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A spectral curve for a matrix model is, in very loose terms, an equation with unknown being the CauchyÌý(a.k.a. Stieltjes) transform of the limiting spectral density. Sometimes also called master loop equation orÌýstring equation, it commonly appears as an algebraic equation, hence the name "curve" as it determines anÌýalgebraic curve. A classical situation is given by the celebrated semicircle law, whose Cauchy transform satisfiesÌýa very simple algebraic equation of degree 2. In this context, it also turns out that this limiting spectral densityÌýis 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 inÌýthe construction of potential-theoretic variational problems that describe the limiting spectral density for theÌýmodel. Our key novel technique is to translate the determination of the solutions to the variational problemÌýinto the problem of geometrically describing trajectories of a canonical quadratic differential that lives on theÌýunderlying algebraic curve.

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

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