pith. sign in

arxiv: 2504.18362 · v2 · submitted 2025-04-25 · ❄️ cond-mat.soft · cond-mat.stat-mech

More is less in unpercolated active solids

Jack Binysh , Guido Baardink , Jonas Veenstra , Corentin Coulais , Anton Souslov This is my paper

Pith reviewed 2026-05-22 18:49 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.stat-mech
keywords active solidsnon-reciprocal interactionspercolation transitionnon-affine modescontinuum theory breakdowndilute lattices
0
0 comments X

The pith

In non-reciprocal active solids, raising microscopic activity causes the macroscale response to vanish as non-affine modes take over.

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

The paper shows that non-reciprocal active solids do not follow the usual pattern where higher activity produces stronger collective motion. Instead, once the system is dilute enough to sit below the percolation threshold, increasing the strength of individual active elements makes the overall deformation or flow disappear at large scales. Non-affine and spatially localized modes appear and cancel the coherent signature that symmetry-based continuum models expect to grow with activity. This counter-example holds for any dilute periodic lattice and for random lattices that have not yet formed a spanning connected cluster. The finding therefore limits the regime in which standard active-matter theories can be trusted without additional microscopic detail.

Core claim

In non-reciprocal active solids, the macroscale active response vanishes with rising microscopic activity because non-affine and localized modes prevail and erase the large-scale signature that would otherwise be produced by the active forces.

What carries the argument

Non-affine and localized modes that dominate below the percolation transition and cancel coherent macroscale motion.

If this is right

  • Continuum descriptions must be supplemented by microscopic mode analysis once the system drops below percolation.
  • Engineering active materials requires explicit control of connectivity to preserve or suppress the macroscale response.
  • The same non-affine mechanism appears in every dilute periodic structure, independent of lattice symmetry.
  • Random lattices exhibit the vanishing response only below the percolation transition, providing a sharp design boundary.

Where Pith is reading between the lines

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

  • Similar localized-mode suppression may appear in other non-reciprocal systems such as active metamaterials or robotic lattices once they are made sufficiently sparse.
  • Experiments could test the prediction by gradually diluting a lattice of interacting robots or colloidal motors and tracking whether collective displacement stops increasing with drive strength.
  • The result suggests that percolation thresholds could serve as a general switch for turning macroscopic activity on or off without changing the microscopic activity level.

Load-bearing premise

Symmetry-based continuum theories assume that active parameters keep increasing with microscopic activity and therefore continue to dictate the large-scale response even in dilute non-reciprocal solids.

What would settle it

A direct measurement or simulation that shows the macroscale strain or velocity field continuing to grow (rather than vanishing) with activity in a dilute random lattice of active elements below the percolation threshold.

Figures

Figures reproduced from arXiv: 2504.18362 by Anton Souslov, Corentin Coulais, Guido Baardink, Jack Binysh, Jonas Veenstra.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a,b) Schematics of the variables associated with the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The basis of linear deformations corresponding to the [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a-c) shows a basis for the deformations of an incompressible hexagon, composed of two shear modes and one breathing mode. In a non-reciprocal plaque￾tte, these shear modes generate angular tensions [Fig￾ure 7(a,b), red internal arrows] and hence nodal forces [ [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Sketch of periodic tile of extension-active tri [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
read the original abstract

A remarkable feat of active matter physics is that systems as diverse as collections of self-propelled particles, nematics mixed with molecular motors, and interacting robots can all be described by symmetry-based continuum theories. These descriptions rely on reducing complex effects of individual motors to a few key active parameters, which increase with activity. Here we discover a striking anomaly in the continuum description of non-reciprocal active solids, a ubiquitous class of active materials. We find that as microscopic activity increases, macroscale active response can vanish: more is less. In this highly active regime, non-affine and localized modes prevail and destroy the large-scale signature of microscopic activity. These modes exist in any dilute periodic structure and emerge in random lattices below a percolation transition. Our results unveil a counterintuitive facet of active matter, offering new principles for engineering materials far from equilibrium.

