Universality

Abstract: Universality of eigenvalue spacings is one of the basic characteristics of random matrices. We give the precise meaning of universality and discuss the standard universality classes (sine, Airy, Bessel) and their appearance in unitary, orthogonal, and symplectic ensembles. The Riemann-Hilbert problem for orthogonal polynomials is one possible tool to derive universality in unitary random matrix ensembles. An overview is presented of the Deift/Zhou steepest descent analysis of the Riemann-Hilbert problem in the one-cut regular case. Non-standard universality classes that arise at singular points in the spectrum are discussed at the end.