Intersection probabilities for flats in hyperbolic space

Abstract

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 $(d-q+\gamma )$-flat in $M_K^d$ 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 γ-flat arising as the intersection of E with L. In contrast to the Euclidean case, the intersection $E\cap L$ can be empty with strictly positive probability. We determine this probability and the full distribution of $E\cap L$. Thereby, we elucidate crucial differences to the Euclidean case. Moreover, we study the limiting behavior 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.