CAREER: Distances in Random Media

CAREER奖支持的项目集中在分析随机环境中的距离。一些例子包括在随机交通的道路系统中寻找旅行时间，在社交媒体图表中人与人之间的最小连接数，以及在微观距离未知的媒介中从一个地方移动到另一个地方的总体成本。这些研究问题与教育活动相结合，比如为年轻研究人员举办的概率论暑期研讨会，为本科生举办的暑期项目，以及为博士后研究人员提供资金。这项提议的工作与数学物理的其他领域有联系，比如无序磁铁的结构，以及计算机科学中的满意度问题。本提案详细介绍了概率论和数学物理，特别是统计力学方面的项目。主要模型有渗流模型和首通道渗流模型（first-passage percolation， FPP），问题是分析高度相关随机变量集合的极值。在FPP中，主要观测值是随机环境中从网络的一个节点到另一个节点的行进时间，我们计划研究该行进时间的渐近性和波动，以及最小值（测地线）和近最小值的结构。大部分研究将涉及Busemann函数，这是度量几何中用于研究测地线射线平行性的工具。预计这些研究结果将对空间生长和流行病模型、其他渗透问题、无序自旋系统和聚合物模型产生影响。拟议的工作包括教育部分，包括为博士后研究员、暑期研讨会和本科生暑期项目提供资金。

The projects to be supported by this CAREER award center on analyzing distances in random environments. Some examples include finding travel times on road systems with random traffic, the minimal number of connections between people in social media graphs, and generally the total cost of moving from one place to another in a medium where microscopic distances are unknown. The research questions are combined with educational activities, like summer workshops on probability theory for young researchers, summer projects with undergraduates, and funding for a postdoctoral researcher. The proposed work has connections to other areas of mathematical physics, like the structure of disordered magnets, and satisfaction problems from computer science.This proposal details projects in probability theory and mathematical physics, specifically in statistical mechanics. The main models are percolation models and first-passage percolation (FPP), and the questions are to analyze the extreme values of collections of random variables that are highly correlated. In FPP, the main observable is the travel time from one node of a network to another one in a random environment, and we plan to study the asymptotics and fluctuations of this travel time, along with the structure of minimizers (geodesics) and near-minimizers. Much of the research will involve Busemann functions, which are tools from metric geometry used to study parallelism of geodesic rays. It is expected that results obtained in these studies will have consequences in spatial growth and epidemic models, other percolation problems, disordered spin systems, and polymer models. The proposed work includes an educational component, with funds for a postdoctoral researcher, summer workshops, and summer projects for undergraduates.