{
  "id": "2311.07472",
  "version": "v1",
  "published": "2023-11-13T17:07:23.000Z",
  "updated": "2023-11-13T17:07:23.000Z",
  "title": "Invariance principle and local limit theorem for a class of random conductance models with long-range jumps",
  "authors": [
    "Sebastian Andres",
    "Martin Slowik"
  ],
  "comment": "39 pages, 1 figure",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We study continuous time random walks on $\\mathbb{Z}^d$ (with $d \\geq 2$) among random conductances $\\{ \\omega(\\{x,y\\}) : x,y \\in \\mathbb{Z}^d\\}$ that permit jumps of arbitrary length. The law of the random variables $\\omega(\\{x,y\\})$, taking values in $[0, \\infty)$, is assumed to be stationary and ergodic with respect to space shifts. Assuming that the first moment of $\\sum_{x \\in \\mathbb{Z}^d} \\omega(\\{0,x\\}) |x|^2$ and the $q$-th moment of $1/\\omega(0,x)$ for $x$ neighbouring the origin are finite for some $ q >d/2$, we show a quenched invariance principle and a quenched local limit theorem, where the moment condition is optimal for the latter. We also obtain H\\\"older regularity estimates for solutions of the heat equation for the associated non-local discrete operator, and deduce that the pointwise spectral dimension equals $d$ almost surely. Our results apply to random walks on long-range percolation graphs with connectivity exponents larger than $d+2$ when all nearest-neighbour edges are present.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-13T17:07:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60F17",
      "82C41",
      "82B43"
    ],
    "keywords": [
      "local limit theorem",
      "random conductance models",
      "invariance principle",
      "long-range jumps",
      "study continuous time random walks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}