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REVIEW 3 major objections 6 minor 68 references

Strongly-connected percolation on directed lattices

T0 review · 3 major / 6 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Strongly-connected percolation on directed lattices forms one universality class distinct from ordinary undirected percolation.

desk verdict Clean high-precision numerics establishing one new 2D universality class for strongly-connected clusters on pure directed lattices; the class is real, the quoted exponent digits are a bit optimistic. read the letter →

arxiv 2607.24975 v1 pith:R6M37KSF submitted 2026-07-27 cond-mat.stat-mech

M. E. J. Newman , P. Grassberger , R. M. Ziff This is my paper · ORCID
classification cond-mat.stat-mech
keywords strongly-connectedpercolationdirectedlatticesuniversalityclasscriticalexponentswrappingprobabilitiesManhattanlatticebondfractaldimension
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

This paper shows that when bonds on a two-dimensional square lattice point in directions, the clusters in which every site can reach every other along directed paths—strongly-connected clusters—percolate with critical exponents and wrapping probabilities that differ sharply from ordinary undirected percolation. High-precision simulations on four globally isotropic arrangements (Manhattan, L-lattice, random diode, and square ice) produce a shared set of exponents, fractal dimensions, crossing probabilities, and thresholds. A reader who cares about networks, traffic, the web, or metabolic graphs cares because those systems are built from directed reachability, yet their large-scale critical geometry was only partly mapped. The work also supplies practical algorithms for locating thresholds and maintaining clusters as bonds are added. The shared numbers across arrangements imply that local direction patterns wash out at large scales, leaving a single new class.

What carries the argument

Strongly-connected clusters (sites mutually reachable by directed paths), located by Tarjan’s algorithm, binary-search threshold finding, and finite-size wrapping detection on periodic lattices. These tools extract universal exponents and crossing numbers that do not depend on the local direction pattern.

What would settle it

On large Manhattan or L-lattices, measure the one-direction wrapping probability at the estimated threshold and test whether it converges to ~0.75, or recompute β from largest-cluster scaling beyond L = 1024 and check whether it stays near 0.262 rather than ordinary percolation’s 5/36.

Watch

Extended reading notes

Core claim

Bond percolation of strongly-connected clusters on the two-dimensional square lattice, for several globally isotropic bond-direction arrangements, belongs to one universality class distinct from ordinary undirected percolation. The measured values are β = 0.2618(3), γ = 2.1423(4), ν ≃ 4/3, fractal dimension df = 1.8035(1), and wrapping probabilities at threshold of about 0.861 (any direction), 0.750 (one direction), and 0.639 (both)—all incompatible with ordinary percolation.

Load-bearing premise

Ordinary two-dimensional finite-size scaling and hyperscaling still hold for these directed clusters, so the correlation-length exponent and fractal dimension can be read directly from the measured β and γ.

Editorial extensions

If this is right

  • Critical exponents and wrapping numbers for strongly-connected percolation can be treated as universal across isotropic directed square lattices.
  • Site percolation on the Manhattan lattice shares the same threshold and exponents as bond percolation on the L-lattice.
  • Hulls of wrapping strongly-connected clusters are consistent with fractal dimension exactly 4/3.
  • High-precision thresholds are established: pc ≈ 0.697160 (Manhattan), 0.740193 (L-lattice), 0.708838 (ice), and exactly 1 for random diodes.

Reading between the lines

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

  • The same class likely covers other isotropic directed models the paper only conjectures about, such as two-neighbor or randomly-oriented Manhattan lattices.
  • The numerics leave open an exact value of 3/4 for one-direction wrapping probability as an analytic target.
  • Higher-dimensional and non-square lattices are the natural next test of whether the class survives or splits.
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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 / 6 minor

Summary. The authors study bond percolation of strongly connected components on four globally isotropic directed square lattices: Manhattan, L-lattice, random diode, and square ice. Using breadth-first search, Tarjan's algorithm, an incremental SCC algorithm, wrapping-based binary searches, and a cluster algorithm for ice configurations, they estimate thresholds, crossing probabilities, critical exponents, cluster-size scaling, and hull dimensions. The principal conclusion is that all four arrangements share a strongly connected percolation universality class distinct from ordinary undirected percolation, with β=0.2618(3), γ=2.1423(4), ν≃4/3, d_f=1.8035(1), and universal wrapping probabilities near 0.861, 0.750, and 0.639. Independent threshold estimates and scaling collapses provide substantial support for the central qualitative claim.

