Random Matrices and Applications

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In random matrix theory, various limiting distribution functions arise and many of them are universal. This project studies the extent of the universality of random matrix distribution functions and also of asymptotic properties of these functions. The project will explore three specific situations in which such functions are expected to appear: random matchings, nonequilibrium interacting particle systems, and Hermitian matrix models with external sources. They illustrate the diversity of the appearance of random matrix theory. The principal investigator also intends to study the asymptotic properties of random matrix distribution functions. The study of the intrinsic properties of random matrix distribution functions is expected to shed light on the universal nature of random matrix theory. The distribution functions from random matrix theory indeed describe a wide variety of objects from both mathematics and other fields of science. In statistics, physics, economics, finance, and electrical engineering, a complicated system is often modeled in terms of random matrices. More importantly, some systems that are not modeled in terms of random matrices do exhibit random-matrix-like behavior when the size of the systems tends to infinity, a curious phenomenon that is known as the "universality" of random matrices. This project will study some of the basic properties of the distribution functions that arise in random matrix theory and also investigate further instances in which such functions arise, all in an effort to understand what it is that makes random matrices so universal. The project will incorporate undergraduates and graduate students into the research activities.