Free Diffusions and Matrix Models with Strictly Convex Interaction

Abstract

We study solutions to the free stochastic differential equation $$dX_{t} = dS_{t} - \frac{1}{2}DV(X_{t})dt$$ , where V is a locally convex polynomial potential in m non-commuting variables and S an m-dimensional free Brownian motion. We prove that such free processes have a unique stationary distribution μV. When the potential V is self-adjoint, we show that the law μV is the limit law of a random matrix model, in which an m-tuple of self-adjoint matrices are chosen according to the law exp $$(-NTr(V(A_{1},\ldots,A_{m})))dA_{1} \cdots dA_{m}/ZN(V)$$ . If V = Vβ depends on complex parameters $$\beta_{1},\ldots ,\beta_{k}$$ , we prove that the moments of the law μV are analytic in β at least for those β for which Vβ is locally convex. In particular, this gives information on the region of convergence of the generating function for the enumeration of related planar maps. We prove that the solution X t has nice convergence properties with respect to the operator norm as t goes to infinity. This allows us to show that the C* and W* algebras generated by an m-tuple with law μV share many properties with those generated by a semi-circular system. Among them is the lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II1 factor. We show that the microstates free entropy χ(μV ) is finite when V is self-adjoint. A corollary of these results is the fact that the support of the law of any self-adjoint polynomial in $$X_{1}, \ldots, X_{n}$$ under the law μV is connected, vastly generalizing the case of a single random matrix. We also deduce from this dynamical approach that the convergence of the operator norms of independent matrices from the GUE proved by Haagerup and Thorbjornsen [HT] extends to the context of matrices interacting via a convex potential.