Random matrices and interacting particle systems

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目探讨了与随机矩阵和相互作用粒子系统的大规模性质有关的问题。β-系综是一个单参数的分布族的随机点在一条线上;对于特定的参数值，它描述了一些最著名的经典随机矩阵系综的特征值。理解这些经典系综的标度极限一直是随机矩阵理论的一个重要问题。该项目旨在建立在PI和合作者的最新结果的基础上，其中描述了β系综的点过程标度限制。该计划是为了提取更多的信息，并达到更好地理解极限过程，同时探索与其他领域的联系。相互作用的粒子系统产生于几个领域的各种应用。例子包括生长和沉积模型，交通模型，趋化性，疾病的传播。近几十年来，人们一直在努力为这类系统提供严格的数学分析。这个建议的目标之一是分析在某一家庭的相互作用粒子系统的平衡涨落。该计划是开发强大的，独立于模型的方法来比较各种粒子系统，并努力证明标度指数在广泛的一类模型中的普适性。该项目涉及与具有大量组件和非平凡相互作用的大型随机系统有关的问题。这些问题的治疗需要广泛的工具，将它们连接到各种其他领域的数学除了概率，例如组合，复杂的分析，算子理论，偏微分方程。除了数学之外，这种系统还出现在许多其他领域，例如物理学，生物学，工程学和经济学。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project explores problems connected to large scale properties of random matrices and interacting particle systems. The beta-ensemble is a one parameter family of distributions of random points on a line; for specific parameter values it describes the eigenvalues of some of the most famous classical random matrix ensembles. Understanding the scaling limit of these classical ensembles has been an important problem of random matrix theory. This project aims to build on the recent results of the PI and a collaborator in which the point process scaling limits of the beta-ensembles are described. The plan is to extract more information and to reach a better understanding of the limit process while exploring connections to other fields.Interacting particle systems arise from various applications in several fields. Examples include growth and deposition models, traffic models, chemotaxis, the spreading of a disease. In recent decades a concerted effort has been made to provide mathematically rigorous analysis of such systems. One of the goals of this proposal is to analyze the equilibrium fluctuations in a certain family of interacting particle systems. The plan is to develop robust, model-independent methods to compare various particle systems and to work towards proving the universality of the scaling exponent in a broad class of models.The project deals with problems related to large random systems with lots of components and non-trivial interactions. The treatment of these problems requires a wide range of tools which connects them to various other fields of mathematics besides probability, e.g. combinatorics, complex analysis, operator theory, partial differential equations. Such systems occur in many other fields in addition to mathematics, for instance in physics, biology, engineering and economics.