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arxiv: 2605.23625 · v1 · pith:3GKSB7T6new · submitted 2026-05-22 · 🪐 quant-ph · cond-mat.mes-hall· physics.optics

Atom-Photon Bound States in Fractal Photonic Lattices: Localization Length and Anomalous Diffusion

Florian B\"onsel , Flore K. Kunst , Federico Roccati This is my paper

Pith reviewed 2026-05-25 04:12 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallphysics.optics
keywords atom-photon bound statesfractal photonic latticeslocalization lengthanomalous diffusionwalk dimensionGreen's functionheat kernelSierpiński gasket
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The pith

The far-field localization length of atom-photon bound states in fractal lattices scales as the inverse detuning to the power of the fractal's walk dimension.

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

The paper establishes that atom-photon bound states in self-similar photonic lattices have a far-field localization length controlled by anomalous diffusion on the fractal. By linking the photonic Green's function to the heat kernel, the localization length follows ξ ∼ Δ^{-1/d_w} where Δ is the detuning from the lower spectral edge and d_w is the walk dimension. This relation holds without translational invariance or an effective-mass approximation near a band edge. Exact diagonalization on Sierpiński gaskets, pyramids, Vicsek graphs, and Sierpiński carpets confirms the scaling once the bath Hamiltonian is adjusted to be Laplacian-like. Near the emitter the amplitude shows an extra algebraic decay whose exponent matches resistance scaling on finitely ramified fractals but deviates on carpets.

Core claim

Expressing the photonic Green's function through the heat kernel shows that the far-field localization length of the bound state obeys ξ ∼ Δ^{-1/d_w}, with the detuning Δ from the lower spectral edge and the walk dimension d_w of the underlying fractal; the scaling is set by anomalous diffusion and is verified by exact diagonalization on several fractals after the Hamiltonian is rendered Laplacian-like.

What carries the argument

The heat-kernel representation of the photonic Green's function, which encodes the walk dimension d_w of anomalous diffusion on the fractal.

If this is right

  • Bound states exist and are localized according to transport exponents even in non-periodic self-similar geometries.
  • The far-field profile is set solely by the heat-kernel decay and is independent of lattice periodicity.
  • Near-field algebraic decay follows classical resistance or first-passage exponents on finitely ramified fractals.
  • Sierpiński carpets exhibit deviations from the simple resistance scaling in the near field.

Where Pith is reading between the lines

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

  • The same heat-kernel approach could be applied to other structured baths whose spectral density is governed by anomalous diffusion.
  • Choosing different fractals would allow tuning of the localization length through their distinct walk dimensions without changing the detuning.
  • The method may extend to time-dependent or driven versions of the emitter-bath problem on fractals.

Load-bearing premise

The bath Hamiltonian must be made Laplacian-like by adding on-site potentials that compensate for local differences in site connectivities.

What would settle it

Measuring a localization-length exponent that differs from -1/d_w on a fractal lattice whose Hamiltonian has been compensated to be Laplacian-like would falsify the scaling.

Figures

Figures reproduced from arXiv: 2605.23625 by Federico Roccati, Flore K. Kunst, Florian B\"onsel.

Figure 1
Figure 1. Figure 1: Atom-photon bound states. Quantum emitters coupled to a regular (a) and fractal (b) photonic lattice. (c) In the regular case, a quadratic band edge ω(k) − ωedge ∼ |k| 2 yields the familiar scaling ξ ∼ ∆−1/2 for the localization length ξ of the atom-photon bound state, with ∆ as the emitter detuning from the lower band edge. (d) In a fractal bath, translational invariance and ordinary bands are absent; ins… view at source ↗
Figure 2
Figure 2. Figure 2: illustrates representative generations of the nested finitely ramified and infinitely ramified lattices studied in this work. For the Sierpi´nski gaskets, we in￾troduce the parameter b. A value of b = 2 means that the generation g of the fractal is created by placing the g −1 generation at the corners of a triangle with an edge made from b = 2 sites, as in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fractal photonic lattices. Graphs of second generation (g = 2) fractals considered in this work, along with the ones displayed in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Far-field localization length. The rescaled localiza￾tion length ξ · ∆1/2 is shown as a function of detuning ∆. A horizontal line corresponds to the regular-lattice bench￾mark ξ ∼ ∆−1/2 , while systematic power-law drifts reveal the fractal exponent 1/dw. Data are extracted along the outer boundary of each fractal, where d(x, x0) coincides with the Euclidean arc length. The following generations are taken … view at source ↗
Figure 5
Figure 5. Figure 5: Near-field scaling of atom-photon bound states. The quantity δψ(r)/δψ(rmin) with δψ(r) = [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

We study atom-photon bound states seeded by two-level emitters coupled to self-similar photonic lattices. By expressing the photonic Green's function through the heat kernel, we show that the far-field localization length obeys $\xi \sim \Delta^{-1/d_w}$, with the detuning $\Delta$ from the lower spectral edge and the walk dimension $d_w$ of the underlying fractal. This scaling is controlled by anomalous diffusion and does not rely on translational invariance or a band-edge effective-mass approximation. Exact diagonalization on Sierpi\'nski gaskets, pyramids, Vicsek graphs, and Sierpi\'nski carpets confirms the far-field prediction once the bath Hamiltonian is rendered Laplacian-like by compensating the local inhomogeneity in the connectivities with on-site potentials. In the near field, the bound-state amplitude exhibits an additional algebraic variation. For nested finitely ramified fractals, the corresponding exponent agrees with the classical resistance/ first-passage scaling, whereas Sierpi\'nski carpets display clear deviations from this simple law. Our results extend structured-bath waveguide QED to self-similar non-periodic geometries and connect bound-state profiles to transport exponents of the underlying fractal lattice.

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 derives that the far-field localization length of atom-photon bound states in fractal photonic lattices scales as ξ ∼ Δ^{-1/d_w} (Δ detuning from lower spectral edge, d_w walk dimension) by expressing the photonic Green's function via the heat kernel. This scaling arises from anomalous diffusion and holds without translational invariance or effective-mass approximations. Exact diagonalization on Sierpiński gaskets, Vicsek graphs, pyramids, and carpets is stated to confirm the prediction after the bath Hamiltonian is modified by on-site potentials to compensate connectivity inhomogeneities and render it Laplacian-like. Near-field amplitude shows additional algebraic decay whose exponent matches resistance/first-passage scaling for finitely ramified fractals but deviates for carpets.

Significance. If the central scaling holds, the work extends waveguide QED to self-similar non-periodic geometries and directly connects bound-state profiles to transport exponents of the underlying fractal. The heat-kernel derivation supplies a parameter-free prediction grounded in the spectral properties of the bath; this is a clear strength relative to effective-mass treatments.

major comments (2)
  1. [Abstract / Numerical section] Abstract and numerical confirmation paragraph: the exact-diagonalization support is reported only after on-site compensation that renders the bath Hamiltonian Laplacian-like. The derivation assumes a Laplacian-like operator whose heat kernel yields the d_w-controlled asymptotics, yet no results or analysis are given for the unmodified graph Laplacian (D−A) whose on-site terms already encode degree variation; this leaves open whether the claimed scaling survives on standard fractal graphs and makes the numerical confirmation conditional rather than general.
  2. [Green's function / heat-kernel derivation] Derivation via heat kernel (Green's function section): while the long-time heat-kernel tail on fractals produces ξ ∼ Δ^{-1/d_w}, the manuscript does not explicitly verify that the compensation potentials preserve the walk dimension d_w and the lower-edge spectral density required for the asymptotic; a short check or reference to the modified operator's spectrum would strengthen the link between analytic prediction and numerics.
minor comments (2)
  1. [Abstract] The term 'nested finitely ramified fractals' is used without definition or citation; a one-sentence clarification or standard reference would improve readability.
  2. [Figures] Figure captions for the localization-length plots should state the fitting range in Δ and the number of disorder realizations or system sizes used, to allow direct assessment of the reported agreement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments help to better delineate the scope of our analytic and numerical results. We address each major comment in turn.

read point-by-point responses

  1. Referee: [Abstract / Numerical section] Abstract and numerical confirmation paragraph: the exact-diagonalization support is reported only after on-site compensation that renders the bath Hamiltonian Laplacian-like. The derivation assumes a Laplacian-like operator whose heat kernel yields the d_w-controlled asymptotics, yet no results or analysis are given for the unmodified graph Laplacian (D−A) whose on-site terms already encode degree variation; this leaves open whether the claimed scaling survives on standard fractal graphs and makes the numerical confirmation conditional rather than general.

    Authors: The analytic derivation is performed for a Laplacian-like operator, as explicitly stated in the manuscript, because this allows us to directly employ the heat-kernel representation and connect the localization length to the walk dimension d_w without additional complications from degree-dependent on-site shifts. The compensation with on-site potentials is introduced precisely to achieve this Laplacian-like form on the fractal graphs, isolating the effect of the self-similar structure. We do not claim the scaling for the unmodified (D−A) operator, where the on-site terms introduce inhomogeneities extrinsic to the fractal diffusion. The numerical confirmation is therefore for the model considered in the derivation, and the abstract already qualifies the results with 'once the bath Hamiltonian is rendered Laplacian-like'. We believe this makes the confirmation appropriate rather than conditional on an unstated assumption. revision: no

  2. Referee: [Green's function / heat-kernel derivation] Derivation via heat kernel (Green's function section): while the long-time heat-kernel tail on fractals produces ξ ∼ Δ^{-1/d_w}, the manuscript does not explicitly verify that the compensation potentials preserve the walk dimension d_w and the lower-edge spectral density required for the asymptotic; a short check or reference to the modified operator's spectrum would strengthen the link between analytic prediction and numerics.

    Authors: The compensation potentials are local and bounded, designed only to cancel the degree variations so that the resulting operator has the same connectivity structure as a standard graph Laplacian. The walk dimension d_w is an asymptotic property determined by the large-scale geometry and the spectral dimension of the fractal, which remain unchanged by local on-site modifications. Similarly, the lower-edge spectral density, which follows from the fractal dimension, is preserved. We will add a short paragraph or appendix in the revised manuscript providing a brief spectral analysis or reference confirming that d_w and the edge density are unaffected by the compensation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; scaling follows from standard heat-kernel asymptotics on fractals

full rationale

The central result ξ ∼ Δ^{-1/d_w} is obtained by writing the photonic Green's function in terms of the heat kernel of the bath and invoking the known long-time asymptotics controlled by the walk dimension d_w of the fractal. This is a direct application of established results on anomalous diffusion and does not reduce to a fitted parameter, a self-definition, or a load-bearing self-citation. Numerical confirmation on the listed fractals is performed after an explicit model modification (on-site compensation) that renders the Hamiltonian Laplacian-like; the paper states this assumption up front and does not claim the scaling holds without it. The derivation chain therefore remains self-contained against external mathematical benchmarks on heat kernels and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the heat-kernel representation of the Green's function (standard in the domain) and the assumption that on-site potentials can render the bath Hamiltonian Laplacian-like for the studied fractals.

axioms (2)
  • domain assumption The photonic Green's function can be expressed through the heat kernel of the lattice.
    Invoked to obtain the far-field localization length scaling without band-edge approximations.
  • domain assumption Compensating local inhomogeneity in connectivities with on-site potentials renders the bath Hamiltonian Laplacian-like.
    Stated as the condition under which exact diagonalization confirms the scaling on Sierpinski gaskets, pyramids, Vicsek graphs, and carpets.

