Universality of Random Matrices and Local Relaxation Flow

Abstract

We consider $N\times N$ symmetric random matrices where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. We prove that the eigenvalue spacing statistics in the bulk of the spectrum for these matrices and for GOE are the same in the limit $N\to \infty$. Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow.

A minor error in the proof of Lemma 2.2 has been corrected and an additional technical condition was added to Cor. 2.4. Final version with some more details in Prop 4.1 was added on Nov 24

References in corpus (5)

• Bulk universality for generalized Wigner matrices
• Random matrices: Universality of local eigenvalue statistics
• Rigidity of Eigenvalues of Generalized Wigner Matrices
• The local relaxation flow approach to universality of the local statistics for random matrices
• Universality for generalized Wigner matrices with Bernoulli distribution

Cited by in corpus (20)

• Random matrices: Universality of local eigenvalue statistics up to the edge
• Universality of covariance matrices
• Random matrices: Universality of local spectral statistics of non-Hermitian matrices
• Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
• The local relaxation flow approach to universality of the local statistics for random matrices
• Eigenvector Distribution of Wigner Matrices
• Random Schrodinger operators on long boxes, noise explosion and the GOE
• Higher Dimensional Coulomb Gases and Renormalized Energy Functionals
• Universality for generalized Wigner matrices with Bernoulli distribution
• Random matrices: The Four Moment Theorem for Wigner ensembles
• Random matrices: Universal properties of eigenvectors
• The Isotropic Semicircle Law and Deformation of Wigner Matrices
• From the Anderson model on a strip to the DMPK equation and random matrix theory
• Random matrices: Sharp concentration of eigenvalues
• Universality of sample covariance matrices: CLT of the smoothed empirical spectral distribution
• Zooming in on local level statistics by supersymmetric extension of free probability
• 1D Log Gases and the Renormalized Energy: Crystallization at Vanishing Temperature
• Spectral Properties of Wigner Matrices
• The asymptotic distribution of a single eigenvalue gap of a Wigner matrix
• The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices