Critical Point and Percolation Probability in a Long Range Site Percolation Model on $\Z^d$

TL;DR

This paper investigates how the percolation threshold in a long-range site percolation model on ^d converges to the threshold of ordinary percolation on ^{2d} as the long-range bond length increases, extending to multiple bond lengths.

Contribution

It proves the convergence of the percolation threshold to that of ^{2d} and generalizes the result for models with multiple long-range bond lengths.

Findings

Percolation threshold converges to that of ^{2d} as bond length increases.

Results extend to models with several long-range bond lengths.

Provides theoretical insight into long-range percolation behavior.

Abstract

Consider an independent site percolation model with parameter $p \in (0,1)$ on $\Z^d,\ d\geq 2$ where there are only nearest neighbor bonds and long range bonds of length $k$ parallel to each coordinate axis. We show that the percolation threshold of such model converges to $p_c(\Z^{2d})$ when $k$ goes to infinity, the percolation threshold for ordinary (nearest neighbour) percolation on $\Z^{2d}$. We also generalize this result for models whose long range bonds have several lengths.