Monomer-monomer and especially dimer-dimer
on a plane bipar-
tite lattice (especially the square and hexagonal lattice) have been studied quite
extensively (see for instance [12], [17], [20] and [21]). Color the vertices of the
lattice white and black so that each edge has one white and one black endpoint.
From the point of view of this paper, there is a fundamental difference between
studying dimer-dimer and monomer-monomer correlations: the former have the
same number of vertices of each color, while the latter have an excess of either a
white or a black vertex.
In this paper we consider correlations of triangular holes on the hexagonal
This type of hole has the convenient feature that the difference between the
number of its white and black constituent vertices is equal to the length of its
side. We will be lead by our results to interpret the white vertices as elemen-
tary negative charges ("electrons"), and the black vertices as elementary positive
charges ("positrons"), so that the triangular plurimers become charges of magni-
tude given by their side-length, and sign given by their orientation (up-pointing
or down-pointing). The main result of this paper, Theorem 2.1 (see also its much
simpler restatement (2.6)), implies that in the fairly general situation in which it
applies (namely, when the holes are symmetrically distributed about an axis, and
all holes have side 2, except for one of side 1, on the symmetry axis) the logarithm
of the joint correlation of triangular holes behaves exactly like the two-dimensional
electrostatic potential energy of the corresponding system of charges: it is obtained
by a Superposition Principle from the interaction of all pairs, and the pair inter-
actions are according to Coulomb's law. (It is now clear why dimer-dimer and
monomer-monomer correlations are fundamentally different: dimers are neutral!)
To present our results in the background of previous related results in the
literature, we point out the following facts.
First, there seem to be very few rigorous results in the literature on the asymp-
totics of the correlation of holes that are not unions of edges. The only ones
the author is aware of are [12], [17], and [9] (they all deal with instances of two
monomers; the first and third paper present explicit conjectures, while the second
proves a special case of a conjecture in the first). It was the work of Fisher and
Stephenson [12] that provided our original motivation for studying joint correla-
tions of holes.
Second, we mention that there is an alternative approach for expressing joint
correlations of holes on the hexagonal (or square) lattice due to Kenyon ([21, The-
orem 2.3]; see also [20]). When it applies, it provides an expression for the joint
lrThe monomers, respectively the dimers, are interacting via a sea of dimers that cover all
lattice sites not occupied by them.
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