Duality, Biorthogonal Polynomials¶and Multi-Matrix Models

Abstract

The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel–Darboux form constructed from sequences of biorthogonal polynomials. For measures involving exponentials of a pair of polynomials V ₁, V ₂ in two different variables, these kernels may be expressed in terms of finite dimensional “windows” spanned by finite subsequences having length equal to the degree of one or the other of the polynomials V ₁, V ₂. The vectors formed by such subsequences satisfy “dual pairs” of first order systems of linear differential equations with polynomial coefficients, having rank equal to one of the degrees of V ₁ or V ₂ and degree equal to the other. They also satisfy recursion relations connecting the consecutive windows, and deformation equations, determining how they change under variations in the coefficients of the polynomials V ₁ and V ₂. Viewed as overdetermined systems of linear difference-differential-deformation equations, these are shown to be compatible, and hence to admit simultaneous fundamental systems of solutions. The main result is the demonstration of a spectral duality property; namely, that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters.