Nonamenable Poisson zoo

TL;DR

This paper investigates percolation phenomena in the Poisson zoo model on nonamenable Cayley graphs, revealing conditions under which infinite clusters form despite low density, and providing examples with unique infinite clusters.

Contribution

It demonstrates that for nonamenable graphs, infinite clusters can occur at arbitrarily low densities with certain lattice animal measures, and introduces a unique infinite cluster example on a specific graph product.

Findings

Infinite clusters form at any positive intensity on nonamenable free products with infinite second moment measures.

Worm-shaped lattice animals also produce infinite clusters at any positive intensity on nonamenable graphs.

An example with a unique infinite cluster exists on a specific graph product with finite first moment measure.

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 $\lambda>0$.