{
  "id": "1111.0741",
  "version": "v2",
  "published": "2011-11-03T07:07:54.000Z",
  "updated": "2012-02-09T06:56:37.000Z",
  "title": "Non-perturbative corrections to mean-field behavior: spherical model on spider-web graph",
  "authors": [
    "Ajit C. Balram",
    "Deepak Dhar"
  ],
  "comment": "substantially revised, a section added",
  "journal": "2012 J. Phys. A: Math. Theor. 45 125006",
  "doi": "10.1088/1751-8113/45/12/125006",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the spherical model on a spider-web graph. This graph is effectively infinite-dimensional, similar to the Bethe lattice, but has loops. We show that these lead to non-trivial corrections to the simple mean-field behavior. We first determine all normal modes of the coupled springs problem on this graph, using its large symmetry group. In the thermodynamic limit, the spectrum is a set of $\\delta$-functions, and all the modes are localized. The fractional number of modes with frequency less than $\\omega$ varies as $\\exp (-C/\\omega)$ for $\\omega$ tending to zero, where $C$ is a constant. For an unbiased random walk on the vertices of this graph, this implies that the probability of return to the origin at time $t$ varies as $\\exp(- C' t^{1/3})$, for large $t$, where $C'$ is a constant. For the spherical model, we show that while the critical exponents take the values expected from the mean-field theory, the free-energy per site at temperature $T$, near and above the critical temperature $T_c$, also has an essential singularity of the type $\\exp[ -K {(T - T_c)}^{-1/2}]$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-02-09T06:56:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spherical model",
      "spider-web graph",
      "non-perturbative corrections",
      "simple mean-field behavior",
      "large symmetry group"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Physics A Mathematical General",
      "year": 2012,
      "month": "Mar",
      "volume": 45,
      "number": 12,
      "pages": 125006
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012JPhA...45l5006B"
    }
  }
}