The Annals of Applied Probability

Invasion percolation on the Poisson-weighted infinite tree

Louigi Addario-Berry, Simon Griffiths, and Ross J. Kang

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Abstract

We study invasion percolation on Aldous’ Poisson-weighted infinite tree, and derive two distinct Markovian representations of the resulting process. One of these is the σ → ∞ limit of a representation discovered by Angel et al. [Ann. Appl. Probab. 36 (2008) 420–466]. We also introduce an exploration process of a randomly weighted Poisson incipient infinite cluster. The dynamics of the new process are much more straightforward to describe than those of invasion percolation, but it turns out that the two processes have extremely similar behavior. Finally, we introduce two new “stationary” representations of the Poisson incipient infinite cluster as random graphs on ℤ which are, in particular, factors of a homogeneous Poisson point process on the upper half-plane ℝ × [0, ∞).

Article information

Source
Ann. Appl. Probab., Volume 22, Number 3 (2012), 931-970.

Dates
First available in Project Euclid: 18 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1337347535

Digital Object Identifier
doi:10.1214/11-AAP761

Mathematical Reviews number (MathSciNet)
MR2977982

Zentralblatt MATH identifier
1262.60091

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60G55: Point processes

Keywords
Invasion percolation Prim’s algorithm Poisson-weighted infinite tree percolation random trees

Citation

Addario-Berry, Louigi; Griffiths, Simon; Kang, Ross J. Invasion percolation on the Poisson-weighted infinite tree. Ann. Appl. Probab. 22 (2012), no. 3, 931--970. doi:10.1214/11-AAP761. https://projecteuclid.org/euclid.aoap/1337347535


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References

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