Mathematical Sciences: Percolation and Random Graphs

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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.