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

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 $\beta \ne 2$, nor general Hermitian matrices with independent entries, let alone more general settings. We include some open problems.

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Cited by in corpus (9)

• Periodic perturbations of unbounded Jacobi matrices III: The soft edge regime
• Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients
• Large gap asymptotics for the generating function of the sine point process
• Asymptotic behaviour of Christoffel-Darboux kernel via three-term recurrence relation II
• Spectral properties of block Jacobi matrices
• Asymptotic behaviour of Christoffel-Darboux kernel via three-term recurrence relation I
• Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case
• Christoffel functions for multiple orthogonal polynomials
• Nevai's condition for measures with unbounded supports