{
  "id": "1204.1180",
  "version": "v4",
  "published": "2012-04-05T11:08:24.000Z",
  "updated": "2015-03-17T06:32:49.000Z",
  "title": "Critical two-point functions for long-range statistical-mechanical models in high dimensions",
  "authors": [
    "Lung-Chi Chen",
    "Akira Sakai"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/13-AOP843 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2015, Vol. 43, 639-681",
  "doi": "10.1214/13-AOP843",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We consider long-range self-avoiding walk, percolation and the Ising model on $\\mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)\\asymp|x|^{-d-\\alpha}$ with $\\alpha>0$. The upper-critical dimension $d_{\\mathrm{c}}$ is $2(\\alpha\\wedge2)$ for self-avoiding walk and the Ising model, and $3(\\alpha\\wedge2)$ for percolation. Let $\\alpha\\ne2$ and assume certain heat-kernel bounds on the $n$-step distribution of the underlying random walk. We prove that, for $d>d_{\\mathrm{c}}$ (and the spread-out parameter sufficiently large), the critical two-point function $G_{p_{\\mathrm{c}}}(x)$ for each model is asymptotically $C|x|^{\\alpha\\wedge2-d}$, where the constant $C\\in(0,\\infty)$ is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between $\\alpha<2$ and $\\alpha>2$. We also provide a class of random walks that satisfy those heat-kernel bounds.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-03-02T10:55:17.000Z",
      "abstract": "We consider long-range self-avoiding walk, percolation and the Ising model on Z^d that are defined by power-law decaying pair potentials with exponent -d-a. The upper-critical dimension d_c is 2min{a,2} for self-avoiding walk and the Ising model, and 3min{a,2} for percolation. Let a<2 or a>2, and assume certain heat-kernel bounds on the n-step distribution of the underlying random walk. We prove that, for d>d_c (and the spread-out parameter sufficiently large), the critical two-point function G(x) for each model is asymptotically |x|^{min{a,2}-d} times a constant which is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between a<2 and a>2. We also provide a class of random walks that satisfy those heat-kernel bounds.",
      "comment": "39 pages, 1 figure; typos corrected, appendix and references added and revised",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-03-17T06:32:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B27"
    ],
    "keywords": [
      "critical two-point function",
      "long-range statistical-mechanical models",
      "high dimensions",
      "random walk",
      "heat-kernel bounds"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.1180C"
    }
  }
}