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 $\Lambda_r$ -- 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 $\dim_{ \mathrm{ H}} \Lambda_r \le (1/2)\dim_{ \mathrm{ H}} \partial \mathbb{ F}$, with equality possible only when $r = R$. In this paper, we further extend this study by conducting a multifractal analysis of the limit set $\Lambda_r$. We obtain the Hausdorff dimensions of the subfractals $\Lambda_r(\alpha) \subset \Lambda_r$, 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 \[ \dim_{ \mathrm{ H}} \Lambda_r = \dim_{ \mathrm{ H}} \Lambda_r(\alpha(r)). \] Moreover, an interesting phase transition occurs: $\alpha(r) > 0$ for $r < R$ while $\alpha(R) = 0$.
Categories: math.PR