Holomorphic matrix integrals

TL;DR

This paper investigates holomorphic matrix models, revealing that eigenvalue distributions in the large-size limit are supported on complex arcs characterized by hyperelliptic integrals, linking matrix theory with complex analysis.

Contribution

It introduces a novel class of holomorphic matrix models and characterizes the eigenvalue support as level sets of hyperelliptic integrals, connecting matrix integrals with complex geometry.

Findings

Eigenvalue distributions form arcs in the complex plane.

Arcs are level sets of the imaginary part of hyperelliptic integrals.

Large matrix size limit leads to a measure supported on these arcs.

Abstract

We study a class of holomorphic matrix models. The integrals are taken over middle dimensional cycles in the space of complex square matrices. As the size of the matrices tends to infinity, the distribution of eigenvalues is given by a measure with support on a collection of arcs in the complex planes. We show that the arcs are level sets of the imaginary part of a hyperelliptic integral connecting branch points.