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.