{
  "id": "2410.22061",
  "version": "v1",
  "published": "2024-10-29T14:11:06.000Z",
  "updated": "2024-10-29T14:11:06.000Z",
  "title": "Non-uniqueness of phase transitions for graphical representations of the Ising model on tree-like graphs",
  "authors": [
    "Ulrik Thinggaard Hansen",
    "Frederik Ravn Klausen",
    "Peter Wildemann"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the graphical representations of the Ising model on tree-like graphs. We construct a class of graphs on which the loop $\\mathrm{O}(1)$ model and then single random current exhibit a non-unique phase transition with respect to the inverse temperature, highlighting the non-monotonicity of both models. It follows from the construction that there exist infinite graphs $\\mathbb{G}\\subseteq \\mathbb{G}'$ such that the uniform even subgraph of $\\mathbb{G}'$ percolates and the uniform even subgraph of $\\mathbb{G}$ does not. We also show that on the wired $d$-regular tree, the phase transitions of the loop $\\mathrm{O}(1)$, the single random current, and the random-cluster models are all unique and coincide.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-29T14:11:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B05",
      "82B20",
      "82B26",
      "82B43",
      "05C80",
      "60K35"
    ],
    "keywords": [
      "tree-like graphs",
      "graphical representations",
      "ising model",
      "single random current",
      "non-uniqueness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}