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 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 $D{V}_N$ 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 ${\chi }^{\ast }(\lambda )$ agree and equal the limit of the normalized classical entropies of ${\mu }_N$.

arXiv comment75 pages

References in corpus (4)

• Free Convolution Operators and Free Hall Transform
• A non-commutative Path Space approach to stationary free Stochastic Differential Equations
• A Laplace Principle for Hermitian Brownian Motion and Free Entropy I: the convex functional case
• A Survey on Connes' Embedding Conjecture

Cited by in corpus (5)

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• A random matrix approach to absorption in free products
• Free noncommutative principal divisors and commutativity of the tracial fundamental group