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 V1, V2 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 V1, V2. 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 V1 or V2 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 V1 and V2. Viewed as overdetermined systems of linear difference-differentialdeformation 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.(1-10)Any other correlation function of m eigenvalues can similarly be written as a determinant involving these four kernels only.The spacing distributions (the probability that two neighbouring eigenvalues are at some given distance) can be computed as Fredholm determinants. For example, the probability that some subset J of the real axis contains no eigenvalue of the first matrix is the Fredholm determinant:( 1-11) where χ J is the characteristic function of the set J.An important feature in the study of the N → ∞ limit is that the kernels K ij may be expressed [20] in terms of sums involving only a fixed number of terms (either d 1 + 1 or d 2 + 1), independently of N , as a consequence of a "generalized Christoffel-Darboux" formula [44,49] following from the recursion relations satisfied by the biorthogonal polynomials. This allows one, in the N → ∞ limit, with suitable scaling in the spectral variables, depending on the region considered, to treat N as just a parameter. Duality Dual isomonodromic deformationsThe notion of duality arises in a number of contexts, both in relation to isospectral flows [1] and isomonodromic [26,27,25] deformations. What is meant here by "duality" in the case of isomonodromic deformations is the existence of a pair of parametric families of meromorphic covariant derivative operators on the Riemann spherewhere L(x, u) and M(y, u) are, respectively, l × l and m × m matrices that are rational functions of the complex variables x ∈ P 1 and y ∈ P 1 , with pole divisors of fixed degrees, depending smoothly on a set of deformation parameters u = (u 1 , u 2 , . . .) in such a way that: 1) The matrices L(x, u) and M(y, u)...

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part g + of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouville-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. The results are formulated in terms of sheaves to allow for singularities due to a degenerate spectrum. Serre duality is used to define a natural symplectic structure on the space of line bundles of suitable degree over a permissible class of spectral curves, and this is shown to be equivalent to the Kostant-Kirillov symplectic structure on rational coadjoint orbits, reduced by the group of constant loops. A similar construction involving a framing at infinity is given for the nonreduced orbits. The general construction is given for g = gl(r) or sl(r), with reductions to orbits of subalgebras determined as invariant fixed point sets under involutive automorphisms. As illustrative examples, the case g = sl(2), together with its real forms, is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, with the Liouville generating function expressed in hyperellipsoidal coordinates, as well as the quasi-periodic solutions of the cubically nonlinear Schrödinger equation. For g = sl(3), the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrödinger equation. This case requires a further symplectic constraining procedure in order to deal with singularities in the spectral data at ∞.

We consider biorthogonal polynomials that arise in the study of a generalization of two-matrix Hermitian models with two polynomial potentials V 1 (x), V 2 (y) of any degree, with arbitrary complex coefficients.Finite consecutive subsequences of biorthogonal polynomials ("windows"), of lengths equal to the degrees of the potentials V 1 and V 2 , satisfy systems of ODE's with polynomial coefficients as well as PDE's (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced.