Eigenvalue Rigidity of Random Unitary Matrices

Description

Random matrices are matrices whose entries are random variables. They play a significant role in both theoretical and applied studies of various different scientific disciplines, such as number theory, integrable systems, free probability, combinatorics, statistics, string theory, condense matter physics, classical and quantum optics, complex networks, mathematical finance, wireless communications, etc. One of the central problems in random matrix theory is to study properties of their eigenvalues, especially when the matrix size becomes large. On one hand, the position of the eigenvalues varies due to the randomness; on the other hand, for a large class of random matrices, the eigenvalues are close to their deterministic limits when the matrix size tends to infinity. The second property is referred as eigenvalue rigidity in random matrix theory, and has attracted a lot of research interests recently. Some bounds for the eigenvalue fluctuation have been established for generalized Wigner matrices. The optimal lower and upper bounds have also been proved for the one-cut regular random unitary matrices. Although breakthrough has been made, there are still many important problems unsolved. In this project, we plan to study eigenvalue rigidity of random unitary matrices. We will show that the eigenvalue counting function can lead to Gaussian multiplicative chaos when the matrix size tends to infinity. Then, by studying the fractal properties of the Gaussian multiplicative chaos measure, we will establish both the upper and lower bounds for the maximum eigenvalue fluctuation. Moreover, we wish to prove that these bounds are optimal.