Electrostatics for type II multiple orthogonal polynomials

Electrostatics for type II multiple orthogonal polynomials
The well-known electrostatic interpretation of the zeros of Hermite, Laguerre or Jacobi polynomials that goes back to the 1885 work of Stieltjes is one of the most popular models in the theory of orthogonal polynomials. Besides its elegance, it has a clear pedagogic value, allowing to predict monotonicity properties of zeros in terms of parameters or their asymptotic distribution. It was picked up and extended to several contexts, such as orthogonal and quasi-orthogonal polynomials on the real line and the unit circle, for classical and semiclassical weights.
Multiple orthogonal (or Hermite–Padé) polynomials satisfy a system of orthogonality conditions with respect to a set of measures. They find applications in number theory, approximation theory, and stochastic processes, and their analytic theory, extremely rich, has been developing since 1980s, in big part by the contribution of the Soviet and Russian school lead by Gonchar and others. However, no electrostatic interpretation of the zeros of such polynomials has been described in the literature.
In this talk, I will describe such a model for the case of type II Hermite–Padé polynomials, will discuss its link with the asymptotic behavior of such zeros, and will illustrate it with some simple cases.
This is a joint work with R. Orive (Universidad de La Laguna, Canary Islands, Spain) and J. Sanchez-Lara (Granada University, Spain).