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

Anderson, Greg W.

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 809-841.

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.

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.

Mots clés : Schwinger–Dyson equation, Wigner matrices, anticommutators, local semicircle law, stability, linearization trick

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