#### Vol. 13, No. 8, 2020

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An elementary approach to free entropy theory for convex potentials

### David Jekel

Vol. 13 (2020), No. 8, 2289–2374
##### 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 ${\mu }_{N}$.

##### Keywords
free entropy, free Fisher information, free Gibbs state, trace polynomials, invariant ensembles
##### Mathematical Subject Classification
Primary: 46L53
Secondary: 35K10, 37A35, 46L52, 46L54, 60B20