{
  "id": "2002.00618",
  "version": "v1",
  "published": "2020-02-03T09:26:44.000Z",
  "updated": "2020-02-03T09:26:44.000Z",
  "title": "Combinatorics of 3D directed animals on a simple cubic lattice",
  "authors": [
    "Sergei Nechaev",
    "Michael Tamm"
  ],
  "comment": "7 pages, 6 figures",
  "categories": [
    "cond-mat.stat-mech",
    "hep-th",
    "math.CO"
  ],
  "abstract": "We provide combinatorial arguments based on a two-dimensional extension of a locally-free semigroup allowing us to compute the growth rate, $\\Lambda$, of the partition function $Z_N=N^{\\theta}\\Lambda^N$ of the $N$-particle directed animals ($N\\gg 1$) on a simple cubic lattice in a three-dimensional space. Establishing the bijection between the particular configuration of the lattice animal and a class of equivalences of words in the 2D projective locally-free semigroup, we find we find $\\ln \\Lambda = \\lim_{N\\to\\infty} \\ln Z_N / N$ with $\\Lambda= 2(\\sqrt{2}+1) \\approx 4.8284$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-03T09:26:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple cubic lattice",
      "3d directed animals",
      "combinatorics",
      "2d projective locally-free semigroup",
      "two-dimensional extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}