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A general framework for crystallization in maximal hard-core models
Alexander Barg, Qidong He, Geyang Wang
TL;DR
Existing Pirogov–Sinai parameter bounds were not directly applicable to maximal hard-core models. This paper uses volume allocation to formulate sufficient conditions and establishes crystallization around stable periodic ground states under those conditions.
Problem
Existing Pirogov–Sinai formulations did not provide numerical parameter bounds directly applicable to maximal hard-core models.
Source: §3 · p. 14Method
The authors use volume allocation to formulate assumptions and apply Pirogov–Sinai contour and cluster-expansion arguments to maximal models.
Source: §5 · p. 24, §4 · p. 24Results
Under the stated assumptions, maximal hard-core models crystallize around stable periodic ground states for sufficiently large β.
Source: §4 · p. 23Takeaways & Limitations
Several maximal hard-core model examples satisfy the assumptions and fall within the theory’s scope.
Source: §5 · p. 24The theorem does not determine which periodic ground states are stable at each β.
Source: §4 · p. 24
Abstract
from arXiv · showhide
We study general maximal hard-core models on periodic lattice-type graphs. While for standard models, phase coexistence may arise for high activity, for maximal models this behavior may occur also for sufficiently low activity values. Relying on the concept of volume allocation, we develop a {\em unified set of assumptions} that imply the Peierls condition, a convergent cluster expansion for the partition function, and hence the conclusions of Pirogov-Sinai theory for this class of models. We further check these assumptions for a number of examples including standard lattice-type periodic graphs, proving crystallization for both high and low activity. We also derive estimates for the values of activity that bound the phase coexistence regions in the phase diagram, rewriting for this purpose the proof of a technical result in the derivation of Pirogov-Sinai theory.
1. Introduction
The paper develops a general Pirogov–Sinai framework for phase coexistence and crystallization in maximal hard-core models on periodic graphs, covering both high- and low-activity regimes. It applies the framework to several lattice-type graphs and derives parameter conditions for its use, while noting that the resulting activity estimates are poor.
Motivation and prior work
Model scope
- The paper generalizes maximal hard-core models on periodic graphs to configurations of multiple tile types with distinct chemical potentials, subject to non-overlap and maximality constraints.Tiles on distinct vertices must form an independent set in the exclusion graph, and no additional tile can be added without overlap.Source: §1 · p. 1
General framework
- Volume allocations and Zahradník’s contour formalism yield assumptions ensuring the Peierls condition, after which Pirogov–Sinai theory gives distinct extremal periodic Gibbs measures from stable periodic ground states.The approach follows the formulation in [18]; the formal hard-core interaction approach was expounded in [29].Source: §1 · p. 2
Examples
- The paper verifies its assumptions for hypercubic lattices Zd (d ≥2), the triangular lattice A2, and the honeycomb lattice H2 in both low- and high-activity regimes.When the assumptions hold, crystallization follows from the paper’s general results.Source: §1 · p. 2
Parameter conditions and limitation
- The paper rederives Zahradník’s theorem to obtain parameter conditions for Pirogov–Sinai theory, but the resulting high- and low-activity estimates are surprisingly poor and improving them remains open.Prior conditions for the standard hard-core model in [19] are not applicable to the maximal model.Source: §1 · p. 2
2. The model
This section defines maximal hard-core models on periodic lattice-type graphs and develops volume-allocation assumptions for identifying perfect configurations with periodic ground states, including the role of local rearrangements. It characterizes periodic ground states as exactly the perfect configurations and derives the Peierls condition under Assumptions 1 and 2, explaining how locality of volume allocation can ensure Assumption 2.
2. The model.
- The model places finitely many translated copies of a lattice in Rd, connects nearest sites, and defines maximality as being unable to add any allowed particle without violating hard-core constraints.Configurations assign each site an empty state or a tile type, with translated tiles required to have disjoint interiors.Source: §2 · pp. 2–3
2.1. The assumptions.
- The framework allocates each vertex’s unit volume among particles and requires a common-sign volume bound, with equality in one or more perfect configurations.Assumption 1 requires σp∗v(x | X) ≥ µx for all particles, with equality for every particle in the specified perfect configurations.Source: §2.1 · p. 5
- In the nearest-neighbor maximal hard-core model on Z2, the high-activity perfect configurations are the two checkerboards, while the low-activity case has ten perfect configurations.The examples give p∗ = 1/2 at high activity and p∗ = 1/5 at low activity; the low-activity configurations are cited as [24] [37].Source: §2.1 · p. 6
