{
  "id": "cond-mat/0401026",
  "version": "v2",
  "published": "2004-01-04T21:25:25.000Z",
  "updated": "2005-08-31T13:36:56.000Z",
  "title": "Spanning forests and the q-state Potts model in the limit q \\to 0",
  "authors": [
    "Jesper Lykke Jacobsen",
    "Jesus Salas",
    "Alan D. Sokal"
  ],
  "comment": "131 pages (LaTeX2e). Includes tex file, three sty files, and 65 Postscript figures. Also included are Mathematica files forests_sq_2-9P.m and forests_tri_2-9P.m. Final journal version",
  "journal": "J.Statist.Phys. 119 (2005) 1153-1281",
  "categories": [
    "cond-mat.stat-mech",
    "hep-lat",
    "hep-th",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "We study the q-state Potts model with nearest-neighbor coupling v=e^{\\beta J}-1 in the limit q,v \\to 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q,v) = (0,0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2 \\le L \\le 10, as well as the limiting curves of partition-function zeros in the complex w-plane. For real w, we find two distinct phases separated by a transition point w=w_0, where w_0 = -1/4 (resp. w_0 = -0.1753 \\pm 0.0002) for the square (resp. triangular) lattice. For w > w_0 we find a non-critical disordered phase, while for w < w_0 our results are compatible with a massless Berker-Kadanoff phase with conformal charge c = -2 and leading thermal scaling dimension x_{T,1} = 2 (marginal operator). At w = w_0 we find a \"first-order critical point\": the first derivative of the free energy is discontinuous at w_0, while the correlation length diverges as w \\downarrow w_0 (and is infinite at w = w_0). The critical behavior at w = w_0 seems to be the same for both lattices and it differs from that of the Berker-Kadanoff phase: our results suggest that the conformal charge is c = -1, the leading thermal scaling dimension is x_{T,1} = 0, and the critical exponents are \\nu = 1/d = 1/2 and \\alpha = 1.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-08-31T13:36:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "q-state potts model",
      "spanning forests",
      "leading thermal scaling dimension",
      "berker-kadanoff phase",
      "conformal charge"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s10955-005-4409-y",
      "journal": "Journal of Statistical Physics",
      "year": 2005,
      "month": "Jun",
      "volume": 119,
      "number": "5-6",
      "pages": 1153
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 131,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 636599,
      "adsabs": "2005JSP...119.1153J"
    }
  }
}