Fine asymptotic behavior in eigenvalues of random normal matrices: Ellipse Case

TL;DR

This paper analyzes the detailed asymptotic behavior of eigenvalues in random normal matrices with elliptical support, revealing boundary corrections, eigenvalue counts outside the droplet, and kernel asymptotics.

Contribution

It provides explicit second-order boundary corrections and eigenvalue distribution estimates for elliptical droplet cases in random normal matrices.

Findings

Eigenvalue density near boundary with second-order corrections

Expected number of eigenvalues outside the droplet computed

Kernel asymptotics show exponentially small corrections in the bulk

Abstract

We consider the random normal matrices with quadratic external potentials where the associated orthogonal polynomials are Hermite polynomials and the limiting support (called droplet) of the eigenvalues is an ellipse. We calculate the density of the eigenvalues near the boundary of the droplet up to the second subleading corrections and express the subleading corrections in terms of the curvature of the droplet boundary. From this result we additionally get the expected number of eigenvalues outside the droplet. We also obtain the asymptotics of the kernel and found that, in the bulk, the correction term is exponentially small. This leads to the vanishing of certain Cauchy transform of the orthogonal polynomial in the bulk of the droplet up to an exponentially small error.