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

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 ${\mathrm{W}}^{\ast }(X_1,\dots ,X_m)\to {\mathrm{W}}^{\ast }(S_1,\dots ,S_m)$ which maps ${\mathrm{W}}^{\ast }(X_1,\dots ,X_k)$ to ${\mathrm{W}}^{\ast }(S_1,\dots ,S_k)$ 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 $(S_1,\dots ,S_m)$.

73 pages