Random Normal Matrices and Polynomial Curves

TL;DR

This paper analyzes the eigenvalue distribution of large normal matrices with specific potentials, showing they fill polynomial curves determined by harmonic moments, and relates orthogonal polynomial zeros to the Schwarz function of these curves.

Contribution

It establishes a connection between eigenvalue distributions of normal matrices and polynomial curves defined by harmonic moments, and links orthogonal polynomial zeros to the Schwarz function of these curves.

Findings

Eigenvalues fill polynomial curves in the large matrix limit.

Zeros of orthogonal polynomials relate to the Schwarz function of the curve.

The distribution is governed by the potential's parameters and harmonic moments.

Abstract

We show that in the large matrix limit, the eigenvalues of the normal matrix model for matrices with spectrum inside a compact domain with a special class of potentials homogeneously fill the interior of a polynomial curve uniquely defined by the area of its interior domain and its exterior harmonic moments which are all given as parameters of the potential. Then we consider the orthogonal polynomials corresponding to this matrix model and show that, under certain assumptions, the density of the zeros of the highest relevant orthogonal polynomial in the large matrix limit is (up to some constant factor) given by the discontinuity of the Schwarz function of this polynomial curve.