Monotonicity of Percolation Probability on $\Z^d$ with Long Range Connections

TL;DR

This paper proves that in a site percolation model on ^d with both nearest neighbor and long-range bonds, the probability of percolation increases monotonically with the length of the long-range bonds.

Contribution

It establishes the monotonicity of percolation probability with respect to long-range bond length and generalizes to multiple bond lengths.

Findings

Percolation probability is non-decreasing with long-range bond length.

Monotonicity holds for models with multiple long-range bond lengths.

Results extend understanding of connectivity in long-range percolation models.

Abstract

Consider an independent site percolation model on $\Z^d,\ d\geq 2$, with parameter $p \in (0,1)$, where there are only nearest neighbor bonds and long range bonds of length $k$ parallel to some coordinate axis. We show that the percolation probability is a non-decreasing function of $k$. We also generalize this result for models whose long range bonds have several lengths.