Skip to main content
SpringerLink
Log in
Book cover

Fractal Geometry and Stochastics IV pp 145–174Cite as

A Survey of Dynamical Percolation

  • Conference paper

Part of the book series: Progress in Probability ((PRPR,volume 61))

Abstract

Percolation is one of the simplest and nicest models in probability theory/statistical mechanics which exhibits critical phenomena. Dynamical percolation is a model where a simple time dynamics is added to the (ordinary) percolation model. This dynamical model exhibits very interesting behavior. Our goal in this survey is to give an overview of the work in dynamical percolation that has been done (and some of which is in the process of being written up).

Research partially supported by the Swedish Natural Science Research Council and the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adelman, O., Burdzy, K. and Pemantle, R. Sets avoided by Brownian motion. Ann. Probab. 26, (1998), 429–464.

    Article  MATH  MathSciNet  Google Scholar 

  2. Aizenman, M., Kesten, H. and Newman, C.M. Uniqueness of the infinite cluster and continuity of connectivity functions for short-and long-range percolation. Comm. Math. Phys. 111, (1987), 505–532.

    Article  MATH  MathSciNet  Google Scholar 

  3. Benjamini, I., Häggström, O., Peres, Y. and Steif, J. Which properties of a random sequence are dynamically sensitive? Ann. Probab. 31, (2003), 1–34.

    Article  MATH  MathSciNet  Google Scholar 

  4. Benjamini, I., Kalai, G. and Schramm, O. Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. 90, (1999), 5–43.

    Article  MATH  MathSciNet  Google Scholar 

  5. Benjamini, I. and Schramm, O. Exceptional planes of percolation. Probab. Theory Related Fields 111, (1998), 551–564.

    Article  MATH  MathSciNet  Google Scholar 

  6. Berg, J. van den, Meester, R. and White, D.G. Dynamic Boolean models. Stochastic Process. Appl. 69, (1997), 247–257.

    Article  MATH  MathSciNet  Google Scholar 

  7. Broman, E.I. and Steif, J.E. Dynamical Stability of Percolation for Some Interacting Particle Systems and ε-Movability. Ann. Probab. 34, (2006), 539–576.

    Article  MATH  MathSciNet  Google Scholar 

  8. Burton, R. and Keane, M. Density and uniqueness in percolation. Comm. Math. Phys. 121, (1989), 501–505.

    Article  MATH  MathSciNet  Google Scholar 

  9. Camia, F. and Newman, C.M. Two-dimensional critical percolation: the full scaling limit, Comm. Math. Phys. 268, (2006), 1–38.

    Article  MATH  MathSciNet  Google Scholar 

  10. Camia, F., Fontes, L.R.G. and Newman, C.M. Two-dimensional scaling limits via marked nonsimple loops. Bull. Braz. Math. Soc. (N.S.) 37, (2006), 537–559.

    Article  MATH  MathSciNet  Google Scholar 

  11. Cardy, J.L. Critical percolation in finite geometries, J. Phys. A 25, (1992), L201–L206.

    Article  MATH  MathSciNet  Google Scholar 

  12. Evans, S.N. Local properties of Levy processes on a totally disconnected group. J. Theoret. Probab. 2, (1989), 209–259.

    Article  MATH  MathSciNet  Google Scholar 

  13. Garban, C. Oded Schramm’s contributions to noise sensitivity (preliminary title), in preparation.

    Google Scholar 

  14. Garban, C., Pete, G. and Schramm, O. The Fourier Spectrum of Critical Percolation, preprint, arXiv:0803.3750[math:PR].

    Google Scholar 

  15. Garban, C., Pete, G. and Schramm, O. Scaling limit of near-critical and dynamical percolation, in preparation.

    Google Scholar 

  16. Grimmett, G. Percolation. Second edition, Springer-Verlag, (1999), New York.

    Google Scholar 

  17. Häggström, O. Dynamical percolation: early results and open problems. Microsurveys in discrete probability (Princeton, NJ, 1997), 59–74, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 41, Amer. Math. Soc., Providence, RI, 1998.

