{
  "id": "2307.07461",
  "version": "v1",
  "published": "2023-07-14T16:43:18.000Z",
  "updated": "2023-07-14T16:43:18.000Z",
  "title": "Shattering in the Ising Pure $p$-Spin Model",
  "authors": [
    "David Gamarnik",
    "Aukosh Jagannath",
    "Eren C. Kızıldağ"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the Ising pure $p$-spin model for large $p$. We investigate the landscape of the Hamiltonian of this model. We show that for any $\\gamma>0$ and any large enough $p$, the model exhibits an intricate geometrical property known as the multi Overlap Gap Property above the energy value $\\gamma\\sqrt{2\\ln 2}$. We then show that for any inverse temperature $\\sqrt{\\ln 2}<\\beta<\\sqrt{2\\ln 2}$ and any large $p$, the model exhibits shattering: w.h.p. as $n\\to\\infty$, there exists exponentially many well-separated clusters such that (a) each cluster has exponentially small Gibbs mass, and (b) the clusters collectively contain all but a vanishing fraction of Gibbs mass. Moreover, these clusters consist of configurations with energy near $\\beta$. Range of temperatures for which shattering occurs is within the replica symmetric region. To the best of our knowledge, this is the first shattering result regarding the Ising $p$-spin models. Our proof is elementary, and in particular based on simple applications of the first and the second moment methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-14T16:43:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spin model",
      "ising pure",
      "shattering",
      "multi overlap gap property",
      "replica symmetric region"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}