2.2. Periodic ground states.
- Under Assumptions 1 and 2, Proposition 2.8 states that a maximal configuration X is perfect if and only if it is a periodic ground state.Source: §2.2 · p. 7
- Local rearrangements redistribute allocated volume among particles inside a region without changing the region’s total allocated volume.Source: §2.2 · p. 9
- Perfect configurations are periodic ground states, and every periodic ground state is perfect.Source: §2.2 · pp. 9, 11
2.3. The Peierls condition.
- Under Assumptions 1 and 2, the Peierls condition holds via a uniform positive allocated-volume gap near each incorrect cell.The resulting energy excess is bounded below by a positive constant times the contour support size.Source: §2.3 · pp. 11–12
2.4. Screening and locality.
- If volume allocation is local at every perfect configuration, Assumption 1 implies Assumption 2; for nearest-neighbor models with discrete allocation, Assumption 1 suffices.Source: §2.4 · pp. 12–13
- Without locality, Assumption 1 need not imply Assumption 2: distant changes can alter weights in F while preserving the configuration on F(1).The example satisfies Assumption 1 but fails Assumption 2.Source: §2.4 · pp. 13–14
3. Parameter estimates in Pirogov–Sinai theory
This section gives explicit sufficient parameter conditions for Pirogov–Sinai theory in the maximal model, carrying constants through contour and cluster-expansion calculations. It presents a detailed version of the theory and proves results that imply crystallization.
3. Parameter estimates in Pirogov–Sinai theory.
- The section adapts Zahradník’s version of Pirogov–Sinai theory and reorganizes its results while largely retaining the original proof of Theorem 3.1.The authors explain that existing formulations do not provide directly applicable numerical bounds, motivating their detailed treatment.Source: §3 · p. 14
3.1. Contour representation.
- The contour representation is obtained by recursively applying the partition-function identity to normalized interior partition functions until no further contours fit.The representation uses compatible contours with disjoint support closures and identifies external contours relative to the collection.Source: §3.1 · pp. 14–15
3.2. Truncated contour model.
- Theorem 3.1 gives explicit conditions on β0 under which stability of a periodic ground state implies stability of every contour of its type.The section states that β0 must satisfy Assumption 3; the theorem’s conclusions support the crystallization result in the next section.Source: §3.2 · p. 16
3.3. Cluster expansion.
- The cluster-expansion section relies on Ueltschi’s convergence criterion and applies it to contour models.Contours of a fixed type serve as polymers in the application.Source: §3.3 · p. 17, §3.3.2 · p. 18
3.3.1. Ueltschi’s criterion.
- Under Ueltschi’s criterion, the polymer partition function is positive and well-defined, and its logarithm has an absolutely convergent series expansion.The criterion is introduced for finite or countable polymers with specified weights and compatibility.Source: §3.3.1 · p. 17
3.3.2. Application to contour models.
- For contour weights bounded by e^(-βτ|supp(γ)|/3), the contour pressure exists and is given by an absolutely convergent cluster series.The proof verifies Ueltschi’s criterion using the stated contour-weight bound and conditions on β.Source: §3.3.2 · p. 18
3.4. Proof of Theorem 3.1.
- By induction on contour and region levels, if aq = 0, no contour in Cq of level at most n + 1 is large; if aq > 0, every such large contour is geometrically large.Source: §3.4 · pp. 20–21
4. Crystallization
Under Assumptions 1–3, the model crystallizes at sufficiently low temperatures around stable periodic ground states. The section also explains how symmetry can help identify which ground states are stable.
4. Crystallization
- For β ≥ β0, stable periodic ground states exist, and the model crystallizes around each, with fixed-cell deviation bounded by ε(β) → 0.This formalizes the picture of low-temperature configurations as small perturbations of perfect configurations.Source: §4 · p. 23
- Finite-volume Gibbs measures with stable-ground-state boundary conditions converge to distinct extremal, LF-invariant Gibbs measures.Distinct stable states give different measures; the proof uses ε(β0) < 1/2 to distinguish their cell-correctness conditions.Source: §4 · pp. 23–24
- Model symmetries preserve stability at fixed β, so a single symmetry orbit of periodic ground states implies all are stable for β ≥ β0.When there are multiple orbits, additional methods are needed to distinguish the stable states.Source: §4 · p. 24
5. Examples
The section presents maximal hard-core model examples satisfying Assumptions 1 and 2, which are covered by the theory. Classifying their periodic ground states requires manual construction of a suitable volume allocation.
