Seeing Through Hyperbolic Space: Visibility for $\lambda$-Geodesic Hyperplanes

Abstract

We study visibility from a fixed point in the presence of a Poisson process of $\lambda$--geodesic hyperplanes in a $d$-dimensional hyperbolic space. The family of $\lambda$--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 $\lambda\in[0,1]$. Namely, there is a critical intensity $\gamma_{\mathrm{crit}}>0$ such that the visible region is unbounded with positive probability for $\gamma < \gamma_{\mathrm{crit}}$ and almost surely bounded for $\gamma > \gamma_{\mathrm{crit}}$. For $d=2$ we establish almost sure boundedness also at criticality. The value for $\gamma_{\mathrm{crit}}$ is explicit and does not depend on $\lambda$. In the bounded phase, we show that the mean visible volume is identical with the known formula for $\lambda=0$. The key integral-geometric step is an explicit computation showing that the measure of $\lambda$-geodesic hyperplanes hitting a geodesic segment is a linear function of the length of the segment, independent of~$\lambda$.