Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Universality for certain Hermitian Wigner matrices under weak moment conditions

Kurt Johansson

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Abstract

We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These random matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a Wigner matrix. We prove that Tracy–Widom universality holds at the edge in this class of random matrices under the optimal moment condition that there is a uniform bound on the fourth moment of the matrix elements. Furthermore, we show that universality holds in the bulk for Gaussian divisible Wigner matrices if we just assume finite second moments.

Résumé

Nous étudions l’universalité des statistiques locales du spectre des matrices de Wigner hermitiennes divisibles par une gaussienne. Ces matrices aléatoires sont obtenues en ajoutant à une matrice de Wigner hermitienne avec des coefficients indépendants une matrice du GUE indépendante. Nous montrons que la classe d’universalité de la loi de Tracy–Widom pour les valeurs propres extrêmes est vérifiée sous la condition optimale d’une borne uniforme sur le quatrième moment des coefficients de la matrice. De plus, nous démontrons l’universalité des fluctuations dans l’intérieur du spectre dès lors que le second moment est fini.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 1 (2012), 47-79.

Dates
First available in Project Euclid: 23 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1327328014

Digital Object Identifier
doi:10.1214/11-AIHP429

Mathematical Reviews number (MathSciNet)
MR2919198

Zentralblatt MATH identifier
1279.60014

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Wigner matrix Gaussian divisible Optimal moment condition Universality Tracy–Widom distribution

Citation

Johansson, Kurt. Universality for certain Hermitian Wigner matrices under weak moment conditions. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 1, 47--79. doi:10.1214/11-AIHP429. https://projecteuclid.org/euclid.aihp/1327328014


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