5. Examples
- The examples satisfy Assumptions 1 and 2 and are therefore covered by the theory.Table 1 collects the assumptions for nearest-neighbor models.Source: §5 · p. 24, Table 1 · p. 25
- Classifying periodic ground states is not addressed by the formalism and is NP-hard in general [9, p. 511].The classification must instead be carried out manually via Proposition 2.8, using a suitable volume allocation that distinguishes exactly finitely many configurations as perfect in the sense of Assumption 1.Source: §5 · pp. 24–25
- Constructing such a volume allocation can itself be a formidable endeavor.The task is to distinguish exactly finitely many configurations as perfect in the sense of Assumption 1.Source: §5 · p. 25
5.1. Nearest-neighbor maximal models.
The section examines nearest-neighbor maximal models on A2, Zd, and H2 in low- and high-activity regimes. It tests how volume-allocation bounds and the structure of configurations attaining them establish crystallization assumptions or reveal their failure.
5.1.1. The discrete volume allocation.
- For the lattices considered, both bounds on vacant-vertex occupancy are attainable, yielding p* = α/(α + ∆) or p* = 1/(1 + ∆) under the corresponding sign choices.The lower bound applies when σ = µx = 1; the upper bound applies when σ = µx = −1.Source: §5.1.1 · p. 25
5.1.2. Triangular lattice A2.
- A2 has 3 perfect configurations for σ = 1 and 14 for σ = −1, attaining the lower and upper bounds, respectively [38].An alternative allocation based on Delaunay-triangulation angles satisfies the volume-allocation conditions, using that triangle angles sum to π [38, Lemma 3.4].Source: §5.1.2 · pp. 25–26
5.1.3. The bipartite lattices at high activity.
- On connected bipartite lattices with α = ∆, the high-activity bound is attained only by the two bipartition configurations, so Assumption 1 holds with p* = 1/2.The argument applies to Zd for d ≥ 2 and H2.Source: §5.1.3 · p. 26
5.1.4. The hypercubic lattices Zd at low activity.
- At low activity on Zd, perfect configurations are radius-one perfect Lee codes; Assumption 1 holds when 2d + 1 is prime, but fails otherwise.For Z2, the stated bound gives crystallization for u ≤ exp(−22725).Source: §5.1.4 · pp. 26–27
5.1.5. Non-example: the honeycomb lattice H2 at low activity.
- At low activity on H2, infinitely many periodic perfect configurations arise because zig-zag lines can slide independently, so Assumption 1(b) fails.Choosing a line position every ℓ ≥ 3 gives a configuration with period lattice generated by 2b1 and 2ℓb2.Source: §5.1.5 · p. 27
5.2. Beyond nearest-neighbor models.
This section extends the volume-allocation framework beyond nearest-neighbor models through a two-tile equivalence, a discussion of hard-disk models, and polyform models with finitely many tilings.
5.2. Beyond nearest-neighbor models.
- At low activity, perfect configurations of H2 can slide individual lines, and sliding one line in every ℓ produces a periodic configuration of period 2ℓb2.The illustrated configurations differ only in the position of the shaded line, shifted by b1.Source: Figure 4 · p. 28
5.2.1. A two-tile example.
- The two-tile maximal model is equivalent by relabeling to a nearest-neighbor model on Z2, so the framework yields crystallization at both high and low activity.Its perfect configurations include fully occupied alternating rows at high activity and two orientations of a perfect code, with translates, at low activity.Source: §5.2.1 · p. 29, Figure 6 · p. 30
5.2.2. Hard-disk models.
- For hard-disk models, prior area-redistribution methods suggest a route to high-activity crystallization under maximality, but proving finitely many perfect configurations is nontrivial and fails for some diameters because of sliding.The prior work studies models without maximality [27, 28, 30]; the proposed extension requires verifying Assumption 1 and finiteness of perfect configurations.Source: §5.2.2 · pp. 29–30
5.2.3. Polyforms with a finite number of tilings.
- Finite tilings of a polyform ensure high-activity crystallization in its maximal model, with tilings as the perfect configurations.A continuous Voronoi volume allocation satisfies the framework’s allocation properties and is local at each perfect configuration, allowing Theorem 4.2 to apply.Source: §5.2.3 · p. 31