Publication : t01/047

Abstract:
The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Darboux-Christoffel form constructed from families of bi-orthogonal polynomials. For measures involving exponentials of a pair of polynomials $V_1$, $V_2$ in two different variables, these families may be viewed within finite dimensional ``windows'' spanned by finite sequences of bi-orthogonal polynomials, of lengths equal to the degree of one or the other of the polynomials $V_1$ or $V_2$. These 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 one. They also satisfy sequences of 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 a proof of the 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.

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