{
  "id": "2111.02116",
  "version": "v2",
  "published": "2021-11-03T10:19:27.000Z",
  "updated": "2022-03-22T11:12:47.000Z",
  "title": "Positivity of Gibbs states on distance-regular graphs",
  "authors": [
    "Michael Voit"
  ],
  "comment": "Example 6.3 on the q-Johnson scheme $J_2(2,4)$ added and minor typos corrected",
  "categories": [
    "math.CO",
    "math-ph",
    "math.CA",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We study criteria which ensure that Gibbs states (often also called generalized vacuum states) on distance-regular graphs are positive. Our main criterion assumes that the graph can be embedded into a growing family of distance-regular graphs. For the proof of the positivity we then use polynomial hypergroup theory and translate this positivity into the problem whether for $x\\in[-1,1]$ the function $n\\mapsto x^n$ has a positive integral representation w.r.t. the orthogonal polynomials associated with the graph. We apply our criteria to several examples. For Hamming graphs and the infinite distance-transitive graphs we obtain a complete description of the positive Gibbs states.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-22T11:12:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "33C45",
      "43A62",
      "81R12",
      "20N20",
      "43A90"
    ],
    "keywords": [
      "distance-regular graphs",
      "positivity",
      "polynomial hypergroup theory",
      "main criterion assumes",
      "generalized vacuum states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}