Limit Set of Branching Random Walks on Hyperbolic Groups

作者：Sidoravicius, Vladas; Wang, Longmin^*; Xiang, Kainan

来源：Communications on Pure and Applied Mathematics, 2023, 76(10): 2765-2803.

DOI：10.1002/cpa.22088

• PERCOLATION
• BOUNDARY

Let & UGamma; be a nonelementary hyperbolic group with a word metric d and & part;gamma its hyperbolic boundary equipped with a visual metric da for some parameter a > 1 . Fix a superexponential symmetric probability mu on & UGamma; whose support generates & UGamma; as a semigroup, and denote by rho the spectral radius of the random walk Y on & UGamma; with step distribution mu. Let nu be a probability on 1,2,3, horizontal ellipsis with mean lambda= sigma(infinity)(k=1) k nu(k) < infinity. Let BRW(gamma, nu, mu) be the branching random walk on & UGamma; with offspring distribution nu and base motion Y, and let H lambda be the volume growth rate for the trace of BRW & UGamma;nu mu . We prove for lambda is an element of [1, rho(-1)] that the Hausdorff dimension of the limit set lambda , which is the random subset of (& part;gamma, d(a)) consisting of all accumulation points of the trace of BRW & UGamma;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 is an element of [1, rho(-1)], is bounded by the square root of the volume growth rate of & UGamma;, and has critical exponent 1/2 at rho(-1) in the sense that H(rho(-1)) - H(lambda) & SIM; C root rho(-1) - lambda as lambda & UARR; rho(-1) for some positive constant C. We conjecture that the Hausdorff dimension of lambda in the critical case lambda=rho(-1) is log(a)H(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.

• 单位
  南开大学; 湘潭大学