Significance. If the numerical conclusions hold, the paper establishes a useful benchmark universality class for strongly-connected percolation: a common exponent set across four directed square-lattice arrangements, clearly distinct from ordinary percolation, together with threshold constants, wrapping amplitudes, cluster-size scaling, and hull scaling. Particularly strong features are the independent out-component determination of p_c using exact ordinary-percolation exponents, the exact random-diode threshold, the agreement of several algorithmic estimators, the explicit scaling collapses, and the release of simulation code. The result is also falsifiable through the reported high-precision exponents and wrapping probabilities. The main qualification is that the smallest quoted errors currently appear to exclude potentially relevant finite-size systematics.

major comments (3)
  1. [§IV.C, Fig. 10, Table III, Appendix B] §IV.C and Appendix B, Fig. 10/Table III: the headline uncertainties β=0.2618(3) and γ=2.1423(4) appear to be statistical errors from pure power-law fits over L=64–1024. Appendix B explicitly says that the threshold extrapolations do not allow for corrections to scaling, and no correction term is reported for the exponent fits either. Yet §IV.A invokes corrections to scaling to explain the roughly 2σ discrepancy between p_av and p_c-c. If comparable corrections affect S(L) or q(L), they can shift one-decade log-log slopes at or above the quoted precision. The qualitative distinction from ordinary percolation is unambiguous, but the stated precision requires a systematic-error analysis: vary the fit window, include a correction-to-scaling term, report local-slope estimates, and propagate threshold uncertainty. The final error bars should include both statistical and systematic components.
  2. [§IV.A, §IV.C, Eqs. (7)–(10)] The central universality-class inference uses ordinary 2D hyperscaling, dν=2β+γ, to infer ν in Eq. (9), while §IV.A already assumes ν=4/3 in the p_av and p_c-c extrapolations. The circularity is substantially reduced by the independent out-component determination p_c=0.6971571(5), which agrees with p_c-c=0.697160(2) and uses only exact ordinary-percolation τ and σ. Nevertheless, the manuscript should make the logical ordering explicit and provide a robustness check for the exponent and wrapping data as p_c is varied over the independently supported interval. This would separate the measured hyperscaling consistency from an assumed value of ν.
  3. [§IV.B, Table II] Table II and the final paragraph of §IV.B report inconsistent combined crossing probabilities. The table's average row gives 0.86118(06), 0.75001(07), and 0.63889(10), whereas the text gives 0.86117(6), 0.75001(7), and 0.63867(11). The difference in the both-direction value is about two quoted errors and is larger than a harmless rounding discrepancy. Since these universal amplitudes are among the paper's principal predictions and are explicitly contrasted with Cardy–Pinson values, the correct numbers and their combination procedure must be stated consistently.
minor comments (6)
  1. [§IV.B] §IV.B: the one-direction crossing estimate is described as "suspiciously close to 0.75." If retained, this observation should be presented as exploratory; the present evidence does not establish an exact value, and the phrase may invite overinterpretation.
  2. [§IV.A, Table I] Table I: for consistency with the other preferred estimates and with the out-component result p_c=0.6971571(5), please clarify exactly how the final "Average" cell-to-cell entry combines the three wrapping definitions and whether correlations between horizontal/vertical measurements are included.
  3. [§IV.B, Fig. 8] Fig. 8 uses an L^{-1} extrapolation for R_L(p_c), while threshold fits elsewhere use L^{-1/ν} or L^{-1-1/ν}. Please state the rationale and whether changing the assumed correction exponent materially changes Table II.
  4. [§III.E] §III.E: the Potts-cluster update is said to require a constant number of sweeps and four sweeps are used. A short diagnostic (for example an autocorrelation estimate or independence check) would strengthen confidence that residual ice-configuration correlations are negligible at the quoted precision.
  5. [§IV.D, Fig. 14] §IV.D: the hull fit yielding d_h=1.3331(2) should specify the fitted L range and give the same correction-to-scaling or fit-window sensitivity requested for β and γ. The coincidence with 4/3 is currently suggestive rather than established.
  6. [Appendix A] The reproducible code link and appendix descriptions are valuable. For long-term reproducibility, please also state the random-number generator, seeding policy, code version or archive DOI, and total run statistics for the principal tables.

