A new proof for percolation phase transition on stretched lattices

Abstract

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.