Spacing Problems in Random Matrix Theory

Description

Random matrices are matrices whose entries are random variables. They have many applications in various areas of mathematics, physics and engineering, such as number theory, combinatorics, statistics, string theory, condense matter physics, classical and quantum optics, complex networks, wireless communications, etc.. In random matrix theory, one of statistical quantities of basic interests is the probability that there are certain number of eigenvalues in a given interval. When the number is zero, this is the well-known gap probability. These probabilities give us important information about the spacing of eigenvalues.In this project, we plan to study gap probabilities for various matrix models on different intervals. We will derive the large gap asymptotics of the probability, that is, the asymptotic expansion when the width of the given interval tends to infinity. We will also investigate the transition properties when certain parameter changes together with the width of the gap.