Columnar order in random packings of $2\times2$ squares on the square lattice

Abstract

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.

Figures (10)

• Figure 1.1: Representatives of the four kinds of fully-packed configurations of the $2\times2$ hard-square model. The colors of the tiles correspond to the parities of their $x$ and $y$ coordinates (see Figure \ref{['fig:configs']} for the precise correspondence).
• Figure 1.2: The left panel depicts a fully-packed configuration, arranged in columns. The middle panel depicts a sample from a union of independent "one-dimensional columnar systems" at high fugacity (denoted by $\mu_{(\mathrm{ver},0)}^{\cup\text{1D}}$ in the text). The right panel depicts a sample from the high-fugacity regime of the $2\times2$ hard-square model. We prove the existence of a phase for the high-fugacity $2\times2$ hard-square model with properties resembling a “ small perturbation” of $\mu_{(\mathrm{ver},0)}^{\cup\text{1D}}$.
• Figure 1.3: Sticks (green lines) in a configuration. On the left there is an abundance of vertical sticks while on the right there is an abundance of horizontal sticks. The interface region is not crossed by long sticks (of either orientation), a feature which we rely upon in order to prove that interface regions are rare. The rectangles $R_{1},R_{2},R_{3}$ are drawn with their concentric $R^{-}$ rectangles (with $N=7$). The rectangles $R_{1}$ and $R_{3}$ are properly divided by vertical and horizontal sticks, respectively, while $R_{2}$ is not properly divided.
• Figure 3.1: In the background the rectangle $\Lambda=\mathrm{R}_{8\times6,(0,0)}$ is shown in green, with a configuration in $\Omega_{\Lambda}^{\mathrm{per}}$. The red rectangle $R=\mathrm{R}_{2\times3,(1,1)}$ is a block of $\Lambda$. The red and blue rectangles are mappings of $R$ by $8$ elements of $\refls[{}]$ that form a representative set of $\refls$.
• Figure 4.1: The sticks of the configuration are highlighted in green. No stick divides both $R_{1}$ and $S_{1}$ although each of them is divided by a stick. A stick divides both $R_{2}$ and $S_{2}$. In the terminology of Section \ref{['sec:multiple_gibbs']}, if $R_{1}=\mathrm{R}_{16\times16,(0,0)}$ and $N=4$ then $S_{1}=R_{1}^{-}$, $S_{2}=R_{2}^{-}$ and $R_{2}$ is properly divided by a $(\mathrm{ver},1)$ stick while $R_{1}$ is not properly divided. In symbols, $(0,0)\notin\Psi^{4\times4}$ and $(6,0)\in\Psi_{(\mathrm{ver},1)}^{4\times4}$.

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Proposition 2.1
• Proposition 2.2
• Lemma 3.1: Reflection positivity
• Proposition 3.2: Chessboard estimate
• Proposition 3.3: positive homogeneity, triangle inequality and monotonicity
• Lemma 3.4: "Recursive chessboard estimate"
• Remark 3.5