We consider the Random-Cluster model on $\mathbb{ Z}^d$ with interactions of infinite range of the form $J_x = \psi(x)\mathsf{ e}^{ -\rho(x)}$ with $\rho$ a norm on $\mathbb{ Z}^d$ and $\psi$ a subexponential correction. We first provide an optimal criterion ensuring the existence of a nontrivial saturation regime (that is, the existence of $\beta_{ \rm sat}(s)>0$ such that the inverse correlation length in the direction $s$ is constant on $[0,\beta_{ \rm sat}(s))$), thus removing a regularity assumption used in a previous work of ours. 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 $(0,\beta_{ \rm sat}(s))$. 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 $\beta$ in the regime $(\beta_{ \rm sat}(s), \beta_{ \rm c})$.
Categories: math.PR cond-mat.stat-mech math-ph math.MP