# A random tiling model for two dimensional electrostatics

### About this Title

**Mihai Ciucu**

Publication: Memoirs of the American Mathematical Society

Publication Year
2005: Volume 178, Number 839

ISBNs: 978-0-8218-3794-8 (print); 978-1-4704-0440-6 (online)

DOI: http://dx.doi.org/10.1090/memo/0839

MathSciNet review: 2172582

MSC (2000): Primary 82B23; Secondary 05A15, 05A16, 05B40, 60C05, 60D05, 60F99, 82B44

### Table of Contents

**Chapters**

- A random tiling model for two dimensional electrostatics
- 1. Introduction
- 2. Definitions, statement of results and physical interpretation
- 3. Reduction to boundary-influenced correlations
- 4. A simple product formula for correlations along the boundary
- 5. A $(2m + 2n)$-fold sum for $\omega _b$
- 6. Separation of the $(2m + 2n)$-fold sum for $\omega _b$ in terms of $4mn$-fold integrals
- 7. The asymptotics of the $T^{(n)}$’s and $T’^{(n)}$’s
- 8. Replacement of the $T^{(k)}$’s and $T’^{(k)}$’s by their asymptotics
- 9. Proof of Proposition 7.2
- 10. The asymptotics of a multidimensional Laplace integral
- 11. The asymptotics of $\omega _b$. Proof of Theorem 2.2
- 12. Another simple product formula for correlations along the boundary
- 13. The asymptotics of $\bar {\omega }_b$. Proof of Theorem 2.1
- 14. A conjectured general two dimensional superposition principle
- 15. Three dimensions and concluding remarks
- B. Plane partitions I: A generalization of MacMahon’s formula
- 1. Introduction
- 2. Two families of regions
- 3. Reduction to simply-connected regions
- 4. Recurrences for $\mathrm {M}(R_{\mathbf {l}, \mathbf {q}}(x))$ and $\mathrm {M}(\bar {R}_{\mathbf {l}, \mathbf {q}}(x))$
- 5. Proof of Proposition 2.1
- 6. The guessing of $\mathrm {M}(R_{\mathbf {l}, \mathbf {q}}(x))$ and $\mathrm {M}(\bar {R}_{\mathbf {l}, \mathbf {q}}(x))$