Intersection probabilities for flats in hyperbolic space

Abstract

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.