Lucas Flammant

Hyperbolic Radial Spanning Tree

Keywords: continuum percolation, hyperbolic space, stochastic geometry, random geometric tree, Radial Spanning Tree, Poisson point processes AMS 2010 Subject Classification: Primary 60D05, 60K35, 82B21

We define and analyze an extention to the d-dimensional hyperbolic space of the Radial Spanning Tree (RST) introduced by Baccelli and Bordenave in the two-dimensional Euclidean space (2007). In particular, we will focus on the description of the infinite branches of the tree. The properties shown for the two-dimensional Euclidean RST are extended to the hyperbolic case in every dimension: almost surely, every infinite branch admits an asymptotic direction and each asymptotic direction is reached by at least one infinite branch. Moreover, the branch converging to any deterministic asymptotic direction is unique almost surely. Our strategy mainly relies on the two following ingredients. First, the hyperbolic metric allows us to control fluctuations of the branches in the hyperbolic DSF without using planarity arguments. Then, we couple the hyperbolic RST with the hyperbolic DSF introduced and studied in Flammant 2019.

Introduction

Geometric random trees are well studied in the literature since it interacts with many other fields, such as communication networks, particles systems or population dynamics. Several works have established scaling limits for two-dimensional radial trees [START_REF] Coupier | Directed, cylindric and radial brownian webs[END_REF][START_REF] Coletti | The radial brownian web[END_REF] and translation invariant forests [START_REF] Coupier | The 2d-directed spanning forest converges to the brownian web[END_REF][START_REF] Roy | Random directed forest and the brownian web[END_REF][START_REF] Gangopadhyay | Random oriented trees: a model of drainage networks[END_REF]. In addition, random spanning trees appear in the context or first passage percolation [START_REF] Howard | Geodesics and spanning trees for Euclidean first-passage percolation[END_REF]. A complete introduction to geometric random graphs is given in Penrose 2003 [START_REF] Penrose | Random geometric graphs[END_REF].

The Radial Spanning Tree (RST) is a random tree whose introduction in the two-dimensional Euclidean space has been motivated by applications for communication networks [START_REF] Baccelli | The radial spanning tree of a poisson point process[END_REF]. The set of vertices is given by a homogeneous Poisson Point Process (PPP) N of intensity λ in R 2 . The RST rooted at the origin 0 is the graph obtained by connecting each point z ∈ N to its parent A(z), defined as the closest point to z among all points z ∈ N ∪ {0} that are closer to the origin than z. This defines a random tree rooted at the origin with a radial structure. An infinite backward path is defined as a sequence of Poisson points (z n ) n≥0 ∈ (N ∪ {0})

N with z 0 = 0 and z n = A(z n+1 ) for any n ≥ 0. Given an infinite path, we will say that the forward direction is towards 0 and the backward direction is towards infinity.

The topological properties of the bi-dimensional Euclidean RST are well-understood. Baccelli and Bordenave showed that almost surely, any infinite backward path admits an asympotic direction; moreover, a.s., every asymptotic direction is reached by at least one infinite backward path and there exists a.s. a unique infinite path in any given deterministic asymptotic direction [START_REF] Baccelli | The radial spanning tree of a poisson point process[END_REF]. These results on the infinite paths are completed by Baccelli, Coupier & Tran [START_REF] Baccelli | Semi-infinite paths of the two-dimensional radial spanning tree[END_REF].

For any integer d ≥ 2, the hyperbolic space H d is a d-dimensional Riemannian manifold with constant negative curvature, that can be chosen equal to -1 without loss of generality. It admits a set of ideal boundary points ∂H d , and H d := H d ∪ ∂H d denotes the hyperbolic space endowed with its boundary. It is a non-amenable space, i.e. the measure of the boundary of a large subset is not negligible with respect to its volume. The hyperbolic space is defined in more details in [START_REF] Cannon | Hyperbolic geometry[END_REF] and [START_REF] Paupert | Introduction to hyperbolic geometry[END_REF].

