Mathematical Sciences: Percolation and Random Graphs

首席研究员将继续他在渗流模型和随机图方面的工作。这两门学科的共同点都涉及研究每条边都以概率p出现的图。这种随机图根据这些图的性质提供了渗流模型。Wierman博士主要关心的是确定临界概率，从而根据概率p是否超过临界概率来确定特定属性的存在或不存在。所研究的特殊性质包括子图计数的渐近分布，以及无限开簇的存在性。使用的主要工具是二元性工具。Wierman博士将对偶性的概念扩展到了许多经典二元性不适用的渗流模型。这项研究的一个主要动机是应用于相变和临界现象。

The principal investigator will continue his work in the percolation models and random graphs. The common aspect of both subjects involves the study of graphs in which each edge is present with a probability p. Such random graphs provide as models for percolation depending on the properties of these graphs. Dr. Wierman's major concern is the determination of the critical probabilities such that a particular property is present or absent according to whether the probability p exceeds the critical probability or not. The particular properties investigated include asymptotic distribution of subgraph counts, and existence of infinite open clusters. The major tool used is that of the duality. Dr. Wierman has extended the concept of duality to many percolation models where the classical duality does not apply. A major motivation for this research is applications to phase transitions and critical phenomena.