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

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。其中包括具有特殊宏观性质的格点的重标度计数测度的极限的自然测度的构造和应用。对于合并一维随机游动及其在二维时空中的布朗网标度极限，特殊点是类型（0，2），其中零条路径进入，两条路径离开，以及类型（1，2），其中一条路径进入，两条路径离开。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，得到了有限簇面积测度和测度系综意义下的欧氏随机场，并提出了解决d ≥ 2时粗化动力学的公开问题，在许多情况下，例如互联网的运作、工程材料和经济的行为，许多小的个体（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，在这些系统中，即使一个人也可以为整个系统提供一个临界点。这个研究项目的主要目标是为理解这种临界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方法。

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.