Tags → #universality

Random matrices with slow correlation decay
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.

Correlated random matrices: Band rigidity and edge universality
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.

Cusp universality for random matrices I: local law and the complex Hermitian case
Hermitian random matrices can only feature three universal eigenvalue statistics: Sine in the bulk, Airy at squareroot edges and Pearcey at cubicroot cusps. In this work we complete the universality program for those matrices by proving Pearceykernel universality at all cusp points.

Cusp universality for random matrices, II: The real symmetric case
This is the companion result to the previous paper on cusp universality for complex Hermitian matrices. Lacking BrézinHikamitype integral formulae we resort to the more robust Dyson Brownian Motion approach.

Edge universality for nonHermitian random matrices
We prove that on the unit circle (the asymptotic boundary of the spectrum) the local eigenvalue statistics on nonHermitian random matrices with IID entries are universal. This generalizes previous results on random matrices matching four Gaussian moments.

Central limit theorem for linear eigenvalue statistics of nonHermitian random matrices
We show that the linear statistics of random matrices with IID entries asymptotically are a rankone perturbation of the Gaussian free field on the unit disc.

Quenched universality for deformed Wigner matrices
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 WignerDysonMehta universal statistics with high probability.

On the spectral form factor for random matrices
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.