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We prove that the q-state Potts model and the random-cluster model with cluster weight q>4 undergo a discontinuous phase transition on the square lattice. More precisely, we show

- Existence of multiple infinite-volume measures for the critical Potts and random-cluster models,

- Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and

- Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models.

The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model.

As a byproduct, we rigorously compute the correlation lengths of the critical random-cluster and Potts models, and show that they behave as \exp(\pi^2/\sqrt{q-4}) as q tends to 4.