Universality limits for random matrices and de Branges spaces of entire functions

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].