The critical fugacity of the hard-core model on $\mathbb{Z}^d$: sharp asymptotics and uniqueness of the phase transition

The hard-core model (nearest-neighbor exclusion lattice gas) on the integer lattice $\mathbb{Z}^d$ has a phase transition: at low fugacity the infinite-volume Gibbs measure is unique, while at high fugacity two extremal Gibbs measures appear, ordered on the even and on the odd sublattice. Writing $\lambda_c^-(d)$ and $\lambda_c^+(d)$ for the lower and upper critical fugacities, the best current bounds are $\lambda_c^-(d) \ge \lambda_{2d} \sim e/(2d)$, from Weitz's strong-spatial-mixing tree threshold, and $\lambda_c^+(d) \le C\log d/d$, from Hadas and Peled (2026); they conjecture $\lambda_c^+(d) \sim e/(2d)$ as $d \to \infty$, which would make the lattice threshold coincide at leading order with the regular-tree threshold. The open core of this question is: (i) determine the sharp first-order asymptotics of $\lambda_c^+(d)$ and $\lambda_c^-(d)$, in particular prove or refute $\lambda_c^+(d) \sim e/(2d)$; and (ii) determine whether $\lambda_c^-(d) = \lambda_c^+(d)$, i.e. whether the transition occurs at a single critical fugacity. No resolution is known in the literature to date.

Background

The hard-core model is the simplest lattice gas with an excluded-volume constraint: on a finite graph $G=(V,E)$ its configurations are independent sets $\sigma$ (subsets of vertices with no two adjacent, i.e. no two occupied vertices are neighbors), weighted by $\lambda^{|\sigma|}$, where the parameter $\lambda>0$ is the fugacity (activity) of a particle; the normalizing partition function $Z_G(\lambda)=\sum_\sigma \lambda^{|\sigma|}$ is the weighted generating function of independent sets. On the integer lattice $\mathbb{Z}^d$ the model is defined in infinite volume through Gibbs measures: probability measures on $\{0,1\}^{\mathbb{Z}^d}$ whose finite-volume conditional distributions obey the same exclusion rule with fugacity $\lambda$ (the Dobrushin-Lanford-Ruelle specification).

Since $\mathbb{Z}^d$ is bipartite, with vertex set partitioned by the parity of the coordinate sum into an even and an odd sublattice $\mathbb{V}_e, \mathbb{V}_o$, the model exhibits chessboard-type order at high fugacity: a typical configuration occupies essentially only one sublattice. Dobrushin (1968) proved that for every $d \ge 2$ the model has a phase transition: for sufficiently large $\lambda$ there are two distinct extremal (infinite-volume) Gibbs measures $\mu^{\mathrm{even}}, \mu^{\mathrm{odd}}$, obtained as infinite-volume limits with even and odd boundary conditions (in which every vertex of the even, respectively odd, sublattice outside the finite region is declared occupied), while for sufficiently small $\lambda$ the Gibbs measure is unique. By standard monotonicity of the specification in its boundary conditions, multiple Gibbs measures exist if and only if $\mu^{\mathrm{even}} \ne \mu^{\mathrm{odd}}$ (van den Berg and Steif), so one may define

$$\lambda_c^-(d) := \inf\{\lambda: \text{the model on } \mathbb{Z}^d \text{ has multiple Gibbs measures at fugacity } \lambda\}, \quad \lambda_c^+(d) := \sup\{\lambda: \text{the model has a unique Gibbs measure at fugacity } \lambda\},$$

with $0 < \lambda_c^-(d) \le \lambda_c^+(d) < \infty$.

