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.
