Random matrices, free probability, and the enumeration of maps

本研究属于随机矩阵领域。该建议集中在两个方面的这个域：（i）分析的拓扑扩展。70年代的研究表明，矩阵积分可以被看作是映射计数的生成函数，映射是按亏格排序的连通图，其中维数起亏格参数的作用。 这就是所谓的拓扑扩张。它们最初是在物理学中作为一种聪明的方法来枚举地图。然后，它成为一种仪器，以寻找在许多不同领域的不变量。从另一个角度看，随机矩阵的极限可以用自由概率来描述。 因此，自由概率对于分析拓扑展开是有用的，反过来，拓扑展开和随机矩阵可以用来构造自由概率中的新概念，并在算子代数中产生结果。建议的第一部分涉及这方面的几个问题。 目标之一是 发展一个通用的数学方案来获得拓扑展开。二是利用随机矩阵和经典概率论发展了自由概率中的最优迁移理论，从而证明了算子代数中的同构结果。（ii）建议的第二部分涉及随机矩阵的普适类的研究。事实上，在最近对高斯矩阵普适性类的探索取得突破之后，研究肯定不在这类矩阵之外的矩阵是很自然的。 该建议提供了研究重尾矩阵，更精确地本地化/离域的特征向量，以及本地波动的特征值在散装。这项研究是在随机矩阵的领域，这是与数学和物理的许多领域，例如作为一个大的（随机）阵列的数据，作为近似的运营商，如汉密尔顿的大型物理系统， 或者更奇特的是通过黎曼zeta函数的非平凡零点。该项目的重点是它的应用算子代数，组合数学和物理。

This research is in the area of random matrices. The proposal focuses on two aspects of this domain: (i) Analysis of the topological expansions. It was shown in the seventies that matrix integrals can be viewed as generating functions for the enumeration of maps, that is connected graphs which are sorted by their genus, where the dimension plays the role of genus parameter. This is the so-called topological expansion. They were first used in physics as a clever way to enumerate maps. It then became an apparatus to find invariants in many different fields. On a different point of view, the limit of random matrices is well described by free probability. Hence, free probability is useful to analyze topological expansions and conversely topological expansions and random matrices can be used to construct new concepts in free probability, with consequences in operator algebra. The first part of the proposal addresses several questions in this direction. One of the goals consists in developing a general mathematical scheme to obtain topological expansions. Another one is to use random matrices and classical probability theory to develop the theory of optimal transport in free probability, hence proving isomorphisms results in operator algebra.(ii)The second part of the proposal concerns the study of the universality classes of random matrices. In fact, after the recent breakthroughs exploring the universality class of Gaussian matrices, it is natural to study matrices which are definitely out of this class. The proposal offers to study heavy tailed matrices, and more precisely localization/delocalization of their eigenvectors as well as local fluctuations of the eigenvalues in the bulk. This research is in the area of random matrices, which are connected with many domains of mathematics and physics, for instance as a large (random) array of data, as the approximation of operators such as the Hamiltonian of large physical systems, or more exotically via the non-trivial zeroes of the Riemann zeta function. The project focuses on its applications to operator algebra, combinatorics and physics.