Universality of random matrices and local relaxation flow
László Erdös
Schlein, Benjamin
Yau, Horng-Tzer
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.
Springer
2011
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doc-type:article
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https://research-explorer.app.ist.ac.at/record/2764
Erdös L, Schlein B, Yau H. Universality of random matrices and local relaxation flow. <i>Inventiones Mathematicae</i>. 2011;185(1):75-119. doi:<a href="https://doi.org/10.1007/s00222-010-0302-7">10.1007/s00222-010-0302-7</a>
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00222-010-0302-7
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