论文标题
在离散表面上：通过拓扑递归的枚举几何形状，矩阵模型和通用类
On discrete surfaces: Enumerative geometry, matrix models and universality classes via topological recursion
论文作者
论文摘要
本文中正在考虑的主要对象称为地图，一类嵌入在表面上的图形。我们的问题具有一个相对较新的工具，即Chekhov，Eynard和Orantin引入的所谓拓扑递归（TR）。 如果图具有非自我交流边界，我们将其称为完全简单的地图，如果没有施加这种限制，则将其称为普通的地图。我们研究了完全简单和普通地图与磁盘或圆柱体的拓扑之间的组合关系，该图在自由概率的背景下重现了时刻与自由累积物之间的关系。我们提出了对TR的交换符合性转换的组合解释。我们通过正式的Hermitian矩阵模型提供了一个具有外部字段的矩阵模型解释，用于完全简单的地图，并推断出生成一系列完全简单和普通地图之间的普遍关系，涉及双单调hurwitz数字。特别是，我们获得了一个类似于ELSV的公式，以双$ 2 $ -Orbifold严格单调Hurwitz数字。 我们认为具有$ O（\ mathsf {n}）$ loop模型的普通地图，该模型是统计物理学中的经典模型，并通过查看循环装饰图地图的嵌套属性来确定哪些形状更可能发生。当顶点数量变得很大时，我们希望研究限制对象，这可以通过研究主要奇异性的生成序列来完成。我们在存在大小边界的情况下，在任意拓扑的随机地图上分析了$ o（\ mathsf {n}）$ loop模型中的嵌套统计信息，依赖于磁盘和圆柱体的先前结果以及该模型的TR。我们研究了一系列生成的地图，这些图实现了固定的嵌套图，并在稠密相相中表征了它们的临界行为。
The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by Chekhov, Eynard and Orantin. We call a map fully simple if it has non self-intersecting disjoint boundaries, and ordinary if such a restriction is not imposed. We study the combinatorial relation between fully simple and ordinary maps with the topology of a disk or a cylinder, which reproduces relations between moments and free cumulants established in the context of free probability. We propose a combinatorial interpretation of the exchange symplectic transformation of TR. We provide a matrix model interpretation for fully simple maps via the formal hermitian matrix model with external field and deduce a universal relation between generating series of fully simple and ordinary maps, which involves double monotone Hurwitz numbers. In particular, we obtain an ELSV-like formula for double $2$-orbifold strictly monotone Hurwitz numbers. We consider ordinary maps endowed with an $O(\mathsf{n})$ loop model, which is a classical model in statistical physics, and determine which shapes are more likely to occur by looking at the nesting properties of the loops decorating the maps. We want to study the limiting objects when the number of vertices becomes arbitrarily large, which can be done by studying the generating series at dominant singularities. We analyze the nesting statistics in the $O(\mathsf{n})$ loop model on random maps of arbitrary topologies in the presence of large and small boundaries, relying on previous results for disks and cylinders and TR for this model. We study the generating series of maps which realize a fixed nesting graph and characterize their critical behavior in the dense and dilute phases.