    Google Scholar 

  18. Häggström, O. and Pemantle, R. On near-critical and dynamical percolation in the tree case. Statistical physics methods in discrete probability, combinatorics, and theoretical computer science (Princeto, NJ, 1997), Random Structures Algorithms 15, (1999), 311–318.

    Article  MATH  MathSciNet  Google Scholar 

  19. Häggström, O., Peres, Y. and Steif, J.E. Dynamical percolation. Ann. Inst. Henri Poincaré, Probab. et Stat. 33, (1997), 497–528.

    Article  MATH  Google Scholar 

  20. Hammond, A., Pete, G. and Schramm, O. Local time for exceptional dynamical percolation, and the incipient infinite cluster, in preparation.

    Google Scholar 

  21. Hara, T. and Slade, G. (1994) Mean field behavior and the lace expansion, in Probability Theory and Phase Transitions, (ed. G. Grimmett), Proceedings of the NATO ASI meeting in Cambridge 1993, Kluwer.

    Google Scholar 

  22. Harris, T.E. A lower bound on the critical probability in a certain percolation process. Proc. Cambridge Phil. Soc. 56, (1960), 13–20.

    Article  MATH  Google Scholar 

  23. Jonasson, J. Dynamical circle covering with homogeneous Poisson updating Statist. Probab. Lett., to appear.

    Google Scholar 

  24. Jonasson, J. and Steif, J.E. Dynamical models for circle covering: Brownian motion and Poisson updating. Ann. Probab. 36, (2008), 739–764.

    Article  MATH  MathSciNet  Google Scholar 

  25. Kahn, J., Kalai, G. and Linial, N. The influence of variables on boolean functions. 29th Annual Symposium on Foundations of Computer Science, (1988), 68–80.

    Google Scholar 

  26. Kalai, G. and Safra, S. Threshold phenomena and influence: perspectives from mathematics, computer science, and economics. Computational complexity and statistical physics, 25–60, St. Fe Inst. Stud. Sci. Complex., Oxford Univ, Press, New York, 2006.

    Google Scholar 

  27. Kallenberg, O. Foundations of modern probability. Second edition. Probability and its Applications (New York), Springer-Verlag, New York, 2002.

    Google Scholar 

  28. Kesten, H. The critical probability of bond percolation on the square lattice equals 1/2. Comm. Math. Phys. 74, (1980), 41–59.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  31. Kesten, H. Scaling relations for 2D-percolation. Commun. Math. Phys. 109, (1987), 109–156.

    Article  MATH  MathSciNet  Google Scholar 

  32. Kesten, H. and Zhang, Y. Strict inequalities for some critical exponents in 2D-percolation. J. Statist. Phys. 46, (1987), 1031–1055.

    Article  MATH  MathSciNet  Google Scholar 

  33. Khoshnevisan D. Dynamical percolation on general trees. Probab. Theory Related Fields 140, (2008) 169–193.

    Article  MATH  MathSciNet  Google Scholar 

  34. Khoshnevisan, D. Multiparameter processes. An introduction to random fields. Springer Monographs in Mathematics. Springer-Verlag, New York, 2002.

    Google Scholar 

  35. Khoshnevisan, D., Peres, Y. and Xiao, Y. Limsup random fractals. Electron. J. Probab. 5, (2000), no. 5, 24 pp. (electronic).

    Google Scholar 

  36. Langlands, R., Pouliot, P. and Saint-Aubin, Y. Conformal invariance in two-dimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30, (1994), 1–61.

    Article  MATH  MathSciNet  Google Scholar 

  37. Lawler, G. Dimension and natural parametrization for SLE curves.

    Google Scholar 

  38. Lawler, G., Schramm, O. and Werner, W. One-arm exponent for critical 2D percolation. Electron. J. Probab. 7, (2002), no. 2, 13 pp. (electronic).

