Limit Set of Branching Random Walks on Hyperbolic Groups

Let Γ be a nonelementary hyperbolic group with word metric d and ∂Γ its boundary equipped visual for some parameter > 1 . Fix superexponential symmetric probability μ on whose support generates as semigroup, denote by ρ the spectral radius of random walk Y step distribution μ. ν 1,2,3 , … mean λ = ∑ k ∞ kν < BRW branching offspring base motion let H volume growth rate trace We prove ∈ − that Hausdorff dimension limit set Λ which is subset consisting all accumulation points given log Furthermore, we almost surely deterministic, strictly increasing, continuous function bounded square root has critical exponent 1/2 at in sense ∼ C ↑ positive constant C. conjecture case surely. This been confirmed free groups or product (by amalgamation) finitely many finite defined standard generating set. © 2022 Wiley Periodicals LLC.