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The Scaling Limit of the Correlation of Holes on the Triangular Lattice with Periodic Boundary Conditions
Mihai Ciucu, Indiana University, Bloomington, IN

Memoirs of the American Mathematical Society
2009; 100 pp; softcover
Volume: 199
ISBN-10: 0-8218-4326-5
ISBN-13: 978-0-8218-4326-0
List Price: US$70
Individual Members: US$42
Institutional Members: US$56
Order Code: MEMO/199/935
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The author defines the correlation of holes on the triangular lattice under periodic boundary conditions and studies its asymptotics as the distances between the holes grow to infinity. He proves that the joint correlation of an arbitrary collection of triangular holes of even side-lengths (in lattice spacing units) satisfies, for large separations between the holes, a Coulomb law and a superposition principle that perfectly parallel the laws of two dimensional electrostatics, with physical charges corresponding to holes, and their magnitude to the difference between the number of right-pointing and left-pointing unit triangles in each hole.

The author details this parallel by indicating that, as a consequence of the results, the relative probabilities of finding a fixed collection of holes at given mutual distances (when sampling uniformly at random over all unit rhombus tilings of the complement of the holes) approach, for large separations between the holes, the relative probabilities of finding the corresponding two dimensional physical system of charges at given mutual distances. Physical temperature corresponds to a parameter refining the background triangular lattice.

He also gives an equivalent phrasing of the results in terms of covering surfaces of given holonomy. From this perspective, two dimensional electrostatic potential energy arises by averaging over all possible discrete geometries of the covering surfaces.

Table of Contents

  • Introduction
  • Definition of \(\hat\omega\) and statement of main result
  • Deducing Theorem 1.2 from Theorem 2.1 and Proposition 2.2
  • A determinant formula for \(\hat\omega\)
  • An exact formula for \(U_s(a,b)\)
  • Asymptotic singularity and Newton's divided difference operator
  • The asymptotics of the entries in the \(U\)-part of \(M'\)
  • The asymptotics of the entries in the \(P\)-part of \(M'\)
  • The evaluation of \(\det(M'')\)
  • Divisibility of \(\det(M'')\) by the powers of \(q-\zeta\) and \(q-\zeta^{-1}\)
  • The case \(q=0\) of Theorem 8.1, up to a constant multiple
  • Divisibility of \(\det(dM_0)\) by the powers of \((x_i-x_j)-\zeta^{\pm1}(y_i-y_j)-ah\) and \((z_i-w_j)-\zeta^{\pm1}(z_i-w_j)-ah\)
  • Divisibility of \(\det(dM_0)\) by the powers of \((x_i-z_j)-\zeta^{\pm1}(y_i-w_j)\)
  • The proofs of Theorem 2.1 and Proposition 2.2
  • The case of arbitrary slopes
  • Random covering surfaces and physical interpretation
  • Appendix. A determinant evaluation
  • Bibliography
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