{
  "id": "2004.01971",
  "version": "v1",
  "published": "2020-04-04T16:34:54.000Z",
  "updated": "2020-04-04T16:34:54.000Z",
  "title": "Quenched Invariance Principle for a class of random conductance models with long-range jumps",
  "authors": [
    "Marek Biskup",
    "Xin Chen",
    "Takashi Kumagai",
    "Jian Wang"
  ],
  "comment": "33 pages, subsumes salvageable parts of arXiv:1412.0175",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study random walks on $\\mathbb Z^d$ (with $d\\ge 2$) among stationary ergodic random conductances $\\{C_{x,y}\\colon x,y\\in\\mathbb Z^d\\}$ that permit jumps of arbitrary length. Our focus is on the Quenched Invariance Principle (QIP) which we establish by a combination of corrector methods, functional inequalities and heat-kernel technology assuming that the $p$-th moment of $\\sum_{x\\in\\mathbb Z^d}C_{0,x}|x|^2$ and the $q$-th moment of $1/C_{0,x}$ for $x$ neighboring the origin are finite for some $p,q\\ge1$ with $p^{-1}+q^{-1}<2/d$. In particular, a QIP thus holds for random walks on long-range percolation graphs with connectivity exponents larger than $2d$ in all $d\\ge2$, provided all the nearest-neighbor edges are present. Although still limited by moment conditions, our method of proof is novel in that it avoids proving everywhere-sublinearity of the corrector. This is relevant because we show that, for long-range percolation with exponents between $d+2$ and $2d$, the corrector exists but fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in $d\\ge3$ under the conditions complementary to those of the recent work of P. Bella and M. Sch\\\"affner (arXiv:1902.05793). These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-04T16:34:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "39B62",
      "60K37",
      "60K35"
    ],
    "keywords": [
      "quenched invariance principle",
      "random conductance models",
      "long-range jumps",
      "stationary ergodic random conductances",
      "th moment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}