Random Matrices, Integrable Systems and Related Stochastic Processes

该项目的主要活动是分析出现在各种随机过程中的新的极限定律及其相关的可积微分方程。这些随机过程包括艾里过程、皮尔西过程及其高阶推广、一个不相交的布朗游程模型和一类称为非对称排斥过程的相互作用的粒子系统。所采用的主要方法是来自随机矩阵理论、可积系统和算子理论的思想和技术的结合。将算子理论应用于可积系统，将为一类算子行列式建立新的极限定理，从而在所谓的临界情况下提供柱形Toda方程的解。这些方法与截断Wiener-Hopf算子和Toeplitz算子的研究方法有关。许多物理系统具有如此复杂的行为，以至于精确的预测变得不可能。随机矩阵理论提供了允许模拟此类行为的数学模型和允许与实验进行比较的预测。这是它最初的动机，但自那以后，它在数学、科学和技术的其他领域有了重要的应用，在通信、概率、统计学、数论、凝聚态物理和工程学等不同的学科中。可以预见，部分由PI和他的合作者开发的随机矩阵理论和技术的想法将在上述随机过程的研究中发挥重要作用。此外，该项目应通过可积微分方程提供可实现的数值算法来计算这些过程的性质。这最后一个方面对应用程序来说是最重要的。

The main activity of the project is the analysis of new limit laws and their associated integrable differential equations that appear in a variety of stochastic processes. These stochastic processes include the Airy process, the Pearcey process and its higher order generalizations, a nonintersecting Brownian excursion path model, and a class of interacting particle systems called the asymmetric exclusion process. The main methods to be employed are a combination of ideas and techniques coming from random matrix theory, integrable systems and operator theory. An operator theory application to integrable systems would establish new limit theorems for a class of operator determinants providing solutions to the cylindrical Toda equations in so-called critical cases. The methods are related to those used in the study of truncated Wiener-Hopf and Toeplitz operators.Many physical systems possess such complicated behavior that exact predictions become impossible. Random matrix theory provides mathematical models that allow a simulation of such behavior and predictions that allow comparison with experiment. This was its original motivation, but it has since had significant applications in other areas of mathematics, science and technology, in such diverse subjects as communications, probability, statistics, number theory, condensed matter physics, and engineering. One can anticipate that ideas from random matrix theory and techniques developed in part by the PI and his collaborators will be instrumental in the study of the stochastic processes describe above. Further, the project should provide, through the integrable differential equations, implementable numerical algorithms to compute properties of these processes. This last aspect is most important for applications.