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REVIEW 3 major objections 5 minor 79 references

Random site percolation with complex neighborhoods in five dimension

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper tries to establish that in five-dimensional site percolation, the random occupation threshold p_c for any neighborhood built from the first seven coordination zones is governed by a single quantity—the weighted coordination number

desk verdict A large, useful 5D percolation threshold table, but the headline exponent is internally inconsistent and the new thresholds have no error bars; worth a round of revision. read the letter →

arxiv 2607.18991 v1 pith:Y4FLLGG5 submitted 2026-07-21 physics.comp-ph

Krzysztof Malarz , Maciej Wo{l}oszyn This is my paper · ORCID
classification physics.comp-ph MSC 82B4382B2782B80 PACS 64.60.ah05.10.Ln
keywords sitepercolationthresholdcomplexneighborhoodsfive-dimensionallatticefinite-sizescalingfractaldimensionweightedcoordinationnumberpowerlawuniversality
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 reports 127 site-percolation thresholds for complex neighborhoods on a five-dimensional simple cubic lattice, 120 of them estimated for the first time, and shows that they follow a clean power law in the weighted coordination number ζ. It also measures the fractal dimension of the incipient percolation cluster for seven extended (compact) neighborhoods, finding a mean of 3.5581(70), close to the scaling-relation prediction of 3.5215. A sympathetic reader cares because if the ζ-scaling is genuine, it provides a compact predictive rule for thresholds in a high-dimensional lattice where direct simulation is expensive. The study also demonstrates a parallel implementation of an efficient Monte Carlo algorithm that makes such large-scale threshold calculations feasible.

What carries the argument

The argument rests on two mechanisms: (1) a fast Monte Carlo algorithm that constructs clusters by adding one occupied site at a time and uses binomial convolution to convert site-count data into occupation-probability data, and (2) finite-size scaling, where the rescaled probability of belonging to the largest cluster, P_max L^{β/ν}, plotted against p, gives a size-independent crossing point at p_c. The crossing point is located using R = 10^6 repetitions per system, claimed to give 10^-5 accuracy.

What would settle it

Compare the extracted crossing point for the simplest neighborhood (sc(5)-1) with the high-precision literature value 0.14079633(4); a discrepancy beyond 10^-5 would indicate a systematic bias in the finite-size scaling procedure. Alternatively, recompute one new threshold at much larger lattice sizes (e.g., L = 48 or 64) and observe whether the crossing point drifts outside the claimed 10^-5 accuracy.

Watch

Extended reading notes

Core claim

The central claim is that percolation thresholds for complex neighborhoods in five-dimensional simple cubic lattices obey a universality in the weighted coordination number ζ: plotting p_c against ζ yields a power law p_c ∝ ζ^{-g} with g ≈ 0.7913(43), supported by 127 thresholds (120 new). For the seven extended neighborhoods tested, the fractal dimension of the incipient percolation cluster averages ⟨d_f⟩ = 3.5581(70), with individual values within 0.26%–1.76% of the theoretically predicted d*_f = 3.5215 derived from the best current critical exponents.

Load-bearing premise

The entire threshold table rests on the assumption that the finite-size scaling ansatz with the literature values β_5 = 0.8457 and ν_5 ≈ 0.572–0.575 holds for systems of linear size L = 16–32; if those exponents are wrong, every new p_c inherits a common error, and the fitted g ≈ 0.7913 could change.

Editorial extensions

If this is right

  • If g ≈ 0.79 is universal in five dimensions, thresholds for neighborhoods extending to any coordination zone can be predicted without new simulations.
  • The 120 new thresholds provide a benchmark dataset for evaluating other heuristic formulas, such as p_c = c/(z + b).
  • The mean fractal dimension ⟨d_f⟩ = 3.5581(70) supports the universality of d_f across extended and complex neighborhoods in five dimensions.
  • The parallelized Monte Carlo implementation shows a practical route to high-precision percolation thresholds in higher dimensions with modest wall-clock time.

Reading between the lines

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

  • The observed ζ-scaling suggests a cross-dimensional check: computing similar thresholds in d = 4 or d = 6 would reveal whether the exponent g depends on dimension or is a constant, potentially connecting to continuous-percolation limits.
  • Because all 120 new thresholds share the same finite-size scaling analysis, any bias in the assumed critical exponents (β_5 = 0.8457, ν_5 ≈ 0.572–0.575) will appear as a common systematic shift; an independent high-precision measurement of a single new threshold would bound that bias.
  • The residual scatter around the ζ power law may encode information about the contribution of individual coordination shells; adding a shape-dependent correction term could tighten the fit and refine the predicted thresholds.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper reports a parallel C++ implementation of the Newman–Ziff algorithm and uses it to study random site percolation on the five-dimensional simple cubic lattice with neighborhoods built from coordination zones up to the seventh. The authors present 127 percolation thresholds, 120 of which they state are new, fit the thresholds to the heuristic relations p_c ∝ ζ^{-g} and p_c = c/(z+b), and compute fractal dimensions d_f for seven compact neighborhoods. The abstract and Section IV quote g ≈ 0.7913(43), ⟨d_f⟩ ≈ 3.5581(70), and relative deviations of d_f from the scaling-relation value d*_f ≈ 3.5215 that range from 0.26% to 1.76%.

