Existence and properties of connections decay rate for high temperature percolation models

Abstract

We consider generic finite range percolation models on ${\mathbb{Z}}^d$ under a high temperature/low density assumption (exponential decay of connection probabilities and exponential ratio weak mixing). We prove that the rate of decay of point-to-point connections exists in every directions and show that it naturally extends to a norm on ${\mathbb{R}}^d$. This result is the base input to obtain fine understanding of the high temperature phase (e.g. Ornstein-Zernike asymptotics for point-to-point connexions) and is usually proven using correlation inequalities (such as FKG). The present work makes no use of such model specific properties and is therefore a step towards the universality of Ornstein-Zernike asymptotics.