Abstract: Consider random configurations of disjoint 2x2 tiles positioned in integer coordinates in a bounded domain in R², where the probability for the appearance of a configuration is proportional to λⁿ, with n being the number of tiles in the configuration. This is called the 2x2-Hard-Squares model with fugacity λ. 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 2x2-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 2x2 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.
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