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arxiv: 2504.14080 · v4 · submitted 2025-04-18 · 🧮 math.CO · math.AT· math.GR· math.NT· math.PR

On minimal shapes and isoperimetric constants in hyperbolic lattices

Matteo D'Achille , Vanessa Jacquier , Wioletta M. Ruszel This is my paper

Pith reviewed 2026-05-22 18:15 UTC · model grok-4.3

classification 🧮 math.CO math.ATmath.GRmath.NTmath.PR
keywords hyperbolic latticesminimal perimeterisoperimetric constantregular tilingshyperbolic planegraph balls
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The pith

Minimal-perimeter shapes on hyperbolic lattices are exactly the balls built by adding successive distance layers.

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

This paper fully characterizes the finite shapes that have the smallest possible perimeter for a given number of vertices in regular hyperbolic tilings. It proves that these minimal shapes are those constructed by starting from a vertex and adding complete layers of all vertices at increasing graph distances. As a result, the perimeter-to-area ratio of these shapes converges to the isoperimetric constant from Häggström-Jonasson-Lyons, and the layer balls achieve this constant exactly in the limit for every fixed size.

Core claim

The set of finite minimal-perimeter shapes is fully characterized as the layer-constructed balls, which realize the isoperimetric constant of Häggström-Jonasson-Lyons for any fixed number of vertices.

What carries the argument

Layer-constructed balls obtained by adding successive distance layers from a center vertex.

If this is right

  • The ratio of perimeter to number of vertices for minimal shapes converges to the known isoperimetric constant.
  • Layer balls provide explicit minimizers for every finite vertex count.
  • The characterization holds for all regular hyperbolic lattices with 1/p + 1/q < 1/2.

Where Pith is reading between the lines

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

  • This explicit construction may simplify computations of isoperimetric profiles in hyperbolic graphs.
  • It suggests that radial growth layers capture the optimal shapes without combinatorial tweaks for most sizes.
  • Similar layer methods could apply to isoperimetric problems in other curved discrete spaces.

Load-bearing premise

That the minimal shapes must be connected and that whole distance layers always produce the absolute minimal perimeter without exceptions for small or irregular counts.

What would settle it

Finding a connected set of vertices with a given size that has a strictly smaller perimeter than the corresponding layer ball, or observing that the ratio does not approach the constant.

Figures

Figures reproduced from arXiv: 2504.14080 by Matteo D'Achille, Vanessa Jacquier, Wioletta M. Ruszel.

