Unresolved phase diagram, Gibbs-measure classification, and sharp correlation decay for the $2\times2$ hard-square lattice gas

The $2\times2$ hard-square lattice gas on $\mathbb{Z}^2$ places $2\times2$ closed squares with centers on lattice sites, each site being excluded from occupation together with its eight nearest sites, and weights each packing by a fugacity $\lambda$ raised to the number of squares. Hadas and Peled (published in the Journal of the European Mathematical Society, 2025) proved that at sufficiently large $\lambda$ the model has four extremal periodic Gibbs measures exhibiting columnar order (typical packings break the lattice's $90^\circ$ rotational symmetry and one axis of translational symmetry), that every periodic Gibbs measure is a mixture of these four, and gave anisotropic correlation-decay upper bounds; at small $\lambda$ the Gibbs measure is unique by classical disagreement-percolation arguments. Between these perturbative regimes almost nothing is rigorous. This problem asks for the remaining equilibrium structure of the model: (i) the phase diagram at intermediate fugacity — existence and uniqueness of a critical fugacity $\lambda_c$ separating the disordered and columnar regimes, and the nature (continuous versus first-order) and universality class (Ashkin–Teller, as predicted in the physics literature) of the transition; (ii) the classification of all Gibbs measures at high fugacity — whether non-periodic extremal Gibbs measures exist, and which sequences of boundary conditions converge to which measures; (iii) sharp correlation asymptotics in the columnar phase — matching lower bounds, the true scaling of the along-column correlation length in $\lambda$, and removal of a logarithmic factor in the inter-column covariance bound. No rigorous resolution of any component is known; the problem is open.

Background

A hard-core lattice gas is a statistical-mechanical model of particles with pure exclusion interactions: each lattice site is either occupied or vacant, and configurations in which two particles' exclusion regions overlap are forbidden. The $2\times2$ hard-square model on the square lattice $\mathbb{Z}^2$ is the hard-core gas whose particles are closed $2\times2$ Euclidean squares with centers at lattice sites: two sites $u,v$ may not both be occupied whenever $\max(|u_x-v_x|,|u_y-v_y|)\le1$, so each particle excludes its eight nearest sites (in physics notation, the $k=2$ nearest-neighbor-exclusion gas). Configurations $\sigma:\mathbb{Z}^2\to\{0,1\}$ in a bounded region $\Lambda$ with boundary condition $\rho$ are weighted proportionally to $\lambda^{N(\sigma)}$, where $N(\sigma)$ counts occupied sites in $\Lambda$ and $\lambda>0$ is the fugacity (activity). Infinite-volume Gibbs measures are probability measures on configurations satisfying the Dobrushin–Lanford–Ruelle (DLR) prescription: conditioning on the outside of any bounded region must reproduce the finite-volume measure with the induced boundary condition. A Gibbs measure is periodic if it is invariant under some full-rank sublattice, and extremal if it assigns probability $0$ or $1$ to every tail event; extremal measures are the model's pure phases, and their number is the basic signature of a phase transition.

At small fugacity the model is disordered: uniqueness of the Gibbs measure holds for $\lambda<p_c/(1-p_c)\approx0.687$, where $p_c\approx0.407$ is the site-percolation threshold of $\mathbb{Z}^2$ with nearest- and next-nearest-neighbor adjacency. This follows from disagreement percolation (a technique comparing two Gibbs samples with disagreeing boundary conditions and showing that disagreements cannot percolate, hence cannot force distinct phases), together with Dobrushin-uniqueness-type criteria. At full packing the model has a continuum of ground states, generated by sliding entire columns or rows of tiles past one another — the 'sliding' degeneracy. This degeneracy defeats Pirogov–Sinai theory (the general perturbative theory of phase transitions based on classifying ground states and controlling contours), which is why the high-fugacity regime resisted rigorous analysis for decades: Fröhlich–Israel–Lieb–Simon (1980) recorded the 'conventional wisdom' that the model should have a unique Gibbs state at high fugacity, and Mazel–Stuhl–Suhov (2019) conjectured exactly that as part of a general conjecture for models with sliding. Monte Carlo and series studies in physics indicated the opposite: a columnar phase in which tile centers predominantly share the parity of one coordinate, breaking the lattice's $90^\circ$ rotation and one direction of translation, with the transition predicted to be continuous and in the Ashkin–Teller universality class — the family of critical behaviors interpolating between Ising and four-state-Potts criticality, close to Ising for the pure $2\times2$ model.

