On the Two-Point Function of the Potts Model in the Saturation Regime

We consider the Random-Cluster model on ℤ^d with interactions of infinite range of the form J_x = ψ (x) 𝖾^-ρ (x) with ρ a norm on ℤ^d and ψ a subexponential correction. We first provide an optimal criterion ensuring the existence of a nontrivial saturation regime (that is, the existence of β _sat(s)>0 such that the inverse correlation length in the direction s is constant on [0,β _sat(s) )), thus removing a regularity assumption used in our previous work (Aoun et al. in Commun Math Phys 386:433–467, 2021). Then, under suitable assumptions, we derive sharp asymptotics (which are not of Ornstein–Zernike form) for the two-point function in the whole saturation regime (0,β _sat(s)) . We also obtain a number of additional results for this class of models, including sharpness of the phase transition, mixing above the critical temperature and the strict monotonicity of the inverse correlation length in β in the regime (β _sat(s), β _ c) .