Poisson Stick Model: Hyperbolic & Bayesian Insights

• The Poisson stick model is a probabilistic framework using Poisson processes to place line segments in space, enabling analysis of percolation and random measure construction.
• In hyperbolic geometry, the model exhibits distinct percolation and uniqueness thresholds, with scaling laws of L⁻² and L⁻¹ that contrast with the Euclidean case.
• In Bayesian nonparametrics, the model underpins stick-breaking constructions for Dirichlet, beta, and gamma processes, facilitating scalable inference for random discrete measures.

The Poisson stick model refers to a class of probabilistic models in which geometrical objects—typically "sticks" or line segments—are placed in space according to a Poisson point process, with further structure and interpretation dependent on the context of use. In probability theory, mathematical physics, and Bayesian nonparametrics, the “Poisson stick model” plays two principal roles: (1) as a continuum percolation model—especially in hyperbolic geometry—and (2) as a constructive representation for completely random measures (CRMs) via the stick-breaking method, crucial for Dirichlet-, beta-, and gamma-process based Bayesian models. These constructions link Poisson process theory, random partitions, stochastic geometry, and nonparametric inference.

1. Hyperbolic Poisson Stick Model: Definition and Fundamental Properties

The two-dimensional hyperbolic Poisson stick model, as studied by Broman and Tykesson, is formulated in the Poincaré disc model of the hyperbolic plane $\mathbb{H}^2$ (Broman et al., 17 Dec 2025). The Poincaré disc is $\{x \in \mathbb{R}^2 : \|x\| < 1\}$, equipped with the hyperbolic distance

$$d_h(x, y) = \cosh^{-1} \Bigl(1 + 2\frac{\|x - y\|^2}{(1 - \|x\|^2)(1 - \|y\|^2)}\Bigr).$$

The natural measure is $$v^h(d\rho\, d\theta) = \sinh(\rho)\, d\rho\, d\theta$$, and the area of a ball of radius $\rho$ is $v^h(B^h(o, \rho)) = 2\pi (\cosh\rho - 1)$.

Sticks are parameterized by their center $x \in \mathbb{H}^2$, orientation $\varphi \in [0, \pi)$, and fixed length $L$. Each stick is the geodesic segment $\ell_L(x, \varphi)$ of length $L$, centered at $x$, oriented at angle $\varphi$.

Placement is via a Poisson point process (PPP) of intensity $\lambda$ on $\mathbb{H}^2 \times [0, \pi)$, so that, for measurable $A \subset \mathbb{H}^2 \times [0, \pi)$, the number of sticks in $A$ is $$\operatorname{Poisson}(\lambda\, v^h \otimes \text{Uniform}[0, \pi](A))$$. The union of all sticks, denoted $\mathcal{C}(\lambda, L)$, forms the occupied set, while its complement $\mathcal{V}(\lambda, L)$ is the vacant set (Broman et al., 17 Dec 2025).

2. Phase Transitions and Critical Intensities in Hyperbolic Space

The principal question is the connectivity of $\mathcal{C}(\lambda, L)$ as parameters $\lambda$ and $L$ vary. Two distinct critical intensities are defined:

• Percolation threshold $\lambda_c(L)$: the minimal $\lambda$ such that $\mathcal{C}(\lambda, L)$ contains an unbounded connected component almost surely.
• Uniqueness threshold $\lambda_u(L)$: the minimal $\lambda$ such that $\mathcal{C}(\lambda, L)$ contains a unique unbounded connected component almost surely.

Main results (Broman et al., 17 Dec 2025):

• As $L\to\infty$, $\lambda_c(L) \sim c_1 L^{-2}$ with $c_1=\pi/2$, bounded by $$\pi/2 L^{-2} \le \lambda_c(L) \le (32\pi/(\sqrt{3}-1)) L^{-2}$$.
• For uniqueness: $\lambda_u(L) \sim c_2 L^{-1}$ with $c_2 = \pi/2$, satisfying $$\pi/2 L^{-1} \le \lambda_u(L) \le 5\sqrt{2}\pi L^{-1}$$.

Thus, the model exhibits two bona fide continuum percolation phase transitions in $\mathbb{H}^2$, separating regimes without infinite clusters, with multiple infinite clusters, and with a unique infinite cluster.

3. Comparison to Euclidean and Related Models

In Euclidean $\mathbb{R}^2$, the scaling of thresholds is well-understood: for sticks of length $L$,

$$\lambda^E_c(L) = \lambda^E_c(1) L^{-2}, \quad \lambda^E_u(L) = \lambda^E_c(L) = \Theta(L^{-2}).$$

Here, percolation and uniqueness thresholds coincide—any supercritical regime has a unique infinite cluster almost surely.

