Limit laws arising in random matrix theory

随机矩阵理论起源于数理统计和理论物理的基本考虑，现在已经应用于数论，统计力学，无线通信和生物信息学等不同领域。一个完美的例子，该领域的范围是提供了所谓的特雷西Widom法律。首先发现在某些大型随机矩阵的最大特征值的描述，这些法律现在被理解为服务的“钟形曲线”（或提供适当的中心极限定理）的作用，为广大地区的现代非线性现象。拟议的研究的主要目标是提高我们的理解，这些和其他概率定律介绍通过研究随机矩阵。重点是他们的基本数学属性，以及扩展类的模型，他们application.The拟议的研究建立在部分PI的最近的工作一般β扩展的Tracy-Widom和相关的分布。这些扩展的优点是将更熟悉的三重极限定律（与正交、酉和辛对称类相关联）嵌入到通过随机微分算子定义的单参数分布族中。第一个目标是使用随机算子方法扩展这些分布的普适性结果。第二个目标是建立当前感兴趣的某些布朗函数恒等式（例如Dufresne恒等式）的非交换版本，因为它们在KPZ普适类中扮演着重要角色。最后一个项目是证明普遍的局部极限定理一类可解的对数气体系综的多个收费之间的经典正交，酉，辛系综插值。

Random matrix theory grew out of fundamental considerations in mathematical statistics and theoretical physics, and now has applications in such disparate areas as number theory, statistical mechanics, wireless communications and bioinformatics. A perfect example of the field's reach is offered by the so-called Tracy-Widom laws. First discovered in the description of the largest eigenvalues for certain large random matrices, these laws are now understood to serve the role of the "bell curve" (or provide the appropriate central limit theorem) for a wide area of modern nonlinear phenomena. The main objective of the proposed research is to advance our understanding of these and other probability laws introduced through the study of random matrices. The focus is on their basic mathematical properties, as well as extending the class of models in which they apply.The proposed research builds in part on the PI's recent work on the general beta extensions of the Tracy-Widom and related distributions. These extensions have the advantage of embedding the more familiar triple(s) of limit laws (tied to the orthogonal, unitary and symplectic symmetry classes) into a one-parameter family of distributions defined via random differential operators. The first goal is to extend universality results for these distributions using the random operator approach. A second goal is to establish non-communative versions of certain Brownian functional identities (for instance, the Dufresne identities) of current interest given their role in the KPZ universality class. The final project is to prove universal local limit theorems for a class of solvable logarithmic gas ensembles of multiple charges which interpolate among the classical orthogonal, unitary, and symplectic ensembles.