# Conditional Expectation, Entropy, and Transport for Convex Gibbs Laws in Free Probability

## TL;DR

This paper establishes a connection between free Gibbs laws and semicircular families using conditional expectations, entropy, and transport, providing new isomorphisms and inequalities in free probability.

## Contribution

It introduces a novel approach to construct measure transport and isomorphisms between free Gibbs laws and semicircular families via matrix models.

## Key findings

- Conditional expectations and entropy converge from matrix models to free Gibbs laws.
- Constructed measure transport maps induce isomorphisms between free probability algebras.
- Proved Talagrand inequality for free Gibbs laws relative to semicircular laws.

## Abstract

Let $(X_1,\dots,X_m)$ be self-adjoint non-commutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,\dots,S_m)$ be a free semicircular family. We show that conditional expectations and conditional non-microstates free entropy given $X_1$, \dots, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the random matrix models associated to $V$. Then by studying conditional transport of measure for the matrix models, we construct an isomorphism $\mathrm{W}^*(X_1,\dots,X_m) \to \mathrm{W}^*(S_1,\dots,S_m)$ which maps $\mathrm{W}^*(X_1,\dots,X_k)$ to $\mathrm{W}^*(S_1,\dots,S_k)$ for each $k = 1, \dots, m$, and which also witnesses the Talagrand inequality for the law of $(X_1,\dots,X_m)$ relative to the law of $(S_1,\dots,S_m)$.