{
  "id": "1708.03838",
  "version": "v1",
  "published": "2017-08-13T02:06:51.000Z",
  "updated": "2017-08-13T02:06:51.000Z",
  "title": "Mixing Times for a Constrained Ising Process on the Two-Dimensional Torus at Low Density",
  "authors": [
    "Natesh S Pillai",
    "Aaron Smith"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a kinetically constrained Ising process (KCIP) associated with a graph $G$ and density parameter $p$; this process is an interacting particle system with state space $\\{ 0, 1 \\}^{G}$, the location of the particles. The `constraint' in the name of the process refers to the rule that a vertex cannot change its state unless it has at least one neighbour in state `1'. The KCIP has been proposed by statistical physicists as a model for the glass transition. In this note, we study the mixing time of a KCIP on the 2-dimensional torus $G = \\mathbb{Z}_{L}^{2}$ in the low-density regime $p = \\frac{c}{L^{2}}$ for arbitrary $0 < c < \\infty$, extending our previous results for the analogous process on the torus $\\mathbb{Z}_{L}^{d}$ in dimension $d \\geq 3$. Our general approach is similar, but the extension requires more delicate bounds on the behaviour of the process at intermediate densities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-13T02:06:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10"
    ],
    "keywords": [
      "constrained ising process",
      "mixing time",
      "two-dimensional torus",
      "low density",
      "density parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}