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)
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.