Questions at the Interface of Probability and Geometry

在动力学过程的数学描述中，一个共同的线索是一些重要量的传输--也许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，但在宏观尺度上产生统计规律性。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通常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片如何随机游走在无限同质网络与树状，或“双曲”几何。第二个目标是确定一些特定的流行病模型和信息传播模型在这样的网络中的行为。部分研究课题将涉及研究生，他们将从接触数学概率论及其激励应用领域的前沿问题中受益。该项目旨在解决一些重要的开放性技术问题，这些问题涉及半单李群和其他不服从离散群格上随机游动的长期行为。这些研究集中在“局部极限”行为上：对于哪些群体来说，随机游动具有任意的，但不受支持的，步分布服从普遍的局部极限定律？相关的问题涉及马丁边界的结构，这样的随机游动，出口措施的支持，和长随机循环的几何形状。该项目还将致力于SIR流行病的研究，特别是描述临界缩放行为的目标。

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.