{
  "id": "2311.14644",
  "version": "v1",
  "published": "2023-11-24T18:20:29.000Z",
  "updated": "2023-11-24T18:20:29.000Z",
  "title": "A new proof for percolation phase transition on stretched lattices",
  "authors": [
    "Marcelo R. Hilário",
    "Marcos Sá",
    "Remy Sanchis",
    "Augusto Teixeira"
  ],
  "comment": "38 pages, 11 figures",
  "categories": [
    "math.PR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-24T18:20:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43",
      "05C10"
    ],
    "keywords": [
      "percolation phase transition",
      "stretched lattices",
      "perform bernoulli bond percolation",
      "nontrivial phase transition",
      "dynamic renormalization employed earlier"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}