The local relaxation flow approach to universality of the local statistics for random matrices
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 1-46.

Nous présentons une généralisation de la méthode du flot de relaxation locale servant à établir l'universalité des statistiques spectrales locales d'une vaste classe de grandes matrices aléatoires. Nous démontrons que la distribution locale des valeurs propres coïncide avec celle de l'ensemble gaussien pourvu que la loi des coefficients individuels de la matrice soit lisse et que les valeurs propres { x j } j = 1 N soient près de leurs quantiles classiques { γ j } j = 1 N determinées par la densité limite des valeurs propres. Dans la normalisation où la distance typique entre les valeurs propres voisines est d'ordre 1/N , la borne a priori nécessaire sur la position des valeurs propres nécessite uniquement l'établissement de 𝔼 | x j - γ j | 2 N - 1 - ε en moyenne. Cette information peut être obtenue par des méthodes bien établies pour divers ensembles de matrices. Nous illustrons la méthode en démontrant l'universalité spectrale locale pour des matrices de covariance.

We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues { x j } j = 1 N are close to their classical location { γ j } j = 1 N determined by the limiting density of eigenvalues. Under the scaling where the typical distance between neighboring eigenvalues is of order 1/N , the necessary apriori estimate on the location of eigenvalues requires only to know that 𝔼 | x j - γ j | 2 N - 1 - ε on average. This information can be obtained by well established methods for various matrix ensembles. We demonstrate the method by proving local spectral universality for sample covariance matrices.

DOI : 10.1214/10-AIHP388
Classification : 15B52, 82B44
Mots clés : random matrix, sample covariance matrix, Wishart matrix, Wigner-Dyson statistics
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Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer; Yin, Jun. The local relaxation flow approach to universality of the local statistics for random matrices. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 1-46. doi : 10.1214/10-AIHP388. http://www.numdam.org/articles/10.1214/10-AIHP388/

[1] D. G. Aronson. Removable singularities for linear parabolic equations. Arch. Ration. Mech. Anal. 17 (1964) 79-84. | MR | Zbl

[2] D. Bakry and M. Émery. Diffusions hypercontractives. In Séminaire de probabilités, XIX, 1983/84 177-206. Lecture Notes in Math. 1123. Springer, Berlin, 1985. | EuDML | Numdam | MR | Zbl

[3] G. Ben Arous and S. Péché. Universality of local eigenvalue statistics for some sample covariance matrices. Comm. Pure Appl. Math. LVIII (2005) 1-42. | MR | Zbl

[4] G. Ben Arous and S. Péché. Private communication.

[5] P. Bleher and A. Its. Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. 150 (1999) 185-266. | EuDML | MR | Zbl

[6] E. Brézin and S. Hikami. Correlations of nearby levels induced by a random potential. Nucl. Phys. B 479 (1996) 697-706; Spectral form factor in a random matrix theory. Phys. Rev. E 55 (1997) 4067-4083. | MR | Zbl

[7] P. Deift. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Math. 3. Amer. Math. Soc., Providence, RI, 1999. | MR | Zbl

[8] P. Deift and D. Gioev. Random Matrix Theory: Invariant Ensembles and Universality. Courant Lecture Notes in Math. 18. Amer. Math. Soc., Providence, RI, 2009. | MR | Zbl

[9] P. Deift, T. Kriecherbauer, K. T.-R. Mclaughlin, S. Venakides and X. Zhou. Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (1999) 1335-1425. | MR | Zbl

[10] P. Deift, T. Kriecherbauer, K. T.-R. Mclaughlin, S. Venakides and X. Zhou. Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 (1999) 1491-1552. | MR | Zbl

[11] I. Dumitriu and A. Edelman. Matrix models for beta-ensembles. J. Math. Phys. 43 (2002) 5830-5847. | MR | Zbl

[12] F. J. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 (1962) 1191-1198. | MR | Zbl

[13] F. J. Dyson. Correlations between eigenvalues of a random matrix. Comm. Math. Phys. 19 (1970) 235-250. | MR | Zbl

[14] L. Erdős, G. Péché, J. Ramírez, B. Schlein and H.-T. Yau. Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63 (2010) 895-925. | MR | Zbl

[15] L. Erdős, J. Ramírez, B. Schlein, T. Tao, V. Vu and H.-T. Yau. Bulk universality for Wigner Hermitian matrices with subexponential decay. Int. Math. Res. Not. 2010 (2010) 436-479. | MR | Zbl