Circularity Check

1 steps flagged · score 1.0 of 10

Numerical measurements with independent pc anchors; only a mild, non-load-bearing FSS consistency loop via assumed ν=4/3.

  1. other [Sec. IV.A (pav scaling, Fig. 7a) and Sec. IV.C Eq. (9)]
    "We expect pav to converge to pc with increasing system size as pav − pc ∼ L^{−1/ν}, where ν=4/3 is the standard correlation length exponent for percolation. ... Armed with the values of β and γ we can also estimate ... ν=β+½γ=1.3330(3), in agreement with our earlier assumption that ν=4/3."

    Ordinary-percolation ν=4/3 is imported to extrapolate one wrapping-based pc estimator, then recovered from hyperscaling on β,γ measured near that pc. This is a mild consistency loop, not a definitional identity: out-component pc (using only external ordinary τ,σ) and cell-to-cell estimates are independent of the strongly-connected ν assumption, and the paper’s headline exponents/wrapping probabilities are direct measurements at those anchors rather than quantities forced by the assumed ν.

full rationale

The paper’s central claims (new universality class; shared exponents/wrapping probabilities across four directed lattices) are Monte Carlo measurements, not closed-form derivations forced by normalization or self-citation. Percolation thresholds are anchored three ways: (i) out-component size distributions using the externally known ordinary-percolation τ=187/91 and σ=36/91 (justified because random-diode out-components are ordinary clusters and pc=1 exactly there); (ii) wrapping pav and cell-to-cell estimators; (iii) mutual agreement of those anchors at the 10^{-6} level. Critical exponents β and γ are read from log-log slopes of largest and average strongly-connected cluster sizes at those pc values; ν and df then follow from standard 2D hyperscaling and are checked for internal consistency (ν≃4/3). Self-citations (Newman–Ziff incremental/wrapping algorithms, Barkema–Newman ice sampling, Ziff–Newman threshold convergence) supply methodology, not uniqueness theorems that forbid alternatives. No fitted parameter is renamed a prediction; no ansatz is smuggled in as a theorem. The sole mild loop is the temporary use of ordinary-percolation ν=4/3 when extrapolating pav, later recovered from β+γ/2—an a-posteriori consistency check that is defused by the out-component and cell-to-cell anchors and does not force the reported β, γ, or wrapping numbers. Score 1 reflects that minor FSS assumption, not structural circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim is a numerical universality classification. It rests on standard percolation finite-size scaling, hyperscaling in 2D, correctness of classical SCC algorithms, and the modeling choice that wrapping of strongly-connected clusters defines the transition. No new physical entities are postulated; free parameters are only ordinary fit intercepts/slopes used to quote exponents and pc.

free parameters (2)
  • Fitted pc intercepts (cell-to-cell and pav extrapolations) = Manhattan 0.697160(2); L-lattice 0.740193(4); ice 0.708838(4)
    Infinite-volume pc for Manhattan, L, and ice are obtained from linear extrapolations in L^{-1/ν} or L^{-1-1/ν}; the intercepts are data-driven fit parameters that enter the quoted thresholds.
  • Fitted slopes for β and γ = β=0.2618(3), γ=2.1423(4) (combined)
    Log-log slopes of largest and average cluster size vs L at pc (Fig. 10, left-most five sizes) are the direct source of the quoted exponents.
assumptions (6)
  • domain assumption 2D percolation finite-size scaling forms S=L^{-β/ν}f(L^{1/ν}(p-pc)) and analogous forms for average cluster size and wrapping probabilities.
    Invoked throughout Section IV to extract exponents and collapse data; not re-derived.
  • domain assumption Hyperscaling dν=2β+γ holds with d=2 for this transition.
    Used in Section IV.C (Eq. 9) to obtain ν from β,γ and to justify ν=4/3 in pc extrapolations.
  • domain assumption Out-components of a random site obey ordinary undirected percolation exponents τ=187/91 and σ=36/91 near criticality.
    Section IV.A; used to locate pc via s^{τ-2}P(s) vs s^σ. Exact for random diode; assumed for Manhattan, L, ice.
  • standard math Tarjan’s algorithm correctly partitions directed graphs into strongly-connected components in O(M) time; vector-pointer wrapping detection correctly flags toroidal wrapping.
    Sections III.A and III.D; classical graph algorithms treated as given.
  • ad hoc to paper Cell-to-cell RG estimator converges as L^{-1-1/ν} (conjectured by analogy with ordinary percolation).
    Section IV.A / Fig. 7b; authors note the exponent is not known exactly but fits are stable under reasonable variation.
  • domain assumption Square-ice configurations from four sweeps of the Potts-antiferromagnet cluster algorithm are sufficiently decorrelated for averaging.
    Section III.E, citing Barkema–Newman dynamic exponent ≈0; four sweeps chosen conservatively.

