Invasion percolation on the Poisson-weighted infinite tree

Louigi Addario-Berry, Simon Griffiths, Ross J. Kang
2012 The Annals of Applied Probability  
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 $\sigma\to\infty$ 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
more » ... colation, 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 $\mathbb {R}\times[0,\infty)$.
doi:10.1214/11-aap761 fatcat:jda5sazgn5hqrkforhy5xgkzue