Vector Extremal Measure and the Distribution of Zeros of Biorthogonal Polynomials

In this paper, we study the limit properties of a sequence of polynomials that satisfy biorthogonality relations with the iterated Cauchy kernel. The orthogonality weights may depend on the degree of the polynomials. Such polynomials have arisen in recent applications in random matrix theory and integrable systems. We prove that the limit distribution of their zeros is described by a vector measure that minimizes the energy functional in the presence of an external field. We also discuss some properties of this extremal measure. We show that its support also minimizes the value of some functional on compact sets. In turn, the external field can be reconstructed from a family of supports of extremal measures, parameterized by their total variation.