On the Ornstein-Zernike behaviour for the supercritical Random-Cluster model on $\mathbb{Z}^{d},d\geq3.$

TL;DR

This paper establishes Ornstein-Zernike asymptotic behavior for the supercritical random cluster model on high-dimensional lattices, showing decay properties and geometric features of decay surfaces.

Contribution

It proves Ornstein-Zernike behavior in all directions for the model near full occupation probability, and demonstrates local analyticity and convexity of equi-decay surfaces.

Findings

Ornstein-Zernike decay holds in all directions for the model.

Equi-decay surfaces are locally analytic and strictly convex.

Surfaces have positive Gaussian curvature.

Abstract

We prove Ornstein-Zernike behaviour in every direction for finite connection functions of the random cluster model on $\mathbb{Z}^{d},d\geq3,$ for $q\geq1,$ when occupation probabilities of the bonds are close to $1.$ Moreover, we prove that equi-decay surfaces are locally analytic, strictly convex, with positive Gaussian curvature.