"Volume 178, number 839 (third of 5 numbers)."

Includes bibliographical references (p. 144).

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 $\mathrm {M}(\bar {R}_{\mathbf {l}, \mathbf {q}}(x))$

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Electronic reproduction. Providence, Rhode Island : American Mathematical Society. 2012

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