There is a growing interest for the study of random models in a hyperbolic setting. Benjamini and Schramm establish percolation results on regular tilings and Voronoï tesselation in the hyperbolic plane [START_REF] Benjamini | Percolation in the hyperbolic plane[END_REF]. Mean characteristics of the Poisson-Voronoï tessellation have also been considered in a general Riemannian manifold by Calka et al. [START_REF] Calka | Mean asymptotics for a poisson-voronoi cell on a riemannian manifold[END_REF]. This interest is explained by at least two reasons. First, hyperbolic random graphs are well-fitted to modelize social networks [START_REF] Boguná | Sustaining the internet with hyperbolic mapping[END_REF]. In addition, strong differences have been noticed for properties of random models depending whether they are considered in an Euclidean or hyperbolic setting. Indeed, some hyperbolic random graphs admits a non-degenerate regime with infinitely many unbounded components in the hyperbolic space [START_REF] Tykesson | The number of unbounded components in the poisson boolean model of continuum percolation in hyperbolic space[END_REF][START_REF] Hutchcroft | Percolation on hyperbolic graphs[END_REF], which is generally not the case in the Euclidean space. In addition, behaviours of non-amenable spaces are well studied in a discrete context [START_REF] Benjamini | Percolation beyond zd, many questions and a few answers [mr1423907[END_REF][START_REF] Lyons | Probability on trees and networks[END_REF][START_REF] Pete | Lecture notes for a graduate course[END_REF].

Thus it is natural to consider and study the hyperbolic RST, which we define in the same way as the Euclidean RST. A simulation of the two-dimensional hyperbolic RST is given in Figure 1. In this paper, we extend the results of Baccelli and his coauthors to hyperbolic geometry in every dimension. Here is our main result: Theorem 1.1. For any dimension d ≥ 1 and any intensity λ, the following happens:

(i) almost surely, any infinite backward path (z n ) n∈N admits an asymptotic direction, i.e. there exists z ∞ ∈ ∂H d+1 such that lim n→∞ z n = z ∞ (in the sense of the topology of H d+1 );

(ii) almost surely, for any I ∈ ∂H d+1 , there exists an infinite backward path (z n ) with asymptotic direction I (i.e. such that lim n→∞ z n = I);

(iii) for any deterministic boundary point I ∈ ∂H d+1 , the path with asymptotic direction I is almost surely unique;

(iv) the set of boundary points with two infinite backward paths is dense in ∂H d+1 ;

(v) this set is moreover countable in the bi-dimensional case (i.e. d = 1).

Establishing the results announced in Theorem 1.1 in every dimension constitutes the main originality of this paper. For the two reasons explained further, the proofs of Baccelli and Bordenave in the 2D-Euclidean setting [START_REF] Baccelli | The radial spanning tree of a poisson point process[END_REF] cannot be generalised to higher dimensions.

In both contexts R 2 and H d+1 , for any d ≥ 1, the proofs of (i), (ii), (iv) and (v) of Theorem 1.1 follow the strategy of Howard and Newman [START_REF] Howard | Geodesics and spanning trees for Euclidean first-passage percolation[END_REF], which is to show that the tree is straight, that is, the descendents subtree of a vertex far from the origin is included in a thin cone. To prove that the 2D-Euclidean RST is straight, Baccelli and Bordenave used a translation invariant model derived from the RST: the Directed Spanning Forest (DSF), which constitutes a local approximation of the RST far from the origin [START_REF] Baccelli | The radial spanning tree of a poisson point process[END_REF]. They exploit the theory of Markov chains to upper-bound fluctuations of trajectories in the DSF and then, they deduce the straightness of the RST via planarity. This strategy cannot be generalised to higher dimensions. However, in H d , we manage to control the angular deviations of branches in the RST without resorting to an auxiliary model, which required planarity in the Euclidean setting. The hyperbolic metric guarantees that angular deviations decay exponentially fast with the distance to the origin, which is strong enough to show straightness.

In addition, in the Euclidean context, the uniqueness part (point (iii) in Theorem 1.1) is only proved in dimension 2 since it strongly uses planarity [START_REF] Howard | Geodesics and spanning trees for Euclidean first-passage percolation[END_REF][START_REF] Baccelli | The radial spanning tree of a poisson point process[END_REF], and the strategy of proof cannot be generalised to higher dimensions. To prove (iii) in H d , our strategy consists in exploiting the link existing between the hyperbolic RST and another random graph, the hyperbolic DSF, defined and studied in Flammant 2019 [START_REF] Coupier | The 2d-directed spanning forest is almost surely a tree[END_REF], which is the hyperbolic counterpart of the Euclidean DSF used by Baccelli and Bordenave. Roughly speaking, the hyperbolic DSF can be defined as the limit of the hyperbolic RST when the origin point tends to an ideal boundary point. Similarly to the Euclidean setting, it constitutes a local approximation of the RST far from the origin. The proof of (iii) exploits the coalescence of the hyperbolic DSF (i.e. it is almost surely a tree) [12, Theorem 1.1], which is a non-trivial fact obtained by exploiting the mass-transport principle, and a local coupling between the two models.

