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Flammant, Lucas

The directed spanning forest in the hyperbolic space. (English) Zbl 1536.60013

Ann. Appl. Probab. 34, No. 1A, 46-102 (2024).
Summary: The Euclidean directed spanning forest (DSF) is a random forest in \(\mathbb{R}^{d}\) introduced by F. Baccelli and C. Bordenave in [Ann. Appl. Probab. 17, No. 1, 305–359 (2007; Zbl 1136.60007)] 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 \(d \geq 3\) (see further): it should be a tree for \(d \in \{2, 3\}\) and a countable union of disjoint trees for \(d \geq 4\). 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.

Cited in 1 Document

MSC:

60D05 Geometric probability and stochastic geometry
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
51M10 Hyperbolic and elliptic geometries (general) and generalizations
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

Keywords:

continuum percolation; hyperbolic space; stochastic geometry; random geometric tree; hyperbolic directed spanning forest; mass transport principle; Poisson point processes

Citations:

Zbl 1136.60007
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References:

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