On the domino shuffle and matrix refactorizations

This paper is motivated by computing correlations for domino tilings of the Aztec diamond. It is inspired by two {of the three} distinct methods that have recently been used in the simplest case of a doubly periodic weighting, that is the two-periodic Aztec diamond. {One of the methods}, powered by the domino shuffle, involved inverting the Kasteleyn matrix which gives correlations through the local statistics formula. {Another of the methods}, driven by a Wiener-Hopf factorization for two-by-two matrix valued functions, involved the Eynard-Mehta theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by an LU- and UL-decomposition for the product of the transition matrices based on a matrix refactorization. This paper shows that, for arbitrary weightings of the Aztec diamond, the evolution of the face weights under domino shuffle and the matrix refactorization is the same. In particular, these dynamics can be used to find the inverse of the LGV matrix in the Eynard-Mehta Theorem.