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), p. 809-841 / Harvested from Numdam

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.

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.

Détail

Publié le : 2015-01-01

MR 3365962

DOI : https://doi.org/10.1214/14-AIHP602

Bibliographie

[1] G. Anderson. Convergence of the largest singular value of a polynomial in independent Wigner matrices. Ann. Probab. 41 (2013) 2103–2181. | MR 3098069 | Zbl 1282.60007

[2] G. Anderson. Support properties of spectra of polynomials in Wigner matrices. Lecture notes, June 2012. Available at z.umn.edu/selfadjlintrick. Retrieved December 28, 2013.

[3] G. Anderson, A. Guionnet and O. Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge, 2010. | MR 2760897 | Zbl 1184.15023

[4] S. T. Belinschi, T. Mai and R. Speicher. Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem. Available at arXiv:1303.3196.

[5] A. Deya and I. Nourdin. Convergence of Wigner integrals to the tetilla law. ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012) 101–127. | MR 2893412 | Zbl 1285.46053

[6] L. Erdös, A. Knowles, H.-T. Yau and J. Yin. Spectral statistics of Erdös–Rényi graphs I: Local semicircle law. Ann. Probab. 41 (2013) 2279–2375. | MR 3098073 | Zbl 1272.05111

[7] L. Erdös, A. Knowles, H.-T. Yau and J. Yin. The local semicircle law for a general class of random matrices. Electron. J. Probab. 18 (2013) 1–58. | MR 3068390 | Zbl 06247228

[8] L. Erdös, H.-T. Yau and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Adv. Math. 229 (2012) 1435–1515. | MR 2871147 | Zbl 1238.15017

[9] L. Erdös, H.-T. Yau and J. Yin. Bulk universality for generalized Wigner matrices. Probab. Theory Related Fields 154 (2012) 341–407. | MR 2981427 | Zbl 1277.15026

[10] Z. Füredi and J. Komlós. The eigenvalues of random symmetric matrices. Combinatorica 1 (1981) 233–241. | Zbl 0494.15010

[11] U. Haagerup, H. Schultz and S. Thorbjørnsen. A random matrix approach to the lack of projections in $C^{*}_{\mathrm {red}}(\mathbb {F}_{2})$. Adv. Math. 204 (2006) 1–83. | MR 2233126 | Zbl 1109.15020

[12] U. Haagerup and S. Thorbjørnsen. A new application of random matrices: $\operatorname{Ext}(C^{*}(\mathbb {F}_{2}))$ is not a group. Ann. of Math. (2) 162 (2005) 711–775. | MR 2183281 | Zbl 1103.46032

[13] J. Helton, R. Rashidi Far and R. Speicher. Operator-valued semicircular elements: Solving a quadratic matrix equation with positivity constraints. Int. Math. Res. Not. IMRN 22 (2007) Art. ID rnm086. | MR 2376207 | Zbl 1139.15006

[14] R. Horn and C. Johnson. Matrix Analysis, corrected reprint of the 1985 original. Cambridge Univ. Press, Cambridge, 1990. | MR 1084815 | Zbl 0704.15002

[15] A. Nica and R. Speicher. Commutators of free random variables. Duke Math. J. 92 (1998) 553–592. | MR 1620518 | Zbl 0968.46053

[16] A. Nica and R. Speicher. Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335. Cambridge Univ. Press, Cambridge, 2006. | MR 2266879 | Zbl 1133.60003

[17] M. Rudelson and R. Vershynin. The Hanson–Wright inequality and sub-Gaussian concentration. Electron. Commun. Probab. 18 (2013) 1–9. | MR 3125258 | Zbl 06346931

[18] D. Shlyakhtenko and P. Skoufranis. Freely independent random variables with non-atomic distributions. Available at arXiv:1305.1920. | MR 3356937

[19] B. Simon. Trace Ideals and Their Applications, 2nd edition. Mathematical Surveys and Monographs 120. Amer. Math. Soc., Providence, RI, 2005. | MR 2154153 | Zbl 1074.47001

[20] T. Tao and V. Vu. Random matrices: Universality of local eigenvalue statistics up to the edge. Comm. Math. Phys. 298 (2010) 549–572. | MR 2669449 | Zbl 1202.15038

[21] T. Tao and V. Vu. Random matrices: Universality of local eigenvalue statistics. Acta Math. 206 (2011) 127–204. | MR 2784665 | Zbl 1217.15043

[22] D. V. Voiculescu, K. J. Dykema and A. Nica. Free Random Variables. A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups. CRM Monograph Series 1. Amer. Math. Soc., Providence, RI, 1992. | MR 1217253 | Zbl 0795.46049

[23] V. H. Vu. Spectral norm of random matrices. Combinatorica 27 (2007) 721–736. | MR 2384414 | Zbl 1164.05066

[24] V. Vu and K. Wang. Random weighted projections, random quadratic forms and random eigenvectors. Available at arXiv:1306.3099.

[25] P. Whittle. Bounds for the moments of linear and quadratic forms in independent variables. Teor. Veroyatn. Primen. 5 (1960) 331–335. Transl. Theory Probab. Appl. 5 (1960) 303–305. | MR 133849 | Zbl 0101.12003