The self-dual point of the two-dimensional random-cluster model is critical for q > 1

We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter q > 1 on the square lattice is equal to the self-dual point psd(q) = p q=(1 + p q). This gives a proof that the critical temperature of the q-state Potts model is equal to log(1 + p q) for all q > 2. We further prove that the transition is sharp, meaning that there is exponential decay of correlations in the sub-critical phase. The techniques of this paper are rigorous and valid for all q > 1, in contrast to earlier methods valid only for certain given q. The proof extends to the triangular and the hexagonal lattices as well.