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

We prove sharpness of the phase transition for the random-cluster model with $q\ge 1$ on graphs of the form $\mathcal{S}:=\mathcal{G}\times S$, where $\mathcal{G}$ is a planar lattice with mild symmetry assumptions, and $S$ a finite graph. That is, for any such graph and any $q\ge 1$, there exists some parameter $p_c=p_c(\mathcal{S},q)$, 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.