{
  "id": "1908.06704",
  "version": "v1",
  "published": "2019-08-19T11:25:24.000Z",
  "updated": "2019-08-19T11:25:24.000Z",
  "title": "A CLT for the total energy of the two-dimensional critical Ising model",
  "authors": [
    "Jianping Jiang"
  ],
  "comment": "14 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider the Ising model on $([1,2N]\\times[1,2M])\\cap\\mathbb{Z}^2$ at critical temperature with periodic boundary condition in the horizontal direction and free boundary condition in the vertical direction. Let $E_{M,N}$ be its total energy (or Hamiltonian). Suppose $M$ is a function of $N$ satisfying $M\\geq N/(\\ln N)^{\\alpha}$ for some $\\alpha\\in[0,1)$. In particular, one may take $M=N$. We prove that \\begin{equation*} \\frac{E_{M,N}+4\\sqrt{2}M N-(4/\\pi)N\\ln N}{\\sqrt{(32/\\pi)MN\\ln N}} \\end{equation*} converges weakly to a standard Gaussian distribution as $N\\rightarrow\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-19T11:25:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "two-dimensional critical ising model",
      "total energy",
      "periodic boundary condition",
      "standard gaussian distribution",
      "free boundary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}