We define the correlation of holes on the triangular lattice under periodic
boundary conditions and study its asymptotics as the distances between the holes
grow to infinity. We prove that the joint correlation of an arbitrary collection of
triangular holes of even side-lengths (in lattice spacing units) satisfies, for large
separations between the holes, a Coulomb law and a superposition principle that
perfectly parallel the laws of two dimensional electrostatics, with physical charges
corresponding to holes, and their magnitude to the difference between the number
of right-pointing and left-pointing unit triangles in each hole.
We detail this parallel by indicating that, as a consequence of our result, the
relative probabilities of finding a fixed collection of holes at given mutual distances
(when sampling uniformly at random over all unit rhombus tilings of the com-
plement of the holes) approach, for large separations between the holes, the rela-
tive probabilities of finding the corresponding two dimensional physical system of
charges at given mutual distances. Physical temperature corresponds to a param-
eter refining the background triangular lattice.
We also give an equivalent phrasing of our result in terms of covering surfaces
of given holonomy. From this perspective, two dimensional electrostatic potential
energy arises by averaging over all possible discrete geometries of the covering
2000 Mathematics Subject Classification Numbers: Primary 82B23, 82D99; Secondary 05A16,
41A63, 60F99.
Keywords: dimer model, random tilings, lozenge tilings, perfect matchings, exact enumera-
tion, asymptotic enumeration, determinant evaluations, correlation function, scaling limit, elec-
trostatics, emergence.
Received by the editor February 11, 2005; and in revised form on November 7, 2006.
This research was supported in part by NSF grant DMS 0500616.
Previous Page Next Page