Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 078, 36 pages      arXiv:1604.03133      https://doi.org/10.3842/SIGMA.2016.078
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles

Doron S. Lubinsky
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160 USA

Received April 05, 2016, in final form August 05, 2016; Published online August 10, 2016

Abstract
We survey the current status of universality limits for $m$-point correlation functions in the bulk and at the edge for unitary ensembles, primarily when the limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider underlying measures on compact intervals, and fixed and varying exponential weights, as well as universality limits for a variety of orthogonal systems. The scope of the survey is quite narrow: we do not consider $\beta$ ensembles for $\beta \neq 2$, nor general Hermitian matrices with independent entries, let alone more general settings. We include some open problems.

Key words: orthogonal polynomials; random matrices; unitary ensembles; correlation functions; Christoffel functions.

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