  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 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 $\mathbb {R}\times[0,\infty)$.
