Free diffusions and Matrix models with strictly convex interaction

We study solutions to the free stochastic differential equation $dX_t = dS_t - \half DV(X_t)dt$, where $V$ is a locally convex polynomial potential in $m$ non-commuting variables. We show that for self-adjoint $V$, the law $\mu_V$ of a stationary solution is the limit law of a random matrix model, in which an $m$-tuple of self-adjoint matrices are chosen according to the law $\exp(-N \textrm{Tr}(V(A_1,...,A_m)))dA_1... dA_m$. We show that if $V=V_\beta$ depends on complex parameters $\beta_1,...,\beta_k$, then the law $\mu_V$ is analytic in $\beta$ at least for those $\beta$ for which $V_\beta$ is locally convex. In particular, this gives information on the region of convergence of the generating function for planar maps. We show that the solution $dX_t$ has nice convergence properties with respect to the operator norm. This allows us to derive several properties of $C^*$ and $W^*$ algebras generated by an $m$-tuple with law $\mu_V$. Among them is lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II$_1$ factor. We show that the microstates free entropy $\chi(\tau_V)$ is finite. A corollary of these results is the fact that the support of the law of any self-adjoint polynomial in $X_1,...,X_n$ under the law $\mu_V$ is connected, vastly generalizing the case of a single random matrix.