讲座报告主题：Boundary of branching random walks on hyperbolic groups
专家姓名：向开南
日期：2019-11-08 时间：16:20
地点：理学院206
主办单位：理学院

主讲简介：向开南，教授，湖南湘西自治州人，1993年6月本科毕业于湘潭大学数学系；1993.9-1996.6在北京师范大学数学系跟从王梓坤院士、李占柄教授读硕士；1996.9-1999.6在中国科学院应用数学研究所师从马志明院士读博士；1999.7-2001.6在北京大学数学科学学院做钱敏平教授的博士后；2001年6月博士后出站后进入湖南师范大学数学与计算机科学学院工作；2007年3月调往南开大学；2019年3月回湘潭大学工作。2010年发表了中国大陆概率论学者在顶级数学期刊Comm. Pure Appl. Math.上的第一篇论文。研究专长：从事理论概率论与统计物理的交叉研究。

主讲内容：In p.275 of his classical book {[T. M. Liggett. (1985). {\it Interacting particle systems}. Springer],}. T. M. Liggett remarked that ``The importance of critical exponents is based largely on what is known as the universality principle, which plays an important role in mathematical physics." Here universality principle means that while the value of critical parameter will usually depend on the details of the definition of the model, the value of critical exponent will be the same for large classes of models (called universality classes). This principle has been an important source of problems in mathematical physics and probability theory. This talk is based on a joint work with Shi Zhan, Sidoravicius Vladas and Wang Longmin, and presents a result on universality of critical exponent for Hausdorff dimensions of boundaries of branching random walks on hyperbolic groups. Let $\Gamma$ be a nonamenable finitely generated infinite hyperbolic group with a symmetric generating set $S,$ and $\partial\Gamma$ the hyperbolic boundary of its Cayley graph. Fix a symmetric probability $\mu$ on $\Gamma$ whose support is $S,$ and denote by $\rho=\rho(\mu)$ the spectral radius of the random walk $\xi$ on $\Gamma$ associated to $\mu.$ Let $\nu$ be a probability on $\{1,2,3,\cdots\}$ with a finite mean $\lambda.$ Write $\Lambda\subseteq\partial\Gamma$ for the boundary of the branching random walk with offspring distribution $\nu$ and underlying random walk $\xi,$ and $h(\nu)$ for the Hausdorff dimension of $\Lambda.$ When $\lambda>1/\rho,$ the branching random walk is recurrent, trivially $$\Lambda=\partial\Gamma,\ h(\nu)=\dim(\partial \Gamma).$$ .In this talk, we focus on the transient setting i.e. $\lambda\in [1,1/\rho],$ and prove the following results: $h(\nu)$ is a deterministic function of $\lambda$ and thus denote it by $h(\lambda);$ and $h(\lambda)$ is continuous and strictly increasing in $\lambda\in [1,1/\rho]$ and $h(1/\rho)\leq\frac{1}{2}\dim(\partial \Gamma);$ and there is a positive constant $C$ such that $$h(1/\rho)-h(\lambda)\sim C\sqrt{1/\rho-\lambda}\ \mbox{as}\ \lambda\uparrow 1/\rho.$$.

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