Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices

TL;DR

This paper proves that for a specific subclass of hermitian Wigner matrices, the local eigenvalue spacing distribution converges to the universal distribution observed in GUE, confirming a long-standing conjecture.

Contribution

It establishes the universality of local eigenvalue spacing for a subclass of hermitian Wigner matrices, extending the known results beyond Gaussian ensembles.

Findings

Confirmed the conjecture for a subclass of Wigner matrices

Demonstrated convergence to GUE spacing distribution

Extended universality results to broader matrix classes

Abstract

Consider an $N\times N$ hermitian random matrix with independent entries, not necessarily Gaussian, a so called Wigner matrix. It has been conjectured that the local spacing distribution, i.e. the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is, in the limit as $N\to\infty$, the same as that of hermitian random matrices from GUE. We prove this conjecture for a certain subclass of hermitian Wigner matrices.