Multiple orthogonal polynomials and vector equilibrium problems in random matrix theory

The study of multiple orthogonal polynomials (MOPs) has its origins in the theory of Hermite-Padé approximation. The theory has been well-developed by the Russian school in approximation theory, including important contributions of Guillermo López Lagomasino. I will give an overview of some recent developments in the use of MOPs in stochastic models from random matrix theory and more general determinantal point processes. Models that have been studied this way include random matrices with external source and non-intersecting Brownian motions (leading to multiple Hermite polynomials), non-intersecting squared Bessel processes (leading to MOPs with modified Bessel weights), and coupled random matrices (leading to MOPs with Pearcey integral weights). An interesting feature in these models is the appearance of a vector equilibrium problem with Nikishintype interactions and both an external field and an upper constraint. One component of the solution of the vector equilibrium problem gives the large n distribution of the zeros of the MOPs as well as the limiting mean density of eigenvalues (or positions of paths) in the stochastic model. The equilibrium measures play a prominent role in the steepest descent analysis of the Riemann-Hilbert problem for MOPs.