{
  "id": "2505.07145",
  "version": "v1",
  "published": "2025-05-11T23:08:27.000Z",
  "updated": "2025-05-11T23:08:27.000Z",
  "title": "Nonamenable Poisson zoo",
  "authors": [
    "Gábor Pete",
    "Sándor Rokob"
  ],
  "comment": "40 pages, 4 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.GR",
    "math.MP"
  ],
  "abstract": "In the Poisson zoo on an infinite Cayley graph $G$, we take a probability measure $\\nu$ on rooted finite connected subsets, called lattice animals, and place i.i.d. Poisson($\\lambda$) copies of them at each vertex. If the expected volume of the animals w.r.t. $\\nu$ is infinite, then the whole $G$ is covered for any $\\lambda>0$. If the second moment of the volume is finite, then it is easy to see that for small enough $\\lambda$ the union of the animals has only finite clusters, while for $\\lambda$ large enough there are also infinite clusters. Here we show that: 1. If $G$ is a nonamenable free product, then for ANY $\\nu$ with infinite second but finite first moment and any $\\lambda>0$, there will be infinite clusters, despite having arbitrarily low density. 2. The same result holds for ANY nonamenable $G$, when the lattice animals are worms: random walk pieces of random finite length. It remains open if the result holds for ANY nonamenable Cayley graph with ANY lattice animal measure $\\nu$ with infinite second moment. 3. We also give a Poisson zoo example $\\nu$ on $\\mathbb{T}_d \\times \\mathbb{Z}^5$ with finite first moment and a UNIQUE infinite cluster for any $\\lambda>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-11T23:08:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonamenable poisson zoo",
      "finite first moment",
      "infinite cluster",
      "result holds",
      "second moment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}