On AB Bond Percolation on the Square Lattice and AB Site Percolation on Its Line Graph

Abstract

We prove that AB site percolation occurs on the line graph of the square lattice when \(p \in (1 - \sqrt {1 - p_c } ,\sqrt {1 - p_c } )\), where p c is the critical probability for site percolation in \(\mathbb{Z}^2\). Also, we prove that AB bond percolation does not occur on \(\mathbb{Z}^2\) for p = \(\frac{1}{2}\).