On the Ornstein-Zernike behaviour for the Bernoulli bond percolation on $\mathbb{Z}^{d},d\geq3,$ in the supercitical regime

TL;DR

This paper establishes Ornstein-Zernike asymptotic behavior for bond percolation on high-dimensional integer lattices near full occupation probability, revealing geometric properties of decay surfaces.

Contribution

It proves Ornstein-Zernike behavior in all directions for supercritical bond percolation on $\,\mathbb{Z}^d$ with $d\geq3$, and analyzes the geometric nature of decay surfaces.

Findings

Ornstein-Zernike behavior holds in every direction near $p\to1$

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 bond percolation on $\mathbb{Z}^{d}$ for $d\geq3$ when $p,$ the probability of occupation of a bond, is sufficiently close to $1.$ Moreover, we prove that equi-decay surfaces are locally analytic, strictly convex, with positive Gaussian curvature.