Approximation on slabs and uniqueness for Bernoulli percolation with a sublattice of defects

We establish uniqueness of the infinite cluster and the Grimmett–Marstrand Theorem for a modified Bernoulli percolation. Edges are open independently with probability \(q\) for the edges in \(\mathbb{Z}^s\times\{0\}^{d-s}\) and another parameter \(p\) everywhere else in \(\mathbb{Z}^d\). What makes this study interesting is that the group of symmetry of the model does not act transitively (not even quasi-transitively) as soon as \(s\lt d\).

Latin American Journal of Probability and Mathematical Statistics (ALEA), 19:1767–1797, 2022