Approximation on slabs and uniqueness for inhomogeneous percolation with a plane of defects

Abstract

Let ${\mathbb{L}}^d=({\mathbb{Z}}^d,{\mathbb{E}}^d)$ be the $d$-dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on ${\mathbb{L}}^d$ in which every edge inside the $s$-dimensional hyperplane ${\mathbb{Z}}^s\times \{0{\}}^{d-s}$, $2\le s<d$, is open with probability $q$ and every other edge is open with probability $p$. We prove the uniqueness of the infinite cluster in the supercritical regime whenever $p\ne p_c(d)$, where $p_c(d)$ denotes the threshold for homogeneous percolation, and that the critical point $(p,q_c(p))$ can be approximated on the phase space by the critical points of slabs, for any $p<p_c(d)$.

34 pages, 6 figures