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

Abstract

AbstractWe consider the Random-Cluster model on $${\mathbb {Z}}^d$$ Z d with interactions of infinite range of the form $$J_x = \psi (x){\mathsf {\scriptstyle e}}^{-\rho (x)}$$ J x = ψ ( x ) e - ρ ( x ) with $$\rho $$ ρ a norm on $${\mathbb {Z}}^d$$ Z d and $$\psi $$ ψ a subexponential correction. We first provide an optimal criterion ensuring the existence of a nontrivial saturation regime (that is, the existence of $$\beta _{\textrm{sat}}(s)>0$$ β sat ( s ) > 0 such that the inverse correlation length in the direction s is constant on $$[0,\beta _{\textrm{sat}}(s)$$ [ 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 $$(0,\beta _{\textrm{sat}}(s))$$ ( 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 $$\beta $$ β in the regime $$(\beta _{\textrm{sat}}(s), \beta _{\mathrm {\scriptscriptstyle c}})$$ ( β sat ( s ) , β c ) .