Topology invariance in Percolation Thresholds

TL;DR

This paper introduces a universal invariant relating site and bond percolation thresholds, demonstrating its applicability across various lattice types and dimensions, revealing a fundamental topological property of percolation phenomena.

Contribution

The authors propose a new invariant formula for percolation thresholds that is topology-invariant and validated across diverse lattice structures and dimensions.

Findings

Invariant holds within ±5% error for 20 tested lattices

Applicable to non-Bravais and aperiodic lattices

Valid up to six dimensions for hypercubes

Abstract

An universal invariant for site and bond percolation thresholds (p_{cs} and p_{cb} respectively) is proposed. The invariant writes {p_{cs}}^{1/a_s}{p_{cb}}^{-1/a_b}=\delta/d where a_s, a_b and \delta are positive constants,and d the space dimension. It is independent of the coordination number, thus exhibiting a topology invariance at any d.The formula is checked against a large class of percolation problems, including percolation in non-Bravais lattices and in aperiodic lattices as well as rigid percolation. The invariant is satisfied within a relative error of \pm 5% for all the twenty lattices of our sample at d=2, d=3, plus all hypercubes up to d=6.