{
  "id": "1912.04854",
  "version": "v1",
  "published": "2019-12-10T17:50:26.000Z",
  "updated": "2019-12-10T17:50:26.000Z",
  "title": "Random spanning forests and hyperbolic symmetry",
  "authors": [
    "Roland Bauerschmidt",
    "Nicholas Crawford",
    "Tyler Helmuth",
    "Andrew Swan"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $\\beta>0$ per edge. This is called the arboreal gas model, and the special case when $\\beta=1$ is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $p=\\beta/(1+\\beta)$ conditioned to be acyclic. It is known that on the complete graph $K_{N}$ with $\\beta=\\alpha/N$ there is a phase transition similar to that of the Erd\\H{o}s--R\\'enyi random graph: a giant tree percolates for $\\alpha > 1$ and all trees have bounded size for $\\alpha<1$. In contrast to this, by exploiting an exact relationship between the arboreal gas and a supersymmetric sigma model with hyperbolic target space, we show that the forest constraint is significant in two dimensions: trees do not percolate on $\\mathbb{Z}^2$ for any finite $\\beta>0$. This result is a consequence of a Mermin--Wagner theorem associated to the hyperbolic symmetry of the sigma model. Our proof makes use of two main ingredients: techniques previously developed for hyperbolic sigma models related to linearly reinforced random walks and a version of the principle of dimensional reduction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-10T17:50:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "random spanning forests",
      "hyperbolic symmetry",
      "hyperbolic sigma models",
      "supersymmetric sigma model",
      "hyperbolic target space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}