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

TL;DR

This paper investigates how the probability of connection in mixed short-long range percolation models on integer lattices influences long-distance connectivity, demonstrating that the decay behavior is primarily determined by the edge-opening probabilities.

Contribution

It extends understanding of connectivity decay in mixed long-range percolation models using multi-scale analysis, up to the critical point.

Findings

Connectivity decay is governed by edge probabilities $p_{xy}$.

Results hold up to the critical percolation threshold.

Multi-scale analysis confirms the decay behavior.

Abstract

In this short note we consider mixed short-long range independent bond percolation models on $\Z^{k+d}$. 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.