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December 2020 Applications of mesoscopic CLTs in random matrix theory
Benjamin Landon, Philippe Sosoe
Ann. Appl. Probab. 30(6): 2769-2795 (December 2020). DOI: 10.1214/20-AAP1572

Abstract

We present some applications of central limit theorems on mesoscopic scales for random matrices. When combined with the recent theory of “homogenization” for Dyson Brownian motion, this yields the universality of quantities which depend on the behavior of single eigenvalues of Wigner matrices and $\beta$-ensembles. Among the results we obtain are the Gaussian fluctuations of single eigenvalues for Wigner matrices (without an assumption of 4 matching moments) and classical $\beta$-ensembles ($\beta=1,2,4$), Gaussian fluctuations of the eigenvalue counting function, and an asymptotic expansion up to order $o(N^{-1})$ for the expected value of eigenvalues in the bulk of the spectrum. The latter result solves a conjecture of Tao and Vu.

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Benjamin Landon. Philippe Sosoe. "Applications of mesoscopic CLTs in random matrix theory." Ann. Appl. Probab. 30 (6) 2769 - 2795, December 2020. https://doi.org/10.1214/20-AAP1572

Information

Received: 1 December 2018; Revised: 1 November 2019; Published: December 2020
First available in Project Euclid: 14 December 2020

Digital Object Identifier: 10.1214/20-AAP1572

Subjects:
Primary: 60F05

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 6 • December 2020
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