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.
