Absence of $\mathrm{O} (2)$ symmetry in the Vicsek model

Hideyuki Miyahara; Kota Mitsui; Tsuyoshi Mizohata; Yushin Takahashi

REVIEW 3 major objections 4 minor 17 references

Absence of $\mathrm{O} (2)$ symmetry in the Vicsek model

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The original Vicsek model, with its principal-value arctangent update rule, is not invariant under global phase shifts, and its reported phase transition vanishes when the global phase is chosen adaptively.

desk verdict The paper's central claim is a coordinate artifact—the Vicsek update is O(2)-equivariant on the unit circle, and the adaptive-phase disorder is a gauge choice. read the letter →

arxiv 2604.00930 v2 pith:EAB65KYP submitted 2026-04-01 cond-mat.stat-mech physics.bio-ph

Yushin Takahashi , Kota Mitsui , Tsuyoshi Mizohata , Hideyuki Miyahara This is my paper · ORCID

classification cond-mat.stat-mechphysics.bio-ph

keywords VicsekmodelO(2)symmetryphasetransitionactivemattercircularmeanprincipal-valuearctangentbranchcutorderparameter

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 revisits the 1995 Vicsek model and argues that its angle update rule—taking the mean direction as the principal-value arctangent of averaged sine and cosine, equivalently the argument of the averaged complex direction—is not invariant under a global rotation of all angles. At the level of the real-valued angular increment, a phase shift θ→θ+φ adds an extra 2πn(φ) term because the principal-value branch cut forces angles into (−π,π]. The authors conclude that the model lacks O(2) symmetry, counter to the standard story, and they show numerically that adaptively shifting the global phase—keeping the mean angle near π—destroys flocking even at low noise and large interaction radius. By contrast, the arithmetic-mean variant averages angles directly, is O(2)-symmetric, and its ordered phase is insensitive to the global phase. If correct, this would reframe a long-studied nonequilibrium phase transition as at least partly a coordinate artifact.

What carries the argument

The load-bearing object is the circular mean ⟨θ⟩ = arctan(⟨sin θ⟩/⟨cos θ⟩) = arg(⟨e^{iθ}⟩) used in the update rule. Its principal-value arctangent has a branch cut at ±π; under the global translation θ→θ+φ the increment Δθ_i acquires the extra term 2πn(φ). This branch-cut term is the entire mechanism: it breaks the invariance of the real-valued increment, and it is what the adaptive-phase simulation exploits by moving the branch cut through the flocking state. The arithmetic-mean update, Δθ_i = ⟨θ_j⟩−θ_i, has no branch cut and therefore is exactly invariant.

What would settle it

Repeat the adaptive-phase simulation for the original arctan update with φ(t)=π−⟨θ_i(t)⟩, N=1600, L=8.0, v_abs=0.01, r_V=0.1, η=0.75, and Δt=1.0, and monitor the order parameter v_op(t). If v_op saturates toward 1 rather than staying near 0, the paper's central numerical claim is wrong; if it remains near 0, the claim is confirmed.

Watch

Extended reading notes

Core claim

The paper's central claim is that the original angle-based Vicsek update is not equivariant under the global transformation θ_i→θ_i+φ when the model is implemented as in the original paper. Because the mean direction is defined as arg(⟨e^{iθ_j}⟩) and computed with the principal value of arctan, a global shift changes the angular increment Δθ_i by 2πn(φ), a jump that is invisible when angles are treated modulo 2π but real when Δθ is read as a real number. The authors take this as a breaking of O(2) symmetry and demonstrate numerically that choosing the global phase adaptively—so that the branch cut sits near the mean angle—suppresses the order parameter entirely. In contrast, replacing the ci

Load-bearing premise

The conclusion rests on treating the unwrapped real-valued angle difference Δθ_i(t) as the physical quantity that must be invariant under θ→θ+φ; if the physical state is instead the unit vector e^{iθ}, the extra 2πn term changes nothing and the original update is equivariant under global rotations.

