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Introduction

Monomer-monomer and especially dimer-dimer

correlations1

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

lattice.

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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