Strict inequality for bond percolation on a dilute lattice with columnar disorder

Abstract

We consider a dilute lattice obtained from the usual ${\mathbb{Z}}^3$ lattice by removing independently each of its columns with probability $1-\rho$. In the remaining dilute lattice independent Bernoulli bond percolation with parameter $p$ is performed. Let $\rho \mapsto p_c(\rho )$ be the critical curve which divides the subcritical and supercritical phases. We study the behavior of this curve near the disconnection threshold ${\rho }_c=p_c^{\text{site}}({\mathbb{Z}}^2)$ and prove that, uniformly over $\rho$ it remains strictly below $1/2$ (the critical point for bond percolation on the square lattice ${\mathbb{Z}}^2)$.

arXiv comment15 pages