Contents

Abstract ix

Part A.

A Random Tiling Model for Two Dimensional Electrostatics 1

1. Introduction 3

2. Definitions, statement of results and physical interpretation 6

3. Reduction to boundary-influenced correlations 16

4. A simple product formula for correlations along the boundary 18

5. A (2m -f 2n)-fold sum for Ub 25

6. Separation of the (2ra+2n)-fold sum for UJ^ in terms of 4ran-fold integrals 37

7. The asymptotics of the T^'s and T'(n)'s 42

8. Replacement of the T^k^s and T^ ^'s by their asymptotics 52

9. Proof of Proposition 7.2 56

10. The asymptotics of a multidimensional Laplace integral 65

11. The asymptotics of UJ^. Proof of Theorem 2.2 69

12. Another simple product formula for correlations along the boundary 80

13. The asymptotics of £&. Proof of Theorem 2.1 84

14. A conjectured general two dimensional Superposition Principle 94

15. Three dimensions and concluding remarks 99

Bibliography 104

Part B.

Plane Partitions I: A Generalization of MacMahon's Formula 107

1. Introduction 109

2. Two families of regions 116