On the domino shuffle and matrix refactorizations

TL;DR

This paper explores the relationship between domino shuffles and matrix refactorizations in Aztec diamond tilings, showing their equivalence and applications to matrix inversions for arbitrary weights.

Contribution

It demonstrates that domino shuffle dynamics and matrix refactorizations are equivalent for arbitrary Aztec diamond weights, enabling new matrix inversion methods.

Findings

Domino shuffle and matrix refactorization dynamics are equivalent.

The methods can be used to invert the LGV matrix in the Eynard-Mehta theorem.

Applicable to arbitrary weightings of the Aztec diamond.

Abstract

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, involves inverting the Kasteleyn matrix giving correlations through the local statistics formula. Another of the methods, driven by a Wiener-Hopf factorization for two-by-two matrix valued functions, involves the Eynard-Mehta theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by an LU- and UL-decomposition, based on a matrix refactorization, for the product of the transition matrices. This paper shows that, for arbitrary weightings of the Aztec diamond, the evolution of the face weights under the 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.