{
  "id": "1812.10275",
  "version": "v1",
  "published": "2018-12-26T09:54:34.000Z",
  "updated": "2018-12-26T09:54:34.000Z",
  "title": "Crossover phenomena in the critical behavior for long-range models with power-law couplings",
  "authors": [
    "Akira Sakai"
  ],
  "comment": "11 pages, 6 diagram pictures in equations",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "This is a short review of the two papers on the $x$-space asymptotics of the critical two-point function $G_{p_c}(x)$ for the long-range models of self-avoiding walk, percolation and the Ising model on $\\mathbb{Z}^d$, defined by the translation-invariant power-law step-distribution/coupling $D(x)\\propto|x|^{-d-\\alpha}$ for some $\\alpha>0$. Let $S_1(x)$ be the random-walk Green function generated by $D$. We have shown that $\\bullet~~S_1(x)$ changes its asymptotic behavior from Newton ($\\alpha>2$) to Riesz ($\\alpha<2$), with log correction at $\\alpha=2$; $\\bullet~~G_{p_c}(x)\\sim\\frac{A}{p_c}S_1(x)$ as $|x|\\to\\infty$ in dimensions higher than (or equal to, if $\\alpha=2$) the upper critical dimension $d_c$ (with sufficiently large spread-out parameter $L$). The model-dependent $A$ and $d_c$ exhibit crossover at $\\alpha=2$. The keys to the proof are (i) detailed analysis on the underlying random walk to derive sharp asymptotics of $S_1$, (ii) bounds on convolutions of power functions (with log corrections, if $\\alpha=2$) to optimally control the lace-expansion coefficients $\\pi_p^{(n)}$, and (iii) probabilistic interpretation (valid only when $\\alpha\\le2$) of the convolution of $D$ and a function $\\varPi_p$ of the alternating series $\\sum_{n=0}^\\infty(-1)^n\\pi_p^{(n)}$. We outline the proof, emphasizing the above key elements for percolation in particular.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-26T09:54:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B20",
      "82B27",
      "82B41",
      "82B43"
    ],
    "keywords": [
      "long-range models",
      "power-law couplings",
      "crossover phenomena",
      "critical behavior",
      "log correction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}