Universal Randomness in Dimension 2

这项研究是概率论方面的。它将尝试在统计物理学和粒子物理学中出现的许多不同的概率模型之间建立联系。理解这些联系将有助于提高对物理模型的理解。PI将培养从本科到研究生阶段的学生。他还将协助培养博士后研究人员。本研究涉及各种随机分形，它们在某种意义上是平面的（即，自然嵌入在平面子集中或由平面子集参数化）。这些分形包括随机路径、随机曲面、随机循环集合、随机树、随机函数和随机生长过程。所研究的对象是“普遍的”，因为它们是许多离散模型的极限，而“规范的”是指它们具有特殊对称性的唯一特征。其中一些对象的动机是统计力学、弦理论和规范理论，以及对自然生长过程（地衣、矿物沉积、雪花、闪电等）的研究。研究的具体技术目标包括：理解离散平面映射的极限共形结构，推广量子Loewner进化增长模型，使其超越定义的范围，赋予一般Liouville量子引力表面以度量结构，将关于随机表面缩放极限的已知结果扩展到嵌入在高维空间中的随机表面，以及更多地了解晶格杨-米尔斯规范理论（及其变体）与威尔逊循环期望公式中出现的随机表面模型之间的关系。更广泛的目标是为我们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牢固的数学理解。

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.