Scaling Limits and Phase Transitions in Spatial Random Processes

该项目致力于相变和临界现象的各个方面的研究。这些起源于物理学，但研究方法需要数学的重要投入。项目中讨论的问题的一个共同特点是，它们涉及许多组成部分的系统。正是在这里，概率论提供了许多不可或缺的工具。具体的问题通常是在一个复杂的微观系统的背景下设置的，该系统是根据数学概念定义的，如随机游走或随机场。其目的是表明，随着系统规模的增加，一个新的结构出现。这种结构，有时被称为标度极限，往往承认一个独立的表征，使其研究使用数学分析的方法。特别注意的现象的普遍性，这是指独立的标度限制的细节的底层微观系统。以数学精确的形式理解普遍性是该项目的首要目标之一。该项目为研究生提供研究培训机会。该项目大致分为两个部分。第一部分主要研究空间随机场的极值问题。借鉴早期成功的二维高斯自由场的分析，目的是建立普遍性的结论的其他例子的物理相关的领域，如随机游走本地时间或离散高斯模型下的Kosterlitz无边界过渡。这里的兴趣是对现象和控制这种系统所需的技术的发展。该项目的第二部分致力于分析一些感兴趣的系统中的相变。其中之一是一个系统的相互作用玻色子，这是建议使用随机循环表示进行分析。这里的目的是给出一个严格的证明玻色-爱因斯坦凝聚在正温度下在适当的缩放限制。另一个要研究的模型是相互作用的随机游动;这里的目的是使用变分问题的行走范围来描述极限形状。该项目自然借鉴了该领域的最新进展，其中一些是由于PI。所涉及的技术跨越了概率和分析中的许多主题;例如，极端顺序统计，均匀化理论，排列统计，不动点理论。该奖项反映了NSF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。

This project is devoted to the study of various aspects of phase transitions and critical phenomena. These originate in physics but the methods of study require significant input from mathematics. A common feature of the problems discussed in the project is that they involve systems of many constituents. It is here where probability theory offers a number of indispensable tools. The specific questions are typically set in the context of a complex microscopic system defined in terms of mathematical concepts such as random walks or random fields. The aim is to show that, as the system size increases, a new structure emerges. This structure, sometimes called a scaling limit, often admits an independent characterization that enables its study using the methods of mathematical analysis. Particular attention is paid to the phenomenon of universality, which refers to independence of the scaling limit to the details of the underlying microscopic system. Understanding universality in its mathematically precise form is one of the overarching goals of the project. The project provides research training opportunities for graduate students. The project is divided roughly into two parts. The first one is focused on the study of extremal problems for spatial random fields. Drawing on earlier success of the analysis of the two-dimensional Gaussian Free Field, the aim is to establish universality of the conclusions for other examples of logarithmically-correlated fields such as the random walk local time or the discrete-Gaussian model below the Kosterlitz-Thouless transition. The interest here is on both the phenomena and the development of techniques required to control such systems. The second part of the project is devoted to the analysis of phase transitions in a number of systems of interest. One of these is a system of interacting bosons, which is proposed to be analyzed using a random-cycle representation. The aim here is to give a rigorous proof of Bose-Einstein condensation at positive temperatures in suitable scaling limits. Another model to be studied is that of interacting random walks; the aim here is to describe the limit shape using a variational problem for the walk range. The project naturally draws on recent advances in the field, some due to the PI. The techniques involved span a number of topics in probability and analysis; for instance, extreme order statistics, homogenization theory, permutation statistics, fixed point theory. A number of problems have been designed with the aim to include graduate students and postdocs in research.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.