Intersection probabilities for flats in hyperbolic space

TL;DR

This paper investigates the probability and distribution of intersections between random flats in hyperbolic space, revealing differences from Euclidean geometry and identifying phase transitions as dimensions and curvature vary.

Contribution

It provides explicit formulas for intersection probabilities and distributions in hyperbolic space, highlighting key differences from Euclidean cases and analyzing asymptotic behaviors.

Findings

Intersection can be empty with positive probability in hyperbolic space.

Derived the full distribution of the intersection flat.

Identified a phase transition with three regimes as dimension and curvature change.

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.