Phase transition for the dilute clock model

We prove that phase transition occurs in the dilute ferromagnetic nearest-neighbour qq-state clock model in ZdZd, for every q≥2q≥2 and d≥2d≥2. This follows from the fact that the Edwards–Sokal random-cluster representation of the clock model stochastically dominates a supercritical Bernoulli bond percolation probability, a technique that has been applied to show phase transition for the low-temperature Potts model. The domination involves a combinatorial lemma which is one of the main points of this article.