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Pith. "Pith review of Strongly-connected percolation on directed lattices." pith.science (2026). https://pith.science/paper/R6M37KSF

@misc{pith2026260724975,
  author       = {Pith},
  title        = {Pith review of: Strongly-connected percolation on directed lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6M37KSF}},
  note         = {Machine review of arXiv:2607.24975}
}
read the original abstract

We study percolation on lattices with directed bonds, focusing on the behavior of strongly-connected percolation clusters -- clusters in which every site is reachable from every other along a directed path. We consider the two-dimensional square lattice and various globally isotropic arrangements of the directions of the bonds. Performing simulations using a range of algorithmic approaches, we calculate high-precision values for critical exponents, fractal dimensions, crossing probabilities, and percolation thresholds for bond percolation with each bond arrangement. We find that the critical behavior is in a distinctly different universality class from that of traditional undirected percolation, but that all bond arrangements appear to fall in the same universality class.

Figures

Figures reproduced from arXiv: 2607.24975 by the authors.

Figure 1
Figure 1. FIG. 1: Strongly-connected percolation on a directed square [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Strongly-connected clusters for the random diode [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Mapping from the L-lattice to the Manhattan lattice. [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figures from the paper (9 more)
Figure 5
Figure 5. Figure 5: FIG. 5: (a) The cumulative distribution of out-component [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The probability [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Estimates of percolation thresholds on [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Scaling of the wrapping probability at thresh [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: a shows a plot of the size S of the largest strongly-connected cluster on a Manhattan lattice as a fraction of the total size L 2 of the system, calculated us￾ing the incremental algorithm described in Section III B and Eq. (2). The five curves are for system sizes L =…
Figure 10
Figure 10. Figure 10: FIG. 10: (a) Log-log plot of the the size of the largest strongly [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Comparison of the values for the exponents [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13: A “hull search walk” that traces the external (or [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Average length of a hull walk on a wrapping cluster [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

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Reference graph

Works this paper leans on

68 extracted references · 2 linked inside Pith

  1. [1]

    Stauffer and A

    D. Stauffer and A. Aharony,Introduction to Percolation Theory. Taylor and Francis, London, 2nd edition (1992)

    1992

  2. [2]

    P. G. de Gennes and E. Guyon, Lois g´ en´ erales pour l’injection d’un fluide dans un milieu poreux al´ eatoire. J. de M´ ecanique3, 403–432 (1978)

    1978

  3. [3]

    R. G. Larson, L. E. Scriven, and H. T. Davis, Percolation theory of two phase flow in porous media.Chem. Eng. Sci.36, 57–73 (1981)

    1981

  4. [4]

    Sahimi, Long-range correlated percolation and flow and transport in heterogeneous porous media.J

    M. Sahimi, Long-range correlated percolation and flow and transport in heterogeneous porous media.J. Physique I4, 1263–1288 (1994)

    1994

  5. [5]

    Odagaki and S

    T. Odagaki and S. Toyofuku, Properties of percolation clusters in a model granular system in two dimensions. J. Phys. Cond. Mat.10, 6447–6452 (1998)

    1998

  6. [6]

    Tobochnik, Granular collapse as a percolation transi- tion.Phys

    J. Tobochnik, Granular collapse as a percolation transi- tion.Phys. Rev. E60, 7137–7142 (1999)

    1999

  7. [7]

    de Bondt, L

    S. de Bondt, L. Froyen, and A. Deruyttere, Electrical conductivity of composites: A percolation approach.J. Mater. Sci.27, 1983–1988 (1992)

    1983

  8. [8]

