We prove universality for a large class of random matrices with correlated entries. This very general result has been used numerous times, also in more applied research.
We prove that also the edge statistics of general correlated Hermitian random matrices are universal. Our result implies that the eigenvalues in each support interval of the asymptotic density is deterministic with high probability.
Hermitian random matrices can only feature three universal eigenvalue statistics: Sine in the bulk, Airy at square-root edges and Pearcey at cubic-root cusps. In this work we complete the universality program for those matrices by proving Pearcey-kernel universality at all cusp points.
This is the companion result to the previous paper on cusp universality for complex Hermitian matrices. Lacking Brézin-Hikami-type integral formulae we resort to the more robust Dyson Brownian Motion approach.
We prove that on the unit circle (the asymptotic boundary of the spectrum) the local eigenvalue statistics on non-Hermitian random matrices with IID entries are universal. This generalizes previous results on random matrices matching four Gaussian moments.
We show that the linear statistics of random matrices with IID entries asymptotically are a rank-one perturbation of the Gaussian free field on the unit disc.
Following Eugene Wigner's original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability.
In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have been restricted only to two exactly solvable models. We rigorously prove the physics prediction on SFF up to an intermediate time scale for a large class of random matrices.
