Skip to main content
SpringerLink
Log in

Further Results for Random Walk on IIC

  • Chapter
  • First Online:
Random Walks on Disordered Media and their Scaling Limits

Part of the book series: Lecture Notes in Mathematics ((LNMECOLE,volume 2101))

  • 1625 Accesses

Abstract

In this chapter, we will summarize results for random walks on various IICs. (Some of the models discussed in this chapter are not directly related to IIC, though.)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 44.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. L. Addario-Berry, N. Broutin, C. Goldschmidt, The continuum limit of critical random graphs. Probab. Theory Relat. Fields 152, 367–406 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Aldous, The continuum random tree. III. Ann. Probab. 21, 248–289 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Aldous, Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25, 812–854 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. O. Angel, Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13, 935–974 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. O. Angel, J. Goodman, F. den Hollander, G. Slade, Invasion percolation on regular trees. Ann. Probab. 36, 420–466 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. O. Angel, O. Schramm, Uniform infinite planar triangulations. Commun. Math. Phys. 241, 191–213 (2003)

    MATH  MathSciNet  Google Scholar 

  7. M.T. Barlow, J. Ding, A. Nachmias, Y. Peres, The evolution of the cover time. Comb. Probab. Comput. 20, 331–345 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. M.T. Barlow, A.A. Járai, T. Kumagai, G. Slade, Random walk on the incipient infinite cluster for oriented percolation in high dimensions. Commun. Math. Phys. 278, 385–431 (2008)

    Article  MATH  Google Scholar 

  9. M.T. Barlow, T. Kumagai, Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50, 33–65 (2006) (electronic)

    Google Scholar 

  10. M.T. Barlow, R. Masson, Spectral dimension and random walks on the two dimensional uniform spanning tree. Commun. Math. Phys. 305, 23–57 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. D. Ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 2000)

    Book  MATH  Google Scholar 

  12. I. Benjamini, N. Curien, Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points. Geom. Funct. Anal. 23, 501–531 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. I. Benjamini, O. Gurel-Gurevich, R. Lyons, Recurrence of random walk traces. Ann. Probab. 35, 732–738 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. I. Benjamini, O. Schramm, Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6(23), 13 pp. (2001)

    Google Scholar 

  15. B. Bollobás, Random Graphs, 2nd edn. (Cambridge University Press, Cambridge, 2001)

    Book  MATH  Google Scholar 

  16. K. Burdzy, G.F. Lawler, Rigorous exponent inequalities for random walks. J. Phys. A Math. Gen. 23, L23–L28 (1999)

    Article  MathSciNet  Google Scholar 

  17. D.A. Croydon, Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. Ann. Inst. Henri Poincaré Probab. Stat. 44, 987–1019 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. D.A. Croydon, Random walk on the range of random walk. J. Stat. Phys. 136, 349–372 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. D.A. Croydon, Scaling limit for the random walk on the largest connected component of the critical random graph. Publ. Res. Inst. Math. Sci. 48, 279–338 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. D.A. Croydon, B.M. Hambly, T. Kumagai, Convergence of mixing times for sequences of random walks on finite graphs. Electron. J. Probab. 17(3), 32 pp. (2012)

    Google Scholar 

  21. D. Croydon, T. Kumagai, Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive. Electron. J. Probab. 13, 1419–1441 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Damron, J. Hanson, P. Sosoe, Subdiffusivity of random walk on the 2D invasion percolation cluster. Stoch. Process. Their Appl. 123, 3588–3621 (2013)

    Article  MathSciNet  Google Scholar 

  23. J. Ding, J.R. Lee, Y. Peres, Cover times, blanket times and majorizing measures. Ann. Math. 175, 1409–1471 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. P. Erdős, A. Rényi, On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5, 17–61 (1960)

    Google Scholar 

  25. I. Fujii, T. Kumagai, Heat kernel estimates on the incipient infinite cluster for critical branching processes, in Proceedings of German-Japanese Symposium in Kyoto 2006, RIMS Kôkyûroku Bessatsu, vol. B6, 2008, pp. 85–95

