DISCONTINUITY OF THE PHASE TRANSITION FOR THE PLANAR RANDOM-CLUSTER AND POTTS MODELS WITH q > 4

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 (1) Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, (2) Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and (3) 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 randomcluster 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)/root q-4) as q tends to 4.