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First-passage times in complex scale-invariant media


How long does it take a random walker to reach a given target point? This quantity, known as a first-passage time (FPT), has led to a growing number of theoretical investigations over the past decade1. The importance of FPTs originates from the crucial role played by first encounter properties in various real situations, including transport in disordered media2,3, neuron firing dynamics4, spreading of diseases5 or target search processes6,7,8,9. Most methods of determining FPT properties in confining domains have been limited to effectively one-dimensional geometries, or to higher spatial dimensions only in homogeneous media1. Here we develop a general theory that allows accurate evaluation of the mean FPT in complex media. Our analytical approach provides a universal scaling dependence of the mean FPT on both the volume of the confining domain and the source–target distance. The analysis is applicable to a broad range of stochastic processes characterized by length-scale-invariant properties. Our theoretical predictions are confirmed by numerical simulations for several representative models of disordered media10, fractals3, anomalous diffusion11 and scale-free networks12.

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Figure 1: Length-scale-invariant networks.
Figure 2: Mean FPT in complex media.


  1. 1

    Redner, S. A Guide to First-Passage Processes (Cambridge Univ. Press, Cambridge, UK, 2001); Errata 〈

  2. 2

    Havlin, S. & ben-Avraham, D. Diffusion in disordered media. Adv. Phys. 36, 695–798 (1987)

    CAS  ADS  Article  Google Scholar 

  3. 3

    ben-Avraham, D. & Havlin, S. Diffusion and Reactions in Fractals and Disordered Systems (Cambridge Univ. Press, Cambridge, UK, 2000)

    Book  Google Scholar 

  4. 4

    Tuckwell, H. C. Introduction to Theoretical Neurobiology (Cambridge Univ. Press, Cambridge, UK, 1988)

    Book  Google Scholar 

  5. 5

    Lloyd, A. L. & May, R. M. Epidemiology — how viruses spread among computers and people. Science 292, 1316–1317 (2001)

    CAS  Article  Google Scholar 

  6. 6

    Bénichou, O., Coppey, M., Moreau, M., Suet, P. H. & Voituriez, R. Optimal search strategies for hidden targets. Phys. Rev. Lett. 94, 198101 (2005)

    ADS  Article  Google Scholar 

  7. 7

    Bénichou, O., Loverdo, C., Moreau, M. & Voituriez, R. Two-dimensional intermittent search processes: An alternative to Lévy flight strategies. Phys. Rev. E 74, 020102 (2006)

    ADS  Article  Google Scholar 

  8. 8

    Shlesinger, M. F. Mathematical physics: Search research. Nature 443, 281–282 (2006)

    CAS  ADS  Article  Google Scholar 

  9. 9

    Eliazar, I., Koren, T. & Klafter, J. Searching circular DNA strands. J. Phys. Condens. Matter 19, 065140 (2007)

    ADS  Article  Google Scholar 

  10. 10

    Bouchaud, J.-P. & Georges, A. Anomalous diffusion in disordered media: statistical mechanisms, models and applications. Phys. Rep. 195, 127–293 (1990)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11

    Metzler, R. & Klafter, J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  12. 12

    Gallos, L. K., Song, C., Havlin, S. & Makse, H. A. Scaling theory of transport in complex biological networks. Proc. Natl Acad. Sci. USA 104, 7746–7751 (2007)

    CAS  ADS  Article  Google Scholar 

  13. 13

    Van den Broeck, C. Renormalization of first-passage times for random walks on deterministic fractals. Phys. Rev. A 40, 7334–7345 (1989)

    CAS  ADS  Article  Google Scholar 

  14. 14

    Yuste, S. B. First-passage time, survival probability and propagator on deterministic fractals. J. Phys. A 28, 7027–7038 (1995)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15

    Song, C., Havlin, S. & Makse, H. A. Self-similarity of complex networks. Nature 443, 392–395 (2005)

    ADS  Article  Google Scholar 

  16. 16

    Rice, S. A. Diffusion-Limited Reactions (Elsevier, Amsterdam, 1985)

    Google Scholar 

  17. 17

    Berg, O. G., Winter, R. B. & von Hippel, P. H. Diffusion-driven mechanisms of protein translocation on nucleic acids. 1. Models and theory. Biochemistry 20, 6929–6948 (1981)

    CAS  Article  Google Scholar 

  18. 18

    Coppey, M., Bénichou, O., Voituriez, R. & Moreau, M. Kinetics of target site localization of a protein on DNA: A stochastic approach. Biophys. J. 87, 1640–1649 (2004)

    CAS  ADS  Article  Google Scholar 

  19. 19

    Holcman, D. Modeling DNA and virus trafficking in the cell cytoplasm. J. Stat. Phys. 127, 471–494 (2007)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  20. 20

    Barabasi, A. L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  21. 21

    Han, J.-D. J. et al. Evidence for dynamically organized modularity in the yeast protein–protein interaction network. Nature 430, 88–93 (2004)

    CAS  ADS  Article  Google Scholar 

  22. 22

    Almaas, E., Kovacs, B., Vicsek, T., Oltvai, Z. N. & Barabasi, A. L. Global organization of metabolic fluxes in the bacterium Escherichia coli . Nature 427, 839–843 (2004)

    CAS  ADS  Article  Google Scholar 

  23. 23

    Hughes, B. D. Random Walks and Random Environments (Oxford Univ. Press, New York, 1995)

    MATH  Google Scholar 

  24. 24

    Noh, J. D. & Rieger, H. Random walks on complex networks. Phys. Rev. Lett. 92, 118701 (2004)

    ADS  Article  Google Scholar 

  25. 25

    Barton, G. Elements of Green Functions and Propagation: Potentials, Diffusion and Waves (Oxford Univ. Press, New York, 1989)

    MATH  Google Scholar 

  26. 26

    Condamin, S., Bénichou, O. & Moreau, M. First-passage times for random walks in bounded domains. Phys. Rev. Lett. 95, 260601 (2005)

    CAS  ADS  Article  Google Scholar 

  27. 27

    Montroll, E. W. Random walks on lattices. iii. Calculation of first-passage times with application to exciton trapping on photosynthetic units. J. Math. Phys. 10, 753–765 (1969)

    CAS  ADS  Article  Google Scholar 

  28. 28

    Bollt, E. M. & ben-Avraham, D. What is special about diffusion on scale-free nets? N. J. Phys. 7, 26–47 (2005)

    Article  Google Scholar 

  29. 29

    Albert, R., Jeong, H. & Barabasi, A. L. Internet: Diameter of the world-wide web. Nature 401, 130–131 (1999)

    CAS  ADS  Article  Google Scholar 

  30. 30

    Song, C., Havlin, S. & Makse, H. A. Origins of fractality in the growth of complex networks. Nature Phys. 2, 275–281 (2006)

    CAS  ADS  Article  Google Scholar 

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We are grateful to J. M. Victor for discussions.

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Correspondence to O. Bénichou.

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The authors declare no competing financial interests.

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Supplementary Equations

The file contains Supplementary Equations with details on calculations involved in the main text. (PDF 523 kb)

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Condamin, S., Bénichou, O., Tejedor, V. et al. First-passage times in complex scale-invariant media. Nature 450, 77–80 (2007).

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