CHAPTER 1

Definition of ˆ ω and statement of main result

Draw the triangular lattice so that some of the lattice lines are vertical. For

terminological brevity, we will often refer to unit lattice triangles as monomers;

a left-pointing (resp., right-pointing) unit lattice triangle is called a left-monomer

(resp., right-monomer).

For any finite union Q of unit holes on the triangular lattice, define the charge

ch(Q) of Q to be the number of right-monomers in Q minus the number of left-

monomers in Q.

Let U1, . . . , Un be arbitrary unit holes on the triangular lattice. Following

Kenyon [15], provided

∑n

i=1

ch(Ui) = 0, we define the joint correlation of U1, . . . , Un

by

(1.1) ω1(U1, . . . , Un) := lim

N→∞

M(HN,N \ U1 ∪ ··· ∪ Un)

M(HN,N )

,

where HN,N is a large lattice rhombus of side N whose opposite sides are identified

so as to create a torus, and M(R) denotes the number of lozenge tilings of the

lattice region R (a lozenge, or unit rhombus, is the union of any two unit lattice

triangles that share an edge)2.

An important advantage of this definition is that, provided U1 ∪ ··· ∪ Un is

a union of non-overlapping lattice triangles of side 2, [15] and [16] provide an

expression for ω1(Q1, . . . , Qn) as a determinant.

To state this explicitly, we introduce the following system of coordinates on the

monomers of the triangular lattice. Choose the origin O to be at the center of a

vertical unit lattice segment, and pick the x- and y-coordinate axes to be straight

lines through O of polar arguments −π/3 and π/3, respectively. Coordinatize

left-monomers by the (integer) coordinates of the midpoints of their vertical sides;

coordinatize each right-monomer likewise, by the coordinates of the midpoint of its

vertical side (thus any pair of integers specifies a unique left-monomer and a unique

right-monomer, sharing a vertical side).

Let (l1, l1), . . . , (lk, lk) be the coordinates of the left-monomers contained in the

union U1 ∪ ··· ∪ Un of our holes, and let the coordinates of the right-monomers

contained in this union be (r1, r1), . . . , (rk, rk) (the latter are the same in number

as the former, since we are assuming the total charge of the unit holes to be zero).

Then, provided U1 ∪ ··· ∪ Un is a union of non-overlapping lattice triangles of side

2, it follows by results of Kenyon (see [15] and [16]) that

(1.2) ω1(U1, . . . , Un) = | det (P (ri − lj , ri − lj ))1≤i,j≤k|,

2The

total charge of the the holes needs to be zero in order for the region in the numerator

of (1.1) to contain the same number of each type of unit triangles—a necessary condition for the

existence of lozenge tilings. For any situation when the total charge is not zero, (1.1) would assign

value 0 to the joint correlation of the holes, irrespective of their relative positions.

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