Existence, sharpness, and location of the uniqueness threshold of the hard-core lattice gas on the square lattice $\mathbb{Z}^2$ (conjectured $\lambda_c(\mathbb{Z}^2)\approx 3.796$)

The hard-core model (the nearest-neighbor hard-square lattice gas) on the square lattice $\mathbb{Z}^2$ places probability on independent sets weighted by an activity $\lambda$ per occupied site. Series analyses going back to Gaunt and Fisher (1965) and Baxter, Enting and Tsang (1980) suggest a single critical activity $\lambda_c(\mathbb{Z}^2)\approx 3.796$ separating uniqueness of the infinite-volume Gibbs measure from phase coexistence of the two checkerboard-ordered phases, but neither the existence of such a sharp threshold nor its location has been proved. The source paper (Restrepo, Shin, Tetali, Vigoda and Yang, FOCS 2011) improved the rigorously provable uniqueness regime to $\lambda<2.3882$ using branching matrices and a finite decay criterion (the DMS condition), and explicitly posed the underlying open problems. Later work improved the lower bound to $2.48$ (Vera, Vigoda and Yang, 2015) and then $2.538$ (Sinclair, Srivastava, Štefankovič and Yin, 2017), while coexistence is proved only for $\lambda>5.3506$ (Blanca, Chen, Galvin, Randall and Tetali, 2019); moreover Vera, Vigoda and Yang proved that for every fixed per-vertex ordering the self-avoiding-walk-tree comparison fails beyond $\lambda=3.4$, so the branching-matrix form of this methodology underlying the earlier uniqueness bounds cannot reach the conjectured threshold. The open problem is to prove or disprove that the hard-core model on $\mathbb{Z}^2$ has a single critical activity separating uniqueness from coexistence, and to determine its value (in particular, prove or refute the conjecture $\lambda_c\approx 3.796$). The current rigorous envelope $2.538<\lambda_c(\mathbb{Z}^2)\le 5.3506$ around the conjectured $3.796$ is unchanged since 2017-2018, and the problem remains open.

Background

The hard-core model is a lattice gas with hard exclusion constraints, one of the two or three most studied models of discrete statistical mechanics. On a finite graph $G=(V,E)$ its configurations are the independent sets $I$ (sets of vertices with no two adjacent, i.e. no two particles at neighboring sites), and $I$ receives weight $\lambda^{|I|}$, where the activity (fugacity) $\lambda>0$ controls the density; the normalizing constant is the partition function $Z(G,\lambda)$. On the infinite square lattice $\mathbb{Z}^2$ one studies infinite-volume Gibbs measures, i.e. probability measures on configurations with the correct conditional probabilities in every finite region. The central question is whether the Gibbs measure is unique or whether there is phase coexistence (multiple Gibbs states, distinguished by far-away boundary conditions). For the hard-core model on $\mathbb{Z}^2$ the two competing extremal states at high activity are the two checkerboard configurations (full occupation of the even or of the odd sublattice, the maximum independent sets), so coexistence means the system spontaneously selects a parity class.

Series analyses of the hard-square model (Gaunt and Fisher 1965; Baxter, Enting and Tsang 1980; Rácz 1980) have long suggested that uniqueness and coexistence are separated by a single critical activity $\lambda_c(\mathbb{Z}^2)\approx 3.796$, with a continuous, Ising-like transition; unlike the Ising model or Baxter's exactly solved hard-hexagon model on the triangular lattice, the square-lattice model has no known exact solution. On the rigorous side, uniqueness is certified through decay of correlations: strong spatial mixing (SSM) means the influence of boundary conditions on the marginal probability of a vertex decays exponentially in the distance to the set where the boundary conditions differ, uniformly over boundary conditions and finite subgraphs; SSM implies uniqueness of the Gibbs measure and underlies deterministic fully polynomial-time approximation schemes (FPTAS) for the partition function and rapid mixing of the Glauber dynamics (the single-site update Markov chain) on finite subregions.

