标题： 信息几何在随机矩阵的大-$n$极限中的类型 显示英文标题

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

摘要： 我们研究自由概率论中熵与Wasserstein距离之间的相互作用。 特别是，我们给出了沿Wasserstein测地线的几种自由熵维数的下界，同时还研究了它们在Wasserstein距离下的拓扑性质。 我们还研究了多变量自由情形下的矩测度，证明了Santambrogio变分问题的一个正则化版本的解的存在性和唯一性。 在这些结果中，概率分布的作用由类型扮演，这些类型不仅对多项式检验函数赋值，还对通过上确界和下确界从它们构建的所有实值逻辑公式赋值。 我们给出一个显式的反例，表明在非交换律的框架下，仅使用非交换多项式检验函数的通常概率分布概念，在随机多矩阵模型中无法同时获得Wasserstein距离和熵的期望大-$n$限制行为。 显示英文摘要

摘要： 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.