Invasion percolation on the Poisson-weighted infinite tree

Details Invasion percolation on the Poisson-weighted infinite tree Invasion percolation on the Poisson-weighted infinite tree

2009-12-02

arxiv.org/abs/0912.0335v1

Mathematics

Probability

30 pages, 6 figures

Scientific paper

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 sigma to infinity limit of a representation discovered by Angel, Goodman, den Hollander and Slade (arXiv:math/0608132v2). 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 Rx[0,infinity).

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