:36 UTC (75 KB)
[v2] Wed, 5 Nov 2025 11:37:01 UTC (73 KB)
[v3] Thu, 20 Aug 2026 11:04:39 UTC (84 KB)