Universality of random matrices and local relaxation flow

By L. Erdős, B. Schlein, Horng-Tzer Yau. 2009

280 citations recorded by the source corpus.

Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we prove that the relaxation time to local equilibrium for the Dyson Brownian motion is bounded above by N−ζ for some ζ>0. The proof is based on an estimate of the entropy flow of the Dyson Brownian motion w.r.t. a “pseudo equilibrium measure”. As an application of this estimate, we prove that the eigenvalue spacing statistics in the bulk of the spectrum for N×N symmetric Wigner ensemble is the same as that of the Gaussian Orthogonal Ensemble (GOE) in the limit N→∞. The assumptions on the probability distribution of the matrix elements of the Wigner ensemble are a subexponential decay and some minor restriction on the support.

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

View Paper: https://www.semanticscholar.org/paper/87246ee80c05a7f5d48a1b444436c7659dc40583

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Citations in the corpus

Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues (2011, methodology): “Moreover, we prove the edge universality of Wigner matrices under the assumption that the matrix entries have 12 + ε moments.”
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.”
Anisotropic local laws for random matrices (2014, background): “1007/s00220-008-0636-9 [17] Erdős, L; Schlein, B; Yau, H-T, Universality of random matrices and local relaxation flow, Invent.”
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.”
A necessary and sufficient condition for edge universality of Wigner matrices (2012, methodology): “) For general Wigner matrices, a new approach was introduced to prove the bulk universality in a series of papers by Erdős, Schlein, Yau, and others in [17, 18, 19, 20, 21, 24, 25, 26, 15, 16].”
Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge (2009, background): “See [27,13,10] for some examples of this type of extension.”
Edge universality for deformed Wigner matrices (2014, background): “Another advantage of the method of the present paper is that it does not require that the eigenvalues evolve autonomously under DBM, i.e., that the stochastic differential equations for the eigenvalue and the eigenvectors decouple under DBM.”