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Generalizations for Strongly Repelling Particles on the Unit Circle and Linear Statistics for Circular Beta-Ensembles

Student-Run Applied & Math Seminar

Speaker: Joshua Sumpter, UC Davis
Location: 3106 MSB
Start time: Fri, May 18 2018, 12:10PM

The study of random matrices first gained attention in the 1950s when Eugene Wigner proposed that the behavior of heavy nuclei could be modeled by the eigenvalues of a large random Hermitian matrix. Since then, a variety of important ensembles in random matrix theory have been introduced such as the Gaussian Ensembles(GOE,GUE, and GSE) and the Circular Ensembles(COE, CUE, and, CSE). In particular, GUE and CUE gained major attention among mathematicians when F. Dyson pointed out a strong connection between eigenphase statistics for GUE/CUE and statistics for the conjectured zeros of the Riemann zeta function. During this talk, I will give an introduction to some of these classical ensembles and introduce some recent results by A. Soshnikov and Y. Xu regarding strongly repelling particles on the unit circle. I will also discuss some important results regarding linear statistics for Circular Beta-Ensembles. Finally, I will discuss my current research involving generalizing the results given by Soshnikov and Xu as well as a proposed plan to prove a generalized Central Limit Theorem(CLT) for linear statistics in the Circular Beta-Ensembles with respect to the mesoscopic regime.



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