The Poisson stick model in hyperbolic space

TL;DR

The article analyzes the Poisson stick model in ${\mathbb{H}}^2$ with sticks of fixed length $L$, establishing two distinct phase-transition scalings: the percolation threshold scales as $\lambda_c(L)=\Theta(L^{-2})$ and the uniqueness threshold scales as $\lambda_u(L)=\Theta(L^{-1})$ as $L\to\infty$. The authors derive these results via a combination of techniques, including coupling to Galton–Watson branching processes for both upper and lower bounds, a restricted Poisson process framework on a half-line, and percolation arguments in a hyperbolic half-plane, complemented by geometric analysis of hyperbolic distances and geodesics. They show that in contrast to Euclidean space where $\lambda_c^E(L)$ scales like $L^{-2}$ and $\lambda_u^E(L)$ is conjectured to coincide, the hyperbolic setting preserves the Euclidean scaling for the percolation transition but alters the scaling for the uniqueness transition. The work provides explicit constants in the bounds and situates the hyperbolic results within a broader comparison to Euclidean stick percolation, offering a foundational framework for hyperbolic continuum percolation with line-segment objects.

Abstract

In this paper we study the Poisson stick model in two dimensional hyperbolic space $\mathbb{H}^2,$ where the sticks all have length $L.$ Typically, percolation models in hyperbolic space undergo two phase transitions as the intensity $λ$ varies, namely the percolation phase transition and the uniqueness phase transition. For the Poisson stick model, the critical intensities at which these transitions occur will depend on $L$, and in this paper we study the asymptotic behavior of these critical points as $L\to \infty.$ Our main results show that the critical point for the percolation phase transition scales like $L^{-2},$ while the critical point for the uniqueness phase transition scales like $L^{-1}.$ Comparing these results to the analogous results in Euclidean space show that the behavior of the percolation phase transition is the same in these two settings, while the uniqueness phase transition scales differently.

Figures (7)

• Figure 3.1: Figure \ref{['fig:restrictA']} shows $g_0^+$ and the first three intersection points of sticks from $\omega^{\lambda,L}$ which intersect it. The distance to the first intersection point is $\rho'_1$ and this is marked along with the three angles $\varphi_1,\varphi_2$ and $\varphi_3$. The solid blue line segments in Figure \ref{['fig:restrictA']} are the sets of points along the geodesics (dashed) which are at distance at most $L/2$ from the intersection points. The centerpoints $x_1,x_2$ and $x_3$ are chosen uniformly along these blue line segments. Figure \ref{['fig:restrictB']} shows the resulting sticks in red.
• Figure 3.2: The triangle ${\mathcal{T}}$.
• Figure 3.3: An illustration of why $\Delta\leq \epsilon$.
• Figure 4.1: The horizontal black line is $l_0.$ In red is $l_1,$ while $g_1$ and $l[o,e_2]$ are marked in blue, and $l[(\rho',0),e_1]$ in green. We can also see the half-planes $H_0$ and $H_1$ and the angles $\varphi,\theta_2,\alpha$ mentioned in the proof of Lemma \ref{['lemma:half-planes']}. The large black arc is part of $\partial {\mathbb{H}}^2.$ The line $l[o,e_1]$ is not in this picture.
• Figure 4.2: In this picture we can see $l_1,l_{-1},l_{1,1}$ and $l_{1,-1}$ all marked in red. In addition, we see the corresponding half-planes $H_1,H_{1,1}$ and $H_{1,-1}.$ The half-plane $H_{-1}$ is outside of the picture. The large black arc is part of $\partial {\mathbb{H}}^2.$

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Proposition 4.1
• proof