Universality of random matrices and local relaxation flow

L. Erdös, B. Schlein, H. Yau, Inventiones Mathematicae 185 (2011) 75–119.

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Abstract
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.
Publishing Year
Date Published
2011-07-01
Journal Title
Inventiones Mathematicae
Volume
185
Issue
1
Page
75 - 119
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Erdös L, Schlein B, Yau H. Universality of random matrices and local relaxation flow. Inventiones Mathematicae. 2011;185(1):75-119. doi:10.1007/s00222-010-0302-7
Erdös, L., Schlein, B., & Yau, H. (2011). Universality of random matrices and local relaxation flow. Inventiones Mathematicae, 185(1), 75–119. https://doi.org/10.1007/s00222-010-0302-7
Erdös, László, Benjamin Schlein, and Horng Yau. “Universality of Random Matrices and Local Relaxation Flow.” Inventiones Mathematicae 185, no. 1 (2011): 75–119. https://doi.org/10.1007/s00222-010-0302-7.
L. Erdös, B. Schlein, and H. Yau, “Universality of random matrices and local relaxation flow,” Inventiones Mathematicae, vol. 185, no. 1, pp. 75–119, 2011.
Erdös L, Schlein B, Yau H. 2011. Universality of random matrices and local relaxation flow. Inventiones Mathematicae. 185(1), 75–119.
Erdös, László, et al. “Universality of Random Matrices and Local Relaxation Flow.” Inventiones Mathematicae, vol. 185, no. 1, Springer, 2011, pp. 75–119, doi:10.1007/s00222-010-0302-7.

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