Universal Randomness in Dimension 2

2 维中的普遍随机性

• 批准号: 1712862
• 负责人: Scott Sheffield
• 金额: $ 62万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712862&HistoricalAwards=false
• 关键词: Universal; Randomness; Dimension

This research is in probability theory. It will attempt to make connections between a number of different probabilistic models that arise in statistical physics and particle physics. Understanding these connections will result in improved understanding of the physical models. The PI will train students from the undergraduate level through the graduate level. He will also assist in the training of postdoctoral researchers. This research concerns a variety of random fractals that are in some sense planar (i.e., naturally embedded in or parameterized by subsets of the plane). These fractals include random paths, random surfaces, random collections of loops, random trees, random functions, and random growth processes. The objects under study are "universal" in the sense that they arise as limits of many discrete models and "canonical" in the sense that they are uniquely characterized by special symmetries. Several of these objects are motivated by statistical mechanics, string theory, and gauge theory, as well as the study of natural growth processes (lichen, mineral depositions, snowflakes, lightning bolts, etc.) The specific technical goals of the research include the following: understanding the limiting conformal structure of discrete planar maps, generalizing quantum Loewner evolution growth models beyond the regime in which they have been defined, endowing general Liouville quantum gravity surfaces with metric structure, extending known results about random surface scaling limits to random surfaces embedded in higher dimensional spaces, and understanding more about the relationship between lattice Yang-Mills gauge theory (and its variants) and the random surface models that arise in the formulas for Wilson loop expectations. The broader aim is to provide a firmer mathematical understanding of some of our most fundamental models for physical phenomena, ranging from microscopic to macroscopic scales.

这项研究属于概率论。它将尝试在统计物理学和粒子物理学中出现的许多不同概率模型之间建立联系。了解这些联系将有助于加深对物理模型的理解。 PI 将培训从本科到研究生的学生。他还将协助博士后研究人员的培训。这项研究涉及各种随机分形，这些分形在某种意义上是平面的（即自然嵌入平面的子集或由平面的子集参数化）。这些分形包括随机路径、随机表面、随机循环集合、随机树、随机函数和随机增长过程。所研究的对象是“通用的”，因为它们是作为许多离散模型的限制而出现的；并且是“规范的”，因为它们具有特殊对称性的独特特征。其中一些对象是由统计力学、弦理论和规范理论以及自然生长过程的研究（地衣、矿物沉积、雪花、闪电等）激发的。研究的具体技术目标包括以下内容：理解离散平面映射的限制共形结构，将量子 Loewner 演化增长模型推广到其定义范围之外，赋予一般刘维尔量子引力表面度量结构，将关于随机表面缩放限制的已知结果扩展到嵌入的随机表面更高维空间，并更多地了解格子杨-米尔斯规范理论（及其变体）与威尔逊环期望公式中出现的随机表面模型之间的关系。更广泛的目标是对从微观到宏观尺度的物理现象的一些最基本的模型提供更坚定的数学理解。