Mathematical Sciences: Percolation and Random Graphs

首席研究员将进行数学 研究渗流和随机图。 这两个主题都涉及 聚类过程的数学模型，并有 相变和临界现象的应用 统计力学 逾渗理论的研究重点 两个问题：在经典的渗流模型中， 严格的数学精确解 概率（相变）值是罕见的。 这个项目 继续开发一种计算精确上限的方法， 和临界概率的下限。 AB渗流是 经典渗流模型的一个变体，为 反铁磁体和凝胶化过程的研究。 AB渗流 模型的行为与经典渗透完全不同 模型，通过在一些模型中展示多个相变， 晶格中，并且在其他中没有相变。 在这个项目中 研究人员将研究为什么会出现这些差异。 的 随机图论研究继续探索 相变附近的Erdos-Renyi随机图模型 值

The principal investigator will conduct mathematical research in percolation and random graphs. Both topics deal with mathematical models of clustering processes, and have applications to phase transitions and critical phenomena in statistical mechanics. The percolation theory research focuses on two problems: In classical percolation models, mathematically rigorous exact solutions for the critical probability (phase transition) value are rare. This project continues development of a method for computing accurate upper and lower bounds for the critical probability. AB percolation is a variant of the classical percolation model, introduced for the study of antiferromagnets and gelation processes. AB percolation models behave quite differently from classical percolation models, by exhibiting multiple phase transitions in some lattices, and no phase transitions in others. In this project the investigator will study why these differences occur. The random graph theory research continues to explore the behavior of the Erdos-Renyi random graph models near the phase transition value.