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Fast construction of self-avoiding polygons and efficient evaluation of closed walk fractions on the square lattice

T0 review · 0 major / 3 minor · reviewed 2026-05-23 · grok-4.3

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Pith's one-line read New algorithms compute the fractions of closed walks for each of over 762 billion self-avoiding polygons of length at most 38 on the square lattice.

desk verdict The paper delivers a massive new set of F_p fractions via novel algorithms for self-avoiding polygons and lattice Green's functions. read the letter →

arxiv 2412.12655 v1 submitted 2024-12-17 math.CO

Jean Fromentin , Pierre-Louis Giscard , Yohan Hosten This is my paper
classification math.CO
keywords self-avoidingpolygonsclosedwalkssquarelatticeGreen'sfunctionenumerationalgorithmsloop-erasedcombinatorics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper develops novel algorithms to construct self-avoiding polygons and evaluate the lattice Green's function precisely. These enable calculation of the fraction F_p for each of the 762,207,869,373 self-avoiding polygons p with length at most 38, where F_p measures the share of closed walks whose last erased loop is p. Only six such fractions were known previously. The resulting data supports two conjectures on the asymptotic sums of F_p and the specific value of F_p for large squares, with theoretical backing for the second. The algorithms extend to any vertex-transitive lattice and resolve open questions for the triangular lattice Green's function.

What carries the argument

Efficient algorithms for constructing self-avoiding polygons and for precise evaluation of the lattice Green's function on vertex-transitive lattices.

What would settle it

Direct numerical comparison of the computed F_p values against the six previously published fractions in the literature for the smallest polygons.

Watch

Extended reading notes

Core claim

We accurately compute the fractions F_p of all closed walks on the infinite square lattice whose last erased loop corresponds to any one of the 762,207,869,373 self-avoiding polygons p of length at most 38. Prior to this work, only 6 values of F_p had been calculated in the literature. The main computational engine uses efficient algorithms for both the construction of self-avoiding polygons and the precise evaluation of the lattice Green's function. Based on our results, we propose two conjectures: one regarding the asymptotic behavior of sums of F_p, and another concerning the value of F_p when p is a large square. We provide strong theoretical arguments supporting the second conjecture.

Load-bearing premise

The novel algorithms for construction of self-avoiding polygons and precise evaluation of the lattice Green's function scale correctly to 762 billion polygons without introducing errors.

Editorial extensions

If this is right

  • The sum of F_p over all polygons of a given length obeys a specific asymptotic form.
  • F_p for a large square polygon equals a definite closed-form expression supported by theory.
  • The same algorithms apply in principle to compute analogous fractions on any vertex-transitive infinite lattice.
  • Two open questions on the triangular lattice Green's function are settled by the extension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The computed F_p values could be used to test numerical predictions for loop-erased random walk statistics on the square lattice.
  • The construction algorithms may be adapted to enumerate or weight polygons on other regular lattices such as the hexagonal lattice.
  • If the conjectures hold, they would simplify the large-scale probability that a closed walk's final loop is a square.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces novel algorithms for efficient construction of self-avoiding polygons (SAPs) and precise evaluation of the lattice Green's function. These enable computation of the exact fractions F_p for all 762,207,869,373 SAPs of length at most 38 on the infinite square lattice, where F_p is the fraction of closed walks whose last erased loop is the polygon p. Prior literature had only 6 such values. The authors propose two conjectures (one on asymptotics of sums of F_p, one on F_p for large squares with strong theoretical support), and show the algorithms extend to other vertex-transitive lattices, resolving two open questions on the triangular lattice Green's function.

Significance. If the reported computations hold, the work supplies an unprecedented volume of exact data on erased-loop fractions for self-avoiding walks, opening new avenues for asymptotic analysis and conjecture testing in enumerative combinatorics. The algorithmic framework and its explicit extension to general lattices constitute reusable tools that could accelerate exact computations beyond the square lattice.

