报告题目：Discrete probability models under viewpoints of quasi-isometry and universality：Starting from concrete examples

报告人：向开南教授

报告时间：2023年2月24日14：00-16：00

报告地点：龙洞校区行政楼610

主持人：陈智奇

摘要：Vladas Sidoravicius, Longmin Wang, Kainan Xiang (2023, Comm. Pure Appl. Math.) proved the universality that for branching random walks on non-elementary hyperbolic groups, critical exponents for their volume growths and Hausdorff dimensions of their boundaries are 1/2 (made a breakthrough of a related conjecture in S. Lalley ICM2006; and conjectured the dimension formula in the critical case is still true, which was confirmed in 2022 as a corollary by a remarkable paper of M. Dussaule, Longmin Wang and Wenyuan Yang (arXiv:2211.07213)). Kainan Xiang, Lang Zou (2022. Stat. Probab. Lett. 184, 109378, 6pp.) proved that surviving end number in Bernoulli percolation on graphs has a certain quasi-isometry invariance. Starting from these, we illustrate universality and rigidity of discrete probability models.

Recall a central prediction of renormalization group theory that critical behaviors of statistical physics models on polynomial growth groups (finite dimensional graphs) depend only dimensions rather than choices of lattices and large-scale geometries. Do critical behaviors of statistical physics models on infinite dimensional graphs (for example, nonamenable graphs) depend only on some large-scale geometry (for instance, hyperbolicity)? These are universalities of statistical physics models on groups and graphs. From a viewpoint of quasi-isometry, it is believed that to a certain degree, some properties of some fundamental stochastic processes (random walk, Bernoulli percolation, uniform spanning forest and minimal spanning forest etc.) on groups and graphs are quasi-isometric invariant, which are rigidities of discrete probability models.

专家简介：向开南，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读硕士，1996.9-1999.6在中国科学院应用数学研究所读博士；1999.7-2001.6在北京大学数学科学学院做博士后；2001年6月博士后出站后进入湖南师范大学工作，2007年3月调往南开大学，2019年3月回湘潭大学工作；对群和图上的概率与几何（随机游走、随机图、渗流、几何群论、无穷图论等）感兴趣；是科学网的博客写手：

https://blog.sciencenet.cn/u/MinGong1