Free probability: combinatorial and analytic aspects, and interactions with other notions of independence

该建议是在该地区的自由概率论-一条线的研究平行方面的经典概率，在一个上下文中，张量产品取代自由产品，和独立的随机变量取代自由独立的非交换随机变量。 自由概率起源于20世纪80年代，与算子代数的自由积问题有关。 从那时起，它已经成为一个独立的学科，与数学的其他部分，如古典概率，代数组合学和随机矩阵理论有联系。 通过与随机矩阵的联系，自由概率在量子信息理论和电子工程中的无线通信领域中得到了应用。 本研究的目的是在自由概率现阶段相当活跃的一个方向上取得进展，并关注随机变量的自由独立性和其他类型的独立性（经典独立性或其他类型的非交换独立性）之间的相互作用。 拟议的研究也有一个特点，它结合了分析和组合方法的基本方式，在正在考虑的问题的治疗。

The proposal is in the area of free probability theory - a line of research which parallels aspects of classical probability, in a context where tensor products are replaced by free products, and independent random variables are replaced by freely independent noncommutative random variables. Free probability originated in the 1980s, in connection to problems about free products of operator algebras. Since then, it has emerged as a subject in its own right, with connections to other parts of mathematics such as classical probability, algebraic combinatorics, and the theory of random matrices. Through its connection to random matrices, free probability has found applications in quantum information theory and in the area of wireless communications in electrical engineering. The objective of the proposed research is to make progress in a direction which is quite active in the present stage of free probability, and concerns the interaction between free independence and other types of independence for random variables (either classical independence, or other types of noncommutative independence). The proposed research also has the feature that it combines in an essential way analytic and combinatorial methods, in the treatment of the problems that are being considered.