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。该项目探索与随机矩阵和相互作用的粒子系统的大规模性质有关的问题。贝塔集合是直线上随机点的单参数分布族；对于特定的参数值，它描述了一些最著名的经典随机矩阵集合的特征值。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。这个项目的目的是建立在PI和一个合作者的最新结果的基础上，在这个合作者中，描述了Beta合奏的点过程比例限制。该计划是为了提取更多的信息，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在几个领域的不同应用中产生。例如生长和沉积模型、交通模型、趋化性、疾病传播。近几十年来，人们做出了协调一致的努力，为此类系统提供了严格的数学分析。这个提议的目标之一是分析一类相互作用的粒子系统中的平衡涨落。该项目计划开发稳健的、独立于模型的方法来比较各种粒子系统，并致力于在广泛的模型类中证明标度指数的普适性。该项目处理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。这些问题的处理需要广泛的工具，这些工具将它们连接到除概率之外的其他数学领域，如组合学、复分析、算子论、偏微分方程式。除了数学之外，这样的系统还存在于许多其他领域，例如物理、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.