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   - 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. 
Hugo Duminil-Copin, Maxime Gagnebin, Matan Harel, Ioan Manolescu,   Vincent Tassion
math.PR math-ph math.MP