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Percolation Perturbations

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Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 2100)

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).

Keywords

  • Incipient Infinite Cluster (IIC)
  • Critical Percolation
  • Bernoulli Percolation
  • Cayley Graph
  • Isoperimetric Profile

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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

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