论文标题
在多米诺骨牌洗牌和基质重构
On the domino shuffle and matrix refactorizations
论文作者
论文摘要
本文是通过计算阿兹台克钻石多米诺骨牌的计算相关性的。它的灵感来自最近在最简单的周期性加权的最简单情况下使用的三种不同的方法中的两种，即两期周期性的阿兹台克钻石。其中一种由多米诺式洗牌提供动力的方法涉及反转Kasteleyn矩阵通过局部统计公式提供相关性。由Wiener-HOPF分解的另一种方法涉及双二个基质值函数，涉及Eynard-Mehta定理。对于任意权重，Wiener-HOPF分解可以用基于基质重构的LU和UL分解代替过渡矩阵的乘积。本文表明，对于阿兹台克钻石的任意权重，在多米诺骨牌洗牌下的面部权重演变，而基质重构是相同的。特别是，这些动力学可用于在Eynard-Mehta定理中找到LGV基质的倒数。
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.
