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 $\lambda$ to the number of squares. We prove that for large $\lambda$,
    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.

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