Seeing Through Hyperbolic Space: Visibility for $\hat{I}»$-Geodesic Hyperplanes

Abstract

We study visibility from a fixed point in the presence of a Poisson process of $\hat{I}»$--geodesic hyperplanes in a $d$-dimensional hyperbolic space. The family of $\hat{I}»$--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 $\hat{I}»\in [0,1]$. Namely, there is a critical intensity ${\hat{I}}_{\mathrm{crit}}^3>0$ such that the visible region is unbounded with positive probability for ${\hat{I}}^3<{\hat{I}}_{\mathrm{crit}}^3$ and almost surely bounded for ${\hat{I}}^3>{\hat{I}}_{\mathrm{crit}}^3$. For $d=2$ we establish almost sure boundedness also at criticality. The value for ${\hat{I}}_{\mathrm{crit}}^3$ is explicit and does not depend on $\hat{I}»$. In the bounded phase, we show that the mean visible volume is identical with the known formula for $\hat{I}»=0$. The key integral-geometric step is an explicit computation showing that the measure of $\hat{I}»$-geodesic hyperplanes hitting a geodesic segment is a linear function of the length of the segment, independent of~$\hat{I}»$.