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

Abstract

. Let L d = ( Z d , E d ) be the d -dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on L d in which every edge inside the s -dimensional sublattice Z s × { 0 } d − s , 2 ≤ 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 (cid:54) = p c ( d ) and 2 ≤ s < d − 1 , full uniqueness when s = d − 1 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 ) , where p c ( d ) denotes the threshold for homogeneous percolation.