Another algebraic variational principle for the spectral curve of matrix models

TL;DR

This paper introduces a new variational principle for determining the spectral curve of matrix models, focusing on an entropy-based extremization rather than energy minimization, applicable to algebraic plane curves with specified asymptotics and cycle integrals.

Contribution

It presents an alternative variational principle for spectral curves in matrix models, emphasizing entropy extremization over traditional energy-based methods.

Findings

The variational principle accurately reproduces the spectral curve in matrix models.

It generalizes to curves with prescribed asymptotics and cycle integrals.

Provides a new perspective on the algebraic structure of spectral curves.

Abstract

We propose an alternative variational principle whose critical point is the algebraic plane curve associated to a matrix model (the spectral curve, i.e. the large $N$ limit of the resolvent). More generally, we consider a variational principle that is equivalent to the problem of finding a plane curve with given asymptotics and given cycle integrals. This variational principle is not given by extremization of the energy, but by the extremization of an "entropy".