SCHWINGER–DYSON EQUATIONS: CLASSICAL AND QUANTUM
James A. Mingo
Roland Speicher
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.
2000 AMS Mathematics Subject Classification: Primary: 60B20, 46L54.
Keywords and phrases: Free probability, random matrices, Schwinger–Dyson
equation.