The aim of this proposal is to study various aspects of random matrix theory and integrable systems. One pat of the project is to study asymptotic properties of various distribution functions arising in random matrix theory using their connection to integrable systems. Another part of the project concerns on the universality questions. These are motivated by problems in statistics and integrable differential equations. The long term goal is to understand a clear picture of the universality property of random matrix theory.
Random matrix theory describes a wide array of objects in mathematics such as number theory, combinatorics, probability, as well as in other disciplines of science like statistics, physics, economics, finance and electrical engineering. This proposal is aimed to study some intrinsic properties of random matrices motivated from such applications to improve our understanding of the field.
This project is about a fundamental research on mathematics, especially on analysis and probability. The random matrix theory was initiated in the 1950's by physicists but over the last many years was found to be relevant in many areas in mathematics and engineering from both theoretical and applicational aspects. The wide arrange of fields in science and engineering to which the random matrix theory has found applications include number theory, combinatorics, random growth models, statistics and wireless communications. This project was aimed at improving our fundamental understanding of random matrix theory and its universal nature. I worked on three topics. The first one was motivated by a statistics set-up in which certain information is hidden under a lot of nosy data. In a previous work, I had worked on the fundamental limit of detecting the information using sample covariance eigenvalues in the so-called spiked model. The first topic was a further generalization of this case and I found certain universal and non-universal limit laws. The second topic I worked on was about a probabilistic model of particles randomly moving on a line under the restriction that they cannot overtake preceding particles. This specific model is called TASEP (totally asymmetric simple exclusion process), and this is one of the most fundamental interacting particle models. There have been great excitement in this area over the last decade since surprisingly it is found to be connected to random matrix theory. I worked on this model when the initial locations of the particles are random and found what happened when one looks at the system in the so-called characteristic line. The third topic was about so-called Painleve equations. These equations are nonlinear differential equations and they are known to be related to random matrix theory. I worked on a certain integral involving Painleve equations that appear in random matrix theory context, and found a simple way to evaluate such integrals. In addition to the above three research topics, I supervised two REU (Research Experiences for Undergraduates) projects. Two undergraduate students undertook separate research projects related to random matrix theory. One used computer simulations in order to understand the behavior of last passage percolation models better and the other solved a certain equilibrium measure problem.
