An Elementary Approach to Free Entropy for Log-Concave Random Matrix Models

David Jekel University of California, Los Angeles (UCLA) Mathematics

We consider a sequence of random matrix models with distribution $\mu_N$ given by a potential $V_N: M_N^{sa}(\C)^m \to \R$. We assume that the $V_N$'s are uniformly convex and semi-concave and that the sequence $N \nabla V_N$ is asymptotically approximable by trace polynomials in a certain sense. In this case, we show that the sequence $\mu_N$ exponentially concentrates around some non-commutative law $\lambda$. The $\limsup$ microstates free entropy $\chi(\lambda)$, the $\liminf$ microstates free entropy $\underline{\chi}(\lambda)$, and the non-microstates free entropy $\chi^*(\lambda)$ all agree and are equal to the limit of the normalized classical entropies $N^{-2} h(\mu_N) + (m/2) \log N$. Our results overlap with those of Dabrowski, but the proof here is based on deterministic PDE techniques rather than stochastic control theory.