On the Gibbs states of the non-critical Potts model on $Z^2$

All Gibbs states of the supercritical q-state Potts model on Z² are convex combinations of the q pure phases; in particular, they are all translation invariant. We recently proved this theorem with Hugo Duminil-Copin (Geneva), Dima Ioffe (Haifa) and Yvan Velenik (Geneva). I will explain the basic concepts underlying this result and present the heuristics of the proof, which consists of considering the model in large finite boxes with arbitrary boundary condition, and proving that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature beta > beta_c(q)=log(1+sqrt(q)).