LEAPS-MPS: Some Applications of Free Probability and Random Matrix Theory

Matrices with random entries arise naturally in physics, statistics, and engineering, and they are designed to describe complicated systems. As the dimensions of these matrices are usually very large, classical tools in linear algebra are inadequate to tackle this situation. One common theme of random matrix theory is that a large family of random matrices shares the same limiting distribution due to universality phenomena. This is an analogue of the central limit theorem in classical probability, where the only requirements for the i.i.d. random variables are some moment conditions. Hence, random matrix theory can make sense of large-scale data under very mild assumptions. Random matrices of large dimension can often be modeled by nonrandom operators living in some abstract operator algebras, where these operators satisfy some highly nontrivial relations characterized by Voiculescu’s free independence. These nonrandom operators are free random variables in free probability theory. The principal investigator will study probability distributions of free random variables and the convergence of suitable random matrix models. The project provides research opportunities for both undergraduate and graduate students.

This project, supported by a LEAPS-MPS award, aims to develop analytic tools for studying fundamental questions regarding the limiting distributions of important random matrix models. These questions are motivated by questions from mathematics, statistics, combinatorics, and quantum information. The Brown measure of a free random variable is a spectral measure that generalizes the eigenvalue distribution of square matrices. One major objective is to develop new techniques for calculating Brown measures, which provide predictions for the limits of non-Hermitian random matrices. The Hermitian reduction method and subordination functions are powerful tools for deriving Brown measure formulas. The new results on Brown measures open the door to the study of random matrix models that were previously inaccessible.

Free probability theory offers a conceptual approach to studying random matrices of large dimensions. The principal investigator will identify the limiting free random variables for various random matrix models arising from high-dimensional statistics and quantum information theory. In particular, the principal investigator will examine the spectrum of the full rank deformed single-ring random matrix model, the autocovariance matrix of time series, and the k-positivity of random tensor networks. Additionally, the PI will explore the theory of epsilon-freeness and investigate its applications to quantum information. This project is funded in part by the NSF Established Program to Stimulate Competitive Research (EPSCoR).

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.

Award Number: 2516951
Principal Investigator: Ping Zhong
Funds Obligated: $148,072
State: TX