Columnar order in random packings of $2\times 2$ squares on the square lattice

Abstract

We study random packings of $2\times 2$ squares with centers on the square lattice ${\mathbb{Z}}^2$, in which the probability of a packing is proportional to $\lambda$ to the power of the number of squares. We prove that for large $\lambda$, typical packings exhibit columnar order, in which either the centers of most tiles agree in the parity of their $x$-coordinate or the centers of most tiles agree in the parity of their $y$-coordinate. This manifests itself 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.