High-intensity convergence to the parabolic hull tessellation

Determine whether, under the high-intensity scaling \(\gamma=\lambda^{d+1}\) as \(\lambda\to\infty\), the transformed downward-paraboloid process associated with the hyperbolic horoball model converges to the tessellation induced by the parabolic hull process.

Background

The paper establishes the low-intensity scaling limit $\gamma=\lambda^{d-1}$ and proves convergence of the associated projected Poisson–Laguerre tessellations to a classical Poisson–Delaunay tessellation. In contrast, the authors identify a different high-intensity regime, $\gamma=\lambda^{d+1}$, which is expected to recover the Euclidean parabolic-hull geometry associated with random polytopes near a smooth boundary.

The authors provide limiting cell-intensity constants in this high-intensity regime, including $c_2=(2/3)^{1/3}\Gamma(5/3)$ and $c_3=35/24$, but do not establish convergence of the underlying transformed process or tessellation. Thus, proving the stated process-level convergence remains unresolved in the paper.