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 MN()sam to prove the following. Suppose μN is a probability measure on MN()sam given by uniformly convex and semiconcave potentials V N, and suppose that the sequence DV N is asymptotically approximable by trace polynomials. Then the moments of μN converge to a noncommutative law λ. Moreover, the free entropies χ(λ), χ ¯(λ), and χ(λ) agree and equal the limit of the normalized classical entropies of μN.

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