{
  "id": "math/0301160",
  "version": "v2",
  "published": "2003-01-15T18:48:09.000Z",
  "updated": "2015-10-16T21:49:09.000Z",
  "title": "A singularity at the criticality for the free energy in percolation",
  "authors": [
    "Yu Zhang"
  ],
  "comment": "37 pages 4 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider percolation on the triangular lattice. Let $\\kappa(p)$ be the free energy at the zero field. We show that $$|\\kappa'''(p)| \\leq |p-p_c|^{-1/3+o(1)} \\mbox{ if } p \\neq p_c.$$ Furthermore, we show that there is a sequence $\\epsilon_n \\downarrow 0$ such that $$\\kappa''' (p_c+\\epsilon_n )\\leq -\\epsilon_n^{-1/3+o(1)} \\mbox{ and } \\kappa''' (p_c-\\epsilon_n )\\geq \\epsilon_n^{-1/3+o(1)}. $$ Note that these inequalities imply that $\\kappa(p)$ is not third differentiable. This answers affirmatively a conjecture, asked by Sykes and Essam a half century ago, whether $\\kappa(p)$ has a singularity at the criticality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-01-15T18:48:09.000Z",
      "title": "A power law for the free energy in two dimensional percolation",
      "abstract": "Consider bond percolation on the square lattice and site percolation on the triangular lattice. Let $\\kappa(p)$ be the free energy at the zero field. If we assume the existence of the critical exponents for the three arm and four arm paths and these critical exponents are -2/3 and -5/4, respectively, then we can show the following power law for the free energy function $\\kappa(p)$: {eqnarray*} &&\\kappa'''(p)= +(1/2-p)^{-1/3+\\delta(|1/2-p|)}{for} p < 1/2 &&\\kappa'''(p)= -(1/2-p)^{-1/3+\\delta(|1/2-p|)}{for} p > 1/2, {eqnarray*} where $\\delta(x)$ goes to zero as $x\\to 0$. Note that the critical exponents for four arm and three arm paths indeed are proven to exist and equal -5/4 and -2/3 on the triangular lattice and the above power law for $\\kappa(p)$ therefore holds for the triangular lattice. Note that the above power law for $\\kappa(p)$ implies $\\kappa(p)$ is not third differentiable at the critical point of the triangular lattice. This answers a long time conjecture that $\\kappa(p)$ has a singularity at 1/2 since 1964 affirmatively.",
      "comment": "49 pages 5 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-10-16T21:49:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "power law",
      "dimensional percolation",
      "triangular lattice",
      "critical exponents",
      "arm paths"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......1160Z"
    }
  }
}