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Percolation on stretched lattices: a static renormalization approach
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DOI：10.1007/s00440-026-01470-4.png)
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Abstract
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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 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} 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 of other rows 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 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.
Keywords:
Percolation
Renormalization
Dependent environments
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