Two facts frame the high-dimensional asymptotics. First, on any graph of maximum degree $\Delta$ the model exhibits strong spatial mixing (the influence of a boundary condition on a vertex decays with its distance from the region where boundary conditions differ) whenever $\lambda \le \lambda_\Delta := (\Delta-1)^{\Delta-1}/(\Delta-2)^\Delta$; this is Weitz's theorem, and the bound is sharp since the threshold $\lambda_\Delta$ is exact for the $\Delta$-regular tree. Strong spatial mixing implies uniqueness of the Gibbs measure, so since $\mathbb{Z}^d$ has degree $2d$ and $\lambda_\Delta \sim e/\Delta$, one has the lower bound $\lambda_c^-(d) \ge \lambda_{2d} \sim e/(2d)$. Second, matching upper bounds proved far harder. Dobrushin's argument gives only an exponentially large bound on $\lambda_c^+$. Galvin and Kahn (2004), building on Kahn's entropy method (2001) for bipartite graphs, showed $\lambda_c^+(d) \le C \log^{3/4} d / d^{1/4}$ and wrote that natural guesses for the correct order are $\log d/d$ or $1/d$; Peled and Samotij (2014) improved this to $C \log^2 d / d^{1/3}$. Hadas and Peled (2026) proved long-range order for the hard-core model on finite regular bipartite graphs satisfying mild expansion conditions - in particular on the $d$-dimensional hypercube and discrete tori at fugacities $\lambda \ge c\log d/d$ for some constant $c > 0$ - and transferred the conclusion to $\mathbb{Z}^d$ using reflection positivity (an argument exploiting invariance of the lattice under reflections in coordinate hyperplanes). They thereby established that multiple Gibbs measures exist on $\mathbb{Z}^d$ for every $\lambda > C \log d/d$, i.e. $\lambda_c^+(d) = d^{-1+o(1)}$, leaving a factor-$\log d$ gap to the tree lower bound, and they conjecture $\lambda_c^+(d) \sim e/(2d)$.

Independently of the constant, it is not known whether the transition is unique. While $\lambda_c^-(d) = \lambda_c^+(d)$ is expected (the analogous uniqueness of the critical point holds for the Ising model), this is open for every $d \ge 2$, and Brightwell, Häggström and Winkler (1999) constructed other infinite graphs on which the hard-core model has several critical points, so a split transition is a genuine logical possibility outside amenable graphs; separately, first-order transitions have recently been proved on biregular trees (Davies, Sandhu and Tan 2026), at a single critical fugacity rather than a split one.

Problem Statement

Determine the phase transition of the hard-core model on the integer lattice $\mathbb{Z}^d$ in the high-dimensional limit. With $\lambda_c^-(d)$ and $\lambda_c^+(d)$ defined as above (the infimum of fugacities with multiple Gibbs measures and the supremum of fugacities with a unique Gibbs measure, respectively), resolve the following two linked questions, which together fix both the location and the structure of the transition:

1. Sharp asymptotics. Prove or refute the conjecture of Hadas and Peled (2026) that

$$\lambda_c^+(d) \sim \frac{e}{2d} \quad \text{as } d \to \infty,$$

together with the corresponding sharp first-order asymptotics of $\lambda_c^-(d)$, which is known to satisfy $\lambda_c^-(d) \ge \lambda_{2d} = (2d-1)^{2d-1}/(2d-2)^{2d} \sim e/(2d)$. In particular, determine whether $\lim_{d \to \infty} 2d\,\lambda_c^+(d)$ exists and equals $e$, and analogously for $\lambda_c^-(d)$.

2. Uniqueness of the transition. Determine whether $\lambda_c^-(d) = \lambda_c^+(d)$, i.e. whether there is a single critical fugacity separating the uniqueness regime from the phase-coexistence regime on $\mathbb{Z}^d$, or whether intermediate fugacities with a re-emergent unique phase or additional transitions can occur, as they do on some other infinite graphs.

A complete answer establishes the first-order asymptotic behavior of both critical fugacities and settles the coincidence question in either direction; in keeping with the high-dimensional framing, these statements are required to hold for all sufficiently large $d$ (the coincidence question of part 2 is likewise open for every fixed $d \ge 2$, and a proof valid for every $d \ge 2$ is also complete). Results restricted to other graphs (regular or biregular trees, general finite regular bipartite graphs, the hypercube, or discrete tori without a transfer to $\mathbb{Z}^d$) do not by themselves resolve the problem; an answer for the lattice may of course be obtained through such graphs if the transfer is rigorously executed.

References

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