Nonamenable Poisson zoo

Abstract

In the Poisson zoo on an infinite Cayley graph $G$, we take a probability measure $\nu$ on rooted finite connected subsets, called lattice animals, and place i.i.d. Poisson($\lambda$) copies of them at each vertex. If the expected volume of the animals w.r.t. $\nu$ is infinite, then the whole $G$ is covered for any $\lambda>0$. If the second moment of the volume is finite, then it is easy to see that for small enough $\lambda$ the union of the animals has only finite clusters, while for $\lambda$ large enough there are also infinite clusters. Here we show that: 1. If $G$ is a nonamenable free product, then for ANY $\nu$ with infinite second but finite first moment and any $\lambda>0$, 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 $\nu$ with infinite second moment. 3. We also give a Poisson zoo example $\nu$ on $\mathbb{T}_d \times \mathbb{Z}^5$ with finite first moment and a UNIQUE infinite cluster for any $\lambda>0$.