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We prove that random-cluster models with q larger than 1 on a variety ofplanar lattices have a sharp phase transition, that is that there exists someparameter p_c below which the model exhibits exponential decay and above whichthere exists a.s. an infinite cluster. The result may be extended to the Pottsmodel via the Edwards-Sokal coupling.  Our method is based on sharp threshold techniques and certain symmetries ofthe lattice; in particular it makes no use of self-duality. Part of theargument is not restricted to planar models and may be of some interest for theunderstanding of random-cluster and Potts models in higher dimensions.  Due to its nature, this strategy could be useful in studying other planarmodels satisfying the FKG lattice condition and some additional differentialinequalities.

The phase transitions of the planar random-cluster and Potts models with q larger than 1 are sharp