Significance. If the threshold catalog is accurate, it is a useful benchmark for percolation in d=5 with long-range neighborhoods, and the 120 new values are a substantial numerical contribution. The public code and the use of a standard, independent algorithm (Newman–Ziff) are strengths. However, the central quantitative claims — the ζ-scaling exponent g and the compatibility of the fractal dimensions with scaling relations — are currently not supported by the reported statistics: the value of g is internally inconsistent, the thresholds are given without error bars or a stated choice of the fixed exponents used in the finite-size scaling, and the d_f estimates deviate from d*_f by many times their quoted uncertainties. These issues must be resolved before the numerical conclusions can be relied upon.

major comments (3)
  1. [§III.B, Fig. 3, §IV] The value of g for p_c(ζ) is reported twice: g ≈ 0.8293(24) in the text of Section III.B, and g ≈ 0.7913(43) in the Fig. 3 caption, the abstract, and Section IV. These differ by 0.038, far outside the quoted uncertainties. Since the ζ-scaling exponent is a central result, the manuscript must state which fit is correct, and if the two fits use different data sets or different definitions of ζ, that must be made explicit.
  2. [§II.A.1 and §III.A] The thresholds in Table I are obtained by common crossings of P_max L^{β/ν} with β_5 = 0.8457 and one of three listed ν_5 values (0.5746, 0.5737(33), or 0.5720(43)), but the text never states which ν was used, and no per-threshold statistical uncertainty is reported. The claimed 10^-5 precision in Fig. 1 is read off the sharpness of the crossing, not from a bootstrap/jackknife or from the variance among the L-pair crossings. Any error in the assumed exponents or uncorrected finite-size corrections enters all 120 new thresholds coherently, and the low-p_c/high-ζ points that dominate the log-log fit for g are precisely the ones for which no independent validation exists. The authors should supply per-threshold error bars, state the exponent choice, and test sensitivity to the FSS ansatz (e.g., by using different L ranges or by comparing with the seven previously known thresholds only).
  3. [Table II and §IV] The d_f values are not merely 'not differ[ing] much' from d*_f = 3.5215. Using the quoted uncertainties, the deviations range from ~4σ (sc(5)-1,2, d_f = 3.5307(23)) to ~45σ (sc(5)-1,2,3,4,5,6, d_f = 3.57944(80)) when the 0.0010 uncertainty on d*_f is included. The sign is also uniform: every estimate exceeds d*_f, which suggests a systematic finite-size effect rather than statistical scatter. The text and abstract should either quantify and explain this discrepancy or soften the claim that the fractal dimensions confirm the scaling-relation value.
minor comments (5)
  1. [Title and Abstract] The phrase 'in five dimension' should be 'in five dimensions' or 'in five-dimensional space'.
  2. [Eq. (11) and Table II] Equation (11) defines δ = (d*_f − d_f)/d*_f · 100%, which is negative for all rows in Table II because every d_f exceeds d*_f. The table lists positive values; either use the absolute value or define δ as |d*_f − d_f|/d*_f.
  3. [References] Reference [4] spells 'Birkhäuser' as 'Brikhauser'; reference [59] is a duplicate of reference [40] and should be consolidated.
  4. [§IV] The statement that equation (2) is 'qualitatively not worse' than p_c ∝ (z+1)^{-1} is vague. Since equation (2) has two fitted parameters, the comparison should be quantitative, e.g., residual per data point or AIC.
  5. [Fig. 4 caption] The phrase 'more or less agree' is too vague for a quantitative comparison. The caption should state the difference and the combined uncertainty, especially in view of the deviations noted in the major comments.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: thresholds and fractal dimensions are measured independently, with only minor non-load-bearing self-citations.

full rationale

The central results are Monte Carlo thresholds obtained with the Newman–Ziff algorithm, which is independent of the scaling laws subsequently fitted. The finite-size scaling relation (Eq. 6) is a standard analysis tool and uses literature exponents β5=0.8457 and ν5 from Refs. [64,65,66], not from the data being predicted. The compact-neighborhood thresholds are validated against the external results of Refs. [50,51] to six-digit accuracy, and the measured fractal dimensions are compared with the theoretically predicted d*_f=3.5215 as well as with the independent d_f=3.5260(14) of Ref. [65]. The p_c ∝ ζ^{-g} relation is not used to produce the thresholds; it is a post-hoc fit to measured values, so calling it a 'verification' is methodologically weak but not circular. Self-citations appear (Refs. [42,46,71] for the ζ heuristic and d_f universality in lower dimensions), but they are not load-bearing for the present 5D claims; ζ is a definition and the universality claim is tested on new data. Non-circular weaknesses that should be addressed: the paper does not state which of the three ν5 values was used in the crossings, per-threshold error bars are absent, and Section III.B reports g≈0.8293(24) while Figure 3, the abstract, and Section IV report g≈0.7913(43). These are reporting/correctness concerns, not circularity of the derivation chain.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

