# A Random Tiling Model for Two Dimensional Electrostatics

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*Mihai Ciucu*

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 between 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 energy 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.

#### Table of Contents

# Table of Contents

## A Random Tiling Model for Two Dimensional Electrostatics

- Contents vii8 free
- Abstract ix10 free
- Part A. A Random Tiling Model for Two Dimensional Electrostatics 112 free
- 1. Introduction 314
- 2. Definitions, statement of results and physical interpretation 617
- 3. Reduction to boundary-influenced correlations 1627
- 4. A simple product formula for correlations along the boundary 1829
- 5. A (2m+2n)-fold sum for ω[sub(b)] 2536
- 6. Separation of the (2m+2n)-fold sum for ω[sub(b)] in terms of 4mn-fold integrals 3748
- 7. The asymptotics of the T[sup((n))]'s and T'[sup((n))]'s 4253
- 8. Replacement of the T[sup((k))]'s and T'[sup((k))]'s by their asymptotics 5263
- 9. Proof of Proposition 7.2 5667
- 10. The asymptotics of a multidimensional Laplace integral 6576
- 11. The asymptotics of ω[sub(b)]. Proof of Theorem 2.2 6980
- 12. Another simple product formula for correlations along the boundary 8091
- 13. The asymptotics of ω[sub(b)]. Proof of Theorem 2.1 8495
- 14. A conjectured general two dimensional Superposition Principle 94105
- 15. Three dimensions and concluding remarks 99110
- Bibliography 105116

- Part B. Plane Partitions I: A Generalization of MacMahon's Formula 107118