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arxiv: 2407.20762 · v3 · submitted 2024-07-30 · 🧮 math-ph · math.MP

On crystallization in the plane for pair potentials with an arbitrary norm

Laurent B\'etermin , Camille Furlanetto (Universit\'e Claude Bernard Lyon 1) This is my paper

Pith reviewed 2026-05-23 23:04 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords crystallizationpair potentialsarbitrary normsHeitmann-Radin potentiallattice minimizerskissing numbersticky disk modelp-norms
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The pith

For the Heitmann-Radin sticky disk potential, minimal configurations in the plane are patches of the triangular or square lattice up to affine transform, for any fixed norm, classified by the norm's kissing number.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the Heitmann-Radin sticky disk potential, which is infinite for distances below 1 and zero above, has ground states that are finite patches of either the triangular lattice or the square lattice. The choice depends on the kissing number of the arbitrary norm used to measure distances, and the result follows from applying Brass' 1996 combinatorial geometry theorem directly in this setting. This holds for any norm, including all p-norms, and yields an explicit family of norms for which any prescribed lattice is a minimizer. The work also resolves part of a related minimization problem with points constrained to the integer lattice and reports numerical findings of an unexpected phase transition in the optimal lattice as the p-norm parameter varies for both the Lennard-Jones potential and the Epstein zeta function.

Core claim

For the Heitmann-Radin potential, crystallization occurs for every fixed norm on R^2: the minimizers are, up to affine transform, patches of the triangular lattice when the kissing number is at least 5 and patches of the square lattice otherwise, with the minimal energy likewise classified by that kissing number. The proof invokes Brass' result on minimal-energy configurations for the sticky disk potential, which the paper treats as extending without change to the arbitrary-norm setting.

What carries the argument

Brass' 1996 combinatorial geometry classification of minimal-energy point sets for the sticky disk potential, applied via the kissing number of the arbitrary norm to decide between triangular and square lattice patches.

If this is right

  • Crystallization holds for the sticky disk potential under every norm on the plane.
  • Minimizers are always affine images of triangular or square lattice patches, selected by the norm's kissing number.
  • Any lattice arises as a minimizer for some explicitly constructed family of norms.
  • Part of the constrained minimization problem on the integer lattice is solved.
  • Numerical evidence indicates a phase transition in optimal lattice type as the p-norm parameter changes for both Lennard-Jones and Epstein zeta energies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Anisotropy can be introduced into crystallization models simply by changing the norm rather than the potential function itself.
  • If analogous combinatorial classification results exist for other short-range potentials, the same norm-based approach would immediately yield crystallization statements in non-Euclidean geometries.
  • The observed numerical phase transitions for non-sticky potentials suggest that the optimal lattice type may depend on both the potential range and the norm parameter in ways not captured by the sticky-disk case.

Load-bearing premise

Brass' 1996 result on minimal-energy configurations for the sticky disk potential carries over unchanged when the Euclidean norm is replaced by an arbitrary norm.

What would settle it

A finite point set whose pairwise distances under the given norm yield strictly lower total Heitmann-Radin energy than every patch of the triangular lattice (when kissing number >=5) or square lattice (otherwise) would falsify the classification.

Figures

Figures reproduced from arXiv: 2407.20762 by Camille Furlanetto (Universit\'e Claude Bernard Lyon 1), Laurent B\'etermin.

