Random matrices, free probability, and the enumeration of maps

这项研究是在随机矩阵领域。该方案集中在这一领域的两个方面：(I)拓扑展开的分析。七十年代的研究表明，矩阵积分可以看作是映射计数的母函数，即按亏格排序的连通图，其中维度起亏格参数的作用。这就是所谓的拓扑展开。它们最初是作为列举地图的一种聪明方法在物理学中使用的。然后，它成为了在许多不同领域中寻找不变量的工具。从不同的角度来看，随机矩阵的极限可以用自由概率来很好地描述。因此，自由概率对于分析拓扑展开是有用的，反之，拓扑展开和随机矩阵可以用来构造自由概率中的新概念，从而得到算子代数中的结果。提案的第一部分解决了这一方向的几个问题。其中一个目标是开发一种通用的数学方案来获得拓扑展开。另一种是利用随机矩阵和经典概率论来发展自由概率下的最优传输理论，从而证明了算子代数中的同构结果。(Ii)第二部分是关于随机矩阵普适类的研究。事实上，在最近对高斯矩阵的普适性类的探索取得突破之后，研究肯定不属于这一类的矩阵是很自然的。该方案提出了研究重尾矩阵，以及更精确地研究其特征向量的局部化/非局部化以及整体特征值的局部波动。这项研究的领域是随机矩阵，它与数学和物理的许多领域相连，例如作为一个大型(随机)数据阵列，作为大型物理系统的哈密顿量等算子的近似，或者更奇怪地通过Riemann Zeta函数的非平凡零点。该项目的重点是它在算子代数、组合学和物理学中的应用。

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.