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Medientyp: E-Artikel Titel: Limit Set of Branching Random Walks on Hyperbolic Groups Beteiligte: Sidoravicius, Vladas; Wang, Longmin; Xiang, Kainan Erschienen: Wiley, 2023 Erschienen in: Communications on Pure and Applied Mathematics, 76 (2023) 10, Seite 2765-2803 Sprache: Englisch DOI: 10.1002/cpa.22088 ISSN: 0010-3640; 1097-0312 Schlagwörter: Applied Mathematics ; General Mathematics Entstehung: Anmerkungen: Beschreibung: AbstractLet Γ be a nonelementary hyperbolic group with a word metric d and ∂Γ its hyperbolic boundary equipped with a visual metric for some parameter . 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 with mean . Let be the branching random walk on Γ with offspring distribution ν and base motion Y , and let be the volume growth rate for the trace of . We prove for that the Hausdorff dimension of the limit set Λ , which is the random subset of consisting of all accumulation points of the trace of , is given by . Furthermore, we prove that is almost surely a deterministic, strictly increasing, and continuous function of , is bounded by the square root of the volume growth rate of Γ , and has critical exponent 1/2 at in the sense thatfor some positive constant C. We conjecture that the Hausdorff dimension of Λ in the critical case is 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. © 2022 Wiley Periodicals LLC.