BRANCHING RANDOM WALKS ON RELATIVELY HYPERBOLIC GROUPS

• 成果类型: Article
• 署名作者: Dussaule, Matthieu; Wang, Longmin; Yang, Wenyuan
• 署名单位: Universite d'Angers; Nankai University; Peking University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1708
• 发表日期: 2025
• 页码: 391-452
• 关键词: free-products; morse boundaries; GROWTH; THEOREM; MAPS

摘要

Let Gamma be a nonelementary, relatively hyperbolic group with a finite generating set. Consider a finitely supported admissible and symmetric probability measure mu on Gamma and a probability measure nu on N with mean r. Let BRW(Gamma, nu, mu) be the branching random walk on Gamma with offspring distribution v and base motion given by the random walk with step distribution mu. It is known that, for 1 < r <= R with R the radius of convergence for the Green function of the random walk, the population of BRW(F, nu, ) survives forever but eventually vacates every finite subset of Gamma. We prove that in this regime, the growth rate of the trace of the branching random walk is equal to the growth rate omega(Gamma)(r) of the Green function of the underlying random walk. We also prove that the Hausdorff dimension of the limit set Lambda(r), which is the random subset of the Bowditch boundary consisting of all accumulation points of the trace of BRW(Gamma,nu, mu), is equal to a constant times omega(Gamma)(r).