If $V$ is sufficiently regular real-valued convex function of $m$ non-commutative self-adjoint variables, then there is free Gibbs law $\lambda_V$ for a non-commutative $m$-tuple $X = (X_1,\dots,X_m)$, which describes the large-$N$ behavior of tuples $X^{(N)} = (X_1^{(N)}, \dots, X_m^{(N)})$ of random matrices chosen according to the probability density $(1/Z^{(N)}) e^{-N^2 V(x)}\,dx$. We show that for a function $f$ that is well approximated by trace polynomials, the classical conditional expectation of $f(X^{(N)})$ given $X_1^{(N)}$, ..., $X_k^{(N)}$ for $k < m$ converges to the $W^*$-algebraic conditional expectation of $f(X)$ given $X_1$, ..., $X_k$. We also construct an isomorphism from $W^*(X_1,\dots,X_m)$ to the algebra $W^*(S_1,\dots,S_m)$ generated by a free semicircular family (i.e.\ a free group factor), which arises as the large $N$ limit of transport maps for probability measures on $M_N(\mathbb{C})_{sa}^m$. This transport can be made ``upper-triangular'' in the sense that $W^*(X_k,\dots,X_m)$ is mapped to $W^*(S_k,\dots,S_m)$ for each $k = 1, \dots, m$. At the same time, this transport map witnesses the Talagrand entropy-cost inequality for $\mu$ relative to the law of a semicircular family.

中文翻译：

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自由概率中凸吉布斯定律的条件期望、熵和传输

如果 $V$ 是 $m$ 非交换自伴随变量的足够正则实值凸函数，那么对于非交换 $m$-元组 $X = (X_1, \dots,X_m)$，描述随机元组 $X^{(N)} = (X_1^{(N)}, \dots, X_m^{(N)})$ 的大 $N$ 行为根据概率密度选择的矩阵 $(1/Z^{(N)}) e^{-N^2 V(x)}\,dx$。我们表明，对于一个很好地由迹多项式近似的函数 $f$，给定 $X_1^{(N)}$, ..., $$f(X^{(N)})$ 的经典条件期望X_k^{(N)}$ for $k < m$ 收敛到 $f(X)$ 的 $W^*$-代数条件期望给定 $X_1$, ..., $X_k$。我们还构造了从 $W^*(X_1,\dots,X_m)$ 到由自由半圆族（即自由群因子）生成的代数 $W^*(S_1,\dots,S_m)$ 的同构，这是由于 $M_N(\mathbb{C})_{sa}^m$ 上概率度量的传输图的大 $N$ 限制而出现的。在 $W^*(X_k,\dots,X_m)$ 映射到 $W^*(S_k,\dots,S_m)$ 的意义上，对于每个 $k = 1, \dots, m$。同时，这张运输图见证了 $\mu$ 相对于半圆族定律的 Talagrand 熵-成本不等式。