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 one of the recent
papers of Erd\"os-Yau-Yin. We also use an algebraic description of the law of
the anticommutator of free semicircular variables due to Nica-Speicher, a
self-adjointness-preserving variant of the linearization trick due to
Haagerup-Schultz-Thorbj\o rnsen, and the Schwinger-Dyson equation. A byproduct
of our work is a relatively simple deterministic version of the local
semicircle law.Comment: 33 pages, LaTeX, 2 figures. In v2 (this version) we make minor
  revisions, add references and correct typo

Anderson, Greg W.

English

arXiv

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 byproduct of our work is a relatively simple deterministic version of the local semicircle law. Content

Greg W. Anderson

CiteSeerX

A LOCAL LIMIT LAW FOR THE EMPIRICAL SPECTRAL DISTRIBUTION OF THE ANTICOMMUTATOR OF INDEPENDENT WIGNER MATRICES

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. Résumé. 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. MSC: 60B20; 15B52; 46L5

A local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices

Abstract

Similar works

Full text

Available Versions

CiteSeerX

CiteSeerX

CiteSeerX

Numérisation de Documents Anciens Mathématiques

Crossref