Scaling Limits and Phase Transitions in Spatial Random Processes

这个项目致力于研究相变和临界现象的各个方面。这些都起源于物理学，但研究方法需要大量的数学输入。项目中讨论的问题的一个共同特征是它们涉及许多组成部分的系统。正是在这里，概率论提供了一些不可或缺的工具。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，这个系统是根据随机游走或随机场等数学概念定义的。其目的是表明，随着系统规模的增加，一种新的结构出现了。这种结构，有时被称为标度极限，通常允许一个独立的表征，使其能够使用数学分析的方法进行研究。特别要注意的是普适性现象，这是指尺度限制对微观系统细节的独立性。以精确的数学形式理解普遍性是该项目的首要目标之一。本项目为研究生提供研究训练机会。该项目大致分为两部分。第一部分主要研究空间随机场的极值问题。借鉴早期对二维高斯自由场的成功分析，目的是为对数相关场的其他例子（如随机漫步局部时间或Kosterlitz-Thouless跃迁下的离散高斯模型）建立结论的普遍性。这里的兴趣在于控制这种系统所需的现象和技术的发展。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。其中之一是相互作用玻色子的系统，建议使用随机循环表示来分析。本文的目的是在合适的尺度范围内给出正温度下玻色-爱因斯坦凝聚的严格证明。另一个要研究的模型是相互作用的随机漫步；这里的目的是用行走范围的变分问题来描述极限形状。该项目自然借鉴了该领域的最新进展，其中一些归功于PI。所涉及的技术跨越了概率和分析中的许多主题；例如，极序统计，同质化理论，排列统计，不动点理论。设计了一些问题，目的是让研究生和博士后参与研究。该奖项反映了美国国家科学基金会的法定使命，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.