[16] L. Erdős, J. Ramirez, B. Schlein and H.-T. Yau. Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electron. J. Probab. 15 (2010) 526-604. | MR | Zbl

[17] L. Erdős, B. Schlein and H.-T. Yau. Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37 (2009) 815-852. | MR | Zbl

[18] L. Erdős, B. Schlein and H.-T. Yau. Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287 (2009) 641-655. | MR | Zbl

[19] L. Erdős, B. Schlein and H.-T. Yau. Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. 2010 (2010) 436-479. | MR | Zbl

[20] L. Erdős, B. Schlein and H.-T. Yau. Universality of random matrices and local relaxation flow. Preprint. Available at arXiv:0907.5605. | MR | Zbl

[21] L. Erdős, H.-T. Yau and J. Yin. Bulk universality for generalized Wigner matrices. Preprint. Available at arXiv:1001.3453. | MR | Zbl

[22] O. Feldheim and S. Sodin. A universality result for the smallest eigenvalues of certain sample covariance matrices. Preprint. Available at arXiv:0812.1961. | MR | Zbl

[23] P. Forrester. Log-Gases and Random Matrices. London Mathematical Socity Monographs Series 34. Princeton Univ. Press, Princeton, NJ. | MR | Zbl

[24] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin, 1994. | MR | Zbl

[25] A. Guionnet. Large random matrices: Lectures on macroscopic asymptotics. In École d'Été de Probabilités de Saint-Flour XXXVI-2006. Lecture Notes in Math. 1957. Springer, Berlin, 2009. | MR | Zbl

[26] K. Johansson. Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215 (2001) 683-705. | MR | Zbl

[27] K. Johansson. Universality for certain Hermitian Wigner matrices under weak moment conditions. Preprint. Available at arXiv:0910.4467. | Numdam | MR | Zbl

[28] M. Ledoux. The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI, 2001. | MR | Zbl

[29] V. A. Marchenko and L. Pastur. The distribution of eigenvalues in a certain set of random matrices. Mat. Sb. 72 (1967) 507-536. | MR | Zbl

[30] M. L. Mehta. Random Matrices. Academic Press, New York, 1991. | MR | Zbl

[31] M. L. Mehta and M. Gaudin. On the density of eigenvalues of a random matrix. Nuclear Phys. 18 (1960) 420-427. | MR | Zbl

[32] L. Pastur and M. Shcherbina. Bulk universality and related properties of Hermitian matrix models. J. Stat. Phys. 130 (2008) 205-250. | MR | Zbl

[33] S. Péché. Universality results for largest eigenvalues of some sample covariance matrix ensembles. Probab. Theory Related Fields 143 (2009) 481-516. | MR | Zbl

[34] A. Ruzmaikina. Universality of the edge distribution of eigenvalues of Wigner random matrices with polynomially decaying distributions of entries. Comm. Math. Phys. 261 (2006) 277-296. | MR | Zbl

[35] Y. Sinai and A. Soshnikov. A refinement of Wigner's semicircle law in a neighborhood of the spectrum edge. Funct. Anal. Appl. 32 (1998) 114-131. | MR | Zbl

[36] S. Sodin. The spectral edge of some random band matrices. Preprint. Available at arXiv:0906.4047. | MR | Zbl

[37] A. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999) 697-733. | MR | Zbl

[38] S. Péché and A. Soshnikov. Wigner random matrices with non-symmetrically distributed entries. J. Stat. Phys. 129 (2007) 857-884. | MR | Zbl

[39] T. Tao and V. Vu. Random matrices: Universality of the local eigenvalue statistics. Preprint. Available at arXiv:0906.0510. | MR | Zbl

[40] T. Tao and V. Vu. Random matrices: Universality of local eigenvalue statistics up to the edge. Preprint. Available at arXiv:0908.1982. | MR | Zbl

[41] T. Tao and V. Vu. Random covariance matrices: Universality of local statistics of eigenvalues. Preprint. Available at arXiv:0912.0966. | MR | Zbl

[42] V. Vu. Spectral norm of random matrices. Combinatorica 27 (2007) 721-736. | MR | Zbl

[43] E. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. 62 (1955) 548-564. | MR | Zbl

[44] H. T. Yau. Relative entropy and the hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22 (1991) 63-80. | MR | Zbl

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