Universal Randomness in Dimension 2

这项研究是在概率论中进行的。它将试图在统计物理和粒子物理中出现的一些不同的概率模型之间建立联系。了解这些联系将有助于更好地理解物理模型。PI将培训从本科生到研究生的学生。他还将协助培训博士后研究人员。这项研究涉及各种随机分形，它们在某种意义上是平面的(即，自然嵌入到平面的子集中或被平面的子集参数化)。这些分形图包括随机路径、随机曲面、随机循环集合、随机树、随机函数和随机增长过程。所研究的对象在某种意义上是“普遍的”，因为它们是作为许多离散模型的极限而出现的；从某种意义上说，“规范”是因为它们具有独特的对称性。这些物体中有几个是由统计力学、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)的研究推动的。这项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，将量子Loewner演化增长模型推广到定义了它们的区域之外，赋予一般的Liouville量子引力表面以度规结构，将关于随机表面标度极限的已知结果推广到嵌入高维空间的随机表面，以及更多地了解晶格Yang-Mills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。更广泛的目标是为我们的一些最基本的物理现象模型提供更坚实的数学理解，从微观尺度到宏观尺度。

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.