Hadas and Peled settled this dispute for very large fugacity: for all $\lambda$ above some $\lambda_0$ the model admits four extremal periodic Gibbs measures $\mu_{(\mathrm{ver},0)},\mu_{(\mathrm{ver},1)},\mu_{(\mathrm{hor},0)},\mu_{(\mathrm{hor},1)}$ exhibiting columnar order; every periodic Gibbs measure is a mixture of these four; and correlations in $\mu_{(\mathrm{ver},0)}$ decay exponentially with correlation length at most a universal constant in the horizontal (symmetry-broken) direction and at most $C\sqrt{\lambda}$ in the vertical (along-column) direction, with the inter-column covariance satisfying $\mathrm{Cov}(\sigma(1,0),\sigma(3,0))\le C\log\lambda/\sqrt{\lambda}$. The proof adapts the chessboard estimate — a consequence of reflection positivity (positivity of the measure under lattice reflections, available here since the interaction is invariant under reflection in lattice lines), traditionally stated for finite-volume torus measures — to all infinite-volume periodic Gibbs measures, and combines it with a mesoscopic analysis of 'sticks' (long column segments) and disagreement percolation. What remains open, and what this problem asks, is everything between and beyond the two perturbative regimes: the transition itself, the classification of all (not necessarily periodic) Gibbs measures, and sharp two-sided correlation asymptotics.

Problem Statement

Consider the $2\times2$ hard-square model on $\mathbb{Z}^2$: configurations $\sigma:\mathbb{Z}^2\to\{0,1\}$ in which two sites at Chebyshev distance at most $1$ cannot both be occupied (equivalently, closed $2\times2$ squares centered at occupied sites have disjoint interiors), with finite-volume measures proportional to $\lambda^{N(\sigma)}$ for the number $N(\sigma)$ of occupied sites and fugacity $\lambda>0$, and infinite-volume Gibbs measures defined by the Dobrushin–Lanford–Ruelle prescription. Determine the remaining equilibrium structure of this model — the components below are jointly constitutive of one objective, the complete rigorous phase diagram, Gibbs-measure classification, and quantitative correlation structure of this single model:

1. Intermediate-fugacity phase diagram and critical behavior. Uniqueness of the Gibbs measure is known for $\lambda<p_c/(1-p_c)$ with $p_c\approx0.407$ the site-percolation threshold of $\mathbb{Z}^2$ with nearest- and next-nearest-neighbor adjacency, and coexistence of four columnar phases for all sufficiently large $\lambda$. Establish what happens in between: does a single critical fugacity $\lambda_c$ separate the unique regime from the multiple regime (in particular, prove or disprove uniqueness or multiplicity at intermediate fugacity, and narrow the gap between the two regimes)? Is the transition continuous or first-order? If continuous, is it in the Ashkin–Teller universality class as predicted by the physics literature, and can critical exponents or scaling relations be identified rigorously?

2. Classification of all Gibbs measures and boundary conditions. Every periodic Gibbs measure is known to be a mixture of $\mu_{(\mathrm{ver},0)},\mu_{(\mathrm{ver},1)},\mu_{(\mathrm{hor},0)},\mu_{(\mathrm{hor},1)}$. Decide whether non-periodic extremal Gibbs measures (that is, extremal Gibbs measures not invariant under any full-rank sublattice) exist at high fugacity — Hadas and Peled conjecture they do not — and establish which sequences of finite-volume boundary conditions converge to which infinite-volume Gibbs measures.

3. Sharp correlation asymptotics in the columnar phase. The proven results give exponential decay with correlation length at most a universal constant in the direction of broken translational symmetry and at most $C\sqrt{\lambda}$ in the preserved direction, and $\mathrm{Cov}(\sigma(1,0),\sigma(3,0))\le C\log\lambda/\sqrt{\lambda}$. Provide matching lower bounds and sharp asymptotics: determine the true exponential rate of correlation decay in each direction as $\lambda\to\infty$ (in particular the scaling of the along-column correlation length in $\lambda$, conjecturally $\Theta(\sqrt{\lambda})$), and decide whether $\mathrm{Cov}(\sigma(1,0),\sigma(3,0))=\Theta(\lambda^{-1/2})$, removing the logarithmic factor.

Answers may resolve the components separately; the completion boundary of the problem is the combination of the full rigorous phase diagram, the classification of all Gibbs measures, and sharp two-sided correlation asymptotics for this model.

References

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