Figure 1.2
Figure 1.2. Figure 1.2: Embedding of L3,7 in the hyper￾bolic disc structure was then explicitly used to compute the growth rate of these lattices, see e.g. [15, 10, 13]. Moreover, the recursive structure is useful in [13, 12] to obtain interesting results about the number of animals and extremal p, q-animals on those lattices. Isoperimetric (or Cheeger) constants ie(·) of a graph are important because they quantify how efficien… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Example of B1(o) in pink and the green vertices are indicating the perime￾ter ∂eB1(o). B1(o) is obtained as the union of two layers, L0 and L1 [PITH_FULL_IMAGE:figures/full_fig_p003_1_3.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Example of layers L0 (blue), L1 (green), L2 (pink). Their union is the ball B2(o) [PITH_FULL_IMAGE:figures/full_fig_p006_2_1.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Example of the sets E0;7,3 and E1;7,3 in blue and I1;3,7 in green. o• [PITH_FULL_IMAGE:figures/full_fig_p006_2_3.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Example of B1(o) in pink and the green vertices are indicating ∂eB1(o). We have |∂eB1(o)|/|B1(o)| = 2. o• [PITH_FULL_IMAGE:figures/full_fig_p007_2_5.png] view at source ↗
Figure 2.7
Figure 2.7. Figure 2.7: Example of the set of connected points A, displayed as blue points. There are two tiles present in A. One is centered at x and one at y with x ⪯ y. The ball BA,max(x) is displayed by the pink circles and and the layer BA,M in(x) \ BA,max(x) by the green circles. the set of empty, resp. occupied vertices, in BA,M in(x) \ BA,max(x). Define a strip S of length |No| in some layer LK for K large enough. Denot… view at source ↗
Figure 2.9
Figure 2.9. Figure 2.9: Example of A ∈ M17 with |No| = 10, |Ne| = 18. We have se = 2, m = 3, and 3 + (2 − 1) vertices are in I1;7,3. 3 Results In this section, we present the main results of this paper. The first result is the explicit charac￾terization of the sets of minimal perimeter for a fixed volume N ∈ N for any hyperbolic lattice Lp,q. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2_9.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Example of D ∈ S30 with one empty and one occupied strip in the last layer, |So(I)| = 5. o• [PITH_FULL_IMAGE:figures/full_fig_p011_4_1.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Example for a set D (blue sites) for p = 7 and q = 3. We highlighted the empty sites by pink vertices. We have two empty strips Se,1, Se,2 in L1 and two empty strips Se,3, Se,4 in L2. see also [PITH_FULL_IMAGE:figures/full_fig_p012_4_4.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Example of a constructed set A obtained from the set D in [PITH_FULL_IMAGE:figures/full_fig_p013_4_6.png] view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Example of a set D with two stripes of red occupied vertices, So,1, So,2 in the last layer and one empty stripe of length 18 in the previous layer which will be occupied by |So,1 ∪ So,2| = 11 many vertices. 4.2 Proof of Theorem 3.2 Remark that for a fixed N ∈ N and let Q ∈ SN , M ∈ MN . By Theorem 3.1 we have that |∂eQ| |Q| ≥ |∂eM| |M| ≥ inf  |∂eA| |A| : 0 < |A| < ∞  . On the other hand by Theorem 4.1 … view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Example of a set M satisfying the relation (4.24) for p = 4, q = 5. We have that |No| = 7, |No ∩I2;4,5| = 5, |No ∩E2;4,5| = 2, |I2;4,5| = 48 and |∂eM| = 61 = 48+ 1·(5−1)+ 3·(2+ 1) CASE (I), p ≥ 4. We note that |No ∩ In+1;p,q| ≥ |In+1;p,q| |Ln+1| |No|, see Remark 4.4 for details. By using Equation (4.24), we can bound from above the ratio as follows |∂eM| |M| = |In+1;p,q| + (q − 4)(|No ∩ In+1;p,q| − 1) + … view at source ↗
read the original abstract

We fully characterize the set of finite shapes with minimal perimeter on hyperbolic lattices given by regular tilings of the hyperbolic plane whose tiles are regular $p$-gons meeting at vertices of degree $q$, with $1/p+1/q<\frac{1}{2}$. In particular, we prove that the ratio between the perimeter and the area (i.e., the number of vertices) of this set of minimal shapes converges to the isoperimetric constant computed in H\"aggstr\"om-Jonasson-Lyons. In fact, our balls which are constructed via layers and not combinatorial balls, will realize the isoperimetric constant for any fixed number of vertices.

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 / 2 minor

Summary. The manuscript claims a complete characterization of all finite minimal-perimeter shapes in hyperbolic lattices arising from regular {p,q}-tilings with 1/p + 1/q < 1/2. It proves that the perimeter-to-area ratio of these minimal shapes converges to the isoperimetric constant of Häggström-Jonasson-Lyons and that the authors' layer-constructed balls (distinct from combinatorial balls) attain this constant for every fixed number of vertices.

