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arXiv:1108.4325 (math)
[Submitted on 22 Aug 2011 (v1), last revised 1 Aug 2012 (this version, v2)]

Title:High-dimensional incipient infinite clusters revisited

Authors:Markus Heydenreich, Remco van der Hofstad, Tim Hulshof
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Abstract:The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second author and Jarai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also obtain estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls.
Comments: 47 pages, 7 figures
Subjects: Probability (math.PR)
MSC classes: 60K35 (Primary) 60K37, 82B43 (Secondary)
Cite as: arXiv:1108.4325 [math.PR]
  (or arXiv:1108.4325v2 [math.PR] for this version)
  https://doi.org/10.48550/arXiv.1108.4325

Submission history

From: Tim Hulshof [view email]
[v1] Mon, 22 Aug 2011 14:17:16 UTC (340 KB)
[v2] Wed, 1 Aug 2012 15:08:37 UTC (344 KB)
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