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Limit set of branching random walks on hyperbolic groups

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