After defining the hyperbolic RST and giving its basic properties, we define two quantities that encode angular fluctuations along trajectories, the Cumulative angular Forward Deviations (CFD) and the Maximal Backward Deviations (MBD). We then establish upper-bounds of these quantities: first, we upper-bound the Maximal Backward Deviations in a thin annulus of width δ > 0 (Proposition 2.5) and then we deduce a global control of MBD in the whole space (Proposition 2.6), that roughly says that angular deviations decay exponentially fast with the distance to the origin. From this upper-bound, we deduce that the RST is straight in the sense of Howard & Newman (Proposition 2.7). The points (i), (ii), (iv) and (v) in Theorem 1.1 can be deduced from straightness and the upper-bound of MBD given by Proposition 2.6. The point (iii) (the uniqueness part) is done by exploiting a local coupling existing between the RST and the DSF far from the origin.

The rest of paper is organized as follows. In Section 2, we set some reminders of hyperbolic geometry and we define the hyperbolic RST. Then, we give its basic properties and a road-map of the proofs. We also announce the upper-bounds of angular deviations (Propositions 2.5 and 2.6) and the straightness property (Proposition 2.7). The proof of Theorem 1.1 is done in Section 3. Proposition 2.5 is proved in Section 4 and the proofs of Propositions 2.6 and 2.7 are done in Section 5.

Definitions, notations and basic properties

We denote by N the set of non-negative integers and by N * the set of positive integers. In the rest of the paper, c (resp. C) will be some small (resp. large) constant whose value can change from a line to another.

The hyperbolic space

We refer to [START_REF] Cannon | Hyperbolic geometry[END_REF] or [START_REF] Paupert | Introduction to hyperbolic geometry[END_REF] for a complete introduction to hyperbolic geometry. For d ∈ N * , the (d + 1)dimensional hyperbolic space, denoted by H d+1 , is a (d + 1)-dimensional Riemannian manifold of constant negative curvature -1 that can be defined by several isometric models. One of them is

I = {(x 1 , ..., x d+1 ) ∈ R d+1 , x 2 1 + ... + x 2 d+1 < 1} (2.1)
endow with the following metric:

ds 2 I := 4 dx 2 1 + ... + dx 2 n+1 1 -x 2 1 -... -x 2 d+1 . (2.2)
This model is rotation invariant. The metric becomes smaller as we get closer to the boundary unit sphere ∂I, and this boundary is at infinite distance from the center 0.

The volume measure on (I, ds 2 I ), denoted by Vol I , is given by

dVol I = 2 d+1 dx 1 ...dx d+1 1 -x 2 1 -... -x 2 d+1 . (2.3)
In this model (I, ds 2 I ), the geodesics are of two types: the diameters of I and the arcs that are perpendicular to the boundary unit sphere ∂I. We refer to discussion [7, P.80] for a proof. Moreover, this model is conformal, which means that the hyperbolic angle between two geodesics corresponds to their Euclidean angle in the open-ball representation.

An important fact about hyperbolic geometry is that all points and all directions play the same role. More precisely, H d+1 is homogeneous and isotropic. It means that the group of isometries of H d+1 acts transitively on the unit tangent bundle of H d+1 : given two points x, y ∈ H d+1 and two unit tangent vectors u ∈ T x H d+1 , v ∈ T y H d+1 , there exists an isometry g of H d+1 such that g(x) = y and that pushes forward u on v. The notations T x , T y and the vocabulary relating to Riemannian geometry are defined in [START_REF] Lee | Riemannian manifolds: an introduction to curvature[END_REF]. We refer to [START_REF] Paupert | Introduction to hyperbolic geometry[END_REF], Proposition 1.2.1 p.5, for a proof. ] the geodesic between z 1 and z 2 . Moreover, we set the notations:

