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

TL;DR

This paper proves the existence of a decay rate for connection probabilities in high temperature percolation models on integer lattices, extending it to a norm and providing insights into the high temperature phase without relying on correlation inequalities.

Contribution

It establishes the existence and properties of the decay rate in all directions for high temperature percolation models without using correlation inequalities.

Findings

Decay rate exists in every direction.

Decay rate extends to a norm on .

Provides a basis for understanding the high temperature phase.

Abstract

We consider generic finite range percolation models on $\mathbb{Z}^d$ under a high temperature 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 and is usually proven using correlation inequalities (such as FKG). The present work makes no use of such model specific properties.