99-380 M.Biskup, L.Chayes, R. Kotecky'

On the Continuity of the Magnetization and the Energy Density for Potts Models on Two-dimensional Graphs (41K, AMS-TeX) Oct 9, 99

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Abstract. We consider the $q$-state Potts model on two-dimensional planar graphs. Our only assumptions concerning the graph and interaction are that the associated graphical representations satisfy the conclusion of the theorem of Gandolfi, Keane and Russo \cite{GKR}. In addition to $\Bbb Z^2$, the class of graphs we consider contains, for example, the triangular, honeycomb, and Kagom\'e lattices. Under these conditions we show that the only possible point of discontinuity of the magnetization and the energy density is at the onset of the magnetic ordering transition (i.e., at the threshold for bond percolation in the random-cluster model). The result generalizes to any model with a natural dual, appropriate FKG monotonicity properties and a percolation characterization of the Gibbs uniqueness.

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