{
  "id": "2202.02301",
  "version": "v1",
  "published": "2022-02-04T18:37:37.000Z",
  "updated": "2022-02-04T18:37:37.000Z",
  "title": "Log-Sobolev inequality for near critical Ising models",
  "authors": [
    "Roland Bauerschmidt",
    "Benoit Dagallier"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "For general ferromagnetic Ising models whose coupling matrix has bounded spectral radius, we show that the log-Sobolev constant satisfies a simple bound expressed only in terms of the susceptibility of the model. This bound implies very generally that the log-Sobolev constant is uniform in the system size up to the critical point (including on lattices), without using any mixing conditions. Moreover, if the susceptibility satisfies the mean-field bound as the critical point is approached, our bound implies that the log-Sobolev constant depends polynomially on the distance to the critical point and on the volume. In particular, this applies to the Ising model on subsets of $\\mathbb{Z}^d$ when $d>4$. The proof uses a general criterion for the log-Sobolev inequality in terms of the Polchinski (renormalisation group) equation, a recently proved remarkable correlation inequality for Ising models with general external fields, the Perron--Frobenius theorem, and the log-Sobolev inequality for product Bernoulli measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-04T18:37:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "log-sobolev inequality",
      "critical ising models",
      "critical point",
      "bound implies",
      "product bernoulli measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}