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2020 An elementary approach to free entropy theory for convex potentials
David Jekel
Anal. PDE 13(8): 2289-2374 (2020). 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 MN()sam to prove the following. Suppose μN is a probability measure on MN()sam given by uniformly convex and semiconcave potentials VN, and suppose that the sequence DVN 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.

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David Jekel. "An elementary approach to free entropy theory for convex potentials." Anal. PDE 13 (8) 2289 - 2374, 2020. https://doi.org/10.2140/apde.2020.13.2289

Information

Received: 31 May 2018; Revised: 27 June 2019; Accepted: 25 September 2019; Published: 2020
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.2140/apde.2020.13.2289

Subjects:
Primary: 46L53
Secondary: 35K10 , 37A35 , 46L52 , 46L54 , 60B20

Keywords: Free entropy , free Fisher information , free Gibbs state , invariant ensembles , trace polynomials

Rights: Copyright © 2020 Mathematical Sciences Publishers

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