Topics in Percolation & Particle Models

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• 批准号: 0606696
• 负责人: Charles Newman
• 金额: $ 24.7万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-05-15 至 2010-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0606696&HistoricalAwards=false
• 关键词: Topics; Percolation; & Particle; Models

This project comprises studies that seek to answer basic questions arising in two areas of Probability Theory, Interacting Particle Systems andPercolation. They include the construction and applicationof nucleation processestaking place at the touching points of forward and backwardpaths in the (double) Brownian web and the conceptually relatedmarking process on the ``double points'' of the (critical) continuumnonsimple loop process. This loop process, constructed by the PIand Federico Camia based on the Schramm-Loewner evolution (SLE)and the work of Schramm and Smirnov, is the full scaling limitfor two-dimensional critical percolation. The double points,where the loops touch themselves and each other, represent thescaling limits of lattice sites where a change in microscopic statehas a macroscopic effect on connectivity.There are many situations, such as the functioning of the Internetand 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 tonovel behavior of the entire system. A classic goal of ProbabilityTheory and Statistics is to understand these situations. A particularlyintriguing set of problems concerns so called ``critical'' systemswhere even one individual can provide a tipping point for the wholesystem. The goal of this research project is to lay a firm foundationfor the mathematics needed to understand such critical systems, at leastin some special situations where there have been theoretical breakthroughsin the past several years. It is hoped that progress in thesespecial cases will lead to more general progress.

该项目包括研究，试图回答概率论，相互作用粒子系统和渗流两个领域中出现的基本问题。其中包括在（双）布朗网的前向和后向路径的接触点上标记位置的成核过程的构造和应用，以及在（临界）连续非简单回路过程的“双点”上概念相关的标记过程。这个循环过程是由PI和Federico Camia基于Schramm-Loewner演化（SLE）和Schramm和Smirnov的工作构造的，是二维临界渗流的满标度极限。环相互接触的双点代表了晶格点的尺度极限，在那里微观状态的变化会对连通性产生宏观影响。在许多情况下，比如互联网的运作、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，导致整个系统的行为改变。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，即使是一个人也可以提供一个整体的临界点。这个研究项目的目标是为理解这些关键系统所需的数学奠定坚实的基础，至少在过去几年中有理论突破的一些特殊情况下。希望这些特殊情况的进展将导致更普遍的进展。