First colonization of a hard-edge in random matrix theory

TL;DR

This paper analyzes the initial eigenvalues forming a new spectral band at a hard edge in random Hermitean matrices, showing they follow Laguerre-type statistics through Riemann-Hilbert methods.

Contribution

It introduces a rigorous Riemann-Hilbert analysis approach to describe the first eigenvalues in a new spectral band at the hard edge of random matrix spectra.

Findings

Eigenvalues follow Laguerre-type spectral statistics

Method based on Riemann-Hilbert analysis of orthogonal polynomials

Describes finite-size effects at the spectral edge

Abstract

We describe the spectral statistics of the first finite number of eigenvalues in a newly-forming band on the hard-edge of the spectrum of a random Hermitean matrix model. It is found that in a suitable scaling regime, they are described by the same spectral statistics of a finite-size Laguerre-type matrix model. The method is rigorously based on the Riemann-Hilbert analysis of the corresponding orthogonal polynomials.