Poisson Zoo Process Overview

• Poisson Zoo Process is a stochastic spatial occupancy model using Poisson-distributed lattice animals to study percolation and phase transitions on nonamenable graphs.
• The model employs a framework where each vertex samples a Poisson number of connected lattice animals, unifying classical percolation, Boolean, and random cluster models.
• Analysis using mass-transport principles, branching process embeddings, and capacity arguments reveals how the moments of animal size distributions dictate cluster formation and uniqueness.

The Poisson Zoo Process refers to a broad class of stochastic spatial occupancy models governed by random placement of structured objects ("lattice animals") on graphs, notably in the context of nonamenable Cayley graphs. The term encapsulates models where i.i.d. Poisson-distributed collections of lattice animals are placed at every graph vertex, with the trace or union of these animals’ supports determining the covered subset of the graph. The Poisson Zoo Process generalizes classical site and bond percolation, Boolean models, and random cluster models, providing a unified framework for analyzing percolation, phase transitions, and cluster geometry when the underlying animal measures exhibit various moments or heavy-tail behaviors (Pete et al., 11 May 2025).

1. Mathematical Definition and Construction

Let $G = (V, E)$ be an infinite Cayley graph of a finitely generated group $\Gamma$, assumed transitive and unimodular with degree $d < \infty$ and Cheeger constant $h(G) > 0$ (i.e., nonamenable). At each vertex $x \in V$, consider a probability measure $\nu_x$ on the set of finite connected rooted subsets $H \ni x$ (lattice animals). For given intensity $\lambda > 0$:

• At each vertex $x$, draw $N_x \sim \mathrm{Poisson}(\lambda)$, then independently sample $N_x$ i.i.d. lattice animals $\{H_i^x\}_{i=1}^{N_x}$ from $\nu_x$.
• The Poisson Zoo is the random ensemble $\{(x,H_i^x)\}$, and its trace is

$$S_\nu^\lambda = \bigcup_{x \in V} \bigcup_{i=1}^{N_x} H_i^x.$$

Typical choices for $\nu$ include uniform measures over given size animals, distributions tailored to random walks (“worms”), or spatially structured objects such as rays or multidimensional segments (Pete et al., 11 May 2025).

The moments of the animal size distribution play a central role: $$m_k(\nu) = E_\nu\left[ |H|^k \right] = \sum_{H} |H|^k \nu_o(H), \quad k \geq 1.$$

2. Percolation Thresholds and Phase Transitions

The Poisson Zoo Process exhibits rich percolation behavior, with critical dependence on the first and second moments of $\nu$:

• Full Coverage: If $m_1 = \infty$, then for every $\lambda > 0$ the whole graph is covered, i.e., $S_\nu^\lambda = V$ almost surely.
• Finite Clustering: If $m_2 < \infty$, for small enough $\lambda$ (specifically, $\lambda < 1/[(d+1)m_2]$), the model contains only finite clusters; thus, there exists a percolation threshold $\lambda_c > 0$.
• Critical Parameters: Define

$$\lambda_c(G, \nu) = \inf\{ \lambda : P(S_\nu^\lambda\ \text{has infinite cluster}) > 0\}$$

$$\lambda_u(G, \nu) = \inf\{ \lambda : P(S_\nu^\lambda\ \text{has unique infinite cluster}) > 0\}.$$

The martingale arguments, mass-transport principle, and Galton–Watson-type branching processes underpin the rigorous proofs of these thresholds.

Moment Condition	Phase at Small $\lambda$	Coverage for Large/All $\lambda$
$m_1 = \infty$	Full coverage for any $\lambda > 0$	Trivial phase: $S_\nu^\lambda = V$
$m_2 < \infty$	No infinite clusters at small $\lambda$	Percolation transition at finite $\lambda_c$
$m_1, m_2 < \infty$	Classical percolation picture	Standard site/bond or Boolean percolation analogy

3. Special Constructions and Universality Classes

Distinct universality classes within the Poisson Zoo are distinguished by both the underlying graph and the animal measure:

