Intersection probabilities for flats in hyperbolic space

Consider the d-dimensional hyperbolic space MKᵈ of constant curvature $K<0$ and fix a point o playing the role of an origin. Let L be a uniform random q-dimensional totally geodesic submanifold (called q-flat) in MKᵈ passing through o and, independently of L, let E be a random (d-q+γ)-flat in MKᵈ which is uniformly distributed in the set of all (d-q+γ)-flats intersecting a hyperbolic ball of radius $u>0$ around o. We are interested in the distribution of the random γ-flat arising as the intersection of E with L. In contrast to the Euclidean case, the intersection E∩ L can be empty with strictly positive probability. We determine this probability and the full distribution of E∩ L. Thereby, we elucidate crucial differences to the Euclidean case. Moreover, we study the limiting behaviour as d↑∞ and also K↑ 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.