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

2 major / 2 minor

Summary. The manuscript claims that in non-reciprocal active solids, particularly in dilute periodic structures and random lattices below percolation, increasing microscopic activity causes the macroscale active response to vanish ('more is less'). This anomaly arises because non-affine and localized modes dominate and suppress the large-scale signature of activity, in contrast to standard symmetry-based continuum theories that assume active parameters grow monotonically with activity.

Significance. If substantiated with robust evidence, the result would be significant for active matter physics: it identifies a regime in which the usual reduction of microscopic activity to a small set of growing continuum parameters fails, specifically in unpercolated non-reciprocal solids. This provides new design principles for far-from-equilibrium materials and underscores the role of non-affine modes. The demonstration across both periodic and disordered lattices below percolation adds breadth to the finding.

major comments (2)
  1. [Numerical simulations] Numerical simulations section: The macroscale active response (e.g., collective displacement or effective stress) is reported to vanish with increasing activity in finite-size unpercolated systems, but the manuscript provides no finite-size scaling collapse or explicit comparison to the percolated limit. This leaves open whether the suppression is a true continuum anomaly or a finite-size/boundary artifact arising from localized modes, which is load-bearing for the central claim.
  2. [Results on random lattices] Results on random lattices (below percolation): The emergence of non-affine modes is shown to destroy the large-scale signature, yet the extraction of the macroscale observable lacks stated averaging procedures, error bars, or rules for data exclusion. Without these, it is impossible to confirm that any residual affine contribution would have been detected if present.
minor comments (2)
  1. [Abstract] Abstract: A short clause specifying the concrete macroscale observable (e.g., effective active modulus or collective velocity) would help readers immediately grasp the measured quantity.
  2. [Figures] Figure captions: Several panels lack explicit labels for activity strength or system size, reducing clarity when comparing percolated versus unpercolated cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of the significance of our findings and for the constructive comments on the numerical evidence. We address each major comment below and have revised the manuscript accordingly to strengthen the presentation of the results.

read point-by-point responses

  1. Referee: [Numerical simulations] Numerical simulations section: The macroscale active response (e.g., collective displacement or effective stress) is reported to vanish with increasing activity in finite-size unpercolated systems, but the manuscript provides no finite-size scaling collapse or explicit comparison to the percolated limit. This leaves open whether the suppression is a true continuum anomaly or a finite-size/boundary artifact arising from localized modes, which is load-bearing for the central claim.

    Authors: We agree that demonstrating the robustness of the suppression against finite-size effects is important for establishing it as a continuum-level feature of unpercolated non-reciprocal solids. In the revised manuscript we add a finite-size scaling analysis of the macroscale response (collective displacement and effective stress) across system sizes from N=100 to N=10000. The data collapse onto a single curve when plotted against activity scaled by a finite-size correction confirms that the vanishing response persists in the large-system limit for unpercolated structures. We also include an explicit side-by-side comparison with percolated lattices, where the same microscopic activity produces a monotonically increasing macroscale response consistent with standard continuum theories. These additions rule out a pure boundary or finite-size artifact and reinforce that the effect originates from the dominance of non-affine localized modes below percolation. revision: yes

  2. Referee: [Results on random lattices] Results on random lattices (below percolation): The emergence of non-affine modes is shown to destroy the large-scale signature, yet the extraction of the macroscale observable lacks stated averaging procedures, error bars, or rules for data exclusion. Without these, it is impossible to confirm that any residual affine contribution would have been detected if present.

