Random Planar Geometry

概率论的一个中心主题是理解大型离散模型及其比例限制。极限对象通常具有一定的对称性和空间独立性，它们反映了许多模型的普遍大尺度行为。在过去的几十年里，随机平面几何取得了很大的进展，特别是在理解一些基本的二维离散模型及其尺度极限方面。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。它们还彻底改变了人们对一些核心物理问题的数学理解，包括保形场理论、临界现象和量子引力。这个项目旨在扩大对这门学科的基本数学基础的理解。项目将探索随机平面几何的两个研究方向。在第一个方向，该项目旨在将两种构造随机曲面的方法联系起来：(1)通过随机平面地图的比例限制；(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，即LQG是强意义上的随机平面映射的尺度极限。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，目的是建立均匀随机三角剖分上的临界渗流的标度极限与规则三角格子上的临界渗流的标度极限相同。这两个研究的内容包括：(A)平面映射的组合双射，它编码了平面映射的几何信息；(B)量子几何和欧几里德几何之间的一种关系，称为Knizhnik-Polyakov-Zamolodchikov关系；(C)布尔函数的傅里叶分析。在第二个方向，研究人员的目标是解决随机平面映射几何、计算几何和分形几何中的问题。这些方法的共同主题是在连续随机平面几何中应用两种最新开发的机器，称为虚构几何和树木配对。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.