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arxiv: 2501.04885 · v3 · pith:2Y5D6AUDnew · submitted 2025-01-08 · 🧮 math.GT · cond-mat.stat-mech· cs.CG· math.SG

Direct Sampling of Confined Polygons in Linear Time

Clayton Shonkwiler , Kandin Theis This is my paper

Pith reviewed 2026-05-23 06:06 UTC · model grok-4.3

classification 🧮 math.GT cond-mat.stat-mechcs.CGmath.SG
keywords equilateral polygonsconfined polygonsmoment polytopesymplectic reductionlinear-time samplingtotal curvaturerandom polygonsequilateral closed curves
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The pith

Sampling confined equilateral polygons reduces to sampling a combinatorially natural moment polytope, enabling linear-time direct sampling.

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

The paper gives an algorithm that produces random equilateral closed polygons confined tightly inside a ball in three-space, with running time linear in the number of edges. It reaches this by showing that symplectic geometry turns the sampling task into the problem of sampling points from a moment polytope whose combinatorial structure is especially simple under the chosen confinement. The same reduction supplies closed-form expressions for the expected distance from each vertex to the origin. The algorithm is then run to measure the expected total curvature of such polygons, producing a sharp conjecture for the asymptotic growth of that quantity.

Core claim

For tightly confined equilateral closed polygons the symplectic reduction maps the uniform measure on the configuration space exactly onto the uniform measure on a moment polytope that admits a direct combinatorial description; sampling the polytope therefore samples the polygons uniformly, and the same description yields explicit formulas for the expected radial distances of the vertices.

What carries the argument

The moment polytope obtained by symplectic reduction of the confined equilateral polygon space; its combinatorial structure supplies both the linear-time sampler and the distance formulas.

If this is right

  • The algorithm generates uniformly distributed confined equilateral polygons in time linear in edge count.
  • Explicit formulas give the expected distance of each vertex from the origin.
  • Numerical experiments on total curvature become feasible at large edge counts and produce a precise asymptotic conjecture.
  • The combinatorial description of the polytope directly controls both sampling speed and expectation calculations.

Where Pith is reading between the lines

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

  • The same reduction might remain tractable for other natural confinement regions whose polytopes retain enough combinatorial regularity.
  • Linear-time generation would let researchers simulate long confined polymer chains or knotted DNA at scales previously inaccessible.
  • The curvature conjecture could be turned into a theorem by counting lattice paths or using generating functions on the same polytope.
  • Connections between the polytope vertices and integer partitions or plane partitions might appear once the combinatorial model is written explicitly.

Load-bearing premise

The chosen tight-confinement model makes the moment polytope combinatorially natural so that uniform sampling on the polytope is equivalent to uniform sampling on the polygon space.

What would settle it

Run the claimed linear-time procedure on polygons with a few hundred edges, compute the empirical distribution of a vertex coordinate or curvature statistic, and check whether it matches the distribution predicted by the volume of the corresponding slice of the moment polytope.

Figures

Figures reproduced from arXiv: 2501.04885 by Clayton Shonkwiler, Kandin Theis.

