Invasion percolation on the Poisson-weighted infinite tree. (English) Zbl 1262.60091

Authors’ 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 \(\sigma \rightarrow \infty \) limit of a representation discovered by O. Angel et al. [Ann. Probab. 36, No. 2, 420–466 (2008; Zbl 1145.60050)]. 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 \(\mathbb Z\) which are, in particular, factors of a homogeneous Poisson point process on the upper half-plane \(\mathbb{R} \times [0, \infty \)).


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60C05 Combinatorial probability
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
05C22 Signed and weighted graphs
05C85 Graph algorithms (graph-theoretic aspects)


Zbl 1145.60050
Full Text: DOI arXiv Euclid


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