Editorial extensions

If this is right

The standard interpretation of the Vicsek transition as spontaneous O(2) symmetry breaking does not hold for the original arctan implementation; the symmetry is broken by the angle coordinate before any flocking order develops.
The order-disorder transition reported for the original model is phase-convention dependent: with an adaptive global phase, the system stays disordered even where low noise and large interaction radius should produce strong alignment.
The arithmetic-mean variant is the O(2)-symmetric version of the model, and its flocking transition is robust to global phase choices, making it a more reliable benchmark for the nonequilibrium transition.
The continuous-time limit, θ̇_i = κ⟨sin(θ_j−θ_i)⟩ + ξ, contains no branch cut, so the natural continuum description of alignment belongs to the arithmetic-mean family, not to the original update.

Reading between the lines

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

A consequence the paper does not draw: because the branch-cut term is invisible to the unit-vector state e^{iθ}, the preference for the real-angle definition is what makes O(2) 'broken'; a reader who takes velocities as the physical state may regard the adaptive-phase simulation as a gauge choice rather than a change of physics.
Published numerical studies of the 'Vicsek model' may inherit the same coordinate dependence; rerunning their order-parameter and finite-size analyses under an adaptive global phase could identify which reported exponents are intrinsic and which are artifacts of the fixed branch cut.
The same diagnostic applies to other models using principal-value circular means, such as synchronization and active-rotation models: adaptively recentering the phase before the arctangent provides a simple test for hidden coordinate dependence.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, my take on arXiv:2604.00930 is that the central claim fails on a definitional point. The authors define O(2) symmetry as invariance of the real-valued angle increment Δθ_i(t) under a global shift θ_i→θ_i+φ, and then show that the principal-value arctangent introduces an extra 2πn(φ) term, which breaks that. The algebra is correct, but the premise is wrong. The physical state is the unit vector v_i=(cosθ_i, sinθ_i); angles are a lift. On the circle, θ and θ+2π are the same, so the 2πn term is the identity. The standard notion of O(2) equivariance for the update is v_i(t+Δt)=v_abs(cos(⟨θ_j⟩+Ξ_i), sin(...)), and rotating all θ_j by φ rotates the average direction by φ mod 2π; the noise distribution is unchanged. So the model is O(2)-equivariant on S^1. Their 'absence of symmetry' is a property of the coordinate, not the model.

What is genuinely useful: the paper is a clear reminder that principal-value arctangent has branch cuts, and that wrapped angle updates can be implementation-sensitive. The arithmetic-mean variant, Eq. (2.6), is a sensible alternative, and the continuous-time discussion in Sec. V is fine. The algebra in Sec. III is correct as far as it goes. So the paper is not careless; it is conceptually misframed.

The soft spots are load-bearing. First, the definition of O(2) symmetry via real-valued increments is nonstandard and unphysical. Second, the adaptive phase φ(t)=π−⟨θ_i(t)⟩ deliberately places the branch cut on the instantaneous mean angle; any angle-based update will show a discontinuity there, and the order parameter collapse in Fig. 1(upper) is a gauge artifact, not a lost transition. Third, the numerical evidence is a single parameter set (N=1600, L=8.0, η=0.75, r_V=0.1) with no phase diagram, so the claim that 'the phase transition vanishes completely' is not supported. The remark about Ref. [15] and periodic boundaries is a side argument and does not rescue the main claim.

Who this is for: someone benchmarking or implementing Vicsek-type models in angle coordinates might find the branch-cut warning useful. But the paper as written does not establish what it claims. I would not cite the central result. I would not send this to peer review as a substantive research paper; a serious editor would desk reject because the premise is unphysical. That said, the authors are earnest, the writing is clear, and there is a kernel of a useful numerical caution here. If they reframed it as a technical note on branch-cut handling and dropped the symmetry-breaking interpretation, it could become a worthwhile, much shorter contribution. My recommendation: decline in current form.

Referee Report

3 major / 4 minor

Summary. The paper argues that the original Vicsek model (Eq. (2.1)) lacks O(2) symmetry because the angular increment Δθ_i(t)=θ_i(t+Δt)-θ_i(t) is not invariant under global phase shifts θ_i→θ_i+φ when the mean direction is computed with the principal-value arctangent. The non-invariance arises through an additional 2πn(φ) term (Eq. (3.7)). The paper then presents a numerical simulation with an adaptive global phase φ(t)=π-⟨θ_i(t)⟩ (Eq. (4.1)) and claims that the order parameter collapses, so the phase transition reported in Vicsek et al. vanishes. It contrasts this with an arithmetic-mean variant (Eq. (2.6)) that is O(2)-symmetric and discusses a continuous-time limit (Section V).

