On the two-point function of the Potts model in the saturation regime

TL;DR

This paper analyzes the two-point function of the Potts model's Random-Cluster representation with infinite-range interactions, establishing conditions for saturation regimes and deriving sharp asymptotics beyond Ornstein-Zernike form.

Contribution

It provides an optimal criterion for the existence of saturation regimes and derives sharp asymptotics for the two-point function in these regimes, extending previous results.

Findings

Existence of a nontrivial saturation regime under optimal conditions

Sharp asymptotics for the two-point function in the saturation regime

Monotonicity of inverse correlation length in the super-saturation regime

Abstract

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 $(\beta_{\rm sat}(s), \beta_{\rm c})$.