Limit laws arising in random matrix theory

随机矩阵理论源于数理统计和理论物理中的基本考虑，现在已经在数论、统计力学、无线通信和生物信息学等不同领域得到了应用。所谓的特雷西-威多姆法律就是该领域覆盖范围的一个完美例子。这些定律最早是在描述某些大型随机矩阵的最大特征值时发现的，现在被理解为在广泛的现代非线性现象中起“钟形曲线”(或提供适当的中心极限定理)的作用。拟议研究的主要目的是促进我们对这些和其他通过研究随机矩阵而引入的概率定律的理解。重点是它们的基本数学性质，以及它们应用的模型类的扩展。拟议的研究部分建立在PI最近关于Tracy-Widom和相关分发的通用Beta扩展的工作基础上。这些扩展的优点是将更熟悉的三重(S)极限定律(绑定到正交、酉和辛对称类)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，Duresne恒等式)的非共生版本，因为它们在KPZ普适性类中所扮演的角色。最后一个项目是证明一类可解的多重电荷对数气体系综的普适局部极限定理，该系综内插在经典的正交、酉系综和辛系综之间。

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.