A random tiling model for two dimensional electrostatics

by Ciucu, Mihai

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Full title :A random tiling model for two dimensional electrostatics / Mihai Ciucu.

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Part A.A Random Tiling Model for Two Dimensional Electrostatics: Introduction Definitions, statement of results and physical interpretation Reduction to boundary-influenced correlations A simple product formula for correlations along the boundary A $(2m+2n)$-fold sum for $\omega_b$ Separation of the $(2m+2n)$-fold sum for $\omega_b$ in terms of $4mn$-fold integrals The asymptotics of the $T^{(n)}$'s and ${T'}^{(n)}$'s Replacement of the $T^{(k)}$'s and ${T'}^{(k)}$'s by their asymptotics Proof of Proposition 7.2 The asymptotics of a multidimensional Laplace integral The asymptotics of $\omega_b$. Proof of Theorem 2.2 Another simple product formula for correlations along the boundary The asymptotics of $\bar{\omega}_b$. Proof of Theorem 2.1 A conjectured general two dimensional Superposition Principle Three dimensions and concluding remarks Bibliography Part B. Plane Partitions I: A Generalization of MacMahon's Formula: Introduction Two families of regions Reduction to simply-connected regions Recurrences for $\textup{M}(R_{{\bf l}, {\bf q}}(x))$ and $\textup{M}(\bar{R}_{{\bf l}, {\bf q}}(x))$ Proof of Proposition 2.1 The guessing of $\textup{M}(R_{{\bf l}, {\bf q}}(x))$ and $\textup{M}(\bar{R}_{{\bf l}, {\bf q}}(x))$ Bibliography.