RUI: Some Problems in Percolation and Stochastic Dynamics of Ising Models

RUI：伊辛模型渗流和随机动力学的一些问题

• 批准号: 0103994
• 负责人: Chris Wu
• 金额: $ 7.8万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103994&HistoricalAwards=false
• 关键词: RUI; Some; Problems; Percolation; Stochastic

The aim of this research is to study several random spatial models on various infinite graphs. Four topics concerning stochastic dynamics of Ising spin systems, the random cluster and Ising ferromagnetic model, and the contact process are proposed to be studied. The first topic concerns stochastic dynamics of an Ising spin system on an infinite lattice where spins evolve according to the usual Glauber dynamics. Some typical questions are: does a spin flip infinitely many times or only finitely many times? what is the probability that a spin has not yet flipped at time t? The second topic is about Ising models on hyperbolic lattices. Although Ising models on the hypercubic lattices have been studied intensively and extensively since they were introduced, these models on hyperbolic lattices have just started to receive attention from physicists and mathematicians. They are found, by both numerical studies and mathematical proofs, to exhibit a phenomenon of multiple phase transitions. Although some results have been rigorously proved, many statements suggested by numerical studies are to be proved, and many more are to be explored. Some GHS type inequalities in the random cluster model and a related question of uniqueness of the random cluster measure are the contents of the third topic. The final topic deals with phase transitions of models with low-dimensional inhomogeneity. These models include percolation, Ising ferromagnetic systems and contact processes. Models of the sort to be studied in this research arise naturally from physical sciences. Percolation is a probabilistic model of studying flow through a discrete disordered system, such as particles flowing through the filter of a gas mask, or fluid seeping through the interstices of a porous medium, while the contact process can be regarded as modeling the spread of an epidemic through a population.

本研究的目的是研究各种无限图上的随机空间模型。提出了四个研究方向，即Ising自旋系统的随机动力学、随机团簇和Ising铁磁模型以及接触过程。第一个主题涉及的伊辛自旋系统的随机动力学在一个无限的晶格自旋演变根据通常的Glauber动力学。一些典型的问题是：一个自旋翻转无限多次还是只有2000次？自旋在时间t还没有翻转的概率是多少？第二个主题是关于双曲格上的伊辛模型。虽然超立方格上的伊辛模型自提出以来就得到了广泛而深入的研究，但双曲格上的伊辛模型才刚刚开始受到物理学家和数学家的关注。通过数值研究和数学证明，发现它们表现出多重相变现象。虽然一些结果已经得到严格的证明，许多陈述建议的数值研究是证明，还有更多的是要探索。第三部分是随机聚类模型中的GHS型不等式和与之相关的随机聚类测度的唯一性问题。最后一个主题涉及低维不均匀模型的相变。这些模型包括渗流、伊辛铁磁系统和接触过程。在本研究中要研究的那种模型自然产生于物理科学。渗流是一种研究离散无序系统中流动的概率模型，例如粒子流过防毒面具的过滤器，或者流体从多孔介质的空隙中渗出，而接触过程可以被视为模拟流行病在人群中的传播。