Introduction

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

11.

(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.

1