A spectral curve for a matrix model is, in very loose terms, an equation with unknown being the Cauchy (a.k.a. Stieltjes) transform of the limiting spectral density. Sometimes also called master loop equation or string equation, it commonly appears as an algebraic equation, hence the name "curve" as it determines an algebraic curve. A classical situation is given by the celebrated semicircle law, whose Cauchy transform satisfies a very simple algebraic equation of degree 2. In this context, it also turns out that this limiting spectral density is the minimizer of a weighted log energy on the real line.

In this talk we plan to discuss spectral curves for various matrix models and how they can be used in the construction of potential-theoretic variational problems that describe the limiting spectral density for the model. Our key novel technique is to translate the determination of the solutions to the variational problem into the problem of geometrically describing trajectories of a canonical quadratic differential that lives on the underlying algebraic curve.

We will focus on two different matrix models. The first one is the normal matrix model, where the random eigenvalues accumulate on a domain of the plane (the droplet) which grows according to the Laplacian growth. In this situation, we are able to reconstruct a mother body measure for the droplet, which describes the limiting eigenvalue distribution for the average characteristic polynomial. The second model to be discussed is the hermitian plus external source ensemble. In this situation, the variational problem asks for finding a saddle point of an energy involving three measures. Also as a consequence of our results, we are able to describe all possible critical local behaviors that can arise in this external source model.

This is a joint work with Andrei Martínez-Finkelshtein(Baylor University/Universidad de Almería)