Tutte seminar - David Wagner

The Random Cluster Model

Abstract:

The random cluster model (RCM) is an object of interest in graph theory, probability, and statistical mechanics. Its combinatorial aspects include chromatic polynomials, spanning trees or forests, and connected spanning subgraphs. Physical applications include percolation, network reliability, and the theory of electrical networks. Applied to a complete graph, the RCM produces the Erdös-Rényi random graph as a special case.

I'll survey the origins of the RCM early in the 20th century, work from the 1970s on positive correlation inequalities, and recent work on the much more delicate subject of negative correlation inequalities.