[z 1 , z 2 [:= [z 1 , z 2 ]\{z 2 }, ]z 1 , z 2 ] := [z 1 , z 2 ]\{z 1 }, ]z 1 , z 2 [\({z 2 } ∪ {z 2 }).
Let us denote by [z 1 , z 2 ) (resp. (z 1 , z 2 ]) the semi-geodesic passing threw z 2 (resp. z 1 ) and ending at z 1 (resp. z 2 ). For z 1 , z 2 , z 3 ∈ H d+1 , z 1 z 2 z 3 is the measure of the corresponding (non-oriented) hyperbolic angle. For any subset B ⊂ H d+1 , B denotes the closure of B in H d+1 . For any point z ∈ H d+1 and θ > 0, Cone(z, θ) := {z ∈ H d+1 , z0z ≤ θ} is defined as the cone of apex 0 and aperture θ (if θ ≥ π then Cone(z, θ) is the hole space H d+1 ). In addition, for r > 0 and z ∈ H d+1 , we define

B S(r) (z, θ) := Cone(z, θ) ∩ S(r). (2.4)
Let 0 ∈ H d+1 be some arbitrary origin point (it can be thought as the center of the ball in the open-ball representation), which will plays the role of the root of the RST. For z ∈ H d+1 and r > 0, we denote by B(z, r) := {z ∈ H, d(z, z ) < r} (resp. S(z, r) := {z ∈ H, d(z, z ) = r}) the hyperbolic ball (resp. sphere) centered at z of radius r, and we set B(r) := B(0, r) (resp. S(r) := s(0, r)). For x ∈ R d and r > 0, let us also denote by B R d (x, r) := {x ∈ R d , x -x < r} the Euclidean ball centered at x of radius r.

Let us denote by S d the unit Euclidean sphere in R d+1 and by ν its d-dimensional volume measure. Since the RST is a rooted graph, a convenient way to represent points in H d+1 is to use polar coordinates. Recall that 0 is the origin point. For any point z ∈ H d+1 , we denote by z = (r; u) its polar coordinates w.r.t. 0: r is its distance to 0 and u ∈ U T 0 H d+1 S d is its direction (U T 0 H d+1 is the unitary tangent space of 0 in H d+1 ). In polar coordinates, the volume measure Vol is given by dVol(r; u) = sinh(r) d dr dν(u).

(2.5)

A direct consequence of this is that the volume of a ball of radius r is given by:

Vol(B(r)) = r 0 sinh(r) d dr dν(S d ) = S d sinh(r) d e dr when r → ∞, (2.6)
where S d is the d-dimensional volume of S d .

The hyperbolic law of cosines [24, p.13] is a well adapted tool to compute distances using polar coordinates. Given z 1 = (r 1 ; u 1 ), z 2 = (r 2 , u 2 ) ∈ H d+1 , the hyperbolic law of cosines gives, cosh d(z 1 , z 2 ) = cosh(r 1 ) cosh(r 2 ) -u 1 , u 2 sinh(r 1 ) sinh(r 2 ).

(2.7)

The hyperbolic RST

In the rest of the paper, the dimension d and the intensity λ > 0 are fixed. Let N be a homogeneous PPP of intensity λ in H d+1 . The definition of the hyperbolic RST is similar to the Euclidean case.

The set of vertices is N ∪ {0}. Each vertex z ∈ N is connected to the closest Poisson point among those that are closer to the origin than z:

Definition 2.1 (Radial Spanning Tree in H d+1 ). For any z = (r; u) ∈ N , the parent of z is defined as

A(z) := argmin z ∈N ∩B(r) d(z , z).
We call Radial Spanning Tree (RST) in H d+1 rooted at 0 the oriented graph (V, E) where

V := N ∪ {0}, E := {(z, A(z)), z ∈ N }.
It is possible to assume that N ∪ {0} does not contain isosceles triangles, since this event has probability 1. Thus the ancestor A(z) is well-defined.

For z ∈ N ∪ {0} and k ∈ N, let us define By definition of the parent, B + (z) ∩ N = ∅ for all z ∈ N .

A (k) (z) = A • .. • A k times and A (-k) (z) = {z ∈ N , A (k) (z ) = z} (in particular A (-1) (z) is the set of daughters of z).
Definition 2.1 does not specify the shape of edges, but the results announced in Theorem 1.1 only concern the graph structure of the hyperbolic RST, so their veracity does not depend on the geometry of edges. It is more natural to represent edges with hyperbolic geodesics, but we do another choice which will appear more convenient for the proofs. Given

z 1 = (r 1 ; u 1 ), z 2 = (r 2 ; u 2 ) ∈ H d+1 such that 0 / ∈ [z 1 , z 2 ],
we define a path [z 1 , z 2 ] * , in an isotropic way, verifying the two following conditions: i) the distance to the origin 0 is monotonous along the path [z 1 , z 2 ] * , ii) the distance to z 1 is also monotonous along this path.

