Abstract
The two parts of this Memoir contain two separate but closely related papers.
In the paper in Part A we study the correlation of holes in random lozenge (i.e.,
unit rhombus) tilings of the triangular lattice. More precisely, we analyze the joint
correlation of these triangular holes when their complement is tiled uniformly at
random by lozenges. We determine the asymptotics of the joint correlation (for
large separations between the holes) in the case when one of the holes has side 1,
all remaining holes have side 2, and the holes are distributed symmetrically with
respect to a symmetry axis. Our result has a striking physical interpretation. If
we regard the holes as electrical charges, with charge equal to the difference be-
tween the number of down-pointing and up-pointing unit triangles in a hole, the
logarithm of the joint correlation behaves exactly like the electrostatic potential en-
ergy of this two-dimensional electrostatic system: it is obtained by a Superposition
Principle from the interaction of all pairs, and the pair interactions are according
to Coulomb's law. The starting point of the proof is a pair of exact lozenge tiling
enumeration results for certain regions on the triangular lattice, presented in the
second paper.
The paper in Part B was originally motivated by the desire to find a multi-
parameter deformation of MacMahon's simple product formula for the number of
plane partitions contained in a given box. By a simple bijection, this formula
also enumerates lozenge tilings of hexagons of side-lengths a,b,c,a,b,c (in cyclic
order) and angles of 120 degrees. We present a generalization in the case b = c
by giving simple product formulas enumerating lozenge tilings of regions obtained
from a hexagon of side-lengths a,b + k,b,a-\-k,b,b-{-k (where k is an arbitrary non-
negative integer) and angles of 120 degrees by removing certain triangular regions
along its symmetry axis. The paper in Part A uses these formulas to deduce that
in the scaling limit the correlation of the holes is governed by two dimensional
electrostatics.
2000 Mathematics Subject Classification Numbers, paper in Part A: Primary 82B23, 82D99;
Secondary 05A16, 41A63, 60F99.
Keywords: dimer model, random tilings, lozenge tilings, perfect matchings, exact enumera-
tion, asymptotic enumeration, correlation, scaling limit, electrostatics.
2000 Mathematics Subject Classification Numbers, paper in Part B: Primary 05A15; Sec-
ondary 82B23.
Keywords: plane partitions, exact enumeration, simple product formulas, lozenge tilings,
perfect matchings.
Received by the editor August 23, 2003; and in revised form on July 11, 2004 and October
11, 2004.
The research for the paper in Part A was supported in part by NSF grants DMS 9802390
and DMS 0100950. The research for the paper in Part B was supported by a Membership at the
Institute for Advanced Study.
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