Positive association and global connectivity in dependent percolation

TL;DR

This paper investigates how positive correlations influence the critical thresholds in percolation on a square lattice, revealing that such correlations can either increase or decrease global connectivity depending on the type of percolation.

Contribution

It introduces two algorithms for generating dependent lattices with controlled positive correlations and compares their critical behaviors with independent lattices.

Findings

Positive correlations lower site percolation thresholds.

Positive correlations increase bond percolation thresholds.

Dual lattice also exhibits positive association, affecting connectivity.

Abstract

We study the effect of positive correlations on the critical threshold of site and bond percolation in a square lattice with d = 2. We propose two algorithms for generating dependent lattices with minimal correlation length and non-negative n-point correlations whose critical behavior is then compared with that of independent lattices. For site percolation, we show numerically that the introduction of this specific form of positive correlation results in a lower percolation threshold, i.e., higher connectivity. For bond percolation, the opposite is observed. In this case, however, we show that the dual lattice is also totally positively associated, demonstrating that positive association can result in either an increase or a decrease in global connectivity.