    MATH  MathSciNet  Google Scholar 

  26. G. Grimmett, Percolation, 2nd edn. (Springer, Berlin, 1999)

    Book  MATH  Google Scholar 

  27. G. Grimmett, P. Hiemer, Directed percolation and random walk, in In and Out of Equilibrium (Mambucaba, 2000). Progress in Probability, vol. 51 (Birkhäuser, Boston, 2002), pp. 273–297

    Google Scholar 

  28. O. Gurel-Gurevich, A. Nachmias, Recurrence of planar graph limits. Ann. Math. (2) 177, 761–781 (2013)

    Google Scholar 

  29. B.M. Hambly, T. Kumagai, Diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice. Commun. Math. Phys. 295, 29–69 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Heydenreich, R. van der Hofstad, Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time. Probab. Theory Relat. Fields 149, 397–415 (2011)

    Article  MATH  Google Scholar 

  31. R. van der Hofstad, F. den Hollander, G. Slade, Construction of the incipient infinite cluster for spread-out oriented percolation above 4 + 1 dimensions. Commun. Math. Phys. 231, 435–461 (2002)

    Article  MATH  Google Scholar 

  32. R. van der Hofstad, G. Slade, Convergence of critical oriented percolation to super-Brownian motion above 4 + 1 dimensions. Ann. Inst. Henri Poincaré Probab. Stat. 39, 415–485 (2003)

    Google Scholar 

  33. A.A. Járai, A. Nachmias, Electrical resistance of the low dimensional critical branching random walk. ArXiv:1305.1092 (2013)

    Google Scholar 

  34. H. Kesten, The incipient infinite cluster in two-dimensional percolation. Probab. Theory Relat. Fields 73, 369–394 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  35. H. Kesten, Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré Probab. Stat. 22, 425–487 (1986)

    MATH  MathSciNet  Google Scholar 

  36. M. Krikun, Local structure of random quadrangulations. ArXiv:math/0512304 (2005)

    Google Scholar 

  37. J.-F. Le Gall, G. Miermont, Scaling limits of random trees and planar maps. Lecture Notes for the Clay Mathematical Institute Summer School in Buzios, 2010. ArXiv:1101.4856

    Google Scholar 

  38. D. Levin, Y. Peres, E. Wilmer, Markov Chains and Mixing Times (American Mathematical Society, Providence, 2009)

    MATH  Google Scholar 

  39. L. Lovász, P. Winkler, Mixing times, in Microsurveys in Discrete Probability (Princeton, NJ, 1997). DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 41 (American Mathematical Society, Providence, 1998), pp. 85–133

    Google Scholar 

  40. R. Montenegro, P. Tetali, Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1, x+121 pp. (2006)

    Google Scholar 

  41. A. Nachmias, Y. Peres, Critical random graphs: diameter and mixing time. Ann. Probab. 36, 1267–1286 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  42. D. Shiraishi, Heat kernel for random walk trace on \({\mathbb{Z}}^{3}\) and \({\mathbb{Z}}^{4}\). Ann. Inst. Henri Poincaré Probab. Stat. 46, 1001–1024 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  43. D. Shiraishi, Exact value of the resistance exponent for four dimensional random walk trace. Probab. Theory Relat. Fields 153, 191–232 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  44. D. Shiraishi, Random walk on non-intersecting two-sided random walk trace is subdiffusive in low dimensions. Trans. Amer. Math. Soc. (to appear)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

    Takashi Kumagai

Authors
  1. Takashi Kumagai
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kumagai, T. (2014). Further Results for Random Walk on IIC. In: Random Walks on Disordered Media and their Scaling Limits. Lecture Notes in Mathematics(), vol 2101. Springer, Cham. https://doi.org/10.1007/978-3-319-03152-1_7

Download citation

Publish with us

Policies and ethics

Search

Navigation