# The Scaling Limit of the Correlation of Holes on the Triangular Lattice with Periodic Boundary Conditions

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*Mihai Ciucu*

The author defines the correlation of holes on
the triangular lattice under periodic boundary conditions and studies
its asymptotics as the distances between the holes grow to
infinity. He proves that the joint correlation of an arbitrary
collection of triangular holes of even side-lengths (in lattice
spacing units) satisfies, for large separations between the holes, a
Coulomb law and a superposition principle that perfectly parallel the
laws of two dimensional electrostatics, with physical charges
corresponding to holes, and their magnitude to the difference between
the number of right-pointing and left-pointing unit triangles in each
hole.

The author details this parallel by indicating that, as a
consequence of the results, the relative probabilities of finding a
fixed collection of holes at given mutual distances (when sampling
uniformly at random over all unit rhombus tilings of the complement of
the holes) approach, for large separations between the holes, the
relative probabilities of finding the corresponding two dimensional
physical system of charges at given mutual distances. Physical
temperature corresponds to a parameter refining the background
triangular lattice.

He also gives an equivalent phrasing of the results in terms of
covering surfaces of given holonomy. From this perspective, two
dimensional electrostatic potential energy arises by averaging over
all possible discrete geometries of the covering surfaces.

#### Table of Contents

# Table of Contents

## The Scaling Limit of the Correlation of Holes on the Triangular Lattice with Periodic Boundary Conditions

- Contents vii8 free
- Abstract ix10 free
- Introduction 112 free
- Chapter 1. Definition of ω and statement of main result 516 free
- Chapter 2. Deducing Theorem 1.2 from Theorem 2.1 and Proposition 2.2 1122
- Chapter 3. A determinant formula for ω 1526
- Chapter 4. An exact formula for U[sub(s)](a, b) 1930
- Chapter 5. Asymptotic singularity and Newton's divided difference operator 2738
- Chapter 6. The asymptotics of the entries in the U-part of M' 3748
- Chapter 7. The asymptotics of the entries in the P-part of M' 4152
- Chapter 8. The evaluation of det(M") 4960
- Chapter 9. Divisibility of det(M") by the powers of q … ς and q … ς[sup(-1)] 5364
- Chapter 10. The case q = 0 of Theorem 8.1, up to a constant multiple 5768
- Chapter 11. Divisibility of det(dM[sub(0)]) by the powers of (x[sub(i)] … x[sub(j)]) … ς[sup(±1)](y[sub(i)] … y[sub(j)]) … ah 6172
- Chapter 12. Divisibility of det(dM[sub(0)]) by the powers of (x[sub(i)] … z[sub(j)]) … ς[sup(±1)](y[sub(i)] … ω[sub(j)]) 6778
- Chapter 13. The proofs of Theorem 2.1 and Proposition 2.2 7384
- Chapter 14. The case of arbitrary slopes 7586
- Chapter 15. Random covering surfaces and physical interpretation 8192
- Appendix. A determinant evaluation 8798
- Bibliography 99110