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August 2015 A local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices
Greg W. Anderson
Ann. Inst. H. Poincaré Probab. Statist. 51(3): 809-841 (August 2015). DOI: 10.1214/14-AIHP602

Abstract

Our main result is a local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices, modeled on the local semicircle law. Our approach is to adapt some techniques from recent papers of Erdös–Yau–Yin. We also use an algebraic description of the law of the anticommutator of free semicircular variables due to Nica–Speicher, the linearization trick due to Haagerup–Schultz–Thorbjørnsen in a self-adjointness-preserving variant and the Schwinger–Dyson equation. A by-product of our work is a relatively simple deterministic version of the local semicircle law.

Notre principal résultat établit la loi limite locale pour la distribution spectrale empirique de l’anti-commutateur de matrices de Wigner indépendantes dans l’esprit de la loi semi-circulaire locale. Notre approche adapte les techniques d’articles récents par Erdös–Yau–Yin. Nous utilisons aussi une description algébrique de la loi de l’anti-commutateur pour des variables libres due à Nica–Speicher, une variante de l’astuce de la linéarisation de Haagerup–Schultz–Thorbjørnsen et l’équation de Schwinger–Dyson. Une conséquence de notre travail est une version déterministe assez simple de la loi semi-circulaire locale.

Citation

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Greg W. Anderson. "A local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices." Ann. Inst. H. Poincaré Probab. Statist. 51 (3) 809 - 841, August 2015. https://doi.org/10.1214/14-AIHP602

Information

Received: 28 August 2013; Revised: 8 January 2014; Accepted: 22 January 2014; Published: August 2015
First available in Project Euclid: 1 July 2015

zbMATH: 1360.60013
MathSciNet: MR3365962
Digital Object Identifier: 10.1214/14-AIHP602

Subjects:
Primary: 15B52, 46L54, 60B20

Rights: Copyright © 2015 Institut Henri Poincaré

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Vol.51 • No. 3 • August 2015
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