Random Planar Geometry

概率论的一个中心主题是理解大型离散模型及其缩放极限。限制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，随机平面几何有了很大的进展，特别是对一些基本的二维离散模型及其缩放极限的理解。这些发展极大地扩展了我们对基本对象（如曲线、函数、树和表面）随机性的认识。它们还彻底改变了对物理学一些核心领域的数学理解，包括共形场论、临界现象和量子引力。本项目旨在拓宽对这一学科基础数学基础的理解。本项目将探索随机平面几何的两个研究方向。在第一个方向上，项目旨在连接构建随机曲面的两种方式：(1)通过随机平面图的缩放限制；(2)通过一种称为刘维尔量子引力（LQG）的连续理论。研究者计划证明一个猜想，即LQG在强意义上是随机平面图的缩放极限。作为证明这一猜想的工具，本项目考虑了一种称为临界渗流的统计力学模型，旨在建立临界渗流在均匀随机三角剖分和正则三角晶格上的尺度极限是相同的。这两项研究的成分包括：(a)平面地图的组合双射，对其几何信息进行编码；(b)量子几何和欧几里得几何中的分形关系，称为Knizhnik-Polyakov-Zamolodchikov关系；(c)布尔函数的傅里叶分析。在第二个方向上，研究者的目标是解决随机平面图的几何问题，计算几何和分形几何。这些方法的共同主题是最近开发的两种机器在连续随机平面几何中的应用，称为虚几何和树的配对。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.