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The inverse Kasteleyn matrix of a bipartite graph holds much informationabout the perfect matchings of the system such as local statistics which can beused to compute local and global asymptotics. In this paper, we consider threedifferent weightings of domino tilings of the Aztec diamond and show usingrecurrence relations, we can compute the inverse Kasteleyn matrix. Theseweights are the one-periodic weighting where the horizontal edges have oneweight and the vertical edges have another weight, the q^{vol} weighting whichcorresponds 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 flatregion with defects in the center.

Coupling Functions for Domino tilings of Aztec diamonds