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

A new proof for percolation phase transition on stretched lattices