{
  "id": "2012.09484",
  "version": "v1",
  "published": "2020-12-17T10:19:49.000Z",
  "updated": "2020-12-17T10:19:49.000Z",
  "title": "Ising model on trees and factors of IID",
  "authors": [
    "Danny Nam",
    "Allan Sly",
    "Lingfu Zhang"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.DS",
    "math.MP"
  ],
  "abstract": "We study the ferromagnetic Ising model on the infinite $d$-regular tree under the free boundary condition. This model is known to be a factor of IID in the uniqueness regime, when the inverse temperature $\\beta\\ge 0$ satisfies $\\tanh \\beta \\le (d-1)^{-1}$. However, in the reconstruction regime ($\\tanh \\beta > (d-1)^{-\\frac{1}{2}}$), it is not a factor of IID. We construct a factor of IID for the Ising model beyond the uniqueness regime via a strong solution to an infinite dimensional stochastic differential equation which partially answers a question of Lyons. The solution $\\{X_t(v) \\}$ of the SDE is distributed as \\[ X_t(v) = t\\tau_v + B_t(v), \\] where $\\{\\tau_v \\}$ is an Ising sample and $\\{B_t(v) \\}$ are independent Brownian motions indexed by the vertices in the tree. Our construction holds whenever $\\tanh \\beta \\le c(d-1)^{-\\frac{1}{2}}$, where $c>0$ is an absolute constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-17T10:19:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ising model",
      "infinite dimensional stochastic differential equation",
      "uniqueness regime",
      "independent brownian motions",
      "free boundary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}