Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Universality for random matrix flows with time-dependent density

László Erdős and Kevin Schnelli

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Abstract

We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion of the eigenvalues hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from Wigner’s semicircle law.

Résumé

Nous démontrons que le mouvement Brownien de Dyson établit l’universalité des statistiques spectrales locales après un temps très court, en supposant la rigidité locale et la répulsion de valeurs propres. Ces conditions sont satisfaites, et donc l’universalité spectrale est démontrée au centre du spectre, pour une large classe des matrices aléatoires du type Wigner, y compris les ensembles de Wigner deformés et des ensembles dont la matrice des variances est non-stochastique, dont les densités asymptotiques diffèrent de la loi du demi-cercle de Wigner.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1606-1656.

Dates
Received: 28 September 2015
Revised: 20 April 2016
Accepted: 11 May 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773721

Digital Object Identifier
doi:10.1214/16-AIHP765

Mathematical Reviews number (MathSciNet)
MR3729630

Zentralblatt MATH identifier
06847057

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Random matrix Local eigenvalue statistics Universality Dyson Brownian motion

Citation

Erdős, László; Schnelli, Kevin. Universality for random matrix flows with time-dependent density. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1606--1656. doi:10.1214/16-AIHP765. https://projecteuclid.org/euclid.aihp/1511773721


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