Nonamenable Poisson zoo

TL;DR

The paper analyzes a correlated site-percolation model, the Poisson zoo, on infinite transitive graphs by dropping random rooted lattice animals with Poisson multiplicities. It establishes strong phase-transition results in nonamenable settings: for nonamenable free products and for random-length worms on any nonamenable unimodular graph, the critical intensity λ_c collapses to 0 when the second moment of the animal size is infinite. It also provides an explicit construction yielding a unique infinite cluster at all intensities on a product graph, and discusses connections to cost and FIID phenomena. The methods center on meticulous first/second moment bounds, capacity estimates, and carefully designed exploration processes that embed branching structures or fattening dynamics, often requiring sprinkling to overcome parity obstacles. Overall, the work reveals robust supercritical behavior of the Poisson zoo in broad nonamenable contexts and highlights deep links to percolation, branching processes, and ergodic properties of invariant measures.

Abstract

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 $λ>0$.

Figures (4)

• Figure 1.1: Exploring a cluster: at any given stage $E_{n-1}$, there are many exposed vertices on its boundary where new worms (the green trajectories) can touch it, creating a much larger $E_n$.
• Figure 1.2: Two examples of free products of graphs.
• Figure 2.1: A parity issue. On the free product $\mathbb{Z}\star\mathbb{Z}$, generated by the letters $a$ and $b$, take a $\nu$ with $m_2(\nu)=\infty$ that is supported only on words in one of the generators, say $a$ (the "horizontal" red generator in the picture). It may happen that we have already explored a large connected set, shaded green in the picture, which has of course a large total exterior boundary, but only a small one in the $b$ direction. But only these are the exterior boundary vertices from where the $a$ direction is completely unexplored, where we can have a lot of new animals covering that vertex, exploiting the size-biasing effect and the $m_2(\nu)=\infty$ condition.
• Figure 4.1: The fat $C_{n+1}$ is the growth through $B_{n}$, yielding $E_{n+1}$, and then $B_{n+1}$ is the new part of the exterior boundary of $E_{n+1}$.

Theorems & Definitions (64)

• Theorem 1.1: Random length worms on any nonamenable group
• Theorem 1.2: General animals on free products
• Proposition 1.4
• Definition 2.1: Unimodularity
• Definition 2.2
• Claim 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Corollary 2.7