minor comments (3)
  1. Abstract: the sentence 'whose the last erased loop corresponds is any one of the 762,207,869,373 self-avoiding polygons p' contains a grammatical error ('whose the' and 'corresponds is') that should be corrected for clarity.
  2. The manuscript states that only 6 prior F_p values existed; a brief table or citation list of those 6 values in the introduction would help readers assess the scale of the advance.
  3. Complexity analysis is mentioned for the new algorithms; adding a short explicit statement of the dominant time or space complexity (e.g., O(n^k) for length-n polygons) in the relevant algorithmic section would strengthen the presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work, the recognition of its significance for enumerative combinatorics, and the recommendation of minor revision. The report contains no specific major comments requiring point-by-point rebuttal.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper's core contribution is a large-scale exact computation of the fractions F_p via newly introduced algorithms for enumerating self-avoiding polygons and evaluating the lattice Green's function on the square lattice (with extension to the triangular lattice). These results are presented as the direct output of the stated procedures and complexity analyses rather than any fitted parameter, self-definitional relation, or load-bearing self-citation chain. No equation or claim reduces by construction to its inputs, and the work resolves independent open questions on other lattices.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claims rest on the correctness of the described algorithms and the validity of the lattice Green's function evaluations.

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Cite this review

Pith. "Pith review of Fast construction of self-avoiding polygons and efficient evaluation of closed walk fractions on the square lattice." pith.science (2026). https://pith.science/paper/2412.12655

@misc{pith2026241212655,
  author       = {Pith},
  title        = {Pith review of: Fast construction of self-avoiding polygons and efficient evaluation of closed walk fractions on the square lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2412.12655}},
  note         = {Machine review of arXiv:2412.12655}
}
abstract

We build upon a recent theoretical breakthrough by employing novel algorithms to accurately compute the fractions $F_p$ of all closed walks on the infinite square lattice whose the last erased loop corresponds is any one of the $762, 207, 869, 373$ self-avoiding polygons $p$ of length at most 38. Prior to this work, only 6 values of $F_p$ had been calculated in the literature. The main computational engine uses efficient algorithms for both the construction of self-avoiding polygons and the precise evaluation of the lattice Green's function. Based on our results, we propose two conjectures: one regarding the asymptotic behavior of sums of $F_p$, and another concerning the value of $F_p$ when $p$ is a large square. We provide strong theoretical arguments supporting the second conjecture. Furthermore, the algorithms we introduce are not limited to the square lattice and can, in principle, be extended to any vertex-transitive infinite lattice. In establishing this extension, we resolve two open questions related to the triangular lattice Green's function.

Figures

Figures reproduced from arXiv: 2412.12655 by the authors.

Figure 1
Figure 1. A self-avoiding polygon on the square lattice and its distance-one neighbors. The matrix [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Direction of computations for the calculation of all [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The 1 × 1 square can be represented by eight different words depending on the chosen starting point (in gray). O [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Base line and base point O, of self-avoiding polygon. The arrow denotes chosen path orientation. Proposition 3.1. Each self-avoiding polygon p of length ℓ ≥ 2 with base point at O, is described by a unique word w(p) = R . . . D of length ℓ on the letters S. For example…
Figure 5
Figure 5. Figure 5: Three words that are not admissible. In a), property (P2) is unsatisfied as the first letter is U. In b) and c) this is property (P4). Base points of drawn paths are in grey. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Game board for ℓ = 8. The base cell is depicted with a gray dot. Forbidden cells are in gray. Paths Eℓ, Nℓ and Wℓ are drawn on a), b) and c) respectively. We identify each cell c of the game board by a unique integer nc in {0, . . . , hℓ × wℓ − 1}: we have nc = wℓ × y …
Figure 7
Figure 7. Figure 7: Depth first exploration of the tree T8. The 8-admissible word w = RURU of length 4 is represented by the solid black line on the game board. The base cell is that with the gray bullet, while cell κw(4) is highlighted with a black bullet. Black numbers in cells c stand …
Figure 8
Figure 8. Figure 8: The complete tree T8. To each node n corresponds the 8-admissible word made of the letters met going from the root of the tree down to node n. The seven elements of SAP8 are the seven leaves at depth 8 of T8. 4. Computational results Our experiments were carried out on…
Figure 9
Figure 9. Figure 9: Left figure: S(ℓ) (red dots), the proportion of closed walks on the infinite square lattice whose last erased loop is a self-avoiding polygon of length at most ℓ, as a function of ℓ. The solid blue line is the conjectured fit by 1 − ℓ −3/5 , which is supposed to hold a…
Figure 10
Figure 10. Figure 10: Log-plot of the fraction FSqL×L of closed walks on the infinite square lattice whose last erased loop is the L×L square as a function of the side length L (red dots); and fit based on analytical considerations §4.4 by the function ( √ 2 − 1)4L (solid blue line). Data …
Figure 11
Figure 11. Figure 11: Coordinate system on triangular lattice. [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

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