Pure and Applied Analysis

Cusp universality for random matrices, II: The real symmetric case

Giorgio Cipolloni, László Erdős, Torben Krüger, and Dominik Schröder

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Abstract

We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices near the cusp points of the eigenvalue density are universal. Together with the companion paper by Erdős et al. (2018, arXiv:1809.03971), which proves the same result for the complex Hermitian symmetry class, this completes the last remaining case of the Wigner–Dyson–Mehta universality conjecture after bulk and edge universalities have been established in the last years. We extend the recent Dyson Brownian motion analysis at the edge by Landon and Yau (2017, arXiv:1712.03881) to the cusp regime using the optimal local law by Erdős et al. (2018, arXiv:1809.03971) and the accurate local shape analysis of the density by Ajanki et al. (2015, arXiv:1506.05095) and Alt et al. (2018, arXiv:1804.07752). We also present a novel PDE-based method to improve the estimate on eigenvalue rigidity via the maximum principle of the heat flow related to the Dyson Brownian motion.

Article information

Source
Pure Appl. Anal., Volume 1, Number 4 (2019), 615-707.

Dates
Received: 28 January 2019
Revised: 17 June 2019
Accepted: 22 July 2019
First available in Project Euclid: 29 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.paa/1572314427

Digital Object Identifier
doi:10.2140/paa.2019.1.615

Mathematical Reviews number (MathSciNet)
MR4026551

Zentralblatt MATH identifier
07142203

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 15B52: Random matrices

Keywords
cusp universality Dyson Brownian motion local law

Citation

Cipolloni, Giorgio; Erdős, László; Krüger, Torben; Schröder, Dominik. Cusp universality for random matrices, II: The real symmetric case. Pure Appl. Anal. 1 (2019), no. 4, 615--707. doi:10.2140/paa.2019.1.615. https://projecteuclid.org/euclid.paa/1572314427


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