The phase transitions of the random-cluster and Potts models on slabs with $q \geq 1$ are sharp

We prove sharpness of the phase transition for the random-cluster model with q >= 1 on graphs of the form J := G x S, where G is a planar lattice with mild symmetry assumptions, and S a finite graph. That is, for any such graph and any q >= 1, there exists some parameter p(c) = p(c)(J, q), below which the model exhibits exponential decay and above which there exists a. s. an infinite cluster. The result is also valid for the random-cluster model on planar graphs with long range, compactly supported interaction. It extends to the Potts model via the Edwards-Sokal coupling.