Random Planar Geometry

概率论的一个中心主题是理解大型离散模型及其尺度限制。极限对象通常具有一定的对称性和空间独立性，它们捕捉了许多模型的普遍大尺度行为。在过去的几十年里，在随机平面几何方面有了很大的进展，特别是在一些基本的二维离散模型及其尺度限制的理解方面。这些发展极大地扩展了我们对基本对象（如曲线、函数、树和曲面）随机性的认识。它们也彻底改变了物理学的一些核心部分的数学理解，包括共形场论，临界现象和量子引力。 本计画旨在扩阔对该学科基本数学基础的理解。本计画将探讨随机平面几何的两个研究方向。在第一个方向上，该项目旨在连接两种构建随机表面的方法：（1）通过随机平面映射的缩放极限;（2）通过称为Liouville量子引力（LQG）的连续理论。研究者计划证明一个猜想，断言LQG是随机平面映射在强意义上的标度极限。作为证明这一猜想的工具，该项目考虑了一种称为临界渗流的统计力学模型，旨在建立均匀随机三角剖分和规则三角网格上临界渗流的标度极限相同。这两项研究的内容包括：（a）编码几何信息的平面映射的组合双射;（B）量子和欧几里得几何中分形之间的关系，称为Knizhnik-Polyakov-Zamolodchikov关系;（c）布尔函数的傅里叶分析。在第二个方向，调查的目的是解决问题的几何随机平面地图，计算几何和分形几何。该方法的共同主题是两个最近开发的机器在连续随机平面几何中的应用，称为假想几何和树木交配。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

A central theme in probability theory is to understand large discrete models and their scaling limits. The limiting objects, which are usually characterized by certain symmetries and spatial independence, capture the universal large-scale behavior of many models. In the last few decades, there have been great advances in random planar geometry, especially in the understanding of some fundamental two-dimensional discrete models and their scaling limits. These developments hugely expand our knowledge on the randomness of basic objects such as curves, functions, trees, and surfaces. They also revolutionize the mathematical understanding of some central pieces of physics, including conformal field theory, critical phenomena, and quantum gravity. This project aims to broaden understanding of the fundamental mathematical underpinnings in this subject.The project will explore two research directions in random planar geometry. In the first direction, the project aims at linking two ways of constructing random surfaces: (1) through the scaling limits of random planar maps; (2) through a continuous theory called the Liouville quantum gravity (LQG). The investigator plans to prove a conjecture asserting that LQG is the scaling limit of random planar maps in a strong sense. As a tool for proving this conjecture, the project considers a statistical mechanical model called the critical percolation and aims to establish that the scaling limit of the critical percolation on the uniform random triangulation and on a regular triangular lattice are the same. The ingredients in both investigations include: (a) combinatorial bijections for planar maps that encode their geometric information; (b) a relation between fractals in quantum and Euclidean geometry called the Knizhnik-Polyakov-Zamolodchikov relation; (c) the Fourier analysis of Boolean functions. In the second direction, the investigator aims to resolve questions in the geometry of random planar maps, computational geometry, and fractal geometry. The common theme in the approaches is the application of two recently-developed machineries in continuous random planar geometry called imaginary geometry and mating of trees.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.