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
ch(Ui) = 0, we define the joint correlation of U1, . . . , Un
(1.1) ω1(U1, . . . , Un) := lim
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|,
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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