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

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.

• The paper establishes lower bounds for various free entropy measures along Wasserstein geodesics using non-commutative optimal transport.
• It demonstrates the existence and uniqueness of Gibbs types through a regularized variational approach, ensuring convergence in multi-matrix models.
• The work leverages Monge-Kantorovich duality to reveal topological properties and generic behaviors in the large-n limit of random matrices.

Information Geometry for Types in the Large-$n$ Limit of Random Matrices

Introduction

This paper investigates the interaction between entropy and Wasserstein distance within free probability theory, particularly in the context of large-$n$ limits of random matrices. The research gives lower bounds for variants of free entropy dimension along Wasserstein geodesics and analyzes their topological properties. Additionally, it studies moment measures in the multivariate free setting, demonstrating the existence and uniqueness of solutions for a regularized version of Santambrogio's variational problem. Here, probability distributions are conceptualized as types—functionals assigning values not only to polynomial test functions but also to all real-valued logical formulas constructed using suprema and infima.

Random Matrix Models and Free Probability

The study considers invariant random $m$-tuples of $n \times n$ matrices—multi-matrix models—whose joint distribution is invariant under unitary conjugation. These models have probability densities like $$d\mu^{(n)}(\mathbf{X}) = \frac{1}{Z^{(n)}} e^{-n^2 V^{(n)}(\mathbf{X})}\,d\mathbf{X}$$ where $$V^{(n)}(\mathbf{X}) = \operatorname{Re}\operatorname{Tr}_n(p(\mathbf{X}))$$ represents a potential. To capture the large-$n$ behavior of these models, we need appropriate extensions of entropy and Wasserstein distance in free probability, expressed in terms of non-commutative laws, microsates free entropy, and the Biane--Voiculescu--Wasserstein distance.

Moment Measures and Optimal Transport

The paper explores solutions of Santambrogio's variational problem in non-commutative settings. Here, given a type $\mu$, the task is to find another type $\nu$ that maximizes $\chi_{\full}(\nu) - C_{\full}(\mu,\nu) - t(\nu,q)$ for a quadratic perturbation $q$. This requires establishing the existence and conditions under which minimizers are obtainable within the non-commutative framework.

Monge-Kantorovich Duality in Non-commutative Spaces

The research leverages the Monge-Kantorovich duality, crucial for understanding geodesic behavior in optimal transport frameworks. For non-commutative settings, the challenge lies in defining continuous functions over operator ball spaces—functions not closed under suprema/infima—thus necessitating a shift to considering logical formulas involving these operations.

Results Summary

1. Free Entropy Along Geodesics: Establishes lower bounds for microstates free entropy $\chi_{\full}$, entropy dimension $\delta$, and $1$-bounded entropy $\Ent$ along Wasserstein geodesics.
2. Topological and Generic Properties: Investigates continuity properties and demonstrates genericity of infinite free entropy in Wasserstein space, revealing broad behavior in tracial von Neumann algebras.
3. Gibbs Types Uniqueness: Proves existence and uniqueness of Gibbs types for strongly convex definable predicates and applies Talagrand inequality, crucial for multi-matrix model convergence.
4. Separability and Limits: The paper further shows the separability of types obtained via Gibbs constructions and confirms limits formed under ultrafilters, ensuring convergence in matrix space.

Conclusion

Through advanced mathematical constructs, the research enhances our comprehension of types in free probability, particularly within the landscape of large-$n$ matrix models. The synthesis of model theory, optimal transport, and matrix approximations provides a potent framework for understanding entropy measures along geodesics and optimizing transport over complex, non-commutative spaces. The outcomes point towards a robust theoretical basis for integrating information geometry within random matrix theory, potentially influencing developments in AI and large-scale data systems.