An Elementary Approach to Free Entropy Theory for Convex Potentials

TL;DR

This paper introduces a PDE-based method for free entropy theory in convex potentials, avoiding SDEs, and shows convergence of measures and entropy equalities in the free probability setting.

Contribution

It provides a novel PDE approach to free entropy theory for convex potentials, replacing stochastic differential equations with asymptotic trace polynomial approximations.

Findings

Convergence of moments to a non-commutative law.

Equality of free entropies and classical entropies in the limit.

Validation of the PDE approach for free Gibbs states.

Abstract

We present an alternative approach to the theory of free Gibbs states with convex potentials. 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 $\mu_N$.