Quenched universality for deformed Wigner matrices

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.