Universality for Eigenvalue Correlations at the Origin of the Spectrum

Abstract

We establish universality of local eigenvalue correlations in unitary random matrix ensembles \({{\frac{{1}}{{Z_n}}|\det M|^{{2\alpha}} e^{{-n{{\rm{ tr}}}\, V(M)}} dM}}\) near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal polynomials associated with |x|2α e − nV(x) have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local eigenvalue correlations have universal behavior described in terms of Bessel functions. We extend this to a much wider class of confining potentials V. Our approach is based on the steepest descent method of Deift and Zhou for the asymptotic analysis of Riemann-Hilbert problems. This method was used by Deift et al. to establish universality in the bulk of the spectrum. A main part of the present work is devoted to the analysis of a local Riemann-Hilbert problem near the origin.