Zeros of orthogonal polinomials and universality limits

Abstract

It has been discovered that "old style" techniques from orthogonal polynomials have been very useful in establishing universality results for quite general measures. The main goal of this master thesis is to present some methods recently introduced by D. S. Lubinsky for establishing universality limits of random matrices, in the unitary case, based on orthogonal polynomials and some Hilbert spaces of entire functions. Let be a measure de ned on the real line with compact support. Assume that is absolutely continuous in a neighbourhood of some point x in the support, and that 0 is positive and continuous in a compact subset of that neighbourhood. Theorem 1:1 shows that universality at x is equivalent to universality "along the diagonal". The same equivalence is obtained when the hypothesis involve a Lebesgue type condition, instead of continuity of 0 on a compact subset. Such universality limits can be also described by the reproducing kernel of a de Branges space of entire functions that equals a Paley-Wiener space (Theorem 1:4). In order to study this assertion, we use the theory of entire functions of exponential type and de Branges spaces as background.