  Consider the $d$-dimensional hyperbolic space $\mathbb{M}_K^d$ of constant
curvature $K<0$ and fix a point $o$ playing the role of an origin. Let
$\mathbf{L}$ be a uniform random $q$-dimensional totally geodesic submanifold
(called $q$-flat) in $\mathbb{M}_K^d$ passing through $o$ and, independently of
$\mathbf{L}$, let $\mathbf{E}$ be a random $(d-q+\gamma)$-flat in
$\mathbb{M}_K^d$ which is uniformly distributed in the set of all
$(d-q+\gamma)$-flats intersecting a hyperbolic ball of radius $u>0$ around $o$.
We are interested in the distribution of the random $\gamma$-flat arising as
the intersection of $\mathbf{E}$ with $\mathbf{L}$. In contrast to the
Euclidean case, the intersection $\mathbf{E}\cap \mathbf{L}$ can be empty with
strictly positive probability. We determine this probability and the full
distribution of $\mathbf{E}\cap \mathbf{L}$. Thereby, we elucidate crucial
differences to the Euclidean case. Moreover, we study the limiting behaviour as
$d\uparrow\infty$ and also $K\uparrow 0$. Thereby we obtain a phase transition
with three different phases which we completely characterize, including a
critical phase with distinctive behavior and a phase recovering the Euclidean
results. In the background are methods from hyperbolic integral geometry.
