  We revisit the phase transition for percolation on randomly stretched
lattices. Starting with the usual square grid, keep all vertices untouched
while erasing edges according as follows: for every integer $i$, the entire
column of vertical edges contained in the line $\{ x = i \}$ is removed
independently of other columns with probability $\rho > 0$. Similarly, for
every integer $j$, the entire row of horizontal edges contained in the line $\{
y = j\}$ is removed independently with probability $\rho$. On the remaining
random lattice, we perform Bernoulli bond percolation. Our main contribution is
an alternative proof that the model undergoes a nontrivial phase transition, a
result established earlier by Hoffman. The main novelty lies on the fact that
the dynamic renormalization employed earlier is replaced by a static version,
which is simpler and more robust to extend to different models. We emphasize
the flexibility of our methods by showing the non-triviality of the phase
transition for a new oriented percolation model in a random environment as well
as for a model previously investigated by Kesten, Sidoravicius and Vares. We
also prove a result about the sensitivity of the phase transition with respect
to the stretching mechanism.
