{
  "id": "1705.06528",
  "version": "v1",
  "published": "2017-05-18T11:20:17.000Z",
  "updated": "2017-05-18T11:20:17.000Z",
  "title": "Exactly solved models on planar graphs with vertices in $\\mathbb{Z}^3$",
  "authors": [
    "Andrew P. Kels"
  ],
  "comment": "33 pages, 18 figures",
  "categories": [
    "math-ph",
    "cond-mat.stat-mech",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "It is shown how exactly solved edge interaction models on the square lattice, may be extended onto more general planar graphs, with edges connecting a subset of next nearest neighbour vertices of $\\mathbb{Z}^3$. This is done by using local deformations of the square lattice, that arise through the use of the star-triangle relation. Similar to Baxter's Z-invariance property, these local deformations leave the partition function invariant up to some simple factors coming from the star-triangle relation. The deformations used here extend the usual formulation of Z-invariance, by requiring the introduction of oriented rapidity lines which form directed closed paths in the rapidity graph of the model. The quasi-classical limit is also considered, in which case the deformations imply a classical Z-invariance property, as well as a related local closure relation, for the action functional of a system of classical discrete Laplace equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-18T11:20:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "square lattice",
      "star-triangle relation",
      "baxters z-invariance property",
      "edge interaction models",
      "local deformations leave"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}