Inverting the Kasteleyn matrix for holey hexagons

Abstract

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 to calculate the entries of the inverse Kasteleyn matrix that arises from interpreting tilings as perfect matchings. Our main result gives precisely the inverse Kasteleyn matrix (up to a possible change in sign) for a semi-regular hexagon of side lengths $a,b,c,a,b,c$ (going clockwise from the south-west side). Not only does this theorem generalise a number of known results regarding tilings of hexagons that contain punctures, but it also provides a new formulation through which we may attack problems in statistical physics such as Ciucu's electrostatic conjecture.