Significance. If the central claim were correct, it would overturn the standard interpretation of the Vicsek transition as spontaneous O(2) symmetry breaking and would imply that the reported phase transition is a gauge artifact. The paper is clearly written, the algebra in Section III is internally consistent, and the distinction between the arctan and arithmetic-mean discretizations is a useful pedagogical point. The continuous-time derivation in Section V is also interesting. However, the load-bearing premise is a nonstandard definition of O(2) symmetry based on invariance of real-valued angular increments of an unwrapped coordinate, whereas the physical state is the velocity vector on S^1. Under the standard equivariant definition, the update is O(2)-equivariant and the adaptive-phase collapse is a branch-cut artifact of the principal-value arctangent. The numerical evidence is also limited to a single time series at one parameter point. The paper therefore does not establish the claimed absence of O(2) symmetry or the vanishing of the Vicsek phase transition.

major comments (3)

[Sec. III, Eqs. (3.2)-(3.7)] The definition of O(2) symmetry as exact invariance of the real-valued increment Δθ_i(t) is not the physically appropriate one. The state of particle i is the velocity vector v_i=v_abs(cos θ_i, sin θ_i) (Eq. (2.1b)), so θ_i and θ_i+2π represent the same state. The offending term 2πn(φ) in Eq. (3.7) is the identity on S^1. On the circle the update is equivariant: arg(⟨e^{i(θ_j+φ)}⟩)=arg(⟨e^{iθ_j}⟩)+φ (mod 2π), and the noise distribution is rotationally invariant. Thus the 'absence of O(2) symmetry' is a property of the chosen unwrapped coordinate, not of the model. The central conclusion of the paper rests on this premise.
[Sec. IV, Fig. 1] The numerical demonstration is a direct branch-cut artifact. The adaptive phase φ(t)=π-⟨θ_i(t)⟩ (Eq. (4.1)) places the principal-value branch cut at the instantaneous mean direction, forcing the wrapped mean angle to jump and the order parameter to collapse regardless of the intrinsic dynamics. The paper itself acknowledges this: 'because a branch cut exists near θ=π.' Moreover, the simulation uses a single parameter set (N=1600, L=8.0, η=0.75, r_V=0.1) and shows no phase diagram; a single trajectory cannot support the statement that 'the phase transition reported in Ref. [1] vanishes completely.' At minimum one would need to vary η and r_V and demonstrate the absence of an ordered phase throughout the parameter plane.
[Sec. VI] The remark about periodic boundary conditions does not address the relevant symmetry. A global phase shift θ_i→θ_i+φ acts on the velocity directions and does not require rotating spatial positions, so the claimed incompatibility with periodic boundary conditions is irrelevant to the O(2)-equivariance of the update rule. If the authors intend to challenge the standard interpretation of the Vicsek transition, they must engage with the standard formulation in which the order parameter is a vector on S^1, rather than redefining symmetry through unwrapped angle increments.

minor comments (4)

[Abstract and Sec. IV] The phrase 'the phase transition vanishes completely' is an overclaim relative to the evidence: Fig. 1 shows the time evolution of the order parameter at one parameter point (N=1600, L=8, η=0.75, r_V=0.1), not a phase diagram or a systematic parameter scan.
[Fig. 1] The caption uses φ(t)=0, −⟨θ_i(t)⟩, π−⟨θ_i(t)⟩, while the text refers to θ̄=π; please unify the notation. Also, the inset time range t∈[0,200] is not clearly labeled on the inset axes.
[Eq. (2.8)] Using the Frobenius norm ∥·∥_F for a vector is unconventional; the standard Euclidean norm is sufficient and clearer.
[Sec. V] The continuous-time limit of the arithmetic-mean model is interesting but orthogonal to the main claim about O(2) symmetry. Consider moving it to an appendix or framing it explicitly as an independent contribution.

