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

TL;DR

This paper studies inhomogeneous percolation on a lattice with a defect plane, proving uniqueness of the infinite cluster outside the critical threshold and showing how critical points can be approximated by slab models.

Contribution

It establishes the uniqueness of the infinite cluster for inhomogeneous percolation outside the critical point and demonstrates approximation of critical points via slab models.

Findings

Uniqueness of the infinite cluster when p ≠ p_c(d)

Critical points can be approximated by slab models

Critical threshold behavior depends on inhomogeneity parameters

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 \leq 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 \neq 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) $.