Quenched universality for deformed Wigner matrices

TL;DR

This paper proves that the eigenvalue gaps in deformed Wigner matrices exhibit universal statistics, extending the universality results to quenched settings and a monoparametric family, using minimal randomness.

Contribution

It establishes quenched bulk universality for deformed Wigner matrices and a monoparametric family, advancing beyond previous annealed results.

Findings

Eigenvalue gaps follow universal Wigner-Dyson-Mehta statistics.

Universality holds for a single scalar random variable in deformed matrices.

Results are quenched, not just annealed, in the bulk spectrum.

Abstract

Following E. Wigner's original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix $H$ yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices $H+xA$ with a deterministic Hermitian matrix $A$ and a fixed Wigner matrix $H$, just using the randomness of a single scalar real random variable $x$. Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble.