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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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, ∞).

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

First available in Project Euclid: 18 May 2012

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Zentralblatt MATH identifier

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

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


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.

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