We prove that the phase transition of random-cluster and Potts models on any transitive graph is sharp. That is, we show that for $p < p_c$, the probability that the origin is connected by an open path to distance $n$ decays exponentially fast in $n$. This would also imply the sharpness of phase transition for the Potts model.
A main ingredient of the proof comes from the theory of decision trees. An inequality on decision trees on monotonic measures is obtained which generalises the OSSS inequality on product spaces.
This talk is based on works with H. Duminil-Copin and V. Tassion.