Correlations and Scaling in Disordered and Critical Stochastic Models

无序和临界随机模型中的相关性和标度

• 批准号: 1612921
• 负责人: Jack Hanson
• 金额: $ 10万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612921&HistoricalAwards=false
• 关键词: Correlations; Scaling; Disordered; Critical; Stochastic

In many physical systems, large and orderly structures are formed without distant parts of the system communicating directly. Understanding these structures is a long-standing challenge in probability theory; this project focuses on the mathematical study of these structures. One class of these, percolation, has arisen to study various random networks: for instance, underground rock formations carrying oil, traffic networks in cities, and polymer chains. Another class of these, Abelian networks, arose to describe situations where energy builds up microscopically until it reaches a certain threshold and then causes a large "avalanche," and has been used to model earthquakes. A major issue the work under this project will touch on is how widely-separated (in space or time) parts of the random structure depend on each other: for instance, how do early topplings in an avalanche influence those occurring much later? These questions have connections to other important physical systems as well as to optimization problems from computer science.The research under this award will study correlations and fluctuations in disordered systems (first-passage percolation and related disordered magnets) and critical systems (percolation, and Bak-Tang-Wiesenfeld models also known as sandpiles). A primary focus in first-passage percolation will be the order of fluctuations of the random metric and its relationship to properties of geodesics. In critical percolation, the work will focus on the scaling of the chemical or graph distance in large finite clusters -- in particular, bounding the chemical distance exponent in two dimensions. A final focus will be structural properties of avalanches in the Abelian sandpile model, including the question of stabilizability of sandpiles in two dimensions.

在许多物理系统中，大而有序的结构是在系统的远距离部分没有直接通信的情况下形成的。理解这些结构是概率论中一个长期存在的挑战;该项目侧重于这些结构的数学研究。其中一类是逾渗（percolation），它被用来研究各种随机网络：例如，承载石油的地下岩层、城市交通网络和聚合物链。另一类是阿贝尔网络，用来描述能量在微观上积累，直到达到一定的阈值，然后导致大规模的“雪崩”，并被用来模拟地震。这个项目下的工作将涉及的一个主要问题是随机结构的广泛分离（在空间或时间上）部分如何相互依赖：例如，雪崩中早期的倾倒如何影响那些发生得更晚的倾倒？这些问题与其他重要的物理系统以及计算机科学的优化问题都有联系。该奖项的研究将研究无序系统（首次通过渗流和相关的无序磁体）和临界系统（渗流和Bak-Tang-维森费尔德模型，也称为沙堆）中的相关性和波动。第一次通过渗流的一个主要焦点将是随机度量的波动顺序及其与测地线性质的关系。在临界渗流中，工作将集中在大型有限簇中化学或图形距离的标度上-特别是在二维中界定化学距离指数。最后一个重点将是在阿贝尔沙堆模型雪崩的结构特性，包括沙堆在两个维度的稳定性问题。