Questions at the Interface of Probability and Geometry

在动态过程的数学描述中，一个共同的线索是一些重要数量的传输——可能是有缺陷的晶体中的电子，或者是生物网络中的感染，或者是社会网络中的信息——通过在微观层面上包含随机性的规则，但在宏观尺度上导致统计规律性。在许多这样的过程中，理解这种宏观秩序的关键是仔细分析网络中单个粒子（或信息位）的随机运动。反过来，理解非均匀网络中粒子的随机运动往往需要分析均匀网络中相应的运动。这项研究的一个主要目标是构建一个全面的图像，随机步行者如何在具有树状或“双曲”几何的无限同质网络中表现。第二个目标是确定这种网络中若干特定流行病模型和信息传播模型的行为。一些研究课题将涉及研究生，他们将从接触这些数学概率论的前沿问题及其激励应用领域中受益。该项目旨在解决一些重要的开放性技术问题，这些问题涉及半简单李群和其他不可服从离散群的格上随机行走的长期行为。这些问题以“局部极限”行为为中心：对于哪些群体进行随机行走，具有任意但有限支持的步长分布服从普遍的局部极限定律？相关的问题涉及这种随机游走的马丁边界的结构，出口措施的支持，以及长随机循环的几何形状。该项目还将投入一些精力研究SIR流行病，具体目标是描述临界缩放行为。

A common thread in the mathematical description of dynamical processes is the transport of some important quantity -- perhaps electrons in a crystal with imperfections, or infection in a biological network, or information in a social network -- by rules that incorporate randomness at the microscopic level, but result in statistical regularity at macroscopic scales. In many such processes, the key to understanding this macroscopic order is a careful analysis of the random motions of individual particles (or bits of information) in the network. In turn, understanding the random motions of particles in inhomogeneous networks often requires analysis of the corresponding motions in homogeneous networks. A major goal of this research is to construct a comprehensive picture of how random walkers behave in infinite homogeneous networks with tree-like, or "hyperbolic'' geometry. A secondary goal is to determine the behavior of a number of specific models of epidemic and information-propagation models in such networks. Some of the research topics will involve graduate students, who will benefit from exposure to these cutting-edge problems in mathematical probability theory and their motivating application areas.The project aims to settle a number of important open technical questions concerning the long-time behavior of random walks on lattices of semi-simple Lie groups and other nonamenable discrete groups. These center on "local limit" behavior: for which groups do random walks with arbitrary, but finitely supported, step distributions obey a universal local limit law? Related questions concern the structure of the Martin boundaries for such random walks, the support of the exit measure, and the geometry of long random loops. The project will also devote some effort to the study of SIR epidemics, with the particular object of describing critical scaling behavior.