Sharp Phase Transition for the Random-Cluster Model with Summable External Magnetic Field

In this paper, we prove sharpness of the phase transition for the random-cluster model in summable positive external fields, with cluster weight q=2,3,…, on the hypercubic lattice. That is, there exists some nontrivial critical parameter that depends on the cluster weight and the external field, below which the model exhibits exponential decay and above which there exists almost surely an infinite cluster.