Memoirs of the American Mathematical Society 2009; 100 pp; softcover Volume: 199 ISBN10: 0821843265 ISBN13: 9780821843260 List Price: US$70 Individual Members: US$42 Institutional Members: US$56 Order Code: MEMO/199/935
 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 sidelengths (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 rightpointing and leftpointing 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 \((x_ix_j)\zeta^{\pm1}(y_iy_j)ah\) and \((z_iw_j)\zeta^{\pm1}(z_iw_j)ah\)
 Divisibility of \(\det(dM_0)\) by the powers of \((x_iz_j)\zeta^{\pm1}(y_iw_j)\)
 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
