Random matrices, operators, and analytic functions

The focus of this project is the study of various limits in random matrix theory. Matrices are one of the most basic mathematical objects. They encode multivariate linear functions and can serve as approximations to more complicated multivariate functions. The eigenvalues of a square matrix describe how the corresponding linear transformation shrinks or stretches the space in certain special directions. One of the central problems in random matrix theory is to understand the scaling properties of the eigenvalues of a growing sequence of randomly chosen square matrices. This is a highly active area of probability, with a number of interesting connections to other fields within mathematics. Graduate and undergraduate students will be involved in the research, including projects at the Madison Experimental Mathematics Lab, and the awardee is involved in developing the graduate and undergraduate probability curriculum at the University of Wisconsin. The project builds on recent results of the awardee and collaborators on random operator representations of limits of random matrices and their characteristic polynomials. The idea is to represent random matrices as certain discretized differential operators and to study the limit of their eigenvalues via the limit of the operators themselves. This approach also provides tools to analyze the limits of characteristic polynomials of random matrices using the framework of functional and stochastic analysis. The project investigates various properties of the random analytic functions obtained as such limits. It also studies various deformations and generalizations of random matrix models that are amenable to the random operator approach, with the goals of discovering new and interesting limit objects and learning more about the original models. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.