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 = V_β 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 V_β 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.