Bulk universality for Wigner matrices

By L. Erdős, S. P'ech'e, J. Ramírez, B. Schlein, Horng-Tzer Yau. 2009

234 citations recorded by the source corpus.

We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e−U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C6( \input amssym $\Bbb R$) with at most polynomially growing derivatives and ν(x) ≥ Ce−C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8f8bb6f01c1b1006c31cfa60e9ef4ccf9d2f5ed3

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Citations in the corpus

The local relaxation flow approach to universality of the local statistics for random matrices (2009, methodology): “2, we apply the reverse heat flow argument, presented first in [14] and used also in Corollary 2.”
Universality of random matrices and local relaxation flow (2009, unknown): “∗Partially supported by SFB-TR 12 Grant of the German Research Council †Partially supported by NSF grants DMS-0757425, 0804279”
Universality of local spectral statistics of random matrices (2011, methodology): “The first approximation result was obtained via the reversal heat flow in [27] which required some smoothness of the distribution of matrix elements.”
Large deviations of the extreme eigenvalues of random deformations of matrices (2010, unknown): “* UPMC Univ Paris 6, LPMA, Case courier 188, 4, Place Jussieu, 75252 Paris Cedex 05, France, florent.benaych@upmc.fr, † UMPA, ENS Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France, aguionne@umpa.ens-lyon.fr, ‡ Université Paris-Sud, Laboratoire de Mathématiques, Bâtiment 425, Faculté des…”
Universality for generalized Wigner matrices with Bernoulli distribution (2010, methodology): “The size of the Gaussian component was reduced to N−1+ε in [16] by using an improved formula for correlation functions and the local semicircle law from [11, 12, 13].”
Eigenvector Statistics of Sparse Random Matrices (2016, background): “For Wigner type random matrices, it is proved in a series of papers [7, 14, 15, 16, 17, 18, 19, 20, 21, 27, 28, 39] for the bulk and [22, 34, 38] for the edge, that the eigenvalue statistics are universal; it is proved in [6, 25, 40] that the eigenvectors are asymptotically normal.”
Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge (2009, methodology): “Remark 1.15. In analogy with [13], one should be able to drop the third moment condition in Theorem 1.13 if one can control the distribution of the largest (or smallest) eigenvalues from random matrices obtained from a suitable Ornstein-Uhlenbeck process, as in [ 12 ].”
A short survey of Stein's method (2014, methodology): “Sophisticated forms of this idea have been used by Erdős, Yau and coauthors in their remarkable series of papers [42, 43, 44, 45] proving universality of random matrix eigenvalue distributions, running parallel to the papers of Tao and Vu, who used the Lindeberg approach.”
Bulk universality for generalized Wigner matrices (2010, methodology): “This tiny Gaussian component can then be removed via a reverse heat flow argument and this proves [17] the bulk universality for Hermitian ensembles provided that the distributions of the matrix elements are sufficiently differentiable.”
Rigidity of eigenvalues of generalized Wigner matrices (2010, methodology): “3 was proved in [23], a similar result, with no error estimate, was obtained in [20] for the hermitian case by using an explicit formula related to Johansson’s formula [26].”
Universality of covariance matrices (2011, methodology): “The approach we take in this paper to prove universality is the one developed in a recent series of papers [10, 11, 13, 12, 14, 16, 17, 18], however there are some important differences which we highlight below.”
Sparse regular random graphs: Spectral density and eigenvectors (2009, background): “o and Vu [55] and Erd˝os et al. [24]; SPARSE REGULAR RANDOM GRAPHS 3 - universality of correlation functions in the bulk, under various assumptions (Tao and Vu [56], Erd˝os et al. [24], Erd˝os et al. [23], Erd˝os, Yau and Yin [27]); - partial or complete delocalization of the eigenvectors (by Erd˝os”
Random matrix theory in statistics: A review (2014, methodology): “Significant new developments on the universality phenomena have been made by Erdős et al. (2010, 2009, 2012), Erdős (2010), Erdős and Yau (2012), and Tao and Vu (2010a,b, 2011, 2012a,b) who used analytical techniques to study the question of both bulk and edge universality and managed to remove…”
Fixed Energy Universality for Generalized Wigner Matrices (2014, background): “The Wigner-Dyson-Mehta conjecture has been widely open until the recent work [10], where a general scheme to approach it was outlined and carried out for complex Hermitian matrices.”