  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 $\mathbb{Z}$. 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 $p > p_c(\mathbb{Z}^2)$ (the critical point for homogeneous Bernoulli
bond percolation) then $q$ can be taken strictly smaller then
$p_c(\mathbb{Z}^2)$ in such a way that the probability that the origin
percolates is still positive.
