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

Let $(X_1,\dots ,X_m)$ be self-adjoint noncommutative 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. For $k < m$, we show that conditional expectations and conditional non-microstates free entropy given $X_1$, …, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the $N \times N$ random matrix models associated to $V$. Then, by studying conditional transport of measure for the matrix models, we construct an isomorphism $\mathrm{W}^*(X_1,\dots ,X_m) \to \mathrm{W}^*(S_1,\dots ,S_m)$ that maps $\mathrm{W}^*(X_1,\dots ,X_k)$ to $\mathrm{W}^*(S_1,\dots ,S_k)$ for each $k = 1, \dots , m$ and that also witnesses the Talagrand inequality for the law of $(X_1,\dots ,X_m)$ relative to the law of $(S_1,\dots ,S_m)$.