Particle Systems and Scaling Limits in Two (and More) Dimensions

二维（及更多）维度的粒子系统和缩放限制

• 批准号: 1007524
• 负责人: Charles Newman
• 金额: $ 50万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007524&HistoricalAwards=false
• 关键词: Particle; Systems; Scaling; Limits; Two

This project comprises studies that seek to answer basic questions arising in interacting particle systems, including percolation and Ising models. They include the construction and application of natural measures as limits of rescaled counting measures of lattice points with special macroscopic properties. For coalescing one-dimensional random walks and their Brownian web scaling limit in two-dimensional space-time, the special points are type (0,2) where zero paths enter and two leave and type (1,2) where one path enters and two leave. In d-dimensional Ising-related random cluster percolation models, the special points are where k disjoint paths emerge, with a focus onk=1 and d=2 to obtain limit cluster area measures and Euclidean random fields in terms of measure ensembles.It is also proposed to attack open problems concerning coarsening dynamics for d equal two or more, invasion percolation for d equal three or more and d=3 Ising models.There are many situations, such as the functioning of the Internet and the behavior of engineered materials and of the economy, where many small individuals (web sites, molecules, investors) interact with each other in seemingly random ways leading to novel behavior of the entire system. A classic goal of Probability Theory and Statistics is to understand these situations. A particularly intriguing set of problems concerns so called ``critical'' systems where even one individual can provide a tipping point for the whole system. A major goal of this research project is to lay a firm foundation for the mathematics needed to understand such critical systems.It is hoped that progress for certain special cases will lead to more general approaches.

该项目包括寻求回答相互作用粒子系统中出现的基本问题的研究，包括渗透和伊辛模型。它们包括自然测度作为具有特殊宏观性质的点阵点的重标计数测度的极限的构造和应用。对于二维时空中一维随机游动及其布朗网缩放极限的合并，特殊点为零路径进入且两条离开的（0,2）型和一条路径进入且两条离开的（1,2）型。在d维ising相关的随机簇渗流模型中，特殊点是k条不相交路径出现的地方，重点是k=1和d=2，以获得测度集合方面的极限簇面积测度和欧氏随机场。还提出了针对d等于两个或两个以上的粗化动力学、d等于三个或三个以上的入侵渗流和d=3个Ising模型的开放性问题。在许多情况下，例如因特网的功能、工程材料的行为和经济，许多小个体（网站、分子、投资者）以看似随机的方式相互作用，导致整个系统的新行为。概率论和统计学的一个经典目标就是理解这些情况。一组特别有趣的问题涉及所谓的“关键”系统，其中甚至一个人可以为整个系统提供一个临界点。该研究项目的一个主要目标是为理解这些关键系统所需的数学奠定坚实的基础。希望某些特殊情况的进展将导致更普遍的办法。