# Spectral curves, variational problems and the hermitian matrix model with external source

## TL;DR

This paper links spectral curves from hermitian matrix models with external sources to a variational problem involving vector measures, classifies their geometries, and describes the local behaviors of limiting eigenvalue densities.

## Contribution

It establishes a unique correspondence between spectral curves and vector measures solving a variational problem, extending understanding of eigenvalue distributions in external source models.

## Key findings

- Limiting eigenvalue densities exhibit sine, Airy, Pearcey, or fifth power cubic behaviors.
- All spectral curves correspond to solutions of a variational problem involving three-component measures.
- The analysis applies to models with bounded zeros of characteristic polynomials, linking spectral curves to eigenvalue distributions.

## Abstract

We consider the hermitian random matrix model with external source and general polynomial potential, when the source has two distinct eigenvalues but is otherwise arbitrary. All such models studied so far have a common feature: an associated cubic equation (spectral curve), one of whose solutions can be expressed in terms of the Cauchy transform of the limiting eigenvalue distribution. This is our starting point: we show that to any such spectral curve (not necessarily given by a random matrix ensemble) it corresponds a unique vector-valued measure with three components on the complex plane, characterized as a solution of a variational problem stated in terms of their logarithmic energy. We describe all possible geometries of the supports of these measures; in particular the third component, if non-trivial, lives on a contour on the plane. This general result is applied to the random matrix model with external source, under an additional assumption of uniform boundedness of the zeros of a sequence of average characteristic polynomials in the large matrix limit. It is shown that any limiting zero distribution for such a sequence can be written in terms of a solution of a spectral curve, and thus admits the variational description obtained in the first part of the paper. As a consequence of our analysis, we obtain that the density of this limiting measure can have only a handful of local behaviors: sine, Airy and their higher order type behavior, Pearcey or yet the fifth power of the cubic (but no higher order cubics can appear). We also compare our findings with the most general results available in the literature, showing that once an additional symmetry is imposed, our vector critical measure contains enough information to recover the solutions to the constrained equilibrium problem that was known to describe the limiting eigenvalue distribution in this symmetric situation.