Limiting Eigenvalue Distribution and Outliers of Deformed Single Ring Random Matrices

Explore advanced mathematical concepts in this 39-minute research lecture where Ping Zhong from the University of Houston delves into the analysis of deformed single ring random matrices. Examine the convergence properties of eigenvalue distributions in square random matrices of the form A + Y, where A represents a deterministic matrix and Y exhibits invariance under unitary group actions. Learn about the single ring theorem by Guionnet, Krishnapur and Zeitouni, and understand how eigenvalue distribution of Y converges to the Brown measure of an R-diagonal operator T. Discover the conditions under which the eigenvalue distribution of A+Y converges to the Brown measure of the operator a+T, and investigate the behavior of outliers when A's eigenvalues exist outside the support of the limit probability measure. Study this collaborative research work conducted with Hari Bercovici, Ching-Wei Ho, and Zhi Yin, which extends Benaych-Georges and Rochet's findings on finite rank cases.