It will be necessary for the proofs that the shape of edges satisfy conditions (i) and (ii), and the geodesic [z 1 , z 2 ] does not verify condition (i) in general. Since 0 / ∈ [z 1 , z 2 ], u 1 and u 2 are not antipodal, thus one can consider the unique geodesic path γ u1,u2 : [0, 1] → U T 0 H d+1 on the sphere with constant speed connecting u 1 to u 2 . Hence we define the path [z 1 , z 2 ] * as

[0, 1] → H d+1 t → (1 -t)r 1 + tr 2 ; γ u1,u2 (φ r1,r2, u1,u2 (t)) , (2.8)
where φ r1,r2, u1,u2 : [0, 1] → [0, 1] is defined as:

φ r1,r2, u1,u2 (t) := 1 u 1 , u 2 arccos (1 -t) sinh(r 1 ) + t cos( u 1 u 2 ) sinh(r 2 ) sinh((1 -t)r 1 + tr 2 ) .
This function φ r1,r2, u1,u2 is built to ensure that the distance to the origin z 1 is monotonous along the path [z 1 , z 2 ] * . Indeed, by the hyperbolic law of cosines (2.7),

cosh d z 1 , ((1 -t)r 1 + tr 2 ; γ u1,u2 (φ(t)) = cosh(r 1 ) cosh((1 -t)r 1 + tr 2 ) -cos(φ(t)( u 1 , u 2 )) sinh(r 1 ) sinh((1 -t)r 1 + tr 2 ) = t cosh(r 1 ) cosh(r 2 ) -cos( u 1 , u 2 ) sinh(r 1 ) sinh(r 2 ) is non-decreasing in t. We define [z 1 , z 2 [ * := [z 1 , z 2 ] * \{z 2 } and ]z 1 , z 2 ] * := [z 1 , z 2 ] * \{z 1 }. It is possible to assume that N does not contain two points z 1 , z 2 such that 0 ∈ [z 1 , z 2 ]
since this event has probability 1. Let us now define the random set RST by connecting each point z ∈ N to A(z) by the path [z, A(z)] * :

RST := z∈N [z, A(z)[ * .
It may exists some points z belonging to several paths

[z 1 , A(z 1 )[ * , ..., [z k , A(z k )[ * ; in that case, z is counted with multiplicity k in RST. Formally, RST = z∈N [z, A(z)[ * ×{z} ⊂ H d+1 × H d+1 ., i.e. an element z = (z , z ) ∈ RST is the data of a point z ∈ H d+1 and an edge [z , A(z )] * containing z . For z = (z , z ) ∈ RST, we define z ↓ = z , z ↑ = A(z ).
In the following, we will commit an abuse of notations by considering that RST ⊂ H d+1 and identifying an element z = (z , z ) ∈ RST to the corresponding point z ∈ H d+1 . Given z ∈ RST, let n := min{k ≥ 0, A (k) (z ↑ ) = 0} be the number of steps required to reach the origin from z ↑ ; we define the trajectory from z as

π(z) := [z, z ↑ ] * ∪ n-1 k=0 A (k) (z ↑ ), A (k+1) (z ↑ ) * .
For r > 0, we define the level r as L r := RST ∩ S(r).

For 0 < r ≤ r and for z ∈ L r , the ancestor at level r of z , denoted by A r r (z ) is the intersection point of π(z ) and S(r). For 0 < r ≤ r and for z ∈ L r , the set of descendents at level r is defined as D r r (z) := {z ∈ L r , z ∈ π(z )} (we extend the notation for z / ∈ L r by setting D r r (z) := ∅). For z = (r; u) ∈ RST, the descendents subtree of z is defined as D(z) := r ≥r D r r (z). In addition, we call infinite backward path a sequence

(z i ) i∈N ∈ H d+1 N such that z 0 = 0 and z i = A(z i+1 ) for all i ≥ 0.
Let us end this section with basic properties about RST proved in Appendix 6.

