Operator Limits of Random Matrices

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。这项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。这项研究部分建立在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。这里的主要目标是将这种方法推向更奇异的极限定理的描述，例如高阶Tracy-Widom定律和各种双重标度制度(同样在一般的Beta上下文中)。其他项目围绕着最近建立的用于积分几何布朗运动的经典杜弗伦恒等式的矩阵版本，以及可在高斯、正交和辛系综之间插补的可解矩阵模型，可以被视为多种类型的库仑气体。

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 this 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 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 main goal here is to push this approach toward descriptions of more exotic limit theorems, such as the higher order Tracy-Widom laws and various double scaling regimes (again in the general beta context). Additional projects center around a recently established matrix version of the classical Dufresne identity for integrated geometric Brownian motion as well as solvable matrix models which interpolate between the Gaussian Orthogonal and Symplectic Ensembles and can be viewed as multi-type coulomb gases.