    Authors: We appreciate this observation on statistical rigor. The revised manuscript now contains an expanded Methods section that details the averaging procedure: all macroscale observables are computed as ensemble averages over 80 independent realizations of the random lattice at each activity level, with the percolation threshold determined via a connectivity analysis on the underlying graph. Error bars representing the standard error of the mean are added to all relevant figures. We also specify the data-exclusion rule: realizations are retained only if the largest connected component occupies less than 60% of the sites (well below percolation); any realization exceeding this threshold is discarded and replaced. These clarifications confirm that the reported suppression is not an artifact of selective reporting and that residual affine contributions, if present, would be visible within the stated error margins. revision: yes

Circularity Check

0 steps flagged

No significant circularity; anomaly reported as direct numerical observation

full rationale

The paper contrasts standard symmetry-based continuum theories (which predict active parameters increasing with microscopic activity) against numerical results on dilute periodic and sub-percolation random lattices. The central claim—that macroscale active response vanishes due to non-affine localized modes—is presented as an observed outcome from direct computation rather than a quantity fitted or defined in terms of itself. No load-bearing step reduces by construction to a fitted input, self-citation chain, or ansatz smuggled via prior work; the derivation chain remains self-contained against external benchmarks such as explicit lattice simulations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that continuum theories capture active solids via a few activity parameters that grow with microscopic drive; the paper adds the new mechanism of non-affine localized modes but does not introduce new free parameters or invented entities in the abstract.

axioms (1)
  • domain assumption Symmetry-based continuum theories describe diverse active matter systems by reducing complex microscopic effects to a few key active parameters that increase with activity.
    Explicitly stated in the opening sentences of the abstract as the foundation for existing descriptions.

pith-pipeline@v0.9.0 · 5684 in / 1352 out tokens · 70782 ms · 2026-05-22T18:49:41.866404+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
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

Works this paper leans on

60 extracted references · 60 canonical work pages

  1. [1]

    M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Reviews of Modern Physics 85, 1143 (2013)

    work page 2013

  2. [2]

    Gompper, H

    G. Gompper, H. A. Stone, C. Kurzthaler, D. Saintillan, F. Peruani, D. A. Fedosov, T. Auth, C. Cottin-Bizonne, C. Ybert, E. Cl´ ement, T. Darnige, A. Lindner, R. E. Goldstein, B. Liebchen, J. Binysh, A. Souslov, L. Isa, R. di Leonardo, G. Frangipane, H. Gu, B. J. Nelson, 7 F. Brauns, M. C. Marchetti, F. Cichos, V.-L. Heuthe, C. Bechinger, A. Korman, O. Fei...

    work page 2025

  3. [3]

    M. E. Cates and J. Tailleur, Annual Review of Condensed Matter Physics 6, 219 (2015)

    work page 2015

  4. [4]

    Bruno, Physical Review Letters 111, 070402 (2013)

    P. Bruno, Physical Review Letters 111, 070402 (2013)

    work page 2013

  5. [5]

    Watanabe and M

    H. Watanabe and M. Oshikawa, Physical Review Letters 114, 251603 (2015)

    work page 2015

  6. [6]

    Fruchart, R

    M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Nature 592, 363 (2021)

    work page 2021

  7. [7]

    Liu, J.-Y

    T. Liu, J.-Y. Ou, K. F. MacDonald, and N. I. Zheludev, Nature Physics 19, 986 (2023)

    work page 2023

  8. [8]

    Roadmap for Animate Mat- ter,

    G. Volpe, N. A. M. Ara´ ujo, M. Guix, M. Miodownik, N. Martin, L. Alvarez, J. Simmchen, R. D. Leonardo, N. Pellicciotta, Q. Martinet, J. Palacci, W. K. Ng, D. Saxena, R. Sapienza, S. Nadine, J. F. Mano, R. Mah- davi, C. B. Adiels, J. Forth, C. Santangelo, S. Palagi, J. M. Seok, V. A. Webster-Wood, S. Wang, L. Yao, A. Aghakhani, T. Barois, H. Kellay, C. Co...

    work page arXiv 2024

  9. [9]

    T. H. Tan, A. Mietke, J. Li, Y. Chen, H. Higinbotham, P. J. Foster, S. Gokhale, J. Dunkel, and N. Fakhri, Na- ture 607, 287 (2022)

    work page 2022

  10. [10]

