{
  "id": "2006.14559",
  "version": "v1",
  "published": "2020-06-25T17:02:29.000Z",
  "updated": "2020-06-25T17:02:29.000Z",
  "title": "Ising model and s-embeddings of planar graphs",
  "authors": [
    "Dmitry Chelkak"
  ],
  "comment": "62 pages, 7 figures",
  "categories": [
    "math-ph",
    "math.CV",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We introduce the notion of s-embeddings $\\mathcal{S}=\\mathcal{S}_\\mathcal{F}$ of planar graphs carrying a (critical) nearest-neighbor Ising model; the construction is based upon a choice of a special solution $\\mathcal{F}$ of the three-terms propagation equation for Kadanoff-Ceva fermions, a so-called Dirac spinor. Each Dirac spinor $\\mathcal{F}$ provides an interpretation of all other solutions of the propagation equations as s-holomorphic functions on the s-embedding $\\mathcal{S}_\\mathcal{F}$, the notion of s-holomorphicity generalizes Smirnov's definition on the square grid/isoradial graphs and is a special case of t-holomorphic functions on t-embeddings appearing in the bipartite dimer model context. We set up a general framework for the analysis of s-holomorphic functions on s-embeddings $\\mathcal{S}^\\delta$ with $\\delta\\to 0$ (algebraic identities, a priori regularity theory etc) and then focus on the simplest situation when $\\mathcal{S}^\\delta$ have uniformly bounded lengths/angles and also lead to the horizontal (more precisely, $O(\\delta)$) profiles of the associated functions $\\mathcal{Q}^\\delta$; the latter can be viewed as the origami maps associated to $\\mathcal{S}^\\delta$ in the dimer model terminology. A very particular case when all these assumptions hold is provided by the critical Ising model on a doubly-periodic graph under its canonical s-embedding, another example is the critical Ising model on circle patterns. Under these assumptions we prove the convergence of basic fermionic observables to a conformally covariant limit; note that we develop a new strategy of the proof because of the lack of tools specific for the isoradial setup. Together with the RSW-type crossing estimates, which we prove under the same assumptions, this also implies the convergence of interfaces in the random cluster representation of the Ising model to Schramm's SLE(16/3) curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-25T17:02:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B20",
      "30G25",
      "60J67",
      "81T40"
    ],
    "keywords": [
      "planar graphs",
      "s-embedding",
      "s-holomorphic functions",
      "bipartite dimer model context",
      "s-holomorphicity generalizes smirnovs definition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}