Site Percolation on a Disordered Triangulation of the Square Lattice

Abstract

In this paper we consider independent site percolation in a triangulation of ${\mathbb{R}}^2$ given by adding $\sqrt{2}$-long diagonals to the usual graph ${\mathbb{Z}}^2$. We conjecture that $p_c=\frac{1}{2}$ for any such graph, and prove it for almost every such graph.