    Bunde, S

    A. Bunde, S. Havlin, and M. Porto, Are branched poly- mers in the universality class of percolation?Phys. Rev. Lett.74, 2714–2716 (1995)

    1995

Show all 68 references
  1. [9]

    Machta, Phase transition in fractal porous media

    J. Machta, Phase transition in fractal porous media. Phys. Rev. Lett.66, 169–172 (1991)

    1991

  2. [10]

    Moon and S

    K. Moon and S. M. Girvin, Critical behavior of superfluid 4He in aerogel.Phys. Rev. Lett.75, 1328–1331 (1995)

    1995

  3. [11]

    M. K. Hassan, Recent development on fragmentation, ag- gregation and percolation.J. Phys. A55, 191001 (2022)

    2022

  4. [12]

    de Arcangelis, S

    L. de Arcangelis, S. Redner, and A. Coniglio, Anomalous voltage distribution of random resistor networks and a new model for the backbone at the percolation threshold. Phys. Rev. B31, 4725–4728 (1985)

    1985

  5. [13]

    C. L. Henley, Statics of a self-organized percolation model.Phys. Rev. Lett.71, 2741–2744 (1993)

    1993

  6. [14]

    M. E. J. Newman, Spread of epidemic disease on net- works.Phys. Rev. E66, 016128 (2002)

    2002

  7. [15]

    Cohen, K

    R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Re- silience of the Internet to random breakdowns.Phys. Rev. Lett.85, 4626–4628 (2000)

    2000

  8. [16]

    D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Network robustness and fragility: Percola- tion on random graphs.Phys. Rev. Lett.85, 5468–5471 (2000)

    2000

  9. [17]

    T. S. Ray and N. Jan, Anomalous approach to the self- organized critical state in a model for ‘life at the edge of chaos’.Phys. Rev. Lett.72, 4045–4048 (1994)

    1994

  10. [18]

    Solomon, G

    S. Solomon, G. Weisbuch, L. de Arcangelis, N. Jan, and D. Stauffer, Social percolation models.Physica A277, 239–247 (2000)

    2000

  11. [19]

    Grassberger, Directed percolation: Results and open problems

    P. Grassberger, Directed percolation: Results and open problems. In S. Puri and S. Dattagupta (eds.),Nonlin- earities in Complex Systems: Proceedings of the 1995 Shimla Conference on Complex Systems, Narosa Publish- ing, New Delhi (1997)

    1995

  12. [20]

    Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states.Advances in Physics49, 815–958 (2000)

    H. Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states.Advances in Physics49, 815–958 (2000)

    2000

  13. [21]

    Ara´ ujo, P

    N. Ara´ ujo, P. Grassberger, B. Kahng, K. Schrenk, and R. Ziff, Recent advances and open challenges in percola- tion.Eur. Phys. J.223, 2307–2321 (2014)

    2014

  14. [22]

    R. M. Ziff, E. Gulari, and Y. Barshad, Kinetic phase tran- sitions in an irreversible surface-reaction model.Phys. Rev. Lett.56, 2553–2556 (1986)

    1986

  15. [23]

    Grassberger and A

    P. Grassberger and A. de la Torre, Reggeon field the- ory (Schl¨ ogl’s first model) on a lattice: Monte Carlo cal- culations of critical behaviour.Annals of Physics122, 373–396 (1979)

    1979

  16. [24]

    Verbavatz and M

    V. Verbavatz and M. Barthelemy, From one-way streets to percolation on random mixed graphs.Phys. Rev. E 103, 042313 (2021)

    2021

  17. [25]

    Dhingra, P

    S. Dhingra, P. S. Dodwad, and M. Madan, Find- ing strongly connected components in a social network graph.Int. J. Computer Applications136, 7 (2016)

    2016

  18. [26]

    Broder, R

    A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Ra- jagopalan, R. Stata, A. Tomkins, and J. Wiener, Graph structure in the web.Computer Networks33, 309–320 (2000)

    2000

  19. [27]

    Kumar, S

    S. Kumar, S. Mahajan, and S. Jain, Feedbacks from the metabolic network to the genetic network reveal regula- tory modules in E. coli and B. subtilis.PLOS One13, e0203311 (2018)

    2018

  20. [28]

    Ma and A.-P

    H.-W. Ma and A.-P. Zeng, The connectivity structure, giant strong component and centrality of metabolic net- works.Bioinformatics19, 1423–1430 (2003)

