论文标题
$ 2 \ times2 $正方形的随机包装柱订单在方格上
Columnar order in random packings of $2\times2$ squares on the square lattice
论文作者
论文摘要
我们研究了$ 2 \ times2 $正方形的随机包装，在方格$ \ mathbb {z}^{2} $上，填料的可能性与$λ$与平方数的数量成正比。我们证明，对于大$λ$，典型的包装展示了柱状订单，其中大多数瓷砖的中心都同意其$ x $ - 坐标的均等或大多数瓷砖中心的均等，或者大多数瓷砖的中心都同意其$ y $ - 坐标的奇偶校验。这表现在存在四个极端和周期性的吉布斯测量中，其中晶格的旋转对称性被损坏，而翻译对称性仅沿单个轴线损坏。我们进一步量化了这些度量中相关性的衰减，从而在保留的翻译对称性方向上获得了缓慢的指数衰减速率，并且在破裂的转化对称性方向上的快速速率。最后，我们证明每个周期性的吉布斯度量都是这四种措施的混合物。 此外，我们的证明引入了棋盘估算的明显新颖的扩展，从有限体积的圆环测量到所有无限体积的周期性Gibbs测量。
We study random packings of $2\times2$ squares with centers on the square lattice $\mathbb{Z}^{2}$, in which the probability of a packing is proportional to $λ$ to the number of squares. We prove that for large $λ$, typical packings exhibit columnar order, in which either the centers of most tiles agree on the parity of their $x$-coordinate or the centers of most tiles agree on the parity of their $y$-coordinate. This manifests in the existence of four extremal and periodic Gibbs measures in which the rotational symmetry of the lattice is broken while the translational symmetry is only broken along a single axis. We further quantify the decay of correlations in these measures, obtaining a slow rate of exponential decay in the direction of preserved translational symmetry and a fast rate in the direction of broken translational symmetry. Lastly, we prove that every periodic Gibbs measure is a mixture of these four measures. Additionally, our proof introduces an apparently novel extension of the chessboard estimate, from finite-volume torus measures to all infinite-volume periodic Gibbs measures.
