CAREER: Brown Measure and non-Hermitian Random Matrices

This CAREER award supports a five-year project in free probability, random matrix theory, and their applications. Free probability began as a subfield of the theory of von Neumann algebras, which in turn originated in a series of papers by Murray and von Neumann in the 1930s, as part of an effort to provide mathematical foundations for quantum mechanics.

A key concept in free probability is the notion of free independence, which generalizes the classical notion of independence of random variables, enabling the development of a robust “noncommutative probability theory” in the setting of von Neumann algebras, where rich mathematical structure emerges from the interactions of objects known as free random variables (owing in part to the fact that their multiplication, like that of matrices of complex numbers, is not commutative). Methods from free probability are now a cornerstone of the structure theory of certain von Neumann algebras, and have applications to a variety of other fields, including random matrix theory and quantum information theory.

In this project, the PI will study several important open problems on probability distributions of free random variables and explore their applications to random matrix models. The project integrates research with education and will provide research opportunities for both graduate and undergraduate students, complemented by outreach initiatives to teach and mentor middle school and high school students, with a particular emphasis on supporting students from underrepresented backgrounds.

The Brown measure of a free random variable is analogous to the eigenvalue counting measure of a square matrix. It is an extension of spectral measures of normal operators to non-normal operators. This measure encodes a great deal of information and can predict the limiting distributions of non-Hermitian random matrix models.

In this project, the PI seeks to develop analytic techniques for computing the Brown measures of a diverse range of free random variables, motivated by operator algebras, random matrix theory, and high-dimensional statistics. The boundary values of certain operator-valued subordination functions will be explored using tools in complex analysis, operator algebras, and noncommutative analysis.

The project will investigate random variables formed through addition, multiplication, or polynomials of free random variables. The new Brown measure results will then be used to analyze limiting laws and the convergence of non-Hermitian random matrix 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.