Universality

TL;DR

This paper discusses the concept of universality in eigenvalue spacings of random matrices, exploring standard classes like sine, Airy, Bessel, and their occurrence across different ensembles, along with analytical tools and special cases.

Contribution

It provides a detailed overview of the mathematical framework and tools, such as Riemann-Hilbert problems, for deriving universality in random matrix theory, including non-standard classes at spectral singularities.

Findings

Identification of standard universality classes in random matrices

Application of Riemann-Hilbert analysis to derive universality

Discussion of non-standard universality at spectral singularities

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.