Mathematical Problems in Percolation and Ising Models

小行星9803598 本研究的目的是在有限长轴上研究随机空间系统 一维超立方格和无限维双曲格。 三个主题 关于渗流，伊辛（铁磁）模型，和接触过程将是 研究了第一个主题涉及双曲格上的渗流和伊辛模型。 虽然人们已经研究了超立方晶格上的Ising模型和渗流 自从双曲格上的模型被提出以来， 才刚刚开始受到物理学家和数学家的关注。 他们被发现， 通过数值研究和数学证明， 相变 虽然一些结果已经得到严格的证明，但许多声明 数值研究所提出的建议有待证明;还有更多的有待探索。 的 第二个主题涉及（独立和依赖）渗透的粗糙化过渡 在维数大于或等于3的超立方格上。这是 铁磁伊辛模型的粗糙化过渡的类比。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、伊辛铁磁系统和接触过程。 这项研究由一个本科院校的研究人员进行， 渗透和相互作用的粒子系统。这些是概率模型， 有助于理解物理系统。进一步了解这些概率 模型可以帮助物理学家。

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.