Significance. If the characterization and convergence hold, the work supplies explicit minimizers for the isoperimetric problem on these graphs and confirms that the independently computed constant of Häggström-Jonasson-Lyons is realized by concrete constructions. This strengthens the link between combinatorial isoperimetry and percolation theory on hyperbolic lattices. The paper correctly takes the constant from prior literature rather than deriving it internally and focuses on proving attainment.

major comments (1)
  1. The central claim that layer-constructed balls are (or dominate) all global minimizers for every n, including values not equal to the size of complete layers, requires a proof that adding a vertex according to the layer rule always yields the minimal perimeter increment. The abstract distinguishes these constructions from combinatorial balls and implies ad-hoc adjustments; the argument is most vulnerable precisely where connectedness is enforced or where irregular boundaries or non-connected sets could produce smaller perimeters for small or non-cumulative n.
minor comments (2)
  1. Define the precise inductive construction of the layer balls (including the rule for partial layers) in the introduction or first section so that the distinction from combinatorial balls is unambiguous from the outset.
  2. Add a short table or explicit formula listing the first few layer sizes for representative {p,q} pairs to illustrate the construction for small n.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the accurate summary of our claims, and the recommendation for major revision. We address the single major comment below, providing the requested justification while noting where minor clarifications will be added.

read point-by-point responses

  1. Referee: The central claim that layer-constructed balls are (or dominate) all global minimizers for every n, including values not equal to the size of complete layers, requires a proof that adding a vertex according to the layer rule always yields the minimal perimeter increment. The abstract distinguishes these constructions from combinatorial balls and implies ad-hoc adjustments; the argument is most vulnerable precisely where connectedness is enforced or where irregular boundaries or non-connected sets could produce smaller perimeters for small or non-cumulative n.

    Authors: We thank the referee for isolating the key technical point. The manuscript already contains an inductive argument establishing minimality for every n. Base cases for complete layers follow directly from the regular {p,q} structure and the definition of successive layers (see Definition 2.3 and Lemma 3.1). For intermediate sizes, the layer rule selects, at each step, the unique vertex that realizes the smallest possible perimeter increment among all vertices adjacent to the current shape; the induction step shows that any other choice of vertex (or any deviation from the layer ordering) produces a strictly larger increment, using the negative curvature of the lattice and the fact that boundary edges are counted with multiplicity 1. This comparison is carried out in the proof of Theorem 3.2 by enumerating the possible local configurations around a boundary vertex. Connectedness is enforced throughout: the problem is posed for connected finite subgraphs, as is standard for isoperimetric problems on infinite graphs (disconnected sets incur at least two additional boundary components and cannot be minimal). We will add an explicit sentence in Section 2 clarifying this convention and noting that the isoperimetric constant of Häggström–Jonasson–Lyons is itself defined via connected sets. The distinction from combinatorial balls is not ad hoc; it is the central geometric feature that allows exact attainment of the constant for every n rather than only along a subsequence. A short remark will be inserted in the abstract and introduction to emphasize the recursive, geometry-driven construction rather than any “adjustment.” revision: partial

Circularity Check

0 steps flagged

No circularity: external constant attained via independent combinatorial proof

full rationale

The paper cites the isoperimetric constant directly from the independent prior work of Häggström-Jonasson-Lyons and proves attainment by its own layer-constructed shapes. The characterization of minimal-perimeter sets proceeds via explicit layer addition rules on the hyperbolic lattice, without any reduction of the target constant or minimality claim to a fitted parameter, self-definition, or self-citation chain within the present manuscript. The derivation remains self-contained as a proof of which explicit constructions realize the externally given constant for every vertex count.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard definition of regular hyperbolic tilings and the combinatorial distance layers; no new free parameters or invented entities are introduced beyond the lattice parameters p and q.

axioms (2)
  • domain assumption The hyperbolic lattice is the infinite graph obtained from the regular {p,q} tiling with 1/p + 1/q < 1/2.
    Invoked in the abstract to define the setting.
  • domain assumption Minimal shapes are finite connected vertex sets.
    Implicit in the perimeter-area ratio definition.

pith-pipeline@v0.9.0 · 5651 in / 1320 out tokens · 44460 ms · 2026-05-22T18:15:42.024090+00:00 · methodology

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Reference graph

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