# Site Percolation on a Disordered Triangulation of the Square Lattice

## TL;DR

This paper investigates site percolation on a modified square lattice with added diagonals, conjecturing a critical probability of 1/2 and proving it for most cases, advancing understanding of percolation thresholds in disordered lattices.

## Contribution

It introduces a new class of disordered triangulations of the square lattice and proves the percolation threshold is 1/2 for almost all such graphs, supporting the conjecture.

## Key findings

- Conjecture that the critical probability p_c=1/2 for these graphs.
- Proof of p_c=1/2 for almost every such triangulation.
- Supports the universality of the percolation threshold in disordered lattices.

## 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.