The ferromagnetic q-state Potts model on three-dimensional lattices: a study for real values of q

We study the phase diagram of the ferromagnetic $q$-state Potts model on the various three-dimensional lattices for integer and non-integer values of $q>1$. Our approach is based on a thermodynamically self-consistent Ornstein-Zernike approximation for the two-point correlation functions. We calculate the transition temperatures and, when the transition is first order, the jump discontinuities in the magnetization and the internal energy, as well as the coordinates of the critical endpoint in external field. Our predictions are in very good agreement with best available estimates. From the numerical study of the region of weak first-order transition, we estimate the critical value $q_c$ for which the transition changes from second to first-order. The $q->1$ limit that describes the bond-percolation problem is also investigated.