The intermediate-fugacity phase diagram of the $2\times2$ hard-square model: uniqueness threshold, order, and universality class of the transition

The $2\times2$ hard-square model — random packings of $2\times2$ tiles on the square lattice $\mathbb{Z}^2$, weighted by a fugacity $\lambda$ per tile — is the canonical hard-core lattice gas whose continuum of fully-packed "sliding" ground states defeats Pirogov–Sinai theory, the standard tool for proving high-density ordering transitions. Uniqueness of the Gibbs measure is classical at small $\lambda$, and Hadas and Peled (2022; Journal of the European Mathematical Society 2025) proved that at large $\lambda$ the model exhibits columnar order, with exactly four extremal periodic Gibbs measures and every periodic Gibbs measure a mixture of them. Physics predictions (Ramola–Damle–Dhar 2015; Mandal–Nath–Rajesh 2017) interpolate between the two regimes by a single continuous transition at $\lambda_c\approx 97.5$–$105$, in the Ashkin–Teller universality class close to its Ising endpoint. None of this has been proven: the rigorous uniqueness/multiplicity structure at intermediate fugacity, the order of the transition, and its universality class are all open. The problem asks for a rigorous determination of this phase diagram; no resolution is known in the literature through August 2026.

Background

The $2\times2$ hard-square model is a fundamental example of a hard-core lattice gas: a class of models in which particles exclude a neighborhood around themselves and all ordering is driven by entropy alone. A configuration is a packing of the square lattice $\mathbb{Z}^2$ by axis-aligned tiles, each covering a $2\times2$ block of sites, with no two tiles overlapping; a packing with $N$ tiles has probability proportional to $\lambda^N$, where $\lambda\ge 0$ is the fugacity (also called activity). Equivalently, this is the lattice gas in which the first two neighbor shells of each occupied site are excluded from occupation (the "$k$-NN" model with $k=2$; exclusion diameter $D^2=4$). An infinite-volume Gibbs measure is a probability measure on packings of the infinite lattice whose finite-dimensional distributions are consistent with these finite-volume weights; it is extremal if it cannot be written as a nontrivial mixture of other Gibbs measures. The existence of more than one Gibbs measure at the same fugacity is the mathematical signature of a phase transition; a continuous (second-order) transition is one in which thermodynamic quantities such as the density vary continuously through the critical point, as opposed to a first-order transition with a jump; and a universality class is an equivalence class of continuous transitions sharing the same critical exponents — numbers such as $\nu$ (the divergence rate of the correlation length) and $\eta$ (anomalous dimension of the two-point function) that govern scaling at criticality.

Two regimes of the model are understood. At small fugacity the Gibbs measure is unique and the system is disordered, by standard arguments: the cluster expansion (a convergent perturbative expansion in the particle density), and disagreement percolation — a technique that couples two copies of the system and proves uniqueness whenever the set of sites on which the coupled copies disagree cannot form an infinite connected cluster — which gives uniqueness for all $\lambda<p_c/(1-p_c)\approx 0.69$ (as recounted in the source; $p_c$ is the critical probability of the associated percolation process). At full packing ($\lambda=\infty$) the model has a continuum of ground states: starting from the packing whose tiles fill every other column, each column can be shifted by one lattice site independently — the "sliding phenomenon" — producing uncountably many fully-packed configurations. This degeneracy defeats Pirogov–Sinai theory, the standard machinery for proving high-density phase transitions, which requires an essentially finite set of periodic ground states; consequently the high-fugacity regime was open for decades, with early expectations favoring a unique Gibbs measure and Mazel–Stuhl–Suhov conjecturing uniqueness at high fugacity for sliding models as a class.

