First Colonization of a Spectral Outpost in Random Matrix Theory

TL;DR

This paper rigorously analyzes the distribution and behavior of the first eigenvalues in a new spectral band of random Hermitean matrices, using Riemann-Hilbert methods and connecting to Freud polynomials.

Contribution

It introduces a precise Riemann-Hilbert analysis of eigenvalue emergence in random matrix spectra with improved error estimates and detailed asymptotic behavior.

Findings

Eigenvalues approach the nonregularity point at rate N^(-h).

One eigenvalue (stray zero) approaches more slowly.

Provides kernels and transition analysis for eigenvalue counts.

Abstract

We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitean matrix model. The method is rigorously based on the Riemann-Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N^(-2 h) where 1/h = 2 nu+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature. The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general all these newborn zeroes approach the point of nonregularity at the rate N^(-h) whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term: in particular the transition between K and K+1 eigenvalues is analyzed in detail.