Dyson-Schwinger loop equations of the two-matrix model: eigenvalue correlations in quantum chaos

Abstract

We determine a set of Dyson-Schwinger equations or loop equations for a model of two coupled random matrices belonging to the orthogonal, unitary, or symplectic ensembles. In the large N limit, the loop equations become closed algebraic equations, allowing us to obtain the correlations between the eigenvalues of the two matrices. The expression we obtain is valid near the center as well as the edge of the cut. In particular, this determines how the correlations between the eigenvalues of perturbed and unperturbed chaotic Hamiltonians depend upon the strength of the perturbation, and also the space and time dependence of density-density correlators of the Calogero-Sutherland-Moser model for three values of the coupling constant.