Figure 1
Figure 1. Figure 1: Illustrating the reconstruction map 𝛼 : P𝑛 ×𝑇 𝑛−3 which takes diagonal lengths and dihedral angles to an equilateral polygon. Top left shows the triangulation of an abstract hexagon. Given 𝑑1, 𝑑2, 𝑑3 which obey the triangle inequalities (1), build the four triangles in the triangulation from their side lengths (top right). Given dihedral angles 𝜃1, 𝜃2, 𝜃3, we can build a piecewise-linear surface out of the… view at source ↗
Figure 2
Figure 2. Figure 2: Left: Average rejection probabilities when generating 1,000,000 random points in [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time per million samples for 𝑛-gons from 𝑛 = 500 to 𝑛 = 20,000 in steps of 500. The fitted line has slope 0.102 (with 𝑅 2 > 0.9999). 4.1 Equilibrium Distribution of Chord Lengths As Marchal observes, the invariant measure of the Markov chain 𝑥1, 𝑥2, . . . generated by the conditional density (8) is given by 𝜇(𝐴) = ∫ 𝐴 (1 − cos (𝜋𝑦)) 𝑑𝑦 for any Borel set 𝐴. To see this, recall that 1 − cos (𝜋𝑦) = 2 sin2 [P… view at source ↗
Figure 4
Figure 4. Figure 4: For (left) 𝑛 = 20 and 𝑁 = 1 million and (right) 𝑛 = 20,000 and 𝑁 = 10,000, we generated 𝑁 samples of (𝑑1, . . . , 𝑑𝑛−3) with Algorithm 1. The plots show the histograms of the resulting 𝑁(𝑛 − 3) numbers (i.e., all the 𝑑𝑖 for all the samples, treated as a single pool of numbers), in both cases plotted against the asymptotic density 𝑓 (𝑡) = 1 − cos(𝜋𝑡). 5 A Combinatorial Digression 5.1 Entringer Numbers We no… view at source ↗
Figure 5
Figure 5. Figure 5: Hasse diagram for the poset 𝑍𝑛,𝑖. 1 2 3 0 4 2 1 3 4 0 3 2 1 0 4 2 1 3 4 0 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: We call 𝑍𝑛,𝑖 an augmented zigzag poset. 𝑖 𝑖 − 1 𝑖 − 2 𝑖 − 3 𝑖 + 1 𝑖 + 2 0 𝑖 + 3 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Expected values of the distances from 𝑣𝑖 to 𝑣1 in confined equilateral 𝑛-gons (marked with the solid dot •) compared to the asymptotic values from Corollary 30 (marked with the open triangle △) and the value 1 2 + 2 𝜋 2 ≈ 0.7026 from Corollary 6 (solid line). Note that |𝑣2 − 𝑣1| and |𝑣𝑛 − 𝑣1| are always exactly 1. an integral formula for the exact value of EPol c (𝑛) [𝜅] is given in [22, Theorem 12]. When … view at source ↗
Figure 8
Figure 8. Figure 8: Average turning angle for 1,000,000 random [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The log–log plot of 2.14625 − (average turning angle) for all of our data. The line is the graph of 2.14625 − 𝑓 (𝑛) = 0.46742 𝑛 . We do not show the best-fit line, which has slope ≈ −0.99154, because it aligns so well to the graph that it would be effectively invisible in the plot. 𝑣𝑖+1 𝑣𝑖 𝑣𝑖+2 𝑑𝑖−2 𝑑𝑖−1 𝑑𝑖 𝜙𝑖 𝜃𝑖−1 𝑣1 [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: A portion of an equilateral 𝑛-gon, showing vertices 𝑣1, 𝑣𝑖 , 𝑣𝑖+1, and 𝑣𝑖+2, along with diagonal lengths 𝑑𝑖−2, 𝑑𝑖−1, and 𝑑𝑖 , the dihedral angle 𝜃𝑖−1, and the 𝑖th turning angle 𝜙𝑖 . The random variate 𝜃𝑖−1 is independent of the rest and has a very simple distribution: it is uniform on [0, 2𝜋). In the asymptotic limit, Corollary 6 implies that the diagonal lengths 𝑑𝑖−2, 𝑑𝑖−1, and 𝑑𝑖 will be chosen from the… view at source ↗
read the original abstract

We present an algorithm for sampling tightly confined random equilateral closed polygons in three-space which has runtime linear in the number of edges. Using symplectic geometry, sampling such polygons reduces to sampling a moment polytope, and in our confinement model this polytope turns out to be very natural from a combinatorial point of view. This connection to combinatorics yields both our fast sampling algorithm and explicit formulas for the expected distances of vertices to the origin. We use our algorithm to investigate the expected total curvature of confined polygons, leading to a very precise conjecture for the asymptotics of total curvature.

