Stochastic Systems with Complex Interactions and Random Environments

本项目研究随机环境中随机路径的数学模型和描述随机生长的渗透模型。目标是描述典型的大规模行为，并量化与典型行为的偏差。重点是寻找适用于具有共同基本特征的模型类的普遍原则。有两个主要的研究方向。(1) kardar - paris - zhang通用性类中1+1维精确可解模型的组合学、概率分布和大偏差的研究。本研究的结果包括对物理文献中推测的波动指数的严格验证，组合学与概率和统计物理模型之间的新联系，以及更好地理解精确可解性背后的可积结构。（ii）用概率工具研究更一般的随机路径和增长簇的模型。这项工作将导致极限自由能和极限形状的变分描述，以及这些模型中结构的识别，使我们能够研究它们的普遍涨落特性。宏伟的目标是为模型建立一些普遍的性质，而不是局限于显式可解的情况。该项目研究了描述不规则环境中粒子复杂相互作用、生长和运动的数学模型。这些数学系统结合随机性来模拟不规则性和不可预测性。目标是发现控制这些系统的一般数学规律。这些系统在微观和宏观尺度上表现得截然不同。因此，了解小规模进化的不同规则如何导致不同的大规模系统行为是很重要的。这种数学研究可以阐明的现实世界现象包括车辆的运动、通信网络中的数据包、管道中的流体颗粒、流体在多孔介质中传播的润湿过渡、在人群中传播的流行病或流体中聚合物链的波动。从长远来看，理解复杂的相互作用对科学和工程，从而对社会有着深远的影响。数学家、自然科学家、社会科学家和工程师都在密切地、同时地研究这类模型。

This project studies mathematical models of random paths in random environments and percolation models that describe random growth. The goal is to describe typical large scale behavior and to quantify deviations from the typical behavior. The emphasis is on finding universal principles that apply to classes of models that share fundamental characteristics. There are two main directions of research. (i) The study of the combinatorics, probability distributions and large deviations of 1+1 dimensional exactly solvable models in the Kardar-Parisi-Zhang universality class. Outcomes of this research include rigorous verification of fluctuation exponents conjectured in the physics literature, new connections between combinatorics and models from probability and statistical physics, and better understanding of the integrable structures underlying exact solvability. (ii) The study of more general models of random paths and growing clusters with probabilistic tools. This work will lead to variational descriptions of limiting free energies and limit shapes, and the identification of structures in these models that enable us to study their universal fluctuation properties. The grand goal is to establish some universal properties for models beyond the narrow set of explicitly solvable cases. This project investigates mathematical models that describe complex interactions, growth, and motion of particles in an irregular environment. These mathematical systems incorporate randomness to model irregularity and unpredictability. The goal is to discover general mathematical laws that govern such systems. These systems appear quite different at microscopic and macroscopic scales. So it is important to understand how different rules for small-scale evolution lead to different large-scale systemwide behavior. Real-world phenomena that such mathematical studies can illuminate include the motion of vehicles, packets in a communication network, fluid particles in a tube, wetting transitions where fluid spreads in a porous medium, epidemics advancing in a population, or the fluctuations of a polymer chain in a fluid. Over the long term understanding complex interactions has profound implications for science and engineering and thereby for society. Models of the kind described in the proposal are intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers.