{
  "id": "2201.04047",
  "version": "v2",
  "published": "2022-01-11T16:48:15.000Z",
  "updated": "2022-03-18T17:27:00.000Z",
  "title": "Macroscopic loops in the Bose gas, Spin O(N) and related models",
  "authors": [
    "Alexandra Quitmann",
    "Lorenzo Taggi"
  ],
  "comment": "43 pages, 9 figures, new results in the appendix, paper submitted",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a general system of interacting random loops which includes several models of interest, such as the Spin O(N) model, random lattice permutations, a version of the interacting Bose gas in discrete space and of the loop O(N) model. We consider the system in $\\mathbb{Z}^d$, $d \\geq 3$, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate $\\mathbb{Z}^d$ by finite boxes and, given any two vertices whose distance is proportional to the diameter of the box, we prove that the probability of observing a loop visiting both is uniformly positive. Our results hold under general assumptions on the interaction potential, which may have bounded or unbounded support or introduce hard-core constraints.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-18T17:27:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B27",
      "60K35",
      "82B20"
    ],
    "keywords": [
      "macroscopic loops",
      "related models",
      "random lattice permutations",
      "general system",
      "hard-core constraints"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}