Information geometry for types in the large-$n$ limit of random matrices

TL;DR

This paper explores the interplay of entropy and Wasserstein distance in free probability, introducing types as a new way to analyze non-commutative laws and their large-$n$ limits, with implications for random matrix models.

Contribution

It introduces the concept of types as a new framework for studying free entropy and Wasserstein distance in non-commutative probability, highlighting limitations of polynomial-based distributions.

Findings

Lower bounds for free entropy dimension along Wasserstein geodesics

Existence and uniqueness of solutions for regularized variational problems

Counterexample showing limitations of polynomial-based distributions in large-$n$ limits

Abstract

We study the interaction between entropy and Wasserstein distance in free probability theory. In particular, we give lower bounds for several versions of free entropy dimension along Wasserstein geodesics, as well as study their topological properties with respect to Wasserstein distance. We also study moment measures in the multivariate free setting, showing the existence and uniqueness of solutions for a regularized version of Santambrogio's variational problem. The role of probability distributions in these results is played by types, functionals which assign values not only to polynomial test functions, but to all real-valued logical formulas built from them using suprema and infima. We give an explicit counterexample showing that in the framework of non-commutative laws, the usual notion of probability distributions using only non-commutative polynomial test functions, one cannot obtain the desired large-$n$ limiting behavior for both Wasserstein distance and entropy simultaneously in random multi-matrix models.