Random matrices and interacting particle systems

随机矩阵和相互作用的粒子系统

• 批准号: 0905820
• 负责人: Benedek Valko
• 金额: $ 15万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905820&HistoricalAwards=false
• 关键词: Random; matrices; interacting; particle; systems

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.

该奖项是根据2009年的《美国回收与再投资法》（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属性相关的问题。 β-集成是一个在线上随机点的分布的一个参数。对于特定参数值，它描述了一些最著名的经典随机矩阵集合的特征值。了解这些经典合奏的缩放限制一直是随机矩阵理论的重要问题。该项目旨在基于PI的最新结果和合作者的最新结果，其中描述了Beta浓度的点过程缩放限制。该计划是在探索与其他字段的连接时提取更多信息，并更好地了解极限过程。互动粒子系统来自多个字段中的各种应用。例子包括生长和沉积模型，交通模型，趋化性，疾病的扩散。近几十年来，已经做出了一致的努力，以对此类系统进行数学上的严格分析。该提案的目标之一是分析某个相互作用粒子系统家族中的平衡波动。该计划是开发坚固的，与模型无关的方法，以比较各种粒子系统，并致力于在广泛的模型中证明缩放指数的普遍性。该项目处理与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其他各个数学领域，例如概率，例如 组合学，复杂分析，操作者理论，部分微分方程。除了数学，例如物理学，生物学，工程和经济学外，此类系统还存在于许多其他领域。