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Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages · 2 internal anchors

  1. [1]

    Goban, C.-L

    A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, Phys. Rev. Lett. 115, 063601 (2015)

    work page 2015

  2. [2]

    Lodahl, S

    P. Lodahl, S. Mahmoodian, and S. Stobbe, Rev. Mod. Phys.87, 347 (2015)

    work page 2015

  3. [3]

    J. S. Douglas, H. Habibian, C.-L. Hung, H. J. Kimble, and D. E. Chang, Nat. Photon.9, 326 (2015)

    work page 2015

  4. [4]

    A. F. van Loo, A. Fedorov, K. Lalumi` ere, B. C. Sanders, A. Blais, and A. Wallraff, Science342, 1494 (2013)

    work page 2013

  5. [5]

    A. S. Sheremet, M. I. Petrov, I. V. Iorsh, A. V. Poshakin- skiy, and A. N. Poddubny, Reviews of Modern Physics 95, 015002 (2023)

    work page 2023

  6. [6]

    Ciccarello, P

    F. Ciccarello, P. Lodahl, and D. Schneble, Optics and Photonics News35, 34 (2024)

    work page 2024

  7. [7]

    John and J

    S. John and J. Wang, Physical review letters64, 2418 (1990)

    work page 1990

  8. [8]

    Liu and A

    Y. Liu and A. A. Houck, Nature Physics13, 48 (2017)

    work page 2017

  9. [9]

    Calaj´ o, F

    G. Calaj´ o, F. Ciccarello, D. E. Chang, and P. Rabl, Phys. Rev. A93, 033833 (2016)

    work page 2016

  10. [10]

    Gonz´ alez-Tudela and J

    A. Gonz´ alez-Tudela and J. I. Cirac, Physical Review Let- ters119, 143602 (2017)

    work page 2017

  11. [11]

    R. C. Verstraten, I. H. Knottnerus, Y. C. Tseng, A. Urech, T. S. d. E. Santo, V. Zampronio, F. Schreck, R. J. Spreeuw, and C. M. Smith, arXiv preprint arXiv:2509.03514 (2025), 10.48550/arXiv.2509.03514

    work page doi:10.48550/arxiv.2509.03514 2025

  12. [12]

    Leonforte, D

    L. Leonforte, D. Valenti, B. Spagnolo, A. Carollo, and F. Ciccarello, Nanophotonics10, 4251 (2021)

    work page 2021

  13. [13]

    Shi, Y.-H

    T. Shi, Y.-H. Wu, A. Gonz´ alez-Tudela, and J. I. Cirac, Physical Review X6, 021027 (2016)

    work page 2016

  14. [14]

    Mirhosseini, E

    M. Mirhosseini, E. Kim, V. S. Ferreira, M. Kalaee, A. Sipahigil, A. J. Keller, and O. Painter, Nature588, 599 (2020)

    work page 2020

  15. [15]

    N. M. Sundaresan, R. Sagastizabal, M. Geiger, J. Xu, Y. Chen, M. Khezri, C. J¨ unger, M. Block, and A. A. Houck, Phys. Rev. X9, 011021 (2019)

    work page 2019

  16. [16]

    Asenjo-Garcia, M

    A. Asenjo-Garcia, M. Moreno-Cardoner, S. V. Albrecht, H. J. Kimble, and D. E. Chang, Phys. Rev. Lett.118, 053601 (2017)

    work page 2017

  17. [17]

    Roccati, arXiv preprint arXiv:2505.02892 (2025), 10.48550/arXiv.2505.02892

    F. Roccati, arXiv preprint arXiv:2505.02892 (2025), 10.48550/arXiv.2505.02892

    work page doi:10.48550/arxiv.2505.02892 2025

  18. [18]

    L. M. Hansen, C. Henke, C. Hotter, O. A. Sandberg, T. W. Sandø, V. Angelopoulou, A. Tiranov, C. B. Møller, Z. Liu, L. Midolo,et al., arXiv preprint arXiv:2605.18525 (2026), 10.48550/arXiv.2605.18525

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.18525 2026

  19. [19]

    John and J

    S. John and J. Wang, Physical Review B43, 12772 (1991)

    work page 1991

  20. [20]

    Raman and S

    A. Raman and S. Fan, Physical review letters104, 087401 (2010)

    work page 2010

  21. [21]

    John and T

    S. John and T. Quang, Physical Review A50, 1764 (1994)

    work page 1994

  22. [22]

    Alexander and R

    S. Alexander and R. Orbach, Journal de Physique Lettres 43, 625 (1982)

    work page 1982

  23. [23]

    M. T. Barlow, inLectures on Probability Theory and Statistics: Ecole d’Et´ e de Probabilit´ es de Saint-Flour XXV—1995(Springer, 2006) pp. 1–121

    work page 1995

  24. [24]