Circularity Check

2 steps flagged · score 8.0 of 10

Absence of O(2) symmetry is true by the paper's own definition; adaptive-phase disorder is a branch-cut gauge choice.

self definitional [Section III, Eq. (3.2) and Eqs. (3.3)-(3.7)]
"∆θi(t) := θi(t + ∆t) − θi(t). (3.2) ... Here, we say that the system of interest is O(2) symmetric if Eq. (3.2) is invariant under the translational transformation in Eq. (3.1)."

The physical state is v_i(t)=v_abs(cos θ_i(t), sin θ_i(t))^T (Eq. 2.1b), so θ_i and θ_i+2π describe the same velocity. Under a global shift, Eq. (3.6)-(3.7) produces the extra term 2πn(φ), but on the circle this term is the identity: arg(⟨e^{i(θ_j+φ)}⟩)=arg(⟨e^{iθ_j}⟩)+φ (mod 2π). By defining O(2) symmetry as exact invariance of the real-valued lift Δθ_i, the paper makes the absence of symmetry true by construction: the principal-value branch cut introduces a 2π jump that is an artifact of the chosen coordinate, not a dynamical symmetry breaking. The conclusion follows from the definition, not from the model's physical content.
other [Section IV, Eq. (4.1), Fig. 1 (upper)]
"ϕ(t) := ¯ϕ − ⟨θi(t)⟩. (4.1) ... Figure 1(upper) clearly demonstrates that the macroscopic behavior of the arctan Vicsek model, Eq. (2.1), heavily depends on the choice of the phase factor, Eq. (4.1). More specifically, the agents do not swarm even if r_V is large and η is small when ¯θ=π is chosen ... because a branch cut exists near θ=π."

The adaptive gauge φ(t)=π−⟨θ_i(t)⟩ sends the instantaneous mean angle to π, exactly where the principal-value arctangent branch cut lies. The wrapped mean used in the update then jumps to −π, so the computed order parameter collapses regardless of the underlying orientational order. The 'vanishing phase transition' is therefore not an independent numerical discovery: the disorder is forced by the choice of gauge. The gauge choice is the input, and the reported disorder is the direct output of that choice.

full rationale

The paper's algebraic steps are internally consistent, but the central claim is made true by the paper's own definition rather than derived from the model's physical state space. In Section III, O(2) symmetry is defined as invariance of the real-valued increment Δθ_i under θ→θ+φ (Eq. 3.2). Because the physical velocity is v_i=v_abs(cos θ_i, sin θ_i) (Eq. 2.1b), θ and θ+2π label the same state; consequently the +2πn(φ) term in Eq. (3.7) is the identity on S^1. The standard equivariance check on the circle shows the original Vicsek update commutes with global rotations, so the claimed absence of O(2) symmetry is an artifact of requiring exact equality of real-valued angle representatives. The numerical section then chooses φ(t)=π−⟨θ_i(t)⟩ (Eq. 4.1), placing the branch cut on the instantaneous mean angle; the collapse of the order parameter in Fig. 1(upper) is a direct consequence of that gauge choice, not an independent prediction. Thus both the symmetry statement and the vanishing-transition demonstration reduce, by construction, to the authors' chosen definition and gauge. No load-bearing self-citation is involved; Ref. [16] is used only for contrast. Score 8 reflects that the result is forced by definition and by the adaptive gauge choice.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No fitted constants appear; the central dependence is on an ad hoc gauge parameter and on the authors' nonstandard definition of symmetry. The 'vanishing transition' is not a free-standing result but a consequence of the chosen adaptive phase and branch-cut placement.

free parameters (1)

Adaptive phase target \bar φ = π (and 0)
The numerical claim of vanishing order is produced by choosing \bar φ=π in φ(t)=\bar φ−⟨θ_i(t)⟩, which places the branch cut on the mean angle. This choice is made by hand and is not part of the original model (Eq. 4.1, Fig. 1).

