Universal condition for critical percolation thresholds of kagome-like lattices

TL;DR

This paper establishes a universal percolation criticality condition for kagome-like lattices, relating connection probabilities, and provides bounds and numerical thresholds that align well with empirical data.

Contribution

It introduces a universal relation between connection probabilities in kagome-like lattices and derives bounds and thresholds that improve understanding of percolation criticality.

Findings

Derived a linear approximation for P_3(P_0)

Provided upper bounds for critical thresholds

Calculated thresholds for various lattices matching numerical results

Abstract

Lattices that can be represented in a kagome-like form are shown to satisfy a universal percolation criticality condition, expressed as a relation between P_3, the probability that all three vertices in the triangle connect, and P_0, the probability that none connect. A linear approximation for P_3(P_0) is derived and appears to provide a rigorous upper bound for critical thresholds. A numerically determined relation for P_3(P_0) gives thresholds for the kagome, site-bond honeycomb, (3-12^2), and "stack-of-triangle" lattices that compare favorably with numerical results.