CAREER: Distances in Random Media

该职业奖将支持的项目集中在分析随机环境中的距离。一些例子包括随机交通的道路系统上的旅行时间，社交媒体图中人与人之间的最小连接数量，以及在微观距离未知的介质中从一个地方移动到另一个地方的总体成本。研究问题与教育活动结合在一起，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。这项拟议的工作与数学物理的其他领域有联系，如无序磁铁的结构，以及计算机科学的满意问题。这项提议详细介绍了概率论和数学物理，特别是统计力学的项目。主要的模型是渗流模型和首次通过渗流(FPP)，问题是分析高度相关的随机变量集合的极值。在FPP中，主要的观察量是随机环境中从一个网络的一个节点到另一个节点的旅行时间，我们计划研究这个旅行时间的渐近性和涨落，以及最小化(测地线)和接近最小化的结构。大部分研究将涉及布斯曼函数，这是一种来自度量几何的工具，用于研究测地线的平行度。预计在这些研究中获得的结果将对空间增长和流行病模型、其他渗流问题、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，资金用于博士后研究人员、暑期研讨会和本科生暑期项目。

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.