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.
The study of complicated random system as the system size become large is an important subject in mathematics as well as in other parts of science and engineering. This project was aimed at improving our understanding of such systems by studying a class of systems which can be analyzed by various mathematical tools. The PI conducted researches on fundamental questions toward this goal. The PI collaborated with various colleagues and obtained several results. These results were published in research journals. The PI also presented the results obtained at many conferences and workshops. Some parts of the project were collaborated with postdocs and a graduate student, thereby giving them opportunities to engage in active researches. In more specific terms, the project was about random matrices which are particular types of complicated random systems. The random matrices are natural and fundamental systems of interest in mathematics and other sciences. The PI obtained results on the basic properties of random matrices by studying the interactions between random matrices and non-random matrices, by studying other combinatorial system which give rise random matrix-like behavior, and by studying relationships between certain measurable quantities arising from random matrices. Some of the results give rise new interesting questions to be answered.
