Mathematical Sciences: Percolation and Random Graphs

首席研究员将在渗流和随机图中进行数学研究。这两个主题都涉及聚集过程的数学模型，并应用于统计力学中的相变和临界现象。渗流理论的研究主要集中在两个问题上：在经典的渗流模型中，对于临界概率(相变)值，很少有严格的数学精确解。这个项目继续开发一种计算临界概率的准确上下限的方法。AB渗流是经典渗流模型的一种变体，被引入用于研究反铁磁体和凝胶过程。AB渗流模型的行为与经典渗流模型有很大的不同，在某些晶格中有多个相变，而在另一些晶格中没有相变。在这个项目中，研究人员将研究为什么会出现这些差异。随机图理论研究继续探索鄂尔多斯-仁义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.