We derive the exact actions of the Q-state Potts model valid on any graph, first for the spin degrees of freedom, and second for the Fortuin-Kasteleyn clusters. In both cases the field is a traceless Q-component scalar field Φα. For the Ising model (Q = 2), the field theory for the spins has upper critical dimension \({d}_{{\text{c}}}^{{\text{spin}}}\) = 4, whereas for the clusters it has \({d}_{{\text{c}}}^{{\text{cluster}}}\) = 6. As a consequence, the probability for three points to be in the same cluster is not given by mean-field theory for d within 4 < d < 6. We estimate the associated universal structure constant as \(C=\sqrt{6-d}+\mathcal{O}{\left(6-d\right)}^{3/2}\) . This shows that some observables in the Ising model have an upper critical dimension of 4, while others have an upper critical dimension of 6. Combining perturbative results from the ϵ = 6 – d expansion with a non-perturbative treatment close to dimension d = 4 allows us to locate the shape of the critical domain of the Potts model in the whole (Q, d) plane.
