Branching random walks on hyperbolic groups: volume growth rate and limit set

Teoria prawdopodobieństwa i modelowanie stochastyczne
Osoba referująca: 
Longmin Wang (Uniwersytet w Nankai, Chiny)
Data spotkania seminaryjnego: 
czwartek, 18. Kwiecień 2019 - 12:15
Let $\Gamma$ be a nonelementary hyperbolic group equipped with a word metric. Consider a branching random walk (BRW) on $\Gamma$ with mean offspring $\lambda < \infty$ and let $\rho$ be the spectral radius of the base motion. It is known that, if and only if $\lambda > \rho^{-1}$, the BRW is recurrent in the sense that every point is visited infinitely often by the particles in BRW. In this talk, we focus on the transient regime $\lambda \in [1,\, \rho^{-1}]$ and study the critical behavior for the volume growth rate and the Hausdorff dimension of the limit set of BRW. More precisely, we prove that both of them exhibit phase transitions at $\lambda = \rho^{-1}$ and have critical exponent $1/2$. The talk is based on a joint work with Z. Shi, V. Sidoravicius and K. Xiang.