Holder continuity of interfaces for scale-invariant Poisson stick soup

TL;DR

The paper studies the interface geometry of scale-invariant Poisson stick soups (IPS) in the plane, focusing on the subcritical regime and the Hölder regularity of interfaces between the covered and vacant regions. By building a natural family of exploration paths $\{\gamma_r^0\}_{r>0}$ and leveraging the Aizenman–Burchard framework, the authors establish tightness and Hölder regularity for subsequential limits, with upper bounds on the fractal dimensions of limit curves. A key technical achievement is Hypothesis H1, proven via uniform multi-arm bounds, a careful handling of dependence among sticks, and BK-type inequalities for marked Poisson processes; these results yield a rigorous description of the scaling geometry and regularity of the interface. The work also connects with scaling limits from the scaling ellipses model and with prior results on vacant-crossing properties, 1-arm events, and fractal dimensions in SA2, highlighting the critical geometry of IPS at $\alpha=2$ (critical short-to-long-range balance) and its potential scaling-limit behavior. Overall, the paper advances understanding of the geometry of scale-invariant continuum percolation models and the regularity of their interfaces, with implications for scaling limits and related percolation phenomena.

Abstract

We study the interface of covered and vacant sets in the subcritical phase of a scale-invariant Poisson stick soup on the plane. This model is a natural candidate for scaling limit of some planar models and has connections with long-range percolation on the plane with critical parameter $s=4$. We analyze a family of exploration paths on boxes and prove tightness for this family and Holder continuity for its limiting measures.

Figures (8)

• Figure 1: Depiction of a random curve from family $\{\gamma_r^0\}_{r > 0}$. For $r>0$ fixed, only finitely many sticks from $\mathcal{E}$ intersect the box. The exploration path $\gamma_r^0$ starts at the bottom-left corner and follows the vacant/covered interface without 'crossing' sticks, until reaching either the right or top sides. As $r \to 0$ more sticks are added to the picture and the exploration path is updated. See Section \ref{['sec:exploration_paths_IPS_ellipses']} for a precise definition.
• Figure 2: Regions $A_{0}(R,V)$ (shaded) and $A_{2}(R,V)$ (hatched) represent centers $z$ for which $(z,R,V)$ intersects $\partial B(1)$ precisely $0$ and $2$ times, respectively. The area of $A_0(R,V)$ is non-zero only for $2R < 2$, in which case it is given by $2 \cdot (\theta - \sin(2\theta)/2)$ (twice the area of a segment of the circle with angle $2\theta$, where $\theta = \arccos R$). The area of $A_{2}(R, V)$ when $2R \geq 2$ is $4R-\pi$, but when $2R < 2$ we must account for the superimposed area ($A_0(R,V)$). The area is then $4R-\pi + \bigl(2\theta - \sin(2\theta)\bigr)$.
• Figure 3: Traversing annulus $D(z; l_1, l_2)$ with $5$ separate segments of $\gamma$.
• Figure 4: Constructions on Proposition \ref{['prop:curve_crosses_stick']}
• Figure 5: Exploration paths for an ellipses model and its stick soup. The latter exploration path can be seen as an exploration path for ellipses of minor axis size equal to zero and can have self intersections.

Theorems & Definitions (64)

• Proposition 1.1: Nacu and Werner nacu2011random
• Theorem 1.2
• Theorem 1.3: H1 for exploration paths
• Remark 1.4
• Remark 1.5
• Lemma 2.1: PPP convergence
• proof
• Proposition 2.2
• proof
• Remark 2.3