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Some properties of a random walk on a comb structure

Author

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  • Weiss, George H.
  • Havlin, Shlomo

Abstract

We analyze transport properties of a random walk on a comb structure, which serves as a model for a random walk on the backbone of a percolation cluster. It is shown that the random walk along the x axis, which is the analog of the backbone, exhibits anomalous diffusion in that 〈x2(n)〉 ∼ n12, and the expected number of x sites visited is proportional to n14 for large n. The distribution function is found to be a two-dimensional Gaussian. If a field in the x direction, so that diffusion is asymmetric, the expected displacement is found to be asymptotically proportional to n12.

Suggested Citation

  • Weiss, George H. & Havlin, Shlomo, 1986. "Some properties of a random walk on a comb structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 134(2), pages 474-482.
    • Handle: RePEc:eee:phsmap:v:134:y:1986:i:2:p:474-482
    DOI: 10.1016/0378-4371(86)90060-9
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    Citations

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    Cited by:

    1. Iomin, Alexander, 2011. "Fractional-time Schrödinger equation: Fractional dynamics on a comb," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 348-352.
    2. Arkhincheev, V.E., 2020. "The capture of particles on absorbing traps in the medium with anomalous diffusion: The effective fractional order diffusion equation and the slow temporal asymptotic of survival probability," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 550(C).
    3. Endre Csáki & Antónia Földes, 2022. "Strong Approximation of the Anisotropic Random Walk Revisited," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2879-2895, December.
    4. Dzhanoev, A.R. & Sokolov, I.M., 2018. "The effect of the junction model on the anomalous diffusion in the 3D comb structure," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 330-336.
    5. Balakrishnan, V. & Van den Broeck, C., 1995. "Transport properties on a random comb," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 217(1), pages 1-21.
    6. Kotak, Jesal D. & Barma, Mustansir, 2022. "Bias induced drift and trapping on random combs and the Bethe lattice: Fluctuation regime and first order phase transitions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 597(C).
    7. Baskin, Emmanuel & Iomin, Alexander, 2011. "Electrostatics in fractal geometry: Fractional calculus approach," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 335-341.
    8. Sandev, Trifce & Schulz, Alexander & Kantz, Holger & Iomin, Alexander, 2018. "Heterogeneous diffusion in comb and fractal grid structures," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 551-555.
    9. Endre Csáki & Antónia Földes, 2022. "On the Local Time of the Half-Plane Half-Comb Walk," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1247-1261, June.
    10. Endre Csáki & Antónia Földes, 2020. "Random Walks on Comb-Type Subsets of $$\mathbb {Z}^2$$ Z 2," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2233-2257, December.
    11. Pottier, N., 1994. "Analytic study of a model of biased diffusion on a random comblike structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 208(1), pages 91-123.
    12. Méndez, Vicenç & Iomin, Alexander, 2013. "Comb-like models for transport along spiny dendrites," Chaos, Solitons & Fractals, Elsevier, vol. 53(C), pages 46-51.
    13. Iomin, A. & Méndez, V., 2016. "Does ultra-slow diffusion survive in a three dimensional cylindrical comb?," Chaos, Solitons & Fractals, Elsevier, vol. 82(C), pages 142-147.
    14. Iomin, A. & Zaburdaev, V. & Pfohl, T., 2016. "Reaction front propagation of actin polymerization in a comb-reaction system," Chaos, Solitons & Fractals, Elsevier, vol. 92(C), pages 115-122.
    15. Valerii M Sukhorukov & Jürgen Bereiter-Hahn, 2009. "Anomalous Diffusion Induced by Cristae Geometry in the Inner Mitochondrial Membrane," PLOS ONE, Public Library of Science, vol. 4(2), pages 1-14, February.
    16. Aslangul, C. & Pottier, N. & Chvosta, P., 1994. "Analytic study of a model of diffusion on a random comblike structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 203(3), pages 533-565.
    17. Csáki, Endre & Csörgo, Miklós & Földes, Antónia & Révész, Pál, 2011. "On the local time of random walk on the 2-dimensional comb," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1290-1314, June.
    18. Pece Trajanovski & Petar Jolakoski & Ljupco Kocarev & Trifce Sandev, 2023. "Ornstein–Uhlenbeck Process on Three-Dimensional Comb under Stochastic Resetting," Mathematics, MDPI, vol. 11(16), pages 1-28, August.

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