    2003

  21. [29]

    Palmer-Rodr ´ ıguez, R

    P. Palmer-Rodr ´ ıguez, R. Alberich, M. Reyes-Prieto, J. A. Castro, and M. Llabr´ es, Metadag: A web tool to gener- ate and analyse metabolic networks.BMC Bioinformat- ics26, 31 (2025)

    2025

  22. [30]

    A. W. T. de Noronha, A. A. Moreira, A. P. Vieira, H. J. Herrmann, J. S. Andrade, and H. A. Carmona, Perco- lation on an isotropically directed lattice.Phys. Rev. E 98, 062116 (2018)

    2018

  23. [31]

    Pauling, The structure and entropy of ice and of other crystals with some randomness of atomic arrangement

    L. Pauling, The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Am. Chem. Soc.57, 2680–2684 (1935)

    1935

  24. [32]

    E. H. Lieb, Exact solution of the problem of the en- tropy of two-dimensional ice.Phys. Rev. Lett.18, 692– 694 (1967)

    1967

  25. [33]

    Kasteleyn, A soluble self-avoiding walk problem.Phys- ica29, 1329–1337 (1963)

    P. Kasteleyn, A soluble self-avoiding walk problem.Phys- ica29, 1329–1337 (1963)

    1963

  26. [34]

    Barber, Asymptotic results for self-avoiding walks on a Manhattan lattice.Physica48, 237–241 (1970)

    M. Barber, Asymptotic results for self-avoiding walks on a Manhattan lattice.Physica48, 237–241 (1970)

    1970

  27. [35]

    Redner, Percolation and conduction in a random resistor-diode network.J

    S. Redner, Percolation and conduction in a random resistor-diode network.J. Phys. A14, L349 (1981)

    1981

  28. [36]

    D. Dhar, M. Barma, and M. K. Phani, Duality transfor- mations for two-dimensional directed percolation and re- sistance problems.Phys. Rev. Lett.47, 1238–1241 (1981)

    1981

  29. [37]

    Wang and M

    Q. Wang and M. Li, Percolation transition of strongly connected clusters in finite dimensions and on complete graphs. Preprint arxiv:2605.16987 (2026)

    2026 arXiv

  30. [38]

    Coupier, B

    D. Coupier, B. Henry, B. Jahnel, and J. K¨ oppl, The planar lattice two-neighbor graph percolates. Preprint arxiv:2412.20781 (2024)

    2024 arXiv

  31. [39]

    Z. Zhou, J. Yang, R. M. Ziff, and Y. Deng, Crossover from isotropic to directed percolation.Phys. Rev. E86, 021102 (2012)

    2012

  32. [40]

    Ledger, B

    S. Ledger, B. T´ oth, and B. Valk´ o, Random walk on the randomly-oriented Manhattan lattice.Electron. Com- mun. Probab.23, 1–11 (2018)

    2018

  33. [41]

    Sharir, A strong-connectivity algorithm and its appli- cations in data flow analysis.Computers and Mathemat- ics with Applications7, 67–72 (1981)

    M. Sharir, A strong-connectivity algorithm and its appli- cations in data flow analysis.Computers and Mathemat- ics with Applications7, 67–72 (1981). 16

    1981

  34. [42]

    T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms. MIT Press, Cambridge, MA, 2nd edition (2001)

    2001

  35. [43]

    R. E. Tarjan, Depth-first search and linear graph algo- rithms.SIAM J. Comput.1, 146–160 (1972)

    1972

  36. [44]

    M. E. J. Newman and R. M. Ziff, Efficient Monte Carlo algorithm and high-precision results for percola- tion.Phys. Rev. Lett.85, 4104–4107 (2000)

    2000

  37. [45]

    M. E. J. Newman and R. M. Ziff, Fast Monte Carlo al- gorithm for site or bond percolation.Phys. Rev. E64, 016706 (2001)

    2001

  38. [46]

    S. S. Manna and A. J. Guttmann, Kinetic growth walks and trails on oriented square lattices: Hull percolation and percolation hulls.J. Phys. A22, 3113 (1989)

    1989

  39. [47]

    R. E. Tarjan, Efficiency of a good but not linear set union algorithm.J. ACM22, 215–225 (1975)

    1975

  40. [48]

