Janossy densities for Unitary ensembles at the spectral edge

TL;DR

This paper proves the universal behavior of Janossy densities at the spectral edge of various unitary random matrix ensembles, using orthogonal polynomials and Riemann-Hilbert analysis.

Contribution

It introduces a new formulation of eigenvalue distribution probabilities at the spectral edge for broad classes of unitary ensembles.

Findings

Janossy densities exhibit universal behavior at the spectral edge.

The method employs orthogonal polynomials and Riemann-Hilbert problem asymptotics.

Results apply to a wide class of unitary ensembles.

Abstract

For a broad class of unitary ensembles of random matrices we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting second, third, etc. largest eigenvalues of the ensembles in question. The approach is based on a representation of the Janossy densities in terms of a system of orthogonal polynomials, plus the steepest descent method of Deift and Zhou for the asymptotic analysis of the associated Riemann-Hilbert problem.