    Telcs,The art of random walks(Springer, 2006)

    A. Telcs,The art of random walks(Springer, 2006)

    work page 2006

  25. [25]

    D. S. Quevedo, M. Conte, M. Dijkstra, and C. M. Smith, arXiv preprint arXiv:2512.20205 (2025), 10.48550/arXiv.2512.20205

    work page doi:10.48550/arxiv.2512.20205 2025

  26. [26]

    Mukherjee, A

    S. Mukherjee, A. Spracklen, M. Valiente, E. Andersson, P. ¨Ohberg, N. Goldman, and R. R. Thomson, Phys. Rev. Lett.121, 093902 (2018)

    work page 2018

  27. [27]

    Hanafi, P

    H. Hanafi, P. Menz, and C. Denz, inNonlinear Optics (Optica Publishing Group, 2021) pp. NM2A–6

    work page 2021

  28. [28]

    M. Li, C. Li, L. Yan, Q. Li, Q. Gong, and Y. Li, Light: Science & Applications12, 262 (2023)

    work page 2023

  29. [29]

    Z. Yang, E. Lustig, Y. Lumer, and M. Segev, Light: Science & Applications9, 128 (2020)

    work page 2020

  30. [30]

    Renet al., Nanophotonics12, 3829 (2023)

    Z. Renet al., Nanophotonics12, 3829 (2023)

    work page 2023

  31. [31]

    Xu, X.-W

    X.-Y. Xu, X.-W. Wang, D.-Y. Chen, C. M. Smith, and X.-M. Jin, Nature Photonics15, 703 (2021)

    work page 2021

  32. [32]

    Dar´ azs, A

    Z. Dar´ azs, A. Anishchenko, T. Kiss, A. Blumen, and O. M¨ ulken, Physical Review E90, 032113 (2014)

    work page 2014

  33. [33]

    Rojo-Franc` as, P

    A. Rojo-Franc` as, P. Pansari, U. Bhattacharya, B. Juli´ a- D´ ıaz, and T. Grass, Communications Physics7, 259 (2024)

    work page 2024

  34. [34]

    Z. V. Vardeny, A. Nahata, and A. Agrawal, Nature pho- tonics7, 177 (2013)

    work page 2013

  35. [35]

    S. V. Boriskina, Nature Photonics9, 422 (2015)

    work page 2015

  36. [36]

    Bello, G

    M. Bello, G. Platero, J. I. Cirac, and A. Gonz´ alez- Tudela, Science advances5, eaaw0297 (2019)

    work page 2019

  37. [37]

    Leonforte, X

    L. Leonforte, X. Sun, D. Valenti, B. Spagnolo, F. Illu- minati, A. Carollo, and F. Ciccarello, Quantum Science and Technology10, 015057 (2024)

    work page 2024

  38. [38]

    B¨ onsel, F

    F. B¨ onsel, F. K. Kunst, and F. Roccati, Quantum10, 2081 (2026)

    work page 2081

  39. [39]

    Akkermans and E

    E. Akkermans and E. Gurevich, Europhysics Letters103, 30009 (2013). 11

    work page 2013

  40. [40]

    E. N. Economou,Green’s functions in quantum physics (Springer, 2006)

    work page 2006

  41. [41]

    M. T. Barlow and E. A. Perkins, Probab. Theory Relat. Fields79, 543 (1988)

    work page 1988

  42. [42]

    M. T. Barlow, T. Coulhon, and T. Kumagai, Commu- nications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences 58, 1642 (2005)

    work page 2005

  43. [43]

    Kumagai,Random walks on disordered media and their scaling limits, Vol

    T. Kumagai,Random walks on disordered media and their scaling limits, Vol. 2101 (Springer, 2014)

    work page 2014

  44. [44]

    M. T. Barlow, R. F. Bass, T. Kumagai, and A. Teplyaev, Journal of the European Mathematical Society12, 655 (2010)

    work page 2010

  45. [45]

    J. E. Hutchinson, Indiana University Mathematics Jour- nal30, 713 (1981)

    work page 1981

  46. [46]

