Abstract

On the two-point function of the Potts model in the saturation regime Y. Aoun, S. Ott, Y. Velenik Commun. Math. Phys. 399, 1103-1138 (2023). We consider FK percolation 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 $(\beta_{\rm sat}(s),\beta_{\rm c})$. Key words: Potts model, FK percolation, infinite-range interactions, correlation length, sharp asymptotics of the 2-point function, violation of Ornstein-Zernike asymptotics, sharpness, mixing Files: PDF file, Published version, bibtex