Percolation on stretched lattices: a static renormalization approach

Abstract

In this article, we consider a variety of percolation models on randomly stretched lattices. The first model we study is constructed on the usual square grid $$\mathbb {Z}^2$$ Z 2 , keeping all vertices untouched while erasing edges according to the following procedure: for every integer i , the entire column of vertical edges contained in the line $$\{ x = i \}$$ { x = i } is removed independently of other columns with probability $$\rho > 0$$ ρ > 0 . Similarly, for every integer j , the entire row of horizontal edges contained in the line $$\{ y = j\}$$ { y = j } is removed independently of other rows with probability $$\rho $$ ρ . 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 which was earlier established by Hoffman. The main novelty of our work is that the dynamic renormalization employed earlier is now replaced by a static version, which is easier to master 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. In addition, we prove a result about the sensitivity of the phase transition with respect to the stretching mechanism and provide a list of open problems that could be explored using our techniques.