Random matrices: Universality of local eigenvalue statistics

2009/06/02 by Terence Tao, Van Vu, Tao, Terence +1 · 20 citations
Mathematics · #15A52 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #math.PR #msc:15A52

paper · pdf · doi:10.48550/arxiv.0906.0510

67 pages; to appear, Acta Math. Some additional corrections and references

openalex publication_date 2009/06/02 · arxiv created 2010/06/29 · arxiv updated 2010/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and k-point correlation and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.

Citations

• Universality for mathematical and physical systems
• On the Distribution of the Roots of Certain Symmetric Matrices
• Bulk Universality for Wigner Matrices

Cited by

• Quantum Mean-Fields Spin Systems in a Random External Field
• Universal scaling limits at the spectral singularity of structured random matrices
• A direct approach to soft and hard edge universality for random normal matrices
• Eigenvector Geometry as a New Route to Criticality in Random Multiplicative Systems
• Random Matrices, Intrinsic Freeness, and Sharp Non-Asymptotic Inequalities
• Convergence of local eigenvector processes of generalized Wigner matrices
• Optimal Delocalization for Non--Hermitian Eigenvectors
• Convergence and asymptotic freeness of missing data matrices
• Deformation of semi-circle law for the correlated time series and Phase transition
• The Limiting Spectral Distribution for Sparse Elliptic Random Matrices
• Quantitative gap universality for Wigner matrices
• Limiting spectral distribution for the adjacency matrix of the Watts-Strogatz random graph
• A Random Matrix Theory of Pauli Tomography
• Incremental universality of Wigner random matrices
• Models of Heavy-Tailed Mechanistic Universality
• A Universal Matrix Ensemble that Unifies Eigenspectrum Laws via Neural Network Models
• Eigenvector Distribution of Wigner Matrices
• Rigidity of Eigenvalues of Generalized Wigner Matrices
• SPECTRAL PROPERTIES OF WIGNER MATRICES
• Zooming in on local level statistics by supersymmetric extension of free probability

Related

• Random matrices: Universality of local eigenvalue statistics
• Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge
• Random matrices: Universal properties of eigenvectors
• Universality for certain Hermitian Wigner Matrices under weak moment conditions
• Random matrices: The Universality phenomenon for Wigner ensembles