Entropy of operator-valued random variables:: A variational principle for large N matrix models

We show that, in 't Hooft's large N limit, matrix models can be formulated as a classical theory whose equations of motion are the factorized Schwinger-Dyson equations. We discover an action principle for this classical theory. This action contains a universal term describing the entropy of the noncommutative probability distributions. We show that this entropy is a nontrivial one-cocycle of the noncommutative analog of the diffeomorphism group and derive an explicit formula for it. The action principle allows us to solve matrix models using novel variational approximation methods; in the simple cases where comparisons with other methods are possible, we get reasonable agreement.