Corpus ID: 119316441

Inverting the Kasteleyn matrix for holey hexagons

  title={Inverting the Kasteleyn matrix for holey hexagons},
  author={Tomack Gilmore},
  journal={arXiv: Combinatorics},
  • Tomack Gilmore
  • Published 24 January 2017
  • Mathematics, Physics
  • arXiv: Combinatorics
Consider a semi-regular hexagon on the triangular lattice (that is, the lattice consisting of unit equilateral triangles, drawn so that one family of lines is vertical). Rhombus (or lozenge) tilings of this region may be represented in at least two very different ways: as families of non-intersecting lattice paths; or alternatively as perfect matchings of a certain sub-graph of the hexagonal lattice. In this article we show how the lattice path representation of tilings may be utilised in order… Expand
Lozenge Tiling Function Ratios for Hexagons with Dents on Two Sides
  • D. Condon
  • Mathematics, Computer Science
  • Electron. J. Comb.
  • 2020
We give a formula for the number of lozenge tilings of a hexagon on the triangular lattice with unit triangles removed from arbitrary positions along two non-adjacent, non-opposite sides. Our formulaExpand


Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole
It is shown that the determinants det0?i, j?n?1(??ij+(m+i+jj)), where ? is any 6th root of unity, are variations of a famous result due to Andrews (1979, Invent. Math.53, 193?225). Expand
Interactions between interleaving holes in a sea of unit rhombi
The results presented below show that as the distance between the holes grows large, the interaction between them may be approximated by taking the exponential of the negative of the electrostatic energy of the system obtained by viewing the holes as a set of point charges, each with a signed magnitude given by a certain statistic. Expand
Lozenge tilings of hexagons with arbitrary dents
This paper addresses the general case when an arbitrary set of unit triangles is removed from along the boundary of the hexagon. Expand
Three Interactions of Holes in Two Dimensional Dimer Systems
Asymptotic expressions for the interactions of the triangular holes in these three different regions of the plane are obtained, providing further evidence for Ciucu's ongoing program that seeks to draw parallels between gaps in dimer systems on the hexagonal lattice and electrostatic phenomena. Expand
Dimer packings with gaps and electrostatics
  • Mihai Ciucu
  • Chemistry
  • Proceedings of the National Academy of Sciences
  • 2008
Fisher and Stephenson conjectured in 1963 that the correlation function (defined by dimer packings) of two unit holes on the square lattice is rotationally invariant in the limit of large separationExpand
Rhombus Tilings of a Hexagon with Two Triangles Missing on the Symmetry Axis
The case of the common vertex being the center of the hexagon solves a problem posed by Propp. Expand
The Scaling Limit of the Correlation of Holes on the Triangular Lattice With Periodic Boundary Conditions
The author of this title defines the correlation of holes on the triangular lattice under periodic boundary conditions and studies its asymptotics as the distances between the holes grow to infinity.Expand
Statistical Mechanics of Dimers on a Plane Lattice. II. Dimer Correlations and Monomers
In part I of this paper, exact expressions were obtained for the partition function and thermodynamic properties of an $m\ifmmode\times\else\texttimes\fi{}n$ plane square lattice filled withExpand
A dual of MacMahon’s theorem on plane partitions
A classical theorem of MacMahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula. In thisExpand
Enumeration of Rhombus Tilings of a Hexagon which Contain a Fixed Rhombus in the Centre
  • Ilse Fischer
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 2001
The number of rhombus tilings of a hexagon with side lengths a, b, c, a,b, c which contain the central Rhombus is computed and the “almost central”rhombus above the centre is calculated. Expand