Percolation is Odd

TL;DR

This paper proves a surprising combinatorial symmetry in site percolation, showing that the total number of spanning configurations is always odd across various lattice types and boundary conditions.

Contribution

It establishes a new symmetry in the count of spanning configurations in site percolation, revealing that the total is always an odd number regardless of lattice size or shape.

Findings

Total spanning configurations are always odd.

Symmetry applies to square and hypercubic lattices.

Results hold for various boundary conditions.

Abstract

We prove a remarkable combinatorial symmetry in the number of spanning configurations in site percolation: for a large class of lattices, the number of spanning configurations with an odd or even number of occupied sites differs by $\pm 1$. In particular, this symmetry implies that the total number of spanning configurations is always odd, independent of the size or shape of the lattice. The class of lattices that share this symmetry includes the square lattice and the hypercubic lattice in any dimension, with a wide variety of boundary conditions.