{
  "id": "2111.05146",
  "version": "v2",
  "published": "2021-11-09T13:43:42.000Z",
  "updated": "2022-04-12T14:11:01.000Z",
  "title": "Ising model with Curie-Weiss perturbation",
  "authors": [
    "Federico Camia",
    "Jianping Jiang",
    "Charles M. Newman"
  ],
  "comment": "20 pages, revision after the referee's report",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider the nearest-neighbor Ising model on $\\Lambda_n:=[-n,n]^d\\cap\\mathbb{Z}^d$ at inverse temperature $\\beta\\geq 0$ with free boundary conditions, and let $Y_n(\\sigma):=\\sum_{u\\in\\Lambda_n}\\sigma_u$ be its total magnetization. Let $X_n$ be the total magnetization perturbed by a critical Curie-Weiss interaction, i.e., \\begin{equation*} \\frac{d F_{X_n}}{d F_{Y_n}}(x):=\\frac{\\exp[x^2/\\left(2\\langle Y_n^2 \\rangle_{\\Lambda_n,\\beta}\\right)]}{\\left\\langle\\exp[Y_n^2/\\left(2\\langle Y_n^2\\rangle_{\\Lambda_n,\\beta}\\right)]\\right\\rangle_{\\Lambda_n,\\beta}}, \\end{equation*} where $F_{X_n}$ and $F_{Y_n}$ are the distribution functions for $X_n$ and $Y_n$ respectively. We prove that for any $d\\geq 4$ and $\\beta\\in[0,\\beta_c(d)]$ where $\\beta_c(d)$ is the critical inverse temperature, any subsequential limit (in distribution) of $\\{X_n/\\sqrt{\\mathbb{E}\\left(X_n^2\\right)}:n\\in\\mathbb{N}\\}$ has an analytic density (say, $f_X$) all of whose zeros are pure imaginary, and $f_X$ has an explicit expression in terms of the asymptotic behavior of zeros for the moment generating function of $Y_n$. We also prove that for any $d\\geq 1$ and then for $\\beta$ small, \\begin{equation*} f_X(x)=K\\exp(-C^4x^4), \\end{equation*} where $C=\\sqrt{\\Gamma(3/4)/\\Gamma(1/4)}$ and $K=\\sqrt{\\Gamma(3/4)}/(4\\Gamma(5/4)^{3/2})$. Possible connections between $f_X$ and the high-dimensional critical Ising model with periodic boundary conditions are discussed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-04-12T14:11:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B20",
      "82B27"
    ],
    "keywords": [
      "curie-weiss perturbation",
      "total magnetization",
      "periodic boundary conditions",
      "free boundary conditions",
      "critical inverse temperature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}