Proposition 2.2. The RST is a tree and it has finite degree a.s. Moreover, in the bi-dimensional case (d = 1), the representation of the RST obtained by connecting each vertex z ∈ N to its parent

A(z) by the geodesic [z, A(z)] (instead of [z, A(z)] * ) is planar, i.e. their is no two points z 1 , z 2 ∈ N such that [z 1 , A(z 1 )] ∩ [z 2 , A(z 2 )] = ∅.

Sketch of proofs

In order to prove our main result (Theorem 1.1), the key point is to upper-bound angular deviations of trajectories. We first introduce two quantities, the Cumulative Forward angular Deviations (CFD) and Maximal Backward Deviations (MBD) to quantify those fluctuations.

0 r r z 2 ∈ L r z 1 ∈ L r A r r (z 1 ) D r r (z 2 )
       z 0z if z ↓ = z ↓ , z 0z ↑ + n-1 k=0 A (k) (z ↑ )0A (k+1) (z ↑ ) + z ↓ 0z else,
where n is the unique non negative integer such that

A (n) (z ↑ ) = z ↓ .
Definition 2.4 (Maximal Backward angular Deviations). Let 0 < r ≤ r and z ∈ S(r). We define the Maximal Backward angular Deviations between levels r and r as

MBD r r (z) :=    0 if z / ∈ RST, sup r ∈[r,r ] max z ∈D r r (z) CFD r r (z ) if z ∈ RST.
We extend the definition to r = ∞ by setting:

MBD ∞ r (z) := lim r →∞ MBD r r (z),
the limit exists since r → MBD r r (z) is non-decreasing.

These quantities will be upper-bounded in two steps. First, a percolation argument is used to control angular deviations in any annulus of width δ > 0 for some small δ > 0 (Proposition 2.5) and then we deduce a global control of angular deviations (Proposition 2.6). Recall that B S(r) (•, •) is defined in (2.4). Proposition 2.5. There exists δ > 0 such that, for any p ≥ 1, there exists C = C(d, p) > 0 such that for any r > 0, θ ≥ 0 and any direction u ∈ S d ,

E   z∈B S(r) (u,θ)∩RST MBD r+δ r (z) p   ≤ Cθ d e r(d-p) , (2.9)
Proposition 2.6. For any p large enough, there exists some constant C fl > 0 such that, for any

0 < r 0 < ∞, A > 0 and any direction u ∈ S 1 , E   z∈B S(r 0 ) (u,Ae -r 0 )∩RST MBD ∞ r0 (z) p   ≤ C fl A d e -r0p .
These controls of angular deviations will be first used to show that the RST is straight (Proposition 2.7). The straightness property is the key to show (i), (ii) and (iv) in Theorem 1.1.

Proposition 2.7 (straightness property). Almost surely, the following happens. For any ε > 0, there exists some R 0 > 0, such that, for any radius r 0 ≥ R 0 , for any z ∈ RST with d(0, z) ≥ r 0 , the descendents subtree D(z) is contained in a cone of apex 0 and aperture e -(1-ε)r0 , i.e. for any

z , z ∈ D(z), z 0z ≤ e -(1-ε)r0 .
The proof of (iii) in Theorem 1.1 exploits the controls of angular deviations (Proposition 2.6) and the link existing between the RST and the hyperbolic Directed Spanning Forest introduced in [START_REF] Coupier | The 2d-directed spanning forest is almost surely a tree[END_REF]: the DSF approximates locally the RST far from the origin. The unicity of the infinite backward path with some given deterministic asymptotic direction has been shown for the DSF [START_REF] Flammant | The directed spanning forest in the hyperbolic space[END_REF], and the local coupling existing between the two models permits to show that this property remains true for the RST.

Proof of Theorem 1.1

Here we assume that Propositions 2.6 and 2.7 are proved and we show that it implies Theorem 1.1.

3.1

The existence part: proof of (i),(ii),(iv) and (v)

It will be shown in this section that any infinite backward path admits an asymptotic direction and that any ideal boundary point is the asymptotic direction of an infinite backward path (point (i) and (ii) in Theorem 1.1. The strategy consists, in exploiting the straightness property (Proposition 2.7):

