A new proof for percolation phase transition on stretched lattices

Marcelo R. Hilário, Marcos Sá, Remy Sanchis, Augusto Teixeira

Published 2023-11-24Version 1

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.