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 / 0 minor

Summary. The manuscript claims an algorithm for sampling tightly confined random equilateral closed polygons in three-space with runtime linear in the number of edges. Using symplectic geometry, sampling reduces to sampling a moment polytope that is combinatorially natural in the chosen confinement model; this yields both the fast algorithm and explicit formulas for expected distances of vertices to the origin. The sampler is then used to investigate expected total curvature, producing a precise conjecture on its asymptotics.

Significance. If the claimed symplectic reduction is exact and measure-preserving, the linear-time sampler and explicit distance formulas would constitute a substantial technical advance for computational studies of confined random polygons, with potential applications in geometric topology and statistical models of polymers. The explicit combinatorial description of the polytope and the resulting formulas are strengths if independently verifiable; the numerical investigation leading to a falsifiable asymptotic conjecture is also a positive feature.

major comments (1)
  1. [Abstract and §2] Abstract and §2 (reduction step): the central claim that the tight-confinement inequalities translate precisely into the facets of a combinatorially natural moment polytope, with the induced measure matching the desired distribution on polygons, is load-bearing for both the linear-time algorithm and the explicit distance formulas. The manuscript asserts this translation occurs but supplies no derivation of the polytope inequalities from the geometric constraints, no verification that no facets are missing or extraneous, and no confirmation that the symplectic moment map preserves the measure; without these steps the correctness of the sampler cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for highlighting the importance of rigorously establishing the central reduction. We agree that the translation from geometric constraints to the moment polytope requires explicit derivation, facet verification, and measure-preservation confirmation to make the claims fully verifiable. We will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and §2] Abstract and §2 (reduction step): the central claim that the tight-confinement inequalities translate precisely into the facets of a combinatorially natural moment polytope, with the induced measure matching the desired distribution on polygons, is load-bearing for both the linear-time algorithm and the explicit distance formulas. The manuscript asserts this translation occurs but supplies no derivation of the polytope inequalities from the geometric constraints, no verification that no facets are missing or extraneous, and no confirmation that the symplectic moment map preserves the measure; without these steps the correctness of the sampler cannot be assessed.

    Authors: We accept the referee's assessment that the reduction step in §2 is asserted without sufficient detail. In the revised version we will expand §2 with three additions: (i) a direct derivation mapping each tight-confinement inequality on the edge vectors to a linear inequality on the moment-map coordinates; (ii) a combinatorial enumeration showing that the resulting inequalities are precisely the facet inequalities of the polytope (no missing or redundant facets) by appealing to the toric structure and the explicit vertex description; (iii) a short argument that the symplectic moment map is measure-preserving, obtained by verifying that the Jacobian of the map is constant on the level sets and that the uniform measure on the confined polygon space pushes forward to the Lebesgue measure on the polytope. These additions will occupy roughly two pages and will be placed immediately after the statement of the main theorem. revision: yes

Circularity Check

0 steps flagged

No circularity: symplectic reduction and combinatorial polytope are independent mathematical claims

full rationale

The paper's derivation proceeds by applying symplectic geometry (moment map for the torus action on edge vectors subject to closure) to reduce sampling of tightly confined equilateral polygons to uniform sampling on a moment polytope; the authors then observe that, for their specific confinement model, this polytope admits a direct combinatorial description that enables linear-time sampling and explicit distance formulas. No step defines a quantity in terms of its own output, renames a known empirical pattern, fits a parameter to data and then calls a related quantity a prediction, or invokes a uniqueness theorem or ansatz justified only by prior self-citation. The reduction and the combinatorial naturalness are presented as consequences of the geometry and model choice, not as self-referential or fitted inputs. The chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The reduction to a moment polytope is presented as following from symplectic geometry without further breakdown.

pith-pipeline@v0.9.0 · 5625 in / 1059 out tokens · 49076 ms · 2026-05-23T06:06:21.179146+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    sampling such polygons reduces to sampling a moment polytope... equivalent to the order polytope of the zig-zag poset... triangulated by simplices indexed by alternating permutations

  • Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    P_n(1) = {(d1,...,dn-3) in [0,1]^{n-3} : di + di+1 >=1}

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The paper's claim is directly supported by a theorem in the formal canon.
supports
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extends
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The paper appears to rely on the theorem as machinery.
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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