Brochette percolation

We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on Z. Select vertical columns at random independently with a given positive probability. Keep (respectively remove) vertical edges in the selected columns, with probability p (respectively 1-p). All horizontal edges and vertical edges lying in unselected columns are kept (respectively removed) with probability q (respectively 1 - q). We show that, if p > p (c)(Z (2)) (the critical point for homogeneous Bernoulli bond percolation), then q can be taken strictly smaller than p (c)(Z (2)) in such a way that the probability that the origin percolates is still positive.