4 A RANDOM TILING MODEL FOR T W O DIMENSIONAL ELECTROSTATICS

correlation as a k x k determinant (where 2k is the total number of vertices in the

holes). However, while not requiring symmetry, the set-up of Kenyon's approach

limits its applicability to the case when all holes have even side (thus not accommo-

dating our hole of side 1 on the symmetry axis), and, more restrictively, to the case

when the total "charge" of the holes is

zero2.

The main advantage of our approach

is that it sets no restriction on the the total charge.

Furthermore, our result (14.9) gives the joint correlation of a general distribu-

tion of collinear monomers on the square lattice, a situation in which none of the

holes satisfies the even-sidedness required by Kenyon's approach.

Third, there are other discrete models in the physics literature (see e.g. the

survey [27] by Nienhuis) believed to behave like a Coulomb gas. However, there

are several important differences between them and our model: (1) we do not re-

quire that the total charge is 0, a fact built into the definition of the Coulomb gas

model in the survey by Nienhuis [(2.7), 27]; (2) by studying correlation of holes,

our discrete model seems quite different from the others in the literature; (3) for

those models surveyed in [27] for which the (believed) Coulomb behavior is only as-

ymptotic (as it is the case for our model), the arguments for their equivalence with

the Coulomb gas model are only heuristic, while our results are proved rigorously;

(4) in our model all states have the same energy, so the emergence of the Coulomb

interaction is entirely due to the number of different geometrical configurations

(unit rhombus tilings) compatible with the holes—in the language of physicists,

our model is stabilized by entropy] by contrast, in all models surveyed in [27] dif-

ferent configurations have different energies, specified by a Hamiltonian—they are

stabilized by energy. Furthermore, in our model the electrical charge has a purely

geometric origin: the charges are holes in the lattice, and their magnitude is the

difference between the number of right-pointing and left-pointing unit triangles in

the hole.

Fourth, in a recent paper [22] Krauth and Moessner study numerically (using

Monte Carlo algorithms) a very special case of the problem considered in this

paper, namely just the two-point correlations of monomers on the square lattice.

Their data leads them to conjecture that the two-point correlations on monomers

behave like a Coulomb potential (the case of monomers on different sublattices

appeared already in [12]; the new part is the simulation data for monomers on the

same sublattice). Krauth and Moessner also state that they could not find these

correlations worked out in the literature. Since in the current paper we address

the case of (2m + 2n + l)-point correlations, showing that they satisfy the much

more general Superposition Principle, their remark suggests that our results are

also new.

And fifth, shortly after the current paper was posted on the preprint archive

(web address arxiv.org/abs/math-ph/0303067, March 2003), physicists D. A. Huse,

W. Krauth, R. Moessner and S. L. Sondhi posted an article (arxiv.org/abs/cond-

mat/0305318, May 2003) presenting numerical simulations that suggest a positive

Indeed, Kenyon [20] defines the correlation of holes as the limit of M.(Hm,n )/M(iJ

m

,

n

) when

m,n —* oo, where Hm,n is a toroidal hexagonal grid graph, and Hm,n is its subgraph obtained

by removing the holes (M(G) denotes the number of perfect matchings of the graph G). In order

for M(Hm,n) to be non-zero one must have the same number of vertices in the two bipartition

classes of the vertices of Hm,n. Thus, since M ( ^

m )

n ) also has to be non-zero in order for Kenyon's

correlation to be non-zero, the number of vertices in the two bipartition classes that fall in the

holes must also be the same; i.e., the total charge of the holes must be zero.