Figure 1
Figure 1. Figure 1: Construction of the configuration HN for N = 26. In this case, (s, k, j) = (2, 2, 1) and we verify that the number of unit distances is indeed ⌊3 × 26 − √ 12 × 26 − 3⌋ = 60. Blue dots • correspond to F2, green crosses + to B2,2 and the red cross × to R2,2,1. • ZN is an incomplete full filled octagon with N points on Z 2 . Contrary to the construction of HN given above which is rather easy to explain and fo… view at source ↗
Figure 2
Figure 2. Figure 2: Construction of the configuration ZN for N ∈ {2, ..., z3 = 76}. Solid lines materialize the boundary of the fully filled octagons F˜ i , i ∈ {2, 3, 4} whereas dashed lines correspond to the other edges of the minimal￾distance graph associated to ZN (for ∥ · ∥∞). The location of each new added points, to pass from ZN to ZN+1, is indicated by the labels (i.e. the numbers correspond to the values of N). {A, B… view at source ↗
Figure 3
Figure 3. Figure 3: For N = 16 and the infinite norm ∥ · ∥∞, the perfect 4 × 4 square configuration (on the left) has 42 edges whereas the true minimal configuration Z16 (on the right) has 43 edges. Example 3.8. As examples, we have constructed the minimizers for the p ∈ {1, 3} cases in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Minimizer of E∥·∥p for (p, N) = (1, 12) (on the left) and (p, N) = (3, 19) (on the right) Proposition 3.9 (Norms implying crystallization on a given lattice). Let L ∈ L2 written in its reduced form L = Z(u1, 0) ⊕ Z(v1, v2) as in (2.1), then for any p ∈ [1, ∞], if Np,L : R 2 → R is defined by ∀(x1, x2) ∈ R 2 , Np,L(x1, x2) =    p s [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Minimal configuration for N = 16 for E∥·∥2 (on the left) and EN∞,A2 (on the right). 1. any short-range perturbation of VHR as it was originally done by Radin in [32] or later by Au Yeung, Friesecke and Schmidt [21] in the euclidean case. However, we strongly believe that Theorem 3.2 still holds – but maybe not uniformly in ∥· ∥ – for perturbations of VHR with strong repulsion before 1−α, a minimum equal to… view at source ↗
Figure 6
Figure 6. Figure 6: Level curves of L 7→ ζL,∥·∥4 (6) (top left, with an optimal square lattice), L 7→ ζL,∥·∥8 (6) (top right, with an optimal lattice parametrized by ≈ (1/2, 1.098), e∥·∥1.5 (bottom left, with an optimal triangular lattice) and e∥·∥∞ (bottom right, with an optimal square lattice) on a part of D. References [1] L. B´etermin. Two-dimensional Theta Functions and Crystallization among Bravais Lattices. SIAM J. Mat… view at source ↗
read the original abstract

We investigate two-dimensional crystallization phenomena, i.e. minimality of a lattice's patch for interaction energies, with pair potentials of type $(x,y)\mapsto V(\|x-y\|)$ where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^2$ and $V:\mathbb{R}_+^*\to\mathbb{R}$ is a function. For the Heitmann-Radin sticky disk potential $V=V_{\text{HR}}$, we prove, using Brass' key result from [\textit{Computational Geometry}, 6:195--214, 1996], that crystallization occurs for any fixed norm, with a classification of minimizers and minimal energies according to the kissing number associated to $\|\cdot\|$. The minimizer is proved to be, up to affine transform, a patch of the triangular or the square lattice, which shows how to easily get anisotropy in a crystallization phenomenon. We apply this result to the $p$-norms $\|\cdot\|_p$, $p\geq 1$, which allows us to construct an explicit family of norms for which crystallization holds on any given lattice. We also solve part of a crystallization problem studied in [\textit{Arch. Ration. Mech. Anal.}, 240:987--1053] where points are constrained to be on $\mathbb{Z}^2$. Moreover, we numerically investigate the minimization problem for the energy per point among lattices for the Lennard-Jones potential $V=V_{\text{LJ}}:r\mapsto r^{-12}-2r^{-6}$ as well as the Epstein zeta function associated to a $p$-norm $\|\cdot\|_p$, i.e. when $V=V_s:r\mapsto r^{-s}$, $s>2$. Our simulations show a new and unexpected phase transition for the minimizers with respect to $p$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proves crystallization for the Heitmann-Radin sticky-disk potential V_HR with respect to an arbitrary norm on R^2: ground-state configurations are, up to affine transformation, finite patches of the triangular or square lattice, selected according to the kissing number of the norm. The argument invokes Brass' 1996 combinatorial-geometry theorem directly. The result is specialized to p-norms to realize crystallization on any prescribed lattice and is used to address a constrained minimization problem on Z^2. Separate numerical experiments minimize the energy per point over lattices for the Lennard-Jones potential and for Epstein zeta functions associated to p-norms, reporting a phase transition in the minimizing lattice as a function of p.

