{
  "id": "1512.08234",
  "version": "v1",
  "published": "2015-12-27T15:58:41.000Z",
  "updated": "2015-12-27T15:58:41.000Z",
  "title": "Large Deviations on a Cayley Tree I: Rate Functions",
  "authors": [
    "Anatoly E. Patrick"
  ],
  "comment": "29 pages, 4 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math.PR"
  ],
  "abstract": "We study the spherical model of a ferromagnet on a Cayley tree and show that in the case of empty boundary conditions the ferromagnetic phase transition takes place at the critical temperature $T_c=\\frac{6\\sqrt{2}}{5}J$, where $J$ is the interaction strength. For any temperature the equilibrium magnetization, $m_n$, tends to zero in the thermodynamic limit, and the true order parameter is the renormalized magnetization $r_n=n^{3/2}m_n$, where $n$ is the number of generations in the Cayley tree. Below $T_c$, the equilibrium values of the order parameter are given by \\[ \\rho^* = \\pm\\frac{2\\pi} {(\\sqrt{2}-1)^2} \\sqrt{1-\\frac{T}{T_c}}. \\] There is one more notable temperature, $T_{\\rm p}$, in the model. Below that temperature the influence of homogeneous boundary field penetrates throughout the tree. We call $T_{\\rm p}$ the penetration temperature, and it is given by \\[ T_{\\rm p}= \\frac{J} {W_{\\rm Cayley} (3/2)} \\left(1-\\frac{1}{\\sqrt{2}} \\left( \\frac{h}{2J} \\right)^2 \\right). \\] The main new technical result of the paper is a complete set of orthonormal eigenvectors for the discrete Laplace operator on a Cayley tree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-27T15:58:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cayley tree",
      "large deviations",
      "rate functions",
      "temperature",
      "homogeneous boundary field penetrates throughout"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151208234P"
    }
  }
}