Seeing Through Hyperbolic Space: Visibility for $λ$-Geodesic Hyperplanes

Abstract

We study visibility from a fixed point in the presence of a Poisson process of $λ$--geodesic hyperplanes in a $d$-dimensional hyperbolic space. The family of $λ$--geodesic hyperplanes interpolates between totally geodesic hyperplanes and horospheres. Our main result establishes a universality principle for this model: we prove that the fundamental visibility properties are invariant with respect to the parameter $λ\in[0,1]$. Namely, there is a critical intensity $γ_{\mathrm{crit}}>0$ such that the visible region is unbounded with positive probability for $γ< γ_{\mathrm{crit}}$ and almost surely bounded for $γ> γ_{\mathrm{crit}}$. For $d=2$ we establish almost sure boundedness also at criticality. The value for $γ_{\mathrm{crit}}$ is explicit and does not depend on $λ$. In the bounded phase, we show that the mean visible volume is identical with the known formula for $λ=0$. The key integral-geometric step is an explicit computation showing that the measure of $λ$-geodesic hyperplanes hitting a geodesic segment is a linear function of the length of the segment, independent of~$λ$.

Figures (10)

• Figure 1: $\lambda=0, \gamma=2<\gamma_{\mathrm{crit}}$
• Figure 2: $\lambda=0.5, \gamma=2<\gamma_{\mathrm{crit}}$
• Figure 3: $\lambda=1, \gamma=2<\gamma_{\mathrm{crit}}$
• Figure 4: $\lambda=0, \gamma=\pi=\gamma_{\mathrm{crit}}$
• Figure 5: $\lambda=0.5, \gamma=\pi=\gamma_{\mathrm{crit}}$

Theorems & Definitions (27)

• Theorem 1.1
• Lemma 3.1
• proof
• Proposition 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Remark 1
• Proposition 3.5