  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.
