Abstract:
Relatively hyperbolic groups form a broad class of groups in geometric group theory, encompassing hyperbolic groups, free products, and geometrically finite Kleinian groups. They exhibit rich large-scale geometry and admit a natural boundary, introduced by Bowditch, which captures the asymptotic behavior of geodesics and random processes on the group.
In this talk, we discuss branching random walks on a non-elementary relatively hyperbolic group Γ. The model is obtained by combining a symmetric, finitely supported admissible random walk on Γ with an offspring distribution of mean r. When 1<r≤R, where R is the radius of convergence of the Green function of the underlying random walk, the branching random walk lies in an interesting intermediate regime: the population survives forever, yet the process remains transient in the sense that it eventually leaves every finite subset of the group.
In this two-part talk, we first describe the background and motivation, and then explain two results concerning the large-scale geometry of the trace of the branching random walk. First, we show that the exponential growth rate of the trace is exactly the growth rate ω(r) of the Green function over spheres of the underlying random walk. Second, if Λ(r) denotes the random limit set of the trace in the Bowditch boundary, then its Hausdorff dimension is almost surely equal to an explicit constant multiple of ω(r).
Bio:
杨文元，伊人直播
数学系的教授，研究方向为几何群论和低维拓扑学。
王龙敏，南开大学统计与数据科学伊人直播
教授，主要研究方向为群上的概率模型。