    R. M. Ziff and M. E. J. Newman, Convergence of thresh- old estimates for two-dimensional percolation.Phys. Rev. E66, 016129 (2002)

    2002

  41. [49]

    Haeupler, T

    B. Haeupler, T. Kavitha, R. Mathew, S. Sen, and R. E. Tarjan, Incremental cycle detection, topological ordering, and strong component maintenance.ACM Transactions on Algorithms8, 3 (2012)

    2012

  42. [50]

    Bernstein, A

    A. Bernstein, A. Dudeja, and S. Pettie, Incremental SCC maintenance in sparse graphs. InProceedings of the 29th Annual European Symposium on Algorithms (ESA 2021), p. 14, Dagstuhl Publishing, Germany (2021)

    2021

  43. [51]

    G. T. Barkema and M. E. J. Newman, Monte Carlo simu- lation of ice models.Phys. Rev. E57, 1155–1166 (1998)

    1998

  44. [52]

    P. J. Reynolds, H. E. Stanley, and W. Klein, Percolation by position-space renormalisation group with large cells. J. Phys. A11, L199 (1978)

    1978

  45. [53]

    H. T. Pinson, Critical percolation on the torus.J. Stat. Phys.75, 1167–1177 (1994)

    1994

  46. [54]

    Sapoval, M

    B. Sapoval, M. Rosso, and J. Gouyet, The fractal nature of a diffusion front and the relation to percolation.J. Physique Lett.46, 149–156 (1985)

    1985

  47. [55]

    R. M. Ziff, P. T. Cummings, and G. Stell, Generation of percolation cluster perimeters by a random walk.J. Phys. A17, 3009 (1984)

    1984

  48. [56]

    Saleur and B

    H. Saleur and B. Duplantier, Exact determination of the percolation hull exponent in two dimensions.Phys. Rev. Lett.58, 2325–2328 (1987)

    1987

  49. [57]

    Grossman and A

    T. Grossman and A. Aharony, Accessible external perimeters of percolation clusters.J. Phys. A20, L1193 (1987)

    1987

  50. [58]

    Kolb, Crossover from standard to reduced hull for ran- dom percolation.Phys

    M. Kolb, Crossover from standard to reduced hull for ran- dom percolation.Phys. Rev. A41, 5725–5727(R) (1990)

    1990

  51. [59]

    S. S. Manna, Structure of backbone perimeters of perco- lation clusters.J. Phys. A22, 433 (1989)

    1989

  52. [60]

    B. B. Mandelbrot,The Fractal Geometry of Nature. W. H. Freeman, New York (1983)

    1983

  53. [61]

    Lawler, O

    G. Lawler, O. Schramm, and W. Werner, Test of scaling exponents for percolation-cluster perimeters.J. Amer. Math. Soc.16, 917–955 (2003)

    2003

  54. [62]

    J. M. F. Gunn and M. Ortu˜ no, Percolation and motion in a simple random environment.J. Phys. A18, L1095 (1985)

    1985

  55. [63]

    R. M. Ziff, Hull-generating walks.Physica D38, 377–383 (1989)

    1989

  56. [64]

    Deng and R

    Y. Deng and R. M. Ziff, The elastic and directed perco- lation backbone.J. Phys. A55, 244002 (2022)

    2022

  57. [65]

    Nolin, W

    P. Nolin, W. Qian, X. Sun, and Z. Zhuang, Backbone ex- ponent and annulus crossing probability for planar per- colation.Phys. Rev. Lett.134, 117101 (2025)

    2025

  58. [66]

    A. P. Sheppard, M. A. Knackstedt, W. V. Pinczewski, and M. Sahimi, Invasion percolation: new algorithms and universality classes.J. Phys. A32, L521 (1999)

    1999

  59. [67]

    J. A. Garofalo, N. A. Ara´ ujo, L. de Arcange- lis, A. Sarracino, and E. Lippiello, Janus percola- tion in anisotropic limited-degree networks. Preprint arxiv:2512.10566 (2025)

    2025

  60. [68]

    H. Hu, R. M. Ziff, and Y. Deng, Universal critical behav- ior of percolation in orientationally ordered Janus parti- cles and other anisotropic systems.Phys. Rev. Lett.129, 278002 (2022)

    2022

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