{
  "id": "1412.0175",
  "version": "v1",
  "published": "2014-11-30T03:21:11.000Z",
  "updated": "2014-11-30T03:21:11.000Z",
  "title": "Quenched Invariance Principle for a class of random conductance models with long-range jumps",
  "authors": [
    "Marek Biskup",
    "Takashi Kumagai"
  ],
  "comment": "26 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study random walks on $\\mathbb Z^d$ among random conductances $\\{C_{xy}\\colon x,y\\in\\mathbb Z^d\\}$ that permit jumps of arbitrary length. Apart from joint ergodicity with respect to spatial shifts, we assume only that the nearest-neighbor conductances are uniformly positive and that $\\sum_{x\\in\\mathbb Z^d} C_{0x}|x|^2$ is integrable. Our focus is on the Quenched Invariance Principle (QIP) which we establish in all $d\\ge3$ by a combination of corrector methods and heat-kernel technology. In particular, a QIP thus holds for random walks on long-range percolation graphs with exponents larger than $d+2$ in all $d\\ge3$, provided all nearest-neighbor edges are present. We then show that, for long-range percolation with exponents between $d+2$ and $2d$, the corrector fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in $d\\ge4$ under the conditions close to, albeit not exactly, complementary to those of the recent work of S. Andres, M. Slowik and J.-D. Deuschel.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-30T03:21:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "60K37",
      "60K35"
    ],
    "keywords": [
      "quenched invariance principle",
      "random conductance models",
      "long-range jumps",
      "study random walks",
      "long-range percolation graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.0175B"
    }
  }
}