An elementary approach to free entropy theory for convex potentials

Abstract

We present an alternative approach to the theory of free Gibbs states with convex potentials. Instead of solving SDEs, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N{\left(ℂ\right)}_{sa}^m$ to prove the following. Suppose ${\mu }_N$ is a probability measure on $M_N{\left(ℂ\right)}_{sa}^m$ given by uniformly convex and semiconcave potentials $V_N$, and suppose that the sequence $D{V}_N$ is asymptotically approximable by trace polynomials. Then the moments of ${\mu }_N$ converge to a noncommutative law $\lambda$. Moreover, the free entropies $\chi \left(\lambda \right)$, $\underset{¯}{\chi }\left(\lambda \right)$, and ${\chi }^{\ast }\left(\lambda \right)$ agree and equal the limit of the normalized classical entropies of ${\mu }_N$.