In [3] we considered the joint correlation ω of symmetrically distributed holes
on the triangular lattice, a natural extension of the monomer-monomer correlation
introduced by Fisher and Stephenson in [11]. Under the assumption that one of
the holes is a unit lattice triangle u on the symmetry axis and all remaining holes
are lattice triangles of side 2, oriented so that they point away from the lattice line
perpendicular to that supports u, we proved in [3] that, asymptotically as the
distances between holes are large, ω satisfies a multiplicative superposition principle
that perfectly parallels the superposition principle of two dimensional electrostatics
(with holes corresponding to electrical charges, and charge magnitude given by the
difference between the number of up-pointing and down-pointing unit triangles in
a hole). Our proof was based on explicit product formulas we obtained in [4] for
the number of lozenge tilings of two families of lattice regions.
The correlation we used in [3] was defined by including the holes inside a lattice
hexagon approaching infinite size so that the holes remain around its center. The
presence of the boundary of the hexagon distorts the local dimer statistics. More
precisely, there exists an explicit real valued function f defined on the hexagon (f
is the unique maximum of a certain local entropy integral; see [9][8]) so that in
the scaling limit the local statistics of dimers at each point inside the hexagon is
governed by µs,t, where (s, t) is the tilt of f at that point and µs,t is the unique
invariant Gibbs measure of slope (s, t) (this was conjectured by Cohn, Kenyon and
Propp in [8] and proved and generalized by Sheffield in [23] and Kenyon, Okounkov
and Sheffield in [18]). It follows from the latter two papers that µ0,0 is the unique
invariant Gibbs measure of maximal entropy. Since the function f has tilt (0, 0)
only at the center of the hexagon, the local dimer statistics is distorted away from
maximal entropy everywhere except at the center; see Figure
(for more details, see [9]; this does not happen in the case of a square on the square
lattice considered in [11]). Therefore, an important goal is to define the correlation
of holes in a different way, via regions that do not distort the dimer statistics, and
determine whether it still reduces to the superposition principle of electrostatics in
the limit as the holes grow far apart. Other highly desirable features are to allow
general, not necessarily symmetric distributions of the holes, as well as holes of
arbitrary size.
It is these goals that we accomplish in this paper. We give a new definition
for the correlation of holes by including them in a sequence of lattice tori of size
approaching infinity (see Chapter 1). We prove that this new correlation also
reduces to the superposition principle of electrostatics as the distances between the
holes grow to infinity.
1This figure is courtesy of David Wilson.
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