Quantitative gap universality for Wigner matrices

2025/07/28 by A. Q. Zhang, Zhang, Albert · 1 citation
Mathematics · Computer Science · #Random Matrices and Applications #Spectral Theory in Mathematical Physics #Quantum Information and Cryptography

paper · pdf · doi:10.48550/arxiv.2507.20442

Abstract

We obtain the explicit rate of convergence N^{-1/2 + ε} for the gaps of generalized Wigner matrices in the bulk of the spectrum, for distributions of matrix entries possibly atomic and supported on enough points. The proof proceeds by a Green function comparison, coupled with the relaxation estimate from [5]. In particular, we extend the 4 moment matching method [33] to arbitrary moments, allowing to compare resolvents down to the submicroscopic scale N^{-3/2 + ε}. This method also gives universality of the smallest gaps between eigenvalues for the Hermitian symmetry class, providing a universal, optimal separation of eigenvalues for discrete random matrices with entries supported on Ω(1) points.

Citations

• Quantitative Tracy-Widom laws for sparse random matrices
• Small gaps of GSE
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• Rigidity of Eigenvalues of Generalized Wigner Matrices
• Universality for generalized Wigner matrices with Bernoulli distribution
• Universality of Random Matrices and Local Relaxation Flow
• Random matrices: Universality of local eigenvalue statistics
• Random matrices: Universality of local eigenvalue statistics
• Statistics of Extreme Spacings in Determinantal Random Point Processes

Cited by

• Smallest gaps of the two-dimensional Coulomb gas

Related

• Extreme gaps between eigenvalues of Wigner matrices
• Bulk universality for Wigner hermitian matrices with subexponential decay
• The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices
• Gap Universality of Generalized Wigner and beta-Ensembles
• Wegner estimate and level repulsion for Wigner random matrices