Random high-density packings of 2x2 tiles on the square lattice
Consider random configurations of disjoint $2\times2$ tiles
positioned in integer coordinates in a bounded domain in $R^2$, where the
probability for the appearance of a configuration is proportional to
$\lambda^n$, with $n$ being the number of tiles in the configuration. This
is called the $2\times2$-Hard-Squares model with fugacity $\lambda$. Many
similar hard-core lattice gases undergo a phase transition: at
low-fugacities the random configuration is disordered with exponential
decay of correlations while at high fugacities the random configuration
globally approximates a single optimally-packed configuration. This
paradigm is inapplicable to the $2\times 2$-hard-squares model due to a
sliding degree of freedom in the optimal packing, and the question of its
high-fugacity behavior remained open. We show that at high fugacities, the
$2\times 2$ hard-squares model exhibits a columnar phase: the tiles either
preferentially occupy the even rows, the odd rows, the even columns or the
odd columns.