No new entities are introduced. The free parameters are fit constants for the two empirical scaling laws and the seven fractal-dimension slopes. The assumptions are standard finite-size scaling, literature exponents, and the absence of significant finite-size corrections at the chosen system sizes.

free parameters (4)
  • g (ζ-scaling exponent) = 0.7913(43) (vs 0.8293(24) in §III.B)
    Exponent in the fit p_c ∝ ζ^{-g} to the 127 thresholds; claimed universal.
  • b (in Eq. 2) = 1.01(18)
    Fitted in p_c = c/(z+b).
  • c (in Eq. 2) = 1.553(22)
    Fitted pre-factor in p_c = c/(z+b).
  • d_f (per compact neighborhood) = 3.5453(30), 3.5307(23), 3.5525(16), 3.5579(11), 3.5575(25), 3.57944(80), 3.5836(25)
    Slope of log Smax vs log L at p_c; these fitted values are the heading result of Section III.C.
assumptions (4)
  • domain assumption Finite-size scaling relation (6) holds exactly with the quoted critical exponents.
    Used to locate p_c via common crossing point in Fig. 1.
  • domain assumption Literature values β_5=0.8457, ν_5=0.572–0.575 are accurate.
    Section II.A.1 cites refs [64–66]; the crossing point and d*_f both depend on them.
  • domain assumption L = 16–32 are large enough that corrections to scaling are negligible for all 127 neighborhoods.
    No systematic check of finite-size corrections is reported.
  • standard math The Newman–Ziff algorithm and R=10^6 repetitions yield unbiased Smax for each configuration.
    Standard algorithm [72]; the code is provided but not formally verified.

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Cite this review

Pith. "Pith review of Random site percolation with complex neighborhoods in five dimension." pith.science (2026). https://pith.science/paper/Y4FLLGG5

@misc{pith2026260718991,
  author       = {Pith},
  title        = {Pith review of: Random site percolation with complex neighborhoods in five dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y4FLLGG5}},
  note         = {Machine review of arXiv:2607.18991}
}
abstract

In this paper, the random site percolation problem in a five-dimensional space for complex neighborhoods is studied. The efficient C++ code (with ordinary work division) of the classical Newman--Ziff algorithm is presented. The obtained speed-up of computations reduces 2170 years of single-core computations -- necessary for obtaining the results presented in this paper -- much below the typical half-decay time of the scientist. For complex neighborhoods, it is for neighborhoods composed with sites taken from several coordination zones (up to the seventh coordination zone), the 127 percolation thresholds are calculated (with 120 among them being estimated for the first time). For seven extended (compact) neighborhoods, the fractal dimensions are also calculated. The mean value of these fractal dimensions, averaged over these seven compact neighborhoods, is estimated as $\langle d_f\rangle\approx 3.5581(70)$. The percentage errors of the values obtained for the fractal dimensions for compact neighborhoods vary from 0.26\% to 1.76\% with respect to the theoretically predicted value based on scaling relations and the most recent estimates of critical exponents for five-dimensional space. The universality of the percolation threshold as dependent on the weighted coordination number $\zeta=\sum_i z_i r_i$ (where $z_i$ is the number of sites in the $i$-th coordination zone and $r_i$ is the Euclidean distance from the sites in the $i$-th coordination zone to the central site) is also verified. The latter manifests itself as the power law ($p_c\propto\zeta^{-g}$) with $g\approx 0.7913(43)$

Figures

Figures reproduced from arXiv: 2607.18991 by the authors.

Figure 1
Figure 1. FIG. 1: Example of rescaled probability of belonging to [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Parallel speedup [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Example of dependency of the size of the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (1 more)
Figure 3
Figure 3. Figure 3: FIG. 3: Dependencies (a) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]

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    The mean value of the computed fractal dimensions for seven extended neighborhoods is ⟨df ⟩= 3.5581(70)

    The percentage errors obtained for all the extended neighborhoods (complex and compact) considered here are0.26%≤δ≤1.76%. The mean value of the computed fractal dimensions for seven extended neighborhoods is ⟨df ⟩= 3.5581(70). In summary, in this paper we show the results of mas- sively parallel computations based on the Newman–Ziff algorithm implemented ...

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