Limit set of branching random walks on hyperbolic groups

Let $\Gamma$ be a nonelementary hyperbolic group with a word metric $d$ and $\partial\Gamma$ its hyperbolic boundary equipped with a visual metric $d_a$ for some parameter $a>1$. Fix a superexponential symmetric probability $\mu$ on $\Gamma$ whose support generates $\Gamma$ as a semigroup, and denote by $\rho$ the spectral radius of the random walk $Y$ on $\Gamma$ with step distribution $\mu$. Let $\nu$ be a probability on $\{1,\, 2, \, 3, \, \ldots\}$ with mean $\lambda=\sum\limits_{k=1}^\infty k\nu(k)<\infty$. Let $\mathrm{BRW}(\Gamma, \, \nu, \, \mu)$ be the branching random walk on $\Gamma$ with offspring distribution $\nu$ and base motion $Y$ and $H(\lambda)$ the volume growth rate for the trace of $\mathrm{BRW}(\Gamma, \, \nu, \, \mu)$. We prove for $\lambda \in [1, \, \rho^{-1})$ that the Hausdorff dimension of the limit set $\Lambda$, which is the random subset of $(\partial \Gamma, \, d_a)$ consisting of all accumulation points of the trace of $\mathrm{BRW}(\Gamma, \, \nu, \, \mu)$, is given by $\log_a H(\lambda)$. Furthermore, we prove that $H(\lambda)$ is almost surely a deterministic, strictly increasing and continuous function of $\lambda \in [1, \, \rho^{-1}]$, is bounded by the square root of the volume growth rate of $\Gamma$, and has critical exponent $1/2$ at $\rho^{-1}$ in the sense that $$H(\rho^{-1}) - H(\lambda) \sim C \sqrt{\rho^{-1} - \lambda} \quad \text{as } \lambda \uparrow \rho^{-1}$$ for some positive constant $C$. We conjecture that the Hausdorff dimension of $\Lambda$ in the critical case $\lambda=\rho^{-1}$ is $\log_aH(\rho^{-1})$ almost surely. This has been confirmed on free groups or the free product (by amalgamation) of finitely many finite groups equipped with the word metric $d$ defined by the standard generating set.