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.
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