The local relaxation flow approach to universality of the local statistics for random matrices

Abstract

We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues ${x_j}_{j=1}^N$ are close to their classical location ${{\gamma }_j}_{j=1}^N$ determined by the limiting density of eigenvalues. Under the scaling where the typical distance between neighboring eigenvalues is of order 1/N, the necessary apriori estimate on the location of eigenvalues requires only to know that $\E |x_j - γ_j |^2 \le N^{-1-\e}$ on average. This information can be obtained by well established methods for various matrix ensembles. We demonstrate the method by proving local spectral universality for Wishart matrices.

61 pages, added a few explanatory sentences in the introduction, small typos corrected, larger font used

References in corpus (5)

• Local semicircle law and complete delocalization for Wigner random matrices
• Random matrices: Universality of local eigenvalue statistics up to the edge
• Bulk universality for generalized Wigner matrices
• Random matrices: Universality of local eigenvalue statistics
• Universality of Random Matrices and Local Relaxation Flow

Cited by in corpus (18)

• Random matrices: Universality of local spectral statistics of non-Hermitian matrices
• Bulk universality for generalized Wigner matrices
• Rigidity of Eigenvalues of Generalized Wigner Matrices
• Universality of Random Matrices and Local Relaxation Flow
• Eigenvector Distribution of Wigner Matrices
• Random Schrodinger operators on long boxes, noise explosion and the GOE
• Universality for generalized Wigner matrices with Bernoulli distribution
• Random matrices: The Four Moment Theorem for Wigner ensembles
• The Isotropic Semicircle Law and Deformation of Wigner Matrices
• Random matrices: Sharp concentration of eigenvalues
• Universality of sample covariance matrices: CLT of the smoothed empirical spectral distribution
• Average Density of States for Hermitian Wigner Matrices
• Spectral Properties of Wigner Matrices
• Nonintersecting paths with a staircase initial condition
• A concentration inequality and a local law for the sum of two random matrices
• Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
• The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices
• Bulk scaling limit of the Laguerre ensemble