Hadas and Peled settled the high-fugacity end. There exists $\lambda_0<\infty$ such that for every $\lambda>\lambda_0$ the model has four extremal periodic Gibbs measures $\mu_{(\mathrm{ver},0)},\mu_{(\mathrm{ver},1)},\mu_{(\mathrm{hor},0)},\mu_{(\mathrm{hor},1)}$ exhibiting columnar order — most tiles have corners whose $x$-coordinates share a parity (or whose $y$-coordinates share a parity) — so that the lattice's $90^\circ$ rotational symmetry and one translational symmetry are broken; every periodic Gibbs measure is a convex combination of these four; and correlations decay exponentially, with correlation length of order $1$ across columns and of order $\sqrt{\lambda}$ along columns. The proof introduces a novel extension of the chessboard estimate (a consequence of reflection positivity, the invariance and positivity of the model's weights under reflections in lattice lines) from finite-volume torus measures to all infinite-volume periodic Gibbs measures. Notably, the triangular-lattice analogue — the hard-hexagon model, the hard-core lattice gas of particles excluding their nearest neighbors on the triangular lattice — was exactly solved by Baxter, including its critical fugacity and exponents; the $2\times2$ hard-square model is the standard square-lattice companion for which no comparable solution exists.

The physics literature predicts a sharp interpolation between the two understood regimes: a single transition point $\lambda_c$ separating the disordered fluid from the columnar phase, with the transition continuous and in the Ashkin–Teller universality class — the family of critical behaviors of two coupled Ising models, with exponents varying continuously along a line connecting the four-state Potts model and two decoupled Ising models — at a point close to the Ising member. Ramola, Damle and Dhar (2015) argued this from Monte Carlo and transfer-matrix studies of mixtures of $2\times2$ hard squares and dimers, and Mandal, Nath and Rajesh (2017) estimated $\lambda_c\approx 105$ by an interfacial-tension approximation, in reasonable agreement with Monte Carlo estimates $\lambda_c\approx 97.5$. None of these predictions has been established mathematically: the high-fugacity theorems hold only beyond some $\lambda_0$ whose relation to $\lambda_c$ is not controlled, and uniqueness is known only for $\lambda\lesssim 0.69$.

Problem Statement

Determine, rigorously, the fugacity phase diagram of the $2\times2$ hard-square model on $\mathbb{Z}^2$ at intermediate fugacity. The single overarching objective is the rigorous uniqueness/ordering-transition structure of the model, with three jointly required components:

1. Uniqueness threshold. Characterize the set of fugacities $\lambda$ at which the model has a unique infinite-volume Gibbs measure. In particular, prove or disprove that there is a single critical fugacity $\lambda_c$ such that the Gibbs measure is unique for all $\lambda<\lambda_c$, while for all $\lambda>\lambda_c$ there are multiple Gibbs measures, with the four columnar measures of Hadas–Peled present among them; if the true structure differs — several transition points, an additional intermediate phase, or another multiplicity structure altogether — determine that structure instead.

2. Order of the transition. Determine whether the transition(s) separating the disordered and columnar regimes is continuous (second-order) or first-order (for instance, via continuity or discontinuity of the density or of the columnar order parameter at the transition).

3. Universality class. If the transition is continuous, rigorously identify its universality class — in particular, confirm or refute the physics prediction of Ashkin–Teller criticality, for example by proving the predicted critical exponents or an equivalent rigorous characterization of the scaling behavior at the transition.

The problem is posed for the standard model on the full lattice, with Gibbs-measure multiplicity as the order criterion. Results restricted to finite volumes, special boundary conditions, or modified weights count only if the reduction to the stated model is itself proved. The completion boundary is the full determination above; resolving only some of the three components constitutes a partial answer.

References

1. Daniel Hadas and Ron Peled, Columnar order in random packings of $2\times2$ squares on the square lattice, Journal of the European Mathematical Society (2025). DOI: 10.4171/jems/1680, https://doi.org/10.4171/jems/1680 (preprint arXiv:2206.01276, https://arxiv.org/abs/2206.01276; latest version v3, revised 17 February 2026)
2. Kabir Ramola, Kedar Damle and Deepak Dhar, Columnar Order and Ashkin-Teller Criticality in Mixtures of Hard Squares and Dimers, Physical Review Letters 114, 190601 (2015). DOI: 10.1103/PhysRevLett.114.190601, https://doi.org/10.1103/PhysRevLett.114.190601 (preprint arXiv:1408.4943, https://arxiv.org/abs/1408.4943)
3. Trisha Nath and R. Rajesh, The high density phase of the $k$-NN hard core lattice gas model, Journal of Statistical Mechanics: Theory and Experiment (2016) 073203. DOI: 10.1088/1742-5468/2016/07/073203, https://doi.org/10.1088/1742-5468/2016/07/073203 (preprint arXiv:1604.08376, https://arxiv.org/abs/1604.08376)
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