Kesten’s Incipient Infinite Cluster

  • Markus HeydenreichEmail author
  • Remco van der Hofstad
Part of the CRM Short Courses book series (CRMSC)


We introduce the high-dimensional incipient infinite cluster, which is an infinite cluster at the critical threshold \(p_c\) as constructed by Kesten. Since the results in the previous chapters imply that there is no infinite cluster at the critical point, this construction relies on a limiting argument. We first motivate the incipient infinite cluster, and then discuss its construction and properties in high dimensions. We also discuss its robustness, in the sense that various natural limiting schemes give rise to the same measure.


Incipient Infinite Cluster (IIC) Higher Dimensions Critical BRW Cylinder Event Infinite Self-avoiding Path 
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Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMünchenGermany
  2. 2.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

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