Two-Dimensional Potts Model and Annular Partitions

Using the random cluster expansion, correlations of the Potts model on a graph can be expressed as sums over partitions of the vertices where the spins are fixed. For a planar graph, only certain partitions can occur in these sums. For example, when all fixed spins lie on the boundary of one face, only noncrossing partitions contribute. In this paper we examine the partitions which occur when fixed spins lie on the boundaries of two disjoint faces. We call these the annular partitions, and we establish some of their basic properties. In particular we demonstrate a partial duality for these partitions, and show some implications for correlations of the Potts model.