{
  "id": "1610.04897",
  "version": "v1",
  "published": "2016-10-16T18:28:10.000Z",
  "updated": "2016-10-16T18:28:10.000Z",
  "title": "Exponential decay of connectivity and uniqueness in percolation on finite and infinite graphs",
  "authors": [
    "Kathleen E. Hamilton",
    "Leonid P. Pryadko"
  ],
  "comment": "2 pages. Abstract for the SIAM Workshop on Network Science (NS16), July 15-16, 2016, Boston, Massachusetts",
  "categories": [
    "math-ph",
    "cond-mat.dis-nn",
    "math.MP"
  ],
  "abstract": "We give an upper bound for the uniqueness transition on an arbitrary locally finite graph ${\\cal G}$ in terms of the limit of the spectral radii $\\rho\\left[ H({\\cal G}_t)\\right]$ of the non-backtracking (Hashimoto) matrices for an increasing sequence of subgraphs ${\\cal G}_t\\subset{\\cal G}_{t+1}$ which converge to ${\\cal G}$. With the added assumption of strong local connectivity for the oriented line graph (OLG) of ${\\cal G}$, connectivity on any finite subgraph ${\\cal G}'\\subset{\\cal G}$ decays exponentially for $p<(\\rho\\left[ H({\\cal G}^{\\prime})\\right])^{-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-16T18:28:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exponential decay",
      "infinite graphs",
      "percolation",
      "strong local connectivity",
      "arbitrary locally finite graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 2,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}