An elementary approach to free entropy theory for convex potentials

Jekel, David

doi:10.2140/apde.2020.13.2289

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An elementary approach to free entropy theory for convex potentials

David Jekel

Vol. 13 (2020), No. 8, 2289–2374

DOI: 10.2140/apde.2020.13.2289

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 $M_{N} {(ℂ)}_{sa}^{m}$ to prove the following. Suppose $μ_{N}$ is a probability measure on $M_{N} {(ℂ)}_{sa}^{m}$ 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 $μ_{N}$ converge to a noncommutative law $λ$ . Moreover, the free entropies $χ (λ)$ , $\underline{χ} (λ)$ , and $χ^{*} (λ)$ agree and equal the limit of the normalized classical entropies of $μ_{N}$ .

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Keywords

free entropy, free Fisher information, free Gibbs state, trace polynomials, invariant ensembles

Mathematical Subject Classification

Primary: 46L53

Secondary: 35K10, 37A35, 46L52, 46L54, 60B20

Milestones

Received: 31 May 2018

Revised: 27 June 2019

Accepted: 25 September 2019

Published: 28 December 2020

Authors

David Jekel
Department of Mathematics University of California Los Angeles, CA United States

Vol. 13, No. 8, 2020