Random matrices, free probability, and the enumeration of maps

本研究是在随机矩阵领域。该建议着重于该领域的两个方面：(i)拓扑扩展的分析。早在七十年代就有人指出，矩阵积分可以看作是映射枚举的生成函数，映射是指按属数排序的连通图，其中维数起着属数参数的作用。这就是所谓的拓扑展开。它们最初被用于物理学中，作为一种列举地图的聪明方法。然后，它成为在许多不同领域寻找不变量的工具。从另一个角度看，自由概率很好地描述了随机矩阵的极限。因此，自由概率对于分析拓扑展开式是有用的，反过来，拓扑展开式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。提案的第一部分解决了这方面的几个问题。其中一个目标是开发一个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在自由概率中发展最优输运理论，从而证明算子代数中的同构结果。提案的第二部分涉及随机矩阵的普适性类的研究。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，很自然地要研究那些绝对不在这一类的矩阵。提出了对重尾矩阵的研究，对其特征向量进行更精确的定位/离域以及特征值的局部波动。这项研究是在随机矩阵领域，它与数学和物理的许多领域有关，例如作为一个大的（随机）数据数组，作为算子的近似，如大型物理系统的哈密顿量，或者更奇特的是通过黎曼ζ函数的非平凡零点。该项目侧重于其在算子代数、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.