Abstract
The unique infinite cluster is a perturbation of the underling graph that shares many of its properties (e.g. transience of random walk). Infinite clusters in the non uniqueness regime on the other hand admit some universal features which are not inherited from the underling graph, they have infinitely many ends and thus are very tree like. When performing the operation of contracting Bernoulli percolation clusters different geometric structures emerge. When the clusters are subcritical, we see random perturbation of the underling graphs but when the construction is based on critical percolation new type of spaces emerges. More precise definitions appears below. We end with an invariant percolation viewpoint on the incipient infinite cluster (IIC).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Aizenman, D.J. Barsky, Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108(3), 489–526 (1987)
D. Aldous, R. Lyons, Processes on unimodular random networks. Electron. J. Probab. 12(54), 1454–1508 (2007)
I. Benjamini, H. Duminil-Copin, G. Kozma, A. Yadin, Disorder, entropy and harmonic functions. arXiv preprint arXiv:1111.4853 (2011)
I. Benjamini, R. Lyons, O. Schramm, Percolation perturbations in potential theory and random walks, in Random Walks and Discrete Potential Theory, Cortona, 1997. Symposia Mathematica XXXIX (Cambridge University Press, Cambridge, 1999), pp. 56–84
I. Benjamini, O. Schramm, Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6(23), 13 pp. (2001) (electronic)
G. Grimmett, Percolation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, 2nd edn. (Springer, Berlin, 1999)
O. Häggström, Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25(3), 1423–1436 (1997)
T. Hara, G. Slade, Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128(2), 333–391 (1990)
A.A. Járai, Invasion percolation and the incipient infinite cluster in 2D. Commun. Math. Phys. 236(2), 311–334 (2003)
G. Kozma, A. Nachmias, Arm exponents in high dimensional percolation. J. Am. Math. Soc. 24(2), 375–409 (2011)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Benjamini, I. (2013). Percolation Perturbations. In: Coarse Geometry and Randomness. Lecture Notes in Mathematics(), vol 2100. Springer, Cham. https://doi.org/10.1007/978-3-319-02576-6_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-02576-6_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02575-9
Online ISBN: 978-3-319-02576-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)