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The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions


About this Title

Mihai Ciucu

Publication: Memoirs of the American Mathematical Society
Publication Year 2009: Volume 199, Number 935
ISBNs: 978-0-8218-4326-0 (print); 978-1-4704-0541-0 (online)
DOI: http://dx.doi.org/10.1090/memo/0935
MathSciNet review: 2508012
MSC: Primary 82B23; Secondary 05A16, 60F99, 60K35, 82B20

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Table of Contents


Chapters

  • Introduction
  • Chapter 1. Definition of $\hat {\omega }$ and statement of main result
  • Chapter 2. Deducing Theorem 1.2 from Theorem 2.1 and Proposition 2.2
  • Chapter 3. A determinant formula for $\hat {\omega }$
  • Chapter 4. An exact formula for $U_s(a, b)$
  • Chapter 5. Asymptotic singularity and Newton’s divided difference operator
  • Chapter 6. The asymptotics of the entries in the $U$-part of $M’$
  • Chapter 7. The asymptotics of the entries in the $P$-part of $M’$
  • Chapter 8. The evaluation of $\det (M”)$
  • Chapter 9. Divisibility of $\det (M”)$ by the powers of $q - \zeta $ and $q - \zeta ^{-1}$
  • Chapter 10. The case $q = 0$ of Theorem 8.1, up to a constant multiple
  • Chapter 11. Divisibility of $\det (dM_0)$ by the powers of $(x_i - x_j) - \zeta ^{\pm 1}(y_i - y_j) - ah$
  • Chapter 12. Divisibility of $\det (dM_0)$ by the powers of $(x_i - z_j) - \zeta ^{\pm 1}(y_i - \hat {\omega }_j)$
  • Chapter 13. The proofs of Theorem 2.1 and Proposition 2.2
  • Chapter 14. The case of arbitrary slopes
  • Chapter 15. Random covering surfaces and physical interpretation
  • Appendix. A determinant evaluation