Operator Limits of Random Matrices

点击翻译按钮获取中文摘要

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.