On the oddness of percolation

TL;DR

This paper provides an alternative proof, using geometric symmetry, that the number of site configurations enabling vertical percolation on an MxN lattice is always odd, complementing previous findings.

Contribution

It introduces a new proof method based on geometric symmetry for the oddness of percolation configurations, applicable to both boundary conditions.

Findings

Number of percolation configurations is always odd.

Symmetry-based proof applies to free and periodic boundaries.

Supports previous results with an alternative approach.

Abstract

Recently Mertens and Moore [arXiv:1909.01484v1] showed that site percolation "is odd." By this they mean that on an $M\times N$ square lattice the number of distinct site configurations that allow for vertical percolation is odd. We report here an alternative proof, based on recursive use of geometric symmetry, for both free and periodic boundary conditions.