  This article studies the planar Potts model and its random-cluster
representation. We show that the phase transition of the nearest-neighbor
ferromagnetic $q$-state Potts model on $\mathbb Z^2$ is continuous for
$q\in\{2,3,4\}$, in the sense that there exists a unique Gibbs state, or
equivalently that there is no ordering for the critical Gibbs states with
monochromatic boundary conditions. The proof uses the random-cluster model with
cluster-weight $q\ge1$ (note that $q$ is not necessarily an integer) and is
based on two ingredients:
  1. The fact that the two-point function for the free state decays
sub-exponentially fast for cluster-weights $1\le q\le 4$, which is derived
studying parafermionic observables on a discrete Riemann surface.
  2. A new result proving the equivalence of several properties of critical
random-cluster models:
  - the absence of infinite-cluster for wired boundary conditions,
  - the uniqueness of infinite-volume measures,
  - the sub-exponential decay of the two-point function for free boundary
conditions,
  - a Russo-Seymour-Welsh type result on crossing probabilities in rectangles
with arbitrary boundary conditions.
  The result leads to a number of consequences concerning the scaling limit of
the random-cluster model with $1\le q \le 4$. It shows that the family of
interfaces (for instance for Dobrushin boundary conditions) are tight when
taking the scaling limit and that any sub-sequential limit can be parametrized
by a Loewner chain. We also study the effect of boundary conditions on these
sub-sequential limits. Let us mention that the result should be instrumental in
the study of critical exponents as well.
