We prove that de Branges spaces of entire functions describe universality limits in the bulk for random matrices, in the unitary case. In particular, under mild conditions on a measure with compact support, we show that each possible universality limit is the reproducing kernel of a de Branges space of entire functions that equals a classical Paley-Wiener space. We also show that any such reproducing kernel, suitably dilated, may arise as a universality limit for sequences of measures on [- 1, 1]. © 2009 Elsevier Inc. All rights reserved.
Lubinsky, D. S. (2009). Universality limits for random matrices and de Branges spaces of entire functions. Journal of Functional Analysis, 256(11), 3688–3729. https://doi.org/10.1016/j.jfa.2009.02.021