An elementary approach to free entropy theory for convex potentials

We present an alternative approach to the theory of free Gibbs states with convex potentials developed in several papers of Guionnet, Shlyakhtenko, and Dabrowski. Instead of solving SDE's, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N(\mathbb{C})_{sa}^m$ to prove the following. Suppose $\mu_N$ is a probability measure on on $M_N(\mathbb{C})_{sa}^m$ given by uniformly convex and semi-concave potentials $V_N$, and suppose that the sequence $DV_N$ is asymptotically approximable by trace polynomials. Then the moments of $\mu_N$ converge to a non-commutative law $\lambda$. Moreover, the free entropies $\chi(\lambda)$, $\underline{\chi}(\lambda)$, and $\chi^*(\lambda)$ agree and equal the limit of the normalized classical entropies of $\mu_N$. A key step is to show that the property of asymptotic approximation by trace polynomials is preserved under several operations, including limits, composition, Gaussian convolution, and ultimately evolution under certain parabolic PDE. This allows us to prove convergence of the moments of $\mu_N$ and of the Fisher information of Gaussian perturbations of $\mu_N$.