    Google Scholar 

  39. Lyons, R. Random walks and percolation on trees. Ann. Probab. 18, (1990), 931–958.

    Article  MATH  MathSciNet  Google Scholar 

  40. Lyons, R. Random walks, capacity, and percolation on trees, Ann. Probab. 20, (1992), 2043–2088.

    Article  MATH  MathSciNet  Google Scholar 

  41. Lyons, R. with Peres, Y. (2008). Probability on Trees and Networks. Cambridge University Press. In preparation. Current version available at http://mypage.iu.edu/~rdlyons/.

    Google Scholar 

  42. Mossel, E. and O’Donnell, R. On the noise sensitivity of monotone functions. Random Structures Algorithms 23, (2003), 333–350.

    Article  MATH  MathSciNet  Google Scholar 

  43. Nolin, P. Near-critical percolation in two dimensions. Electron. J. Probab. 13, (2008), no. 55, 1562–1623.

    MATH  MathSciNet  Google Scholar 

  44. O’Donnell, R. Computational applications of noise sensitivity, Ph.D. thesis, MIT (2003). Version available at http://www.cs.cmu.edu/~odonnell/.

    Google Scholar 

  45. Peres, Y. Intersection-equivalence of Brownian paths and certain branching processes. Commun. Math. Phys. 177, (1996), 417–434.

    Article  MATH  MathSciNet  Google Scholar 

  46. Peres, Y. Remarks on intersection-equivalence and capacity-equivalence. Ann. Inst. Henri Poincaré (Physique théorique) 64, (1996), 339–347.

    MATH  MathSciNet  Google Scholar 

  47. Peres, Y., Schramm, O. and Steif, J.E. Dynamical sensitivity of the infinite cluster in critical percolation. Ann. Inst. Henri Poincaré. Probab. et Stat., to appear.

    Google Scholar 

  48. Peres, Y., Schramm, O., Sheffield, S. and Wilson, D.B. Random-turn hex and other selection games. Amer. Math. Monthly 114, (2007), 373–387.

    MATH  MathSciNet  Google Scholar 

  49. Peres, Y. and Steif, J.E. The number of infinite clusters in dynamical percolation. Probab. Theory Related Fields 111, (1998), 141–165.

    Article  MATH  MathSciNet  Google Scholar 

  50. Schramm, O. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, (2000), 221–288.

    Article  MATH  MathSciNet  Google Scholar 

  51. Schramm, O. and Steif, J.E. Quantitative noise sensitivity and exceptional times for percolation. Ann. Math., to appear.

    Google Scholar 

  52. Smirnov, S. Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333, (2001), 239–244.

    MATH  Google Scholar 

  53. Smirnov, S. and Werner, W. Critical exponents for two-dimensional percolation, Math. Res. Lett. 8, (2001), 729–744.

    MATH  MathSciNet  Google Scholar 

  54. Werner, W. Lectures on two-dimensional critical percolation. IAS Park City Graduate Summer School, 2007, arXiv:0710.0856[math:PR].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Sciences, Chalmers University of Technology, SE-41296, Gothenburg, Sweden

    Jeffrey E. Steif

  2. Mathematical Sciences, Göteborg University, SE-41296, Gothenburg, Sweden

    Jeffrey E. Steif

Authors
  1. Jeffrey E. Steif
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität, 17487, Greifswald, Germany

    Christoph Bandt

  2. Mathematisches Institut, Friedrich-Schiller-Universität, 07740, Jena, Germany

    Martina Zähle

  3. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Peter Mörters

Additional information

Dedicated to the memory of Oded Schramm, who has been a great inspiration to me and with whom it has been a great honor and privilege to work.

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Steif, J.E. (2009). A Survey of Dynamical Percolation. In: Bandt, C., Zähle, M., Mörters, P. (eds) Fractal Geometry and Stochastics IV. Progress in Probability, vol 61. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0030-9_5

Download citation

Publish with us

Policies and ethics

Search

Navigation