Significance. If the central claim is valid, the work supplies an explicit, norm-dependent mechanism that produces anisotropic crystallization and constructs a continuous family of norms for which any given lattice is a ground state. The numerical observation of a p-dependent transition between triangular and square lattices is of independent interest for norm-dependent lattice minimization.

major comments (1)
  1. [Proof invoking Brass (1996)] Proof of the main crystallization theorem (invocation of Brass 1996): the manuscript applies Brass' combinatorial result on minimal-energy configurations directly to an arbitrary norm without additional verification that the key intersection lemmas survive the change from Euclidean circles to boundaries of general centrally symmetric convex bodies. Brass' arguments rely on the fact that two circles intersect in at most two points; for a general norm the unit sphere is an arbitrary centrally symmetric convex curve and two such curves may intersect in four or more points. This extension is load-bearing for the classification of minimizers (triangular vs. square patches) and for the kissing-number criterion.
minor comments (1)
  1. [Numerical experiments] The numerical section would benefit from an explicit statement of the lattice-parameter search domain, convergence tolerance, and number of random initializations used for each p-value.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to confirm that Brass' combinatorial lemmas extend beyond the Euclidean case. We address the point below and will revise the manuscript to include the required verification.

read point-by-point responses

  1. Referee: [Proof invoking Brass (1996)] Proof of the main crystallization theorem (invocation of Brass 1996): the manuscript applies Brass' combinatorial result on minimal-energy configurations directly to an arbitrary norm without additional verification that the key intersection lemmas survive the change from Euclidean circles to boundaries of general centrally symmetric convex bodies. Brass' arguments rely on the fact that two circles intersect in at most two points; for a general norm the unit sphere is an arbitrary centrally symmetric convex curve and two such curves may intersect in four or more points. This extension is load-bearing for the classification of minimizers (triangular vs. square patches) and for the kissing-number criterion.

    Authors: We agree that a direct invocation requires explicit justification when the unit ball is an arbitrary centrally symmetric convex body. The key lemmas in Brass (1996) use the at-most-two intersection property to control crossings in the contact graph and to classify rigid minimal configurations. For a general norm the boundaries remain simple closed curves, but convexity and central symmetry imply that any two distinct translates intersect in a way that preserves the planarity and degree bounds needed for the kissing-number classification; multiple intersection points cannot occur without interior overlap, which is forbidden by the sticky-disk condition. Nevertheless, to make this rigorous we will add a short appendix (or expanded remark in Section 2) that adapts the intersection argument using supporting lines of the convex bodies rather than relying on the Euclidean circle property. This will confirm that the triangular- and square-lattice patches remain the only minimizers and that the kissing-number criterion is unchanged. We view the addition as a clarification rather than a change of the result. revision: yes

Circularity Check

0 steps flagged

No circularity; central claim rests on external 1996 theorem plus independent numerics

full rationale

The paper's main analytic result for the Heitmann-Radin potential invokes Brass' 1996 combinatorial geometry theorem directly to classify minimizers according to the norm's kissing number. This is an external citation whose validity for general norms is asserted by the authors but does not reduce any equation or claim inside the paper to a self-fit or self-citation chain. The numerical minimization sections for Lennard-Jones and Epstein zeta functions are performed separately and do not feed back into the analytic classification. No derivation step equates a prediction to its own fitted input or renames an ansatz as a theorem. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the applicability of Brass' 1996 result to arbitrary norms and on the standard definition of the Heitmann-Radin potential; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption Brass' combinatorial geometry theorem on minimal point configurations holds for any norm on R^2.
    Invoked directly to classify minimizers according to kissing number.

pith-pipeline@v0.9.0 · 5873 in / 1243 out tokens · 23176 ms · 2026-05-23T23:04:25.386437+00:00 · methodology

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Reference graph

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