Type: Article
Publication Date: 2022-07-23
Citations: 3
DOI: https://doi.org/10.1007/s00440-022-01156-7
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 $$H+xA$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> 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.