The Potts model is defined on all possible colourings of vertices in q>1 possible values (q=2 corresponds to the Ising model). The distribution depends on the external parameter T>0 (temperature) and is given by the number of pairs of adjacent vertices having different colours. The agreements are favoured, but increasing temperature weakens the interactions.

It is classical that the model undergoes a phase transition: at low temperature, one
of the colours forms an infinite connected component (order); at high temperature, all monochromatic components are finite (disorder). When q=2,3,4, we give a new proof of the continuity of the phase transition (joint work with Lammers): unique Gibbs measure, no spontaneous magnetisation. When q>4, we establish the wetting phenomenon (joint work with Dober and Ott): separation of two monochromatic phases by a disordered layer.

Our arguments rely on couplings between several models: random-cluster, six-vertex and Ashkin-Teller. At the core of the proof of the continuity is a delocalisation result that applies also to random Lipschitz functions (the loop O(2) model).