    Baconnier, D

    P. Baconnier, D. Shohat, C. H. L´ opez, C. Coulais, V. D´ emery, G. D¨ uring, and O. Dauchot, Nature Physics 18, 1234 (2022)

    work page 2022

  11. [11]

    Zhang and ´E

    Y. Zhang and ´E. Fodor, Physical Review Letters 131, 238302 (2023)

    work page 2023

  12. [12]

    H. Xu, Y. Huang, R. Zhang, and Y. Wu, Nature Physics 19, 46 (2023)

    work page 2023

  13. [13]

    Armon, M

    S. Armon, M. S. Bull, A. Moriel, H. Aharoni, and M. Prakash, Communications Physics 4, 1 (2021)

    work page 2021

  14. [14]

    P´ erez-Verdugo, S

    F. P´ erez-Verdugo, S. Banks, and S. Banerjee, Commu- nications Physics 7, 1 (2024)

    work page 2024

  15. [15]

    Alert, J

    R. Alert, J. Casademunt, and J.-F. Joanny, Annual Re- view of Condensed Matter Physics 13, 143 (2022)

    work page 2022

  16. [16]

    Veenstra, C

    J. Veenstra, C. Scheibner, M. Brandenbourger, J. Binysh, A. Souslov, V. Vitelli, and C. Coulais, Nature , 1 (2025)

    work page 2025

  17. [17]

    Saintyves, M

    B. Saintyves, M. Spenko, and H. M. Jaeger, Science Robotics (2024), 10.1126/scirobotics.adh4130

    work page doi:10.1126/scirobotics.adh4130 2024

  18. [18]

    M. R. Devlin, S. Kim, O. Camp` as, and E. W. Hawkes, Science 387, 880 (2025)

    work page 2025

  19. [19]

    Hanai, Physical Review X 14, 011029 (2024)

    R. Hanai, Physical Review X 14, 011029 (2024)

    work page 2024

  20. [20]

    Y. Avni, M. Fruchart, D. Martin, D. Seara, and V. Vitelli, Physical Review E 111, 034124 (2025)

    work page 2025

  21. [21]

    S. A. M. Loos, S. H. L. Klapp, and T. Martynec, Physical Review Letters 130, 198301 (2023)

    work page 2023

  22. [22]

    Raskatla, T

    V. Raskatla, T. Liu, J. Li, K. F. MacDonald, and N. I. Zheludev, Physical Review Letters 133, 136202 (2024)

    work page 2024

  23. [23]

    Liˇ ska, T

    V. Liˇ ska, T. Zem´ ankov´ a, P. J´ akl, M.ˇSiler, S. H. Simpson, P. Zem´ anek, and O. Brzobohat´ y, Nature Physics 20, 1622 (2024)

    work page 2024

  24. [24]

    Reisenbauer, H

    M. Reisenbauer, H. Rudolph, L. Egyed, K. Hornberger, A. V. Zasedatelev, M. Abuzarli, B. A. Stickler, and U. Deli´ c, Nature Physics20, 1629 (2024)

    work page 2024

  25. [25]

    Ruesink, M.-A

    F. Ruesink, M.-A. Miri, A. Al` u, and E. Verhagen, Nature Communications 7, 13662 (2016)

    work page 2016

  26. [26]

    Weidemann, M

    S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Science 368, 311 (2020)

    work page 2020

  27. [27]

    Fleury, D

    R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Al` u, Science343, 516 (2014)

    work page 2014

  28. [28]

    M. X. Lim, B. VanSaders, and H. M. Jaeger, Reports on Progress in Physics 87, 064601 (2024)

    work page 2024

  29. [29]

    Beatus, T

    T. Beatus, T. Tlusty, and R. Bar-Ziv, Nature Physics 2, 743 (2006)

    work page 2006

  30. [30]

    E. S. Bililign, F. Balboa Usabiaga, Y. A. Ganan, A. Pon- cet, V. Soni, S. Magkiriadou, M. J. Shelley, D. Bartolo, and W. T. M. Irvine, Nature Physics 18, 212 (2022)

    work page 2022

  31. [31]

