{
  "id": "2203.09291",
  "version": "v1",
  "published": "2022-03-17T12:48:45.000Z",
  "updated": "2022-03-17T12:48:45.000Z",
  "title": "Convergence of the free energy for spherical spin glasses",
  "authors": [
    "Eliran Subag"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove that the free energy of any spherical mixed $p$-spin model converges as the dimension $N$ tends to infinity. While the convergence is a consequence of the Parisi formula, the proof we give is independent of the formula and uses the well-known Guerra-Toninelli interpolation method. The latter was invented for models with Ising spins to prove that the free energy is super-additive and therefore (normalized by $N$) converges. In the spherical case, however, the configuration space is not a product space and the interpolation cannot be applied directly. We first relate the free energy on the sphere of dimension $N+M$ to a free energy defined on the product of spheres in dimensions $N$ and $M$ to which we then apply the interpolation method. This yields an approximate super-additivity which is sufficient to prove the convergence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-17T12:48:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free energy",
      "spherical spin glasses",
      "convergence",
      "well-known guerra-toninelli interpolation method",
      "spin model converges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}