Biorthogonal ensembles

Abstract

One object of interest in random matrix theory is a family of point ensembles (random point configurations) related to various systems of classical orthogonal polynomials. The paper deals with a one--parametric deformation of these ensembles, which is defined in terms of the biorthogonal polynomials of Jacobi, Laguerre and Hermite type. Our main result is a series of explicit expressions for the correlation functions in the scaling limit (as the number of points goes to infinity). As in the classical case, the correlation functions have determinantal form. They are given by certain new kernels which are described in terms of the Wright's generalized Bessel function and can be viewed as a generalization of the well--known sine and Bessel kernels. In contrast to the conventional kernels, the new kernels are non--symmetric. However, they possess other, rather surprising, symmetry properties. Our approach to finding the limit kernel also differs from the conventional one, because of lack of a simple explicit Christoffel--Darboux formula for the biorthogonal polynomials.