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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adelman, O., Burdzy, K. and Pemantle, R. Sets avoided by Brownian motion. Ann. Probab. 26, (1998), 429–464.
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.
Benjamini, I., Häggström, O., Peres, Y. and Steif, J. Which properties of a random sequence are dynamically sensitive? Ann. Probab. 31, (2003), 1–34.
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.
Benjamini, I. and Schramm, O. Exceptional planes of percolation. Probab. Theory Related Fields 111, (1998), 551–564.
Berg, J. van den, Meester, R. and White, D.G. Dynamic Boolean models. Stochastic Process. Appl. 69, (1997), 247–257.
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.
Burton, R. and Keane, M. Density and uniqueness in percolation. Comm. Math. Phys. 121, (1989), 501–505.
Camia, F. and Newman, C.M. Two-dimensional critical percolation: the full scaling limit, Comm. Math. Phys. 268, (2006), 1–38.
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.
Cardy, J.L. Critical percolation in finite geometries, J. Phys. A 25, (1992), L201–L206.
Evans, S.N. Local properties of Levy processes on a totally disconnected group. J. Theoret. Probab. 2, (1989), 209–259.
Garban, C. Oded Schramm’s contributions to noise sensitivity (preliminary title), in preparation.
Garban, C., Pete, G. and Schramm, O. The Fourier Spectrum of Critical Percolation, preprint, arXiv:0803.3750[math:PR].
Garban, C., Pete, G. and Schramm, O. Scaling limit of near-critical and dynamical percolation, in preparation.
Grimmett, G. Percolation. Second edition, Springer-Verlag, (1999), New York.
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.
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.
Häggström, O., Peres, Y. and Steif, J.E. Dynamical percolation. Ann. Inst. Henri Poincaré, Probab. et Stat. 33, (1997), 497–528.
Hammond, A., Pete, G. and Schramm, O. Local time for exceptional dynamical percolation, and the incipient infinite cluster, in preparation.
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.
Harris, T.E. A lower bound on the critical probability in a certain percolation process. Proc. Cambridge Phil. Soc. 56, (1960), 13–20.
Jonasson, J. Dynamical circle covering with homogeneous Poisson updating Statist. Probab. Lett., to appear.
Jonasson, J. and Steif, J.E. Dynamical models for circle covering: Brownian motion and Poisson updating. Ann. Probab. 36, (2008), 739–764.
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.
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.
Kallenberg, O. Foundations of modern probability. Second edition. Probability and its Applications (New York), Springer-Verlag, New York, 2002.
Kesten, H. The critical probability of bond percolation on the square lattice equals 1/2. Comm. Math. Phys. 74, (1980), 41–59.
Kesten, H. The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73, (1986), 369–394.
Kesten, H. Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré, Probab. et Stat. 22, (1986), 425–487.
Kesten, H. Scaling relations for 2D-percolation. Commun. Math. Phys. 109, (1987), 109–156.
Kesten, H. and Zhang, Y. Strict inequalities for some critical exponents in 2D-percolation. J. Statist. Phys. 46, (1987), 1031–1055.
Khoshnevisan D. Dynamical percolation on general trees. Probab. Theory Related Fields 140, (2008) 169–193.
Khoshnevisan, D. Multiparameter processes. An introduction to random fields. Springer Monographs in Mathematics. Springer-Verlag, New York, 2002.
Khoshnevisan, D., Peres, Y. and Xiao, Y. Limsup random fractals. Electron. J. Probab. 5, (2000), no. 5, 24 pp. (electronic).
Langlands, R., Pouliot, P. and Saint-Aubin, Y. Conformal invariance in two-dimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30, (1994), 1–61.
Lawler, G. Dimension and natural parametrization for SLE curves.
Lawler, G., Schramm, O. and Werner, W. One-arm exponent for critical 2D percolation. Electron. J. Probab. 7, (2002), no. 2, 13 pp. (electronic).
Lyons, R. Random walks and percolation on trees. Ann. Probab. 18, (1990), 931–958.
Lyons, R. Random walks, capacity, and percolation on trees, Ann. Probab. 20, (1992), 2043–2088.
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/.
Mossel, E. and O’Donnell, R. On the noise sensitivity of monotone functions. Random Structures Algorithms 23, (2003), 333–350.
Nolin, P. Near-critical percolation in two dimensions. Electron. J. Probab. 13, (2008), no. 55, 1562–1623.
O’Donnell, R. Computational applications of noise sensitivity, Ph.D. thesis, MIT (2003). Version available at http://www.cs.cmu.edu/~odonnell/.
Peres, Y. Intersection-equivalence of Brownian paths and certain branching processes. Commun. Math. Phys. 177, (1996), 417–434.
Peres, Y. Remarks on intersection-equivalence and capacity-equivalence. Ann. Inst. Henri Poincaré (Physique théorique) 64, (1996), 339–347.
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.
Peres, Y., Schramm, O., Sheffield, S. and Wilson, D.B. Random-turn hex and other selection games. Amer. Math. Monthly 114, (2007), 373–387.
Peres, Y. and Steif, J.E. The number of infinite clusters in dynamical percolation. Probab. Theory Related Fields 111, (1998), 141–165.
Schramm, O. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, (2000), 221–288.
Schramm, O. and Steif, J.E. Quantitative noise sensitivity and exceptional times for percolation. Ann. Math., to appear.
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.
Smirnov, S. and Werner, W. Critical exponents for two-dimensional percolation, Math. Res. Lett. 8, (2001), 729–744.
Werner, W. Lectures on two-dimensional critical percolation. IAS Park City Graduate Summer School, 2007, arXiv:0710.0856[math:PR].
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
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
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
DOI: https://doi.org/10.1007/978-3-0346-0030-9_5
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0029-3
Online ISBN: 978-3-0346-0030-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)