  The Euclidean Directed Spanning Forest is a random forest in $\mathbb{R}^d$
introduced by Baccelli and Bordenave in 2007 and we introduce and study here
the analogous tree in the hyperbolic space. The topological properties of the
Euclidean DSF have been stated for $d=2$ and conjectured for $d \ge 3$ (see
further): it should be a tree for $d \in \{2,3\}$ and a countable union of
disjoint trees for $d \ge 4$. Moreover, it should not contain bi-infinite
branches whatever the dimension $d$. In this paper, we construct the Hyperbolic
Directed Spanning Forest (HDSF) and we give a complete description of its
topological properties, which are radically different from the Euclidean case.
Indeed, for any dimension, the hyperbolic DSF is a tree containing infinitely
many bi-infinite branches, whose asymptotic directions are investigated. The
strategy of our proofs consists in exploiting the Mass Transport Principle,
which is adapted to take advantage of the invariance by isometries. Using
appropriate mass transports is the key to carry over the hyperbolic setting
ideas developed in percolation and for spanning forests. This strategy provides
an upper-bound for horizontal fluctuations of trajectories, which is the key
point of the proofs. To obtain the latter, we exploit the representation of the
forest in the hyperbolic half space.