assumptions (4)

ad hoc to paper O(2) symmetry is judged by invariance of real-valued angular increments Δθ_i under global phase shifts.
Introduced in Sec. III after Eq. (3.2); not the standard definition of rotational symmetry, which applies to unit vectors/velocities.
ad hoc to paper Unwrapped angle values and their 2π jumps are dynamically meaningful.
Update (2.1c)-(2.4) uses only cos/sin of θ_j, so θ is defined modulo 2π; the 2πn term in Eq. (3.7) cannot affect velocities or the order parameter.
ad hoc to paper One time series at (N=1600, L=8, η=0.75, r_V=0.1) suffices to show the phase transition vanishes.
Sec. IV and Fig. 1 show a single run at one parameter set; no order parameter vs noise or interaction radius is given.
domain assumption Principal-value arctangent in (-π,π] is the correct implementation of the original model.
The paper assumes Eq. (2.2)'s arctan is principal-valued as in the original Vicsek paper; reasonable, but the conclusion depends on this coordinate convention.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Absence of $\mathrm{O} (2)$ symmetry in the Vicsek model." pith.science (2026). https://pith.science/paper/EAB65KYP

@misc{pith2026260400930,
  author       = {Pith},
  title        = {Pith review of: Absence of $\mathrmO (2)$ symmetry in the Vicsek model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EAB65KYP}},
  note         = {Machine review of arXiv:2604.00930}
}

abstract

The phase transition in the Vicsek model is widely believed to be associated with spontaneous symmetry breaking of the two-dimensional rotational symmetry $\mathrm{O} (2)$. In this paper, we revisit the original Vicsek model introduced by Vicsek \textit{et al.} [Phys.~Rev.~Lett.~75,~1226~(1995)] and demonstrate that the original angle-based update rule is not invariant under global phase shifts at the level of angular increments when it is implemented using the principal-value arctangent, as in the original definition. As a consequence, we numerically demonstrate that the phase transition reported in the original paper vanishes when the global phase is adaptively chosen.

Figures

Figures reproduced from arXiv: 2604.00930 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 5 canonical work pages

[15]

Baglietto and E

G. Baglietto and E. V. Albano, Phys. Rev. E80, 050103 (2009), URLhttps://link.aps.org/doi/10. 1103/PhysRevE.80.050103

2009
[1]

Vicsek, A

T. Vicsek, A. Czir´ ok, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett.75, 1226 (1995), URL https://link.aps.org/doi/10.1103/PhysRevLett.75. 1226

doi:10.1103/physrevlett.75 1995
[2]

W. J. Rappel, A. Nicol, A. Sarkissian, H. Levine, and W. F. Loomis, Phys. Rev. Lett.83, 1247 (1999), URL https://doi.org/10.1103/PhysRevLett.83.1247

work page doi:10.1103/physrevlett.83.1247 1999
[3]

X. L. Wu and A. Libchaber, Phys. Rev. Lett.84, 3017 (2000), URLhttps://doi.org/10.1103/PhysRevLett. 84.3017

doi:10.1103/physrevlett 2000
[4]

Bonabeau, L

E. Bonabeau, L. Dagorn, and P. Fr´ eon, J. Phys. A31, L731 (1998), URLhttps://doi.org/10.1088/ 0305-4470/31/19/001

1998
[5]

Deneubourg, S

J.-L. Deneubourg, S. Aron, S. Goss, and J. M. Pasteels, Journal of Insect Behavior3, 159 (1990), URLhttps: //doi.org/10.1007/BF01049102

work page doi:10.1007/bf01049102 1990
[6]

Theraulaz, E

G. Theraulaz, E. Bonabeau, S. C. Nicolis, R. V. Sol´ e, V. Fourcassi´ e, S. Blanco, R. Fournier, J. L. Joly, P. Fer- nandez, A. Grimal, et al., Proc. Natl. Acad. Sci. U.S.A. 99, 9645 (2002), URLhttps://doi.org/10.1073/pnas. 142292699

doi:10.1073/pnas 2002
[7]

Becco, N

C. Becco, N. Vandewalle, J. Delcourt, and P. Poncin, Physica A367, 487 (2006), URLhttps://doi.org/10. 1016/j.physa.2005.11.041

2006

Show all 17 references

[8]

Chat´ e, F

H. Chat´ e, F. Ginelli, G. Gr´ egoire, and F. Raynaud, Phys. Rev. E77, 046113 (2008), URLhttps://link.aps.org/ doi/10.1103/PhysRevE.77.046113