Let (z n ) be an infinite backward path, we prove that (z n ) admits an asymptotic direction. For n ≥ 0, let us decompose z n in polar coordinates:

z n = (r n ; u n ). Proposition 2.7 immediately implies that the sequence (u n ) n≥0 is a Cauchy sequence in U T 0 H d+1 S d , thus it converges, so (z n ) converges to some boundary point z ∞ ∈ ∂H d+1 .
Let Ψ = {lim n→∞ z n , (z n ) is an infinite backward path} ⊂ ∂H d+1 be the set of asymptotic directions reached by at least one infinite backward path. In order to prove that Ψ = ∂H d+1 , we proceed in two steps: we first show that Ψ is dense in ∂H d+1 , then we show that Ψ is closed in ∂H d+1 . Since the RST is an infinite tree with finite degree a.s. (Proposition 2.2), there exists an infinite backward path from 0 and the corresponding infinite backward path converges to an ideal boundary point by the previous paragraph, thus Ψ = ∅ almost surely.

We denote by Stab(0) the set of isometries that fix 0, in particular it contains rotations centered at 0. Let B be an open subset of ∂H d+1 . Since ∂H d+1 S d is compact, there exists finitely many isometries γ 1 , ..., γ k ∈ Stab(0) such that i=1,...,k γ i B = ∂H d+1 . The random set RST is invariant in distribution by Stab(0), so the events {Ψ ∩ γ i B = ∅} all have the same probability. Since Ψ = ∅ almost surely, P i=1,...,k {Ψ ∩ γ i B = ∅} = 1 therefore P(Ψ ∩ B = ∅) > 0. In addition, for any neighborhood Φ ⊂ H d+1 of B, the event {Ψ ∩ B = ∅} is entirely determined by N ∩ Φ, therefore it has probability 0 or 1. Thus Ψ ∩ B = ∅ almost surely. Since the topology on ∂H d+1 admits a countable basis, Ψ is almost surely dense in ∂H d+1 .

It remains to show that Ψ is a closed subset of ∂H d+1 . Let I ∈ Ψ (recall that Ψ is the closure of Ψ in H d+1 ). We construct by induction a sequence (z n ) n≥1 ∈ N N * is an infinite backward path such that, for any i ∈ N, I ∈ D(z i ). Suppose 0, ..., z i-1 already defined such that z j = A(z j+1 ) for 0 ≤ j ≤ i -2 and I ∈ D(z i-1 ). Since the vertex z i-1 has finitely many daughters, there exists some z ∈ A (-1) (z i-1 ) such that I ∈ D(z). Thus we define z i as such a z.

We now use straightness to show that the infinite backward path (z n ) constructed above converges to I (and thus I ∈ Ψ). This infinite backward path converges to some I ∈ ∂H d+1 by (i). Let ε > 0, by Proposition 2.7 there exists some i ≥ 0 such that D(z i ) (and thus D(z i )) is contained in a cone of apex 0 and aperture at most ε. Since both I and I belong to D(z i ), I0I ≤ ε. Thus I = I , which achieves the proof of the existence part.

Proof of (iv) and (v) Let us denote by Ψ ⊂ ∂H d+1 the set of asymptotic directions with two infinite backward paths. To show (iv), we first show that, a.s. Ψ = ∅. For z ∈ RST, let us define Ψ z ⊂ ∂H d+1 as the set of asymptotic directions of infinite backward paths from z. By the same argument as in Step 2, Ψ z is a closed subset of ∂H d+1 . By (ii), a.s., there exists at least two infinite backward paths, so there exists a.s. some level r 0 > 0 with two points connected to infinity. Thus {Ψ z , z ∈ L(r 0 )} is a covering of ∂H d+1 by closed subsets, where at least two of them are nonempty. Since ∂H d+1 is connected, it implies that there exists

z 1 , z 2 ∈ L r0 such that Ψ z1 ∩ Ψ z2 = ∅. Thus Ψ = ∅ a.s.
We use the same argument as in Step 2 to deduce that Ψ is dense. Let B be an open subset of ∂H d+1 . Since ∂H d+1 S d is compact, there exists finitely many isometries γ 1 , ..., γ k ∈ Stab(0) such that i=1,...,k γ i B = ∂H d+1 . The random set RST is invariant in distribution by Stab(0), so the events Γ i := {Ψ ∩ γ i B = ∅} all have the same probability. Since Ψ = ∅ almost surely, P i=1,...,k Γ i = 1 therefore P(Γ i ) > 0. In addition, for any neighborhood Φ ⊂ H d+1 of B, the event Γ i is entirely determined by N ∩ Φ, therefore it has probability 0 or 1. Thus Ψ ∩ B = ∅ almost surely. Since the topology