Global Asymptotics for the Christoffel-Darboux Kernel of Random Matrix Theory

Abstract

The investigation of universality questions for local eigenvalue statistics continues to be a driving force in the theory of Random Matrices. For Matrix Models [53] the method of orthogonal polynomials can be used and the asymptotics of the Christoffel-Darboux kernel [59] become the key for studying universality. In this paper the existing results on the CD-kernel will be extended in two directions. Firstly, in order to analyze the transition from the universal to the non-universal regime, we provide leading order asymptotics that are global rather than local. This allows e.g. to describe the moderate deviations for the largest eigenvalues of unitary ensembles ($β$ = 2), where such a transition occurs. Secondly, our asymptotics will be uniform under perturbations of the probability measure that defines the matrix ensemble. Such information is useful for the analysis of a different type of ensembles [25], which is not known to be determinantal and for which the method of orthogonal polynomials cannot be used directly. The just described applications of our results are formulated in this paper but will be proved elsewhere. As a byproduct of our analysis we derive first order corrections for the 1-point correlation functions of unitary ensembles in the bulk. Our proofs are based on the nonlinear steepest descent method [20]. They follow closely [17] and incorporate improvements introduced in [36, 64]. The presentation is self-contained except for a number of general facts from Random Matrix theory and from the theory of singular integral operators.