Contents

Abstract ix

Introduction 1

Chapter 1. Definition of ˆ ω and statement of main result 5

Chapter 2. Deducing Theorem 1.2 from Theorem 2.1 and Proposition 2.2 11

Chapter 3. A determinant formula for ˆ ω 15

Chapter 4. An exact formula for Us(a, b) 19

Chapter 5. Asymptotic singularity and Newton’s divided difference operator 27

Chapter 6. The asymptotics of the entries in the U-part of M 37

Chapter 7. The asymptotics of the entries in the P -part of M

41

Chapter 8. The evaluation of det(M ) 49

Chapter 9. Divisibility of det(M ) by the powers of q − ζ and q − ζ−1 53

Chapter 10. The case q = 0 of Theorem 8.1, up to a constant multiple 57

Chapter 11. Divisibility of det(dM0) by the powers of

(xi − xj ) − ζ±1(yi − yj ) − ah 61

Chapter 12. Divisibility of det(dM0) by the powers of (xi

−zj)−ζ±1(yi

−wj) 67

Chapter 13. The proofs of Theorem 2.1 and Proposition 2.2 73

Chapter 14. The case of arbitrary slopes 75

Chapter 15. Random covering surfaces and physical interpretation 81

Appendix. A determinant evaluation 87

Bibliography 99

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