An Elementary Approach to Free Entropy Theory for Convex Potentials

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. Su...

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Bibliographic Details
Published in:arXiv.org 2019-03
Main Author: Jekel, David
Format: Article
Language:English
Subjects:
Asymptotic properties
Convergence
Convolution
Functions (mathematics)
Mathematical analysis
Parabolic differential equations
Polynomials
Online Access:Get full text
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Summary: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 \(DV_N\) 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 \(\chi^*(\lambda)\) agree and equal the limit of the normalized classical entropies of \(\mu_N\).
ISSN:2331-8422
DOI:10.48550/arxiv.1805.08814