{
  "id": "1501.05263",
  "version": "v1",
  "published": "2015-01-21T19:02:22.000Z",
  "updated": "2015-01-21T19:02:22.000Z",
  "title": "Mixing times for a constrained Ising process on the 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 stationary distribution of the KCIP Markov chain is the Binomial($|G|, p$) distribution on the number of particles, conditioned on having at least one particle. 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, and more recently as a simple algorithm for data storage in computer networks. In this note, we study the mixing time of this process on the torus $G = \\mathbb{Z}_{L}^{d}$, $d \\geq 3$, in the low-density regime $p = \\frac{c}{n}$ for arbitrary $0 < c < 1$; this regime is the subject of a conjecture of Aldous and is natural in the context of computer networks. Our results provide a counterexample to Aldous' conjecture, suggest a natural modifcation of the conjecture, and show that this modifcation is correct up to logarithmic factors. The methods developed in this paper also provide a strategy for tackling Aldous' conjecture for other graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-21T19:02:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "constrained ising process",
      "mixing time",
      "low density",
      "computer networks",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150105263P"
    }
  }
}