Hermite-Pade approximations and multiple orthogonal polynomial ensembles

This paper is concerned with Hermite-Pade rational approximants of analytic functions and their connection with multiple orthogonal polynomial ensembles of random matrices. Results on the analytic theory of such approximants are discussed, namely, convergence and the distribution of the poles of the rational approximants, and a survey is given of results on the distribution of the eigenvalues of the corresponding random matrices and on various regimes of such distributions. An important notion used to describe and to prove these kinds of results is the equilibrium of vector potentials with interaction matrices. This notion was introduced by A. A. Gonchar and E. A. Rakhmanov in 1981.