Coupling Functions for Domino tilings of Aztec diamonds

TL;DR

This paper develops methods to compute the inverse Kasteleyn matrix for domino tilings of Aztec diamonds under three different weightings, revealing detailed local and global statistical properties of the system.

Contribution

It introduces recurrence relations to explicitly compute the inverse Kasteleyn matrix for three new weighting schemes of Aztec diamond domino tilings.

Findings

Explicit formulas for inverse Kasteleyn matrices under various weightings

Analysis of local statistics and phase regions in weighted tilings

Extension of inverse Kasteleyn techniques to new weighting models

Abstract

The inverse Kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics. In this paper, we consider three different weightings of domino tilings of the Aztec diamond and show using recurrence relations, we can compute the inverse Kasteleyn matrix. These weights are the one-periodic weighting where the horizontal edges have one weight and the vertical edges have another weight, the q^{vol} weighting which corresponds to multiplying the product of tile weights by q if we add a `box' to the height function and the two-periodic weighting which exhibits a flat region with defects in the center.