Weitz (STOC 2006) established the previous benchmark: the marginal at any vertex of any graph $G$ is exactly reproduced at the root of a tree of self-avoiding walks (SAW tree), an infinite tree encoding all self-avoiding walks from the vertex with cycle-closing leaves pinned by an ordering $\sigma$ of the neighbors; comparing with the infinite regular tree $T_\Delta$, whose hard-core uniqueness threshold $\lambda_c(T_\Delta)=(\Delta-1)^{\Delta-1}/(\Delta-2)^{\Delta}$ is classical, gives SSM on any graph of maximum degree $\Delta$ for $\lambda<\lambda_c(T_\Delta)$. For $\mathbb{Z}^2$ ($\Delta=4$) this yields $\lambda<27/16=1.6875$ (earlier disagreement-percolation arguments, a coupling technique of van den Berg and Steif, and computer-assisted uniqueness-criterion arguments had given bounds only in the range 1.25-1.51). The source paper refined Weitz's comparison for $\mathbb{Z}^2$: grouping self-avoiding walks by finitely many types that avoid short cycles, it bounds all SAW trees of the lattice by trees generated from a branching matrix $M$ (a nonnegative integer matrix whose entry $M_{j\ell}$ bounds the number of children of type $\ell$ of a type-$j$ vertex), and certifies exponential decay of the message recursions via a finite system of inequalities in parameters $s_j>1$ and $c_j>0$ involving the suprema $D_{jj}$ of explicit one-parameter functions (the DMS condition, Definition 7 of that paper). With 4, 17, 132 and 922 types this proves SSM for $\lambda<1.8801, 2.1625, 2.3335, 2.3882$ respectively; verifying the condition required heuristically guessed parameters (found by random-walk search) together with provable piecewise-linear envelope bounds for the $D_{jj}$.

Subsequent work shaped the problem as it stands today. Vera, Vigoda and Yang replaced the heuristic search by linear programming over piecewise-linear messages, reaching $\lambda<2.48$, and, crucially, proved a limitation: for every per-vertex ordering $\sigma$, SSM fails on the SAW tree $T^{\sigma}_{\mathrm{saw}}(\mathbb{Z}^2)$ as soon as $\lambda>3.4$ (for homogeneous orderings, $\lambda>3$), so the SAW-tree comparison methodology provably cannot certify uniqueness up to the conjectured $3.796$ (the cap is proven for fixed per-vertex orderings; the path-dependent relative orderings that Sinclair, Srivastava, Štefankovič and Yin use fall outside its formal statement, but their achieved bound remains far below $3$). Sinclair, Srivastava, Štefankovič and Yin introduced the connective constant (the exponential growth rate of the number of self-avoiding walks, rigorously bounded above by $2.6792$ on $\mathbb{Z}^2$) as the effective degree, obtaining SSM for $\lambda<\Delta^{\Delta}/(\Delta-1)^{\Delta+1}$ on graphs of connective constant at most $\Delta$, and a tailored analysis bounding the connective constant of a Weitz SAW tree by $2.429$ gives SSM on $\mathbb{Z}^2$ for $\lambda<2.538$, the best current lower bound. On the other side, a classical Peierls-type contour argument (the energy of the long domain wall forced between the two ordered phases dominates its entropy at large $\lambda$) gives $\lambda_c(\mathbb{Z}^2)=O(1)$, and fault-line Peierls contours counted by oriented (taxi-walk) self-avoiding walks prove coexistence for all $\lambda>5.3506$, with a proven barrier at about $4.8771$ for that method. The model lacks monotonicity in $\lambda$ (a counterexample on a non-regular tree is due to Brightwell, Häggström and Winkler), which blocks the sharpness machinery that pins exact critical points for percolation and the Ising model; Häggström constructed other periodic lattices with monotone hard-core transitions, but for $\mathbb{Z}^d$ the question is open.

Problem Statement

Determine the uniqueness phase transition of the hard-core model (nearest-neighbor hard-square lattice gas) on the square lattice $\mathbb{Z}^2$, as posed in the source paper for $\mathbb{Z}^d$, $d\ge 2$, with $\mathbb{Z}^2$ the flagship case: (1) Prove or disprove that there exists a single critical activity $\lambda_c(\mathbb{Z}^2)\in(0,\infty)$ such that the infinite-volume Gibbs measure is unique for every $\lambda<\lambda_c$ and there are multiple Gibbs states (coexistence of the two checkerboard-ordered phases) for every $\lambda>\lambda_c$; in particular rule out, or establish, non-monotone behavior in which the uniqueness region is not an interval. (2) Determine the value of $\lambda_c(\mathbb{Z}^2)$: prove or refute that it equals the empirically conjectured $\approx 3.796$ of the hard-square series analyses, by closing the current rigorous gap $2.538<\lambda_c(\mathbb{Z}^2)\le 5.3506$ to within the precision of that series estimate — that is, by rigorous bounds that either locate $\lambda_c(\mathbb{Z}^2)$ at the conjectured value or else exclude it. These two facets serve one overarching objective, the determination of the phase diagram in the activity parameter, and a complete answer settles the existence, sharpness and location of the transition; improved bounds on either side are progress but not completion unless they locate the conjectured value of $\lambda_c(\mathbb{Z}^2)$ at the precision of that series estimate or else exclude it. The same questions remain open for $\mathbb{Z}^d$ with $d\ge 3$.

References

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