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.
