Schwinger-Dyson equations: classical and quantum

Abstract

In this note we want to have another look on Schwinger-Dyson equations for the eigenvalue distributions and the fluctuations of classical unitarily invariant random matrix models. We are exclusively dealing with one-matrix models, for which the situation is quite well understood. Our point is not to add any new results to this, but to have a more algebraic point of view on these results and to understand from this perspective the universality results for fluctuations of these random matrices. We will also consider corresponding non-commutative or "quantum" random matrix models and contrast the results for fluctuations and Schwinger-Dyson equations in the quantum case with the findings from the classical case.