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

本研究的目的是研究各种无限图上的几种随机空间模型。提出了要研究伊辛自旋系统的随机动力学、随机团簇和伊辛铁磁模型以及接触过程四个主题。第一个主题涉及无限晶格上伊辛自旋系统的随机动力学，其中自旋按照通常的Glauber动力学演化。一些典型的问题是：自旋翻转是无限次还是有限次？在时间t，自旋还没有翻转的概率是多少？第二个主题是关于双曲格子上的伊辛模型。虽然超立方晶格上的Ising模型自提出以来已经得到了广泛而深入的研究，但双曲晶格上的这些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，它们都表现出多重相变现象。虽然一些结果已经得到了严格的证明，但数值研究提出的许多说法仍有待证明，还有更多的结论有待探索。第三个主题是随机聚类模型中的一些GHS型不等式和与之相关的随机聚类测度的唯一性问题。最后一个主题涉及具有低维不均匀的模型的相变。这些模型包括渗流模型、伊辛铁磁系统模型和接触过程模型。这项研究中要研究的那种模型自然而然地产生于物理学。渗流是一种研究离散无序系统流动的概率模型，例如通过防毒面具过滤器的颗粒流动，或通过多孔介质的缝隙渗漏的液体，而接触过程可以被视为模拟流行病在人群中的传播。

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.