双曲空间与双曲群上几类概率模型的临界现象

本项目专注于研究双曲空间和双曲群上概率模型的相变和临界现象，主要包括以下问题：双曲空间上独立分枝布朗运动极限集的相交性；双曲群上临界分枝随机游走极限集的Hausdorff维数；双曲群上具有相依结构分枝随机游走的常返暂留相变和极限集的大小；双曲空间中格点上的Bernoulli渗流团簇极限集的测度以及相变的平均场行为。这些问题的研究须结合来源于几何群论、动力系统和概率论中的工具和方法，并有助于理解双曲结构上其它概率模型的相变行为。

In this project we focus on the phase transition and critical phenomena of probability models on hyperbolic spaces and groups. We will study the following problems: the intersection property of the limit sets of two independent branching Brownian motions on the real hyperbolic spaces; the Hausdorff dimension of the limit set of critical branching random walks on hyperbolic groups; the transience/recurrence phase transition and the size of the limit set of branching random walks on hyperbolic groups with dependent structure between branching and moves; the Lebesgue measure of the limit set of clusters, as well as the mean-field behavior, of Bernoulli percolation on the lattices in the real hyperbolic spaces. We will use the ideas and techniques from geometric group theory, dynamical system and probability theory to study these problems. It may help us to understand the phase transition of other probability models on the hyperbolic spaces and groups.