Nonamenable Poisson zoo
In the Poisson zoo on an infinite Cayley graph $G$, we take a probability measure $ν$ on rooted finite connected subsets, called lattice animals, and place i.i.d. Poisson($λ$) copies of them at each vertex. If the expected volume of the animals w.r.t. $ν$ is infinite, then the whole $G$ is covered for any $λ>0$. If the second moment of the volume is finite, then it is easy to see that for small enough $λ$ the union of the animals has only finite clusters, while for $λ$ large enough there are also infinite clusters. Here we show that: 1. If $G$ is a nonamenable free product, then for ANY $ν$ with infinite second but finite first moment and any $λ>0$, there will be infinite clusters, despite having arbitrarily low density. 2. The same result holds for ANY nonamenable $G$, when the lattice animals are worms: random walk pieces of random finite length. It remains open if the result holds for ANY nonamenable Cayley graph with ANY lattice animal measure $ν$ with infinite second moment. 3. We also give a Poisson zoo example $ν$ on $\mathbb{T}_d \times \mathbb{Z}^5$ with finite first moment and a UNIQUE infinite cluster for any $λ>0$.