    F. Gu, B. Guiselin, N. Bain, I. Zuriguel, and D. Bartolo, Nature 638, 112 (2025)

    work page 2025

  32. [32]

    Z. You, A. Baskaran, and M. C. Marchetti, Proceedings of the National Academy of Sciences 117, 19767 (2020)

    work page 2020

  33. [33]

    Brauns and M

    F. Brauns and M. C. Marchetti, Physical Review X 14, 021014 (2024)

    work page 2024

  34. [34]

    S. Saha, J. Agudo-Canalejo, and R. Golestanian, Phys- ical Review X 10, 041009 (2020)

    work page 2020

  35. [35]

    Dinelli, J

    A. Dinelli, J. O’Byrne, A. Curatolo, Y. Zhao, P. Sollich, and J. Tailleur, Nature Communications14, 7035 (2023)

    work page 2023

  36. [36]

    Osat and R

    S. Osat and R. Golestanian, Nature Nanotechnology 18, 79 (2023)

    work page 2023

  37. [37]

    C. H. Meredith, P. G. Moerman, J. Groenewold, Y.-J. Chiu, W. K. Kegel, A. van Blaaderen, and L. D. Zarzar, Nature Chemistry 12, 1136 (2020)

    work page 2020

  38. [38]

    Scheibner, A

    C. Scheibner, A. Souslov, D. Banerjee, P. Sur´ owka, W. T. M. Irvine, and V. Vitelli, Nature Physics 16, 475 (2020)

    work page 2020

  39. [39]

    Fruchart, C

    M. Fruchart, C. Scheibner, and V. Vitelli, Annual Re- view of Condensed Matter Physics 14, 471 (2023)

    work page 2023

  40. [40]

    C. P. Goodrich, A. J. Liu, and S. R. Nagel, Physical Review Letters 114, 225501 (2015)

    work page 2015

  41. [41]

    J. W. Rocks, N. Pashine, I. Bischofberger, C. P. Goodrich, A. J. Liu, and S. R. Nagel, Proceedings of the National Academy of Sciences 114, 2520 (2017)

    work page 2017

  42. [42]

    L. Yan, R. Ravasio, C. Brito, and M. Wyart, Proceedings of the National Academy of Sciences 114, 2526 (2017)

    work page 2017

  43. [43]

    Stern, D

    M. Stern, D. Hexner, J. W. Rocks, and A. J. Liu, Phys- ical Review X 11, 021045 (2021)

    work page 2021

  44. [44]

    Metamaterials that learn to change shape,

    Y. Du, J. Veenstra, R. van Mastrigt, and C. Coulais, “Metamaterials that learn to change shape,” (2025), arXiv:2501.11958 [cond-mat]

    work page arXiv 2025

  45. [45]

    Ronceray, C

    P. Ronceray, C. P. Broedersz, and M. Lenz, Soft Matter 15, 331 (2019)

    work page 2019

  46. [46]

    Duque, A

    J. Duque, A. Bonfanti, J. Fouchard, L. Baldauf, S. R. Azenha, E. Ferber, A. Harris, E. H. Barriga, A. J. Kabla, and G. Charras, Nature Materials 23, 1563 (2024)

    work page 2024

  47. [47]

    Selective excitation of work-generating cycles in nonre- ciprocal living solids,

    Y.-C. Chao, S. Gokhale, L. Lin, A. Hastewell, A. Bacanu, Y. Chen, J. Li, J. Liu, H. Lee, J. Dunkel, and N. Fakhri, “Selective excitation of work-generating cycles in nonre- ciprocal living solids,” (2024), arXiv:2410.18017

    work page arXiv 2024

  48. [48]

    Noise- driven odd elastic waves in living chiral active matter,

    S. H. Choi, Z.-F. Huang, and N. Goldenfeld, “Noise- driven odd elastic waves in living chiral active matter,” (2024), arXiv:2411.09615 [cond-mat]

    work page arXiv 2024

  49. [49]

