Normal random matrix ensemble as a growth problem

TL;DR

This paper interprets the evolution of eigenvalue distributions in normal random matrix ensembles as a growth process, using algebro-geometrical properties of spectral curves to describe boundary dynamics and wave functions.

Contribution

It introduces a novel geometric framework linking eigenvalue support evolution to spectral curve growth in normal random matrices.

Findings

Eigenvalue support boundaries form algebraic curves.

Spectral curves encode physical properties of ensembles.

Growth can be described via evolution of these curves.

Abstract

In general or normal random matrix ensembles, the support of eigenvalues of large size matrices is a planar domain (or several domains) with a sharp boundary. This domain evolves under a change of parameters of the potential and of the size of matrices. The boundary of the support of eigenvalues is a real section of a complex curve. Algebro-geometrical properties of this curve encode physical properties of random matrix ensembles. This curve can be treated as a limit of a spectral curve which is canonically defined for models of finite matrices. We interpret the evolution of the eigenvalue distribution as a growth problem, and describe the growth in terms of evolution of the spectral curve. We discuss algebro-geometrical properties of the spectral curve and describe the wave functions (normalized characteristic polynomials) in terms of differentials on the curve. General formulae and emergence of the spectral curve are illustrated by three meaningful examples.