{
  "id": "1808.06789",
  "version": "v1",
  "published": "2018-08-21T07:13:31.000Z",
  "updated": "2018-08-21T07:13:31.000Z",
  "title": "Critical two-point function for long-range models with power-law couplings: The marginal case for $d\\ge d_c$",
  "authors": [
    "Lung-Chi Chen",
    "Akira Sakai"
  ],
  "comment": "31 pages, 1 figure, 3 diagrams in equations",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Consider the long-range models on $\\mathbb{Z}^d$ of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as $|x|^{-d-\\alpha}$ for some $\\alpha>0$. In the previous work (Ann. Probab., 43, 639--681, 2015), we have shown in a unified fashion for all $\\alpha\\ne2$ that, assuming a bound on the \"derivative\" of the $n$-step distribution (the compound-zeta distribution satisfies this assumed bound), the critical two-point function $G_{p_c}(x)$ decays as $|x|^{\\alpha\\wedge2-d}$ above the upper-critical dimension $d_c\\equiv(\\alpha\\wedge2)m$, where $m=2$ for self-avoiding walk and the Ising model and $m=3$ for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the $n$-step distribution, that $G_{p_c}(x)$ for the marginal case $\\alpha=2$ decays as $|x|^{2-d}/\\log|x|$ whenever $d\\ge d_c$ (with a large spread-out parameter $L$). This solves the conjecture in the previous work, extended all the way down to $d=d_c$, and confirms a part of predictions in physics (Brezin, Parisi, Ricci-Tersenghi, J. Stat. Phys., 157, 855--868, 2014). The proof is based on the lace expansion and new convolution bounds on power functions with log corrections.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-21T07:13:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B20",
      "82B27",
      "82B41",
      "82B43"
    ],
    "keywords": [
      "critical two-point function",
      "long-range models",
      "marginal case",
      "power-law couplings",
      "step distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}