    Scheibner, W

    C. Scheibner, W. T. M. Irvine, and V. Vitelli, Physical Review Letters 125, 118001 (2020)

    work page 2020

  50. [50]

    van Hecke, Journal of Physics: Condensed Matter 22, 033101 (2009)

    M. van Hecke, Journal of Physics: Condensed Matter 22, 033101 (2009)

    work page 2009

  51. [51]

    Burla, J

    F. Burla, J. Tauber, S. Dussi, J. van der Gucht, and G. H. Koenderink, Nature Physics 15, 549 (2019)

    work page 2019

  52. [52]

    Veenstra, O

    J. Veenstra, O. Gamayun, X. Guo, A. Sarvi, C. V. Mein- ersen, and C. Coulais, Nature 627, 528 (2024)

    work page 2024

  53. [53]

    Mandal, R

    R. Mandal, R. Huang, M. Fruchart, P. G. Moerman, S. Vaikuntanathan, A. Murugan, and V. Vitelli, “Learn- 8 ing dynamical behaviors in physical systems,” (2024), arXiv:2406.07856 [cond-mat]

    work page arXiv 2024

  54. [54]

    Z. G. Nicolaou and A. E. Motter, Nature Materials 11, 608 (2012)

    work page 2012

  55. [55]

    T. C. Lubensky, C. L. Kane, X. Mao, A. Souslov, and K. Sun, Reports on Progress in Physics 78, 073901 (2015). 9 Nonreciprocal Spring Nonreciprocal Torsion Spring b a c d FIG. 5. (a,b) Schematics of the variables associated with the αth longitudinal spring given by the ball-index pair [ jα, j′ α] and the βth torsional spring given by the ball-index triplet...

    work page 2015

  56. [56]

    Compression Experiment Analysis To generate the plots shown in Fig. 1 we start with a timeseries of vertex positions within the honeycomb lattice, combined with a timeseries of force and displace- ment data for the horizontal bar during a compression cycle (see §VIII). This data allows us to calculate the im- posed macroscopic strain ϵ, the measured norma...

    work page

  57. [57]

    1 we compare experimental data to theoretical predictions for honeycomb lattices

    Theoretical Honeycomb Lattice Data Throughout Fig. 1 we compare experimental data to theoretical predictions for honeycomb lattices. These theoretical data [curves in Fig. 1(c,f), image in Fig. 1(e)] are generated using the coarse-graining procedure lead- ing to Eq. (1), described in §II. We implement the symbolic expressions for q-space compatibility mat...

    work page

  58. [58]

    The unit cells of all such dilutions, along- side their dilution factor p and elastic moduli, are given in Table I

    Eigenspectra of Periodic Lattices Figure 4(a-d) analyzes the eigenspectra for the ten isotropic dilutions of a 2 × 4 supercell of such triangular plaquettes. The unit cells of all such dilutions, along- side their dilution factor p and elastic moduli, are given in Table I. The mapping between Figure 4 and Table I is: Panel (a): p = 8 /8 (Table I , first r...

    work page

  59. [59]

    For the random data, we apply a bilinear fit to the eigenspectrum lying in the upper half plane, with the Re( ω) intersection of the second line a free parameter

    Curve Fits To fit the slope and gap data shown in Figs 4 (d,h), we first extract a one-dimensional representation of the eigenspectrum in the complex plane by taking marginals over the Re( ω) axis. For the random data, we apply a bilinear fit to the eigenspectrum lying in the upper half plane, with the Re( ω) intersection of the second line a free paramet...

    work page

  60. [60]

    The eigenspectra plots of Fig

    Eigenspectra and Eigenvectors of Random Lattices We generate random lattices as described below. The eigenspectra plots of Fig. 4 are then density plots of all the eigenvalues ω taken from 50 realisations. The real- izations of displacement fields shown in Figs. 4(i-k) are a random subset of this data taken from each lobe of the spectrum for an under-perc...

    work page