Mathematical Problems in Percolation and Ising Models

9803598吴，本研究的目的是研究有限维双三次格和无限维双曲格上的随机空间系统。我们将研究三个主题：渗流、伊辛(铁磁)模型和接触过程。第一个主题是关于双曲晶格上的渗流和伊辛模型。虽然超立方晶格上的Ising模型和渗流模型自提出以来已经得到了广泛而深入的研究，但这些双曲晶格上的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。虽然有几个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。第二个主题涉及维度大于或等于3的超立方晶格上(独立的和依赖的)渗流的粗化转变。这是铁磁伊辛模型粗糙跃迁的类比。第三个主题是关于低维非均匀模型的相变。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。这项由本科生机构的研究人员进行的研究涉及渗流和相互作用的粒子系统的研究。这些都是概率模型，对理解物理系统很有用。进一步了解这些概率模型可能会对物理学家有所帮助。

9803598 Wu The aim of this research is the study of random spatial systems on the finite dimensional hypercubic lattices and infinite dimensional hyperbolic lattices. Three topics concerning percolation, Ising (ferromagnetic) models, and contact processes will be studied. The first topic concerns percolation and Ising models on hyperbolic lattices. Although Ising models and percolation 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 a few results have been rigorously proved, many statements suggested by numerical studies are to be proved; and many more are to be explored. The second topic concerns a roughening transition of (independent and dependent) percolation on the hypercubic lattices with dimensions larger than or equal to three. This is the analogy of a roughening transition of ferromagnetic Ising models. The third topic is about a phase transition of models with low-dimensional inhomogeneity. These models include percolation, Ising ferromagnetic systems and contact processes. This research by an investigator at an undergraduate institution involves the study of percolation and interacting particle systems. These are probabilistic models, which have been useful in understanding physical systems. Further understanding of these probabilistic models can be helpful to physicists.