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Beta-matrix models and random matrix universality

Raghav Tripathi
Monday, November 9, 2020 - 2:30pm to 3:30pm

A major result in random matrix theory, Wigner's semicircle law, states that the empirical spectral distribution of a random symmetric matrix--with i.i.d. entries (on and above diagonal) with finite variance--converge (almost surely) to a deterministic probability measure, namely, semicircle law. A remarkable conclusion of this result is that the behavior of large random matrix ensembles is independent of the precise description of the entries. This phenomenon is called universality. It is to be noted that universality is not limited to the 'global picture', but the local statistics of eigenvalues also exhibit a universal behavior. For instance, the fluctuation of the largest eigenvalue of Wigner matrices (with suitable scaling) is given by Tracy-Widom distribution. Proving such universality results for a large class of matrix ensembles is one of the key focus of research in random matrix theory. In the case of the Gaussian Ensembles (i.e. Gaussian Unitary Ensembles and Gaussian Orthogonal Ensembles), it is possible to write the joint density of eigenvalues explicitly that makes its study amenable. The beta-ensemble is a family of probability densities that is obtained as a natural generalization of these densities. For beta=1 and 2 (with quadratic potential), we obtain the GOE and GUE, respectively. Naturally, one is interested in understanding the local statistics of the particles under beta-ensemble. Various techniques have been employed by various authors to study the beta-ensembles for the different range of beta and with varying assumptions on the potential. In the first part of the talk, we will survey various universality results known in random matrix theory, and in the second part, we would specialize to beta-ensembles. We will describe a powerful approach--developed by Bekerman, Figalli, and Guionnet--to prove the universality in beta-ensembles that is based on constructing approximate transport maps between two beta-ensembles.

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