Intersection probabilities for flats in hyperbolic space

Consider the d -dimensional hyperbolic space M K d 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 M K d passing through o and, independently of L , let E be a random ( d − q + γ ) -flat in M K d 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 behavior 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.