  It is well known that the orbit of a lattice in hyperbolic $n$-space is
uniformly distributed when projected radially onto the unit sphere. In the
present work, we consider the fine-scale statistics of the projected lattice
points, and express the limit distributions in terms of random hyperbolic
lattices. This provides in particular a new perspective on recent results by
Boca, Popa, and Zaharescu on 2-point correlations for the modular group, and by
Kelmer and Kontorovich for general lattices in dimension $n=2$.