In contrast, the hyperbolic case displays a new phenomenon: supercritical $\lambda$ can yield multiple infinite clusters, and uniqueness only appears above a larger threshold, scaling as $L^{-1}$. The coincidence of percolation and uniqueness thresholds in Euclid is a consequence of polynomial volume growth; the exponential growth in $\mathbb{H}^2$ is fundamental to the separation in critical scalings. Locally, the expected number of intersections per stick remains $\sim \lambda L^2$, explaining the percolation threshold scaling $L^{-2}$ as in Euclid, but the geometry at large scale allows non-uniqueness, related in the $L \to \infty$ limit to the Poisson cylinder model, which is known to admit multiple infinite clusters in hyperbolic space (Broman et al., 17 Dec 2025).

4. Analytical Techniques and Proof Structures

Key analytic tools and proof ideas for these results include:

• Coupling the component growth from a reference stick to a Galton–Watson process with mean offspring $\sim (2\lambda/\pi)L^2$ for lower bounds on $\lambda_c(L)$.
• Embedding binary trees in hyperbolic half-planes to establish existence of infinite clusters for $\lambda\gg L^{-2}$, giving upper bounds on $\lambda_c(L)$.
• Adapting techniques from the theory of bi-infinite geodesics in the vacant set, together with percolation renormalization arguments, to show that below $\pi/(2L)$ the model has at least two disjoint infinite clusters—providing the lower bound for $\lambda_u(L)$.
• Using exponential decay of vacant set connectivity

$$\Pr[B^h(o,1) \stackrel{\mathcal{V}(\lambda,L)}{\longleftrightarrow} B^h(x,1)] \le C e^{-c\,d_h(o,x)}$$

for $\lambda \gtrsim L^{-1}$ to show uniqueness—any two infinite clusters must be separated by a long vacant path whose probability decays exponentially.

• Application of mapping theorems for Poisson processes, hyperbolic trigonometric identities, and branching process couplings.

These techniques leverage fundamental geometric properties of $\mathbb{H}^2$, especially the exponential growth of volume with radius, and interface with methodologies from stochastic geometry and continuum percolation (Broman et al., 17 Dec 2025).

5. Poisson Stick Models in Bayesian Nonparametrics and Process Theory

A distinct but technically related notion of "Poisson stick model" arises in Bayesian nonparametrics, especially as an underpinning of the stick-breaking constructions for random discrete measures—Dirichlet, beta, and gamma processes.

• The Poisson representation of the beta process (Paisley et al., 2011), gamma process (Roychowdhury et al., 2014), and more general Poisson–Kingman laws (James, 2013) is achieved via marked Poisson point processes whose atoms and jump sizes correspond to the "stick breaks." For example, the beta process CRM is constructed from a Poisson process on $\Theta\times(0,1]$ with Lévy intensity $c p^{-1}(1-p)^{c-1} dp\, B_0(d\theta)$.
• This viewpoint allows exact stick-breaking representations: atoms in each "round" are drawn from a Poisson process, each with independently drawn stick-break proportions; concatenating over all rounds and superposing yields the full process (Paisley et al., 2011, Roychowdhury et al., 2014).
• The Poisson stick perspective admits closed-form expressions for the mean measures, improved truncation error analysis, and efficient MCMC or variational inference schemes, essential for large-scale Bayesian models.

More generally, “Poisson stick” methodologies provide constructive, process-based frameworks for a variety of random discrete distributions—those arising from normalized subordinators—enabling tractable inference and theory in machine learning and stochastic process contexts (Paisley et al., 2011, James, 2013, Roychowdhury et al., 2014).

6. Implications, Applications, and Open Problems

The dual role of Poisson stick models—in both continuum percolation (especially in hyperbolic spaces) and the construction of random measures—underlines their centrality in modern probability and statistical modeling.

Principal implications and directions:

• For continuum percolation, the hyperbolic Poisson stick model demonstrates that percolation and uniqueness thresholds can decouple, leading to regimes where multiple infinite clusters coexist—a marked departure from the Euclidean paradigm (Broman et al., 17 Dec 2025).
• The precise critical scaling results have implications for statistical mechanics on non-Euclidean spaces, random geometric graph theory, and the study of non-amenable lattices.
• In Bayesian nonparametrics, Poisson stick constructions yield explicit, scalable algorithms for models requiring random discrete measures, supporting applications in clustering, nonparametric mixture modeling, and latent factor analysis (Paisley et al., 2011, Roychowdhury et al., 2014).
• Open mathematical problems include determining sharp constants for critical intensities in $\mathbb{H}^2$, extension to higher-dimensional or positive-width stick models, and a rigorous analysis of the Poisson–cylinder model's non-uniqueness regime (Broman et al., 17 Dec 2025).

The Poisson stick model remains a focal point at the interface of geometry, stochastic processes, and inference, with ongoing developments driven by both theoretical and applied considerations across mathematics, physics, and statistics.