Mathematical Sciences: Percolation and Random Graphs

首席研究员将对渗流和随机图进行数学研究。这两个主题都涉及聚类过程的数学模型，并应用于统计力学中的相变和临界现象。渗流理论研究主要集中在两个问题上：在经典渗流模型中，很难找到临界概率（相变）值的精确数学解；本项目继续发展一种计算临界概率精确上限和下限的方法。AB渗透是经典渗透模型的一种变体，用于研究反铁磁体和凝胶过程。AB渗流模型与经典渗流模型表现出很大的不同，在一些晶格中表现出多个相变，而在其他晶格中则没有相变。在这个项目中，研究者将研究这些差异发生的原因。随机图理论研究继续探索Erdos-Renyi随机图模型在相变值附近的行为。

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.