• Free Product Graphs: For nonamenable free products $G = G_1 \star G_2$ with $0 < m_1(\nu) < \infty$ and $m_2(\nu) = \infty$, infinite clusters emerge for every $\lambda > 0$, i.e., $\lambda_c(G,\nu) = 0$.
• Random Walk Worms: If animals are traces of random walks of random (finite) length $L$ with $E[L]<\infty$ but $E[L^2]=\infty$, then for any $\lambda > 0$, percolation occurs on any nonamenable $G$.
• Uniqueness on Product Graphs: There exist measures $\nu$ on $G = T_d \times \mathbb{Z}^5$ (tree $\times$ lattice), where for all $\lambda > 0$, there is with high probability a unique infinite cluster at arbitrarily small density.

A key feature in these cases is the use of heavy-tailed animal-volume distributions, which can trigger percolation even at arbitrarily low intensity on nonamenable structures.

4. Techniques and Proof Strategies

Several methodological pillars support the analysis of Poisson Zoo Processes:

• Mass-Transport Principle: For unimodular graphs, exchange between sites allows balancing of local and global expectations, reducing the analysis of cluster covering probabilities to tractable sums.
• Branching Process Embeddings: Within free product graphs, the existence of cut-points enables the construction of embedded Galton–Watson processes whose offspring distributions relate to the moments $m_2(\nu)$, delivering supercriticality when second moments diverge.
• Capacity Arguments: For random-walk-based animals, cluster exploration stages are analyzed using linear capacity properties, leading to provable exponential cluster growth for $E[L]<\infty$, $E[L^2]=\infty$.

These approaches are complemented by classical arguments from percolation theory (e.g., union bounds, domination by subcritical/subtree percolation), with "sprinkling" methods employed to glue near-clusters when necessary (Pete et al., 11 May 2025).

5. Comparison with Other Fractional and Poisson Models

The Poisson Zoo occupies a distinct position within the broader "Poisson Zoo" of stochastic counting and percolation processes. Whereas the space-fractional Poisson process $N^\alpha(t)$, time-fractional Poisson process $N_\nu(t)$, and their space–time hybrid generalizations study non-integer-difference generators for one-dimensional, renewal-type jumps (Orsingher et al., 2011, Uchaikin et al., 2010), the Poisson Zoo percolation focuses on spatial coverage, cluster geometry, and global connectivity questions for collections of random volumes on graphs.

A plausible implication is that heavy-tailed distributions (e.g., infinite second moment of animal volume) can induce percolation at arbitrarily low density on sufficiently "expansive" underlying graphs, which is not possible in classical percolation settings.

6. Open Problems and Research Directions

Several open problems highlight the boundaries of current knowledge for Poisson Zoo Processes:

• Universality for Nonamenable Graphs: Is it true for all nonamenable unimodular transitive graphs and all animal measures with $m_1 < \infty$, $m_2 = \infty$ that $\lambda_c(G, \nu) = 0$?
• Unique Infinite Cluster at Low Density: For which nonamenable Cayley graphs does there exist an FIID percolation (such as a Poisson Zoo process) with arbitrarily small density and a unique infinite cluster, tying into deep questions in measured group theory (fixed-price 1 problem)?
• Criticality in Amenable Geometries: For example, in $\mathbb{Z}^2$ or $\mathbb{R}^2$, does $E[L^2]=\infty$ suffice for zero percolation threshold for random-ray animal measures?

Current research continues to investigate the interplay between graph geometry, animal law, cluster uniqueness, and the relationship to classical random interlacement, Boolean, and continuum percolation models (Pete et al., 11 May 2025).

7. Connections and Further Developments

The Poisson Zoo paradigm unifies numerous stochastic occupancy and percolation models by subsuming as special cases the Poisson Boolean model (balls of random radius), random interlacements, and loop soups. It provides a robust framework for understanding how local heavy-tailed randomness interacts with large-scale geometry to produce phase transitions. The exploration of limit theorems, universality classes, and ergodic properties within this setting links to modern work in group theory, statistical physics, and probabilistic combinatorics.

References for full proofs and deeper context include G. Pete and S. Rokob, "Nonamenable Poisson zoo," and related treatments in Probability on Trees and Networks by Lyons and Peres (Pete et al., 11 May 2025).