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...

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Bibliographic Details
Published in:Communications in mathematical physics 2023-07, Vol.401 (2), p.1417-1467
Main Authors: Chhita, Sunil, Duits, Maurice
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Diamonds
Factorization
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Theorems
Theoretical
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Summary: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.
ISSN:0010-3616
1432-0916
1432-0916
DOI:10.1007/s00220-023-04676-y