Operator Limits of Random Matrices

随机矩阵的算子极限

• 批准号: 1712729
• 负责人: Brian Rider
• 金额: $ 19万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712729&HistoricalAwards=false
• 关键词: Operator; Limits; Random; Matrices

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.

随机矩阵理论产生于数理统计和理论物理的基本考虑，现在在数论、统计力学、无线通信和生物信息学等不同领域都有应用。所谓的特蕾西-威登定律是该领域影响范围的一个完美例子。这些定律最初是在对某些大型随机矩阵的最大特征值的描述中发现的，现在被理解为为广泛的现代非线性现象提供“钟形曲线”的作用（或提供适当的中心极限定理）。本研究的主要目的是通过研究随机矩阵来提高我们对这些和其他概率律的理解。重点是它们的基本数学性质，以及扩展它们所应用的模型类别。该研究部分建立在PI最近对tracy - wisdom和相关发行版的通用beta扩展的工作上。这些扩展的优点是将更熟悉的三重极限定律（与正交、酉和辛对称类有关）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特的极限定理的描述，例如高阶Tracy-Widom定律和各种双尺度制度（再次在一般的beta环境中）。其他项目围绕着最近建立的集成几何布朗运动的经典Dufresne恒等式的矩阵版本，以及可解的矩阵模型，该模型在高斯正交和辛系综之间插值，可以被视为多类型库仑气体。

项目成果

Universality of the Stochastic Bessel Operator

• DOI: 10.1007/s00440-018-0888-z
• 发表时间: 2016-10
• 期刊: Probability Theory and Related Fields
• 影响因子: 2
• 作者: B. Rider;Patrick Waters