    K. J. Falconer,The geometry of fractal sets, 85 (Cam- bridge university press, 1985)

    work page 1985

  47. [47]

    Lindstrøm,Brownian motion on nested fractals (American Mathematical Society Providence, 1990)

    T. Lindstrøm,Brownian motion on nested fractals (American Mathematical Society Providence, 1990)

    work page 1990

  48. [48]

    Hilfer and A

    R. Hilfer and A. Blumen, Journal of Physics A: Mathe- matical and General17, L537 (1984)

    work page 1984

  49. [49]

    Argyrakis, S

    P. Argyrakis, S. Evangelou, and K. Magoutis, Zeitschrift f¨ ur Physik B Condensed Matter87, 257 (1992)

    work page 1992

  50. [50]

    Nakayama, K

    T. Nakayama, K. Yakubo, and R. L. Orbach, Reviews of modern physics66, 381 (1994)

    work page 1994

  51. [51]

    Rammal and G

    R. Rammal and G. Toulouse, Journal de Physique Lettres 44, 13 (1983)

    work page 1983

  52. [52]

    Gefen, A

    Y. Gefen, A. Aharony, and S. Alexander, Physical Re- view Letters50, 77 (1983)

    work page 1983

  53. [53]

    O’Shaughnessy and I

    B. O’Shaughnessy and I. Procaccia, Physical Review A 32, 3073 (1985)

    work page 1985

  54. [54]

    O’Shaughnessy and I

    B. O’Shaughnessy and I. Procaccia, Physical review let- ters54, 455 (1985)

    work page 1985

  55. [55]

    X. R. Wang, Physical Review B51, 9310 (1995)

    work page 1995

  56. [56]

    T. Li, X. Tang, S. Liu, and H. Hu, arXiv preprint arXiv:2605.17953 (2026), 10.48550/arXiv.2605.17953

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.17953 2026

  57. [57]

    Kumagai, Probab

    T. Kumagai, Probab. Theory Relat. Fields96, 205 (1993)

    work page 1993

  58. [58]

    B. M. Hambly and T. Kumagai, Proceedings of the Lon- don Mathematical Society78, 431 (1999)

    work page 1999

  59. [59]

    Havlin and D

    S. Havlin and D. Ben-Avraham, Advances in physics36, 695 (1987)

    work page 1987

  60. [60]

    Ben-Avraham and S

    D. Ben-Avraham and S. Havlin,Diffusion and reactions in fractals and disordered systems(Cambridge university press, 2000)

    work page 2000

  61. [61]

    Levy and B

    Y.-E. Levy and B. Souillard, Europhysics Letters4, 233 (1987)

    work page 1987

  62. [62]

    Akkermans, O

    E. Akkermans, O. Benichou, G. V. Dunne, A. Teplyaev, and R. Voituriez, Physical Review E—Statistical, Non- linear, and Soft Matter Physics86, 061125 (2012)

    work page 2012

  63. [63]

    Condamin, O

    S. Condamin, O. B´ enichou, V. Tejedor, R. Voituriez, and J. Klafter, Nature450, 77 (2007)

    work page 2007

  64. [64]

    P. J. Grabner and W. Woess, Stochastic Processes and their Applications69, 127 (1997)

    work page 1997

  65. [65]

    B´ enichou, B

    O. B´ enichou, B. Meyer, V. Tejedor, and R. Voituriez, Physical review letters101, 130601 (2008)

    work page 2008

  66. [66]

    B´ enichou, C

    O. B´ enichou, C. Chevalier, J. Klafter, B. Meyer, and R. Voituriez, Nature chemistry2, 472 (2010)

    work page 2010

  67. [67]

    M. T. Barlow and R. F. Bass, Proc. R. Soc. Lond. A431, 345 (1990)

    work page 1990

  68. [68]

    Pati˜ no Ortiz, M

    J. Pati˜ no Ortiz, M. Pati˜ no Ortiz, M.-´A. Mart´ ınez-Cruz, and A. S. Balankin, Fractal and Fractional7, 597 (2023)

    work page 2023