2008 doi
[9]

Ramaswamy, Annual Review of Condensed Matter Physics1, 323 (2010), URLhttps://doi.org/10.1146/ annurev-conmatphys-070909-104101

S. Ramaswamy, Annual Review of Condensed Matter Physics1, 323 (2010), URLhttps://doi.org/10.1146/ annurev-conmatphys-070909-104101

2010
[10]

M. J. Bowick, N. Fakhri, M. C. Marchetti, and S. Ra- maswamy, Phys. Rev. X12, 010501 (2022), URLhttps: //link.aps.org/doi/10.1103/PhysRevX.12.010501

2022 doi
[11]

´Etienne Fodor and M. C. Marchetti, Physica A: Statis- tical Mechanics and its Applications504, 106 (2018), ISSN 0378-4371, lecture Notes of the 14th International Summer School on Fundamental Problems in Statis- tical Physics, URLhttps://www.sciencedirect.com/ science/article...

2018
[12]

Te Vrugt and R

M. Te Vrugt and R. Wittkowski, The European Physical Journal E48, 12 (2025)

2025
[13]

Czir´ ok, E

A. Czir´ ok, E. Ben-Jacob, I. Cohen, and T. Vicsek, Phys. 5 Rev. E54, 1791 (1996), URLhttps://doi.org/10. 1103/PhysRevE.54.1791

1996
[14]

Feder, Physics Today60, 28 (2007), URLhttps:// doi.org/10.1063/1.2807298

T. Feder, Physics Today60, 28 (2007), URLhttps:// doi.org/10.1063/1.2807298

2007 doi
[16]

Miyahara, H

H. Miyahara, H. Yoneki, T. Mizohata, and V. Roy- chowdhury, Journal of the Physical Society of Japan94, 084002 (2025), https://doi.org/10.7566/JPSJ.94.084002, URLhttps://doi.org/10.7566/JPSJ.94.084002

2025 doi
[17]

Fruchart, R

M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Nature592, 363 (2021)

2021

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [15]

Baglietto and E

G. Baglietto and E. V. Albano, Phys. Rev. E80, 050103 (2009), URLhttps://link.aps.org/doi/10. 1103/PhysRevE.80.050103

2009

[2] [1]

Vicsek, A

T. Vicsek, A. Czir´ ok, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett.75, 1226 (1995), URL https://link.aps.org/doi/10.1103/PhysRevLett.75. 1226

doi:10.1103/physrevlett.75 1995

[3] [2]

W. J. Rappel, A. Nicol, A. Sarkissian, H. Levine, and W. F. Loomis, Phys. Rev. Lett.83, 1247 (1999), URL https://doi.org/10.1103/PhysRevLett.83.1247

work page doi:10.1103/physrevlett.83.1247 1999

[4] [3]

X. L. Wu and A. Libchaber, Phys. Rev. Lett.84, 3017 (2000), URLhttps://doi.org/10.1103/PhysRevLett. 84.3017

doi:10.1103/physrevlett 2000

[5] [4]

Bonabeau, L

E. Bonabeau, L. Dagorn, and P. Fr´ eon, J. Phys. A31, L731 (1998), URLhttps://doi.org/10.1088/ 0305-4470/31/19/001

1998

[6] [5]

Deneubourg, S

J.-L. Deneubourg, S. Aron, S. Goss, and J. M. Pasteels, Journal of Insect Behavior3, 159 (1990), URLhttps: //doi.org/10.1007/BF01049102

work page doi:10.1007/bf01049102 1990

[7] [6]

Theraulaz, E

G. Theraulaz, E. Bonabeau, S. C. Nicolis, R. V. Sol´ e, V. Fourcassi´ e, S. Blanco, R. Fournier, J. L. Joly, P. Fer- nandez, A. Grimal, et al., Proc. Natl. Acad. Sci. U.S.A. 99, 9645 (2002), URLhttps://doi.org/10.1073/pnas. 142292699

doi:10.1073/pnas 2002

[8] [7]

Becco, N

C. Becco, N. Vandewalle, J. Delcourt, and P. Poncin, Physica A367, 487 (2006), URLhttps://doi.org/10. 1016/j.physa.2005.11.041

2006