Decay Properties of the Connectivity for Mixed Long Range Percolation Models on ${\mathbb{Z}}^d$

Abstract

In this short note we consider mixed short-long range independent bond percolation models on ${\mathbb{Z}}^{k+d}$. Let $p_{uv}$ be the probability that the edge $(u,v)$ will be open. Allowing a $x,y$-dependent length scale and using a multi-scale analysis due to Aizenman and Newman, we show that the long distance behavior of the connectivity ${\tau }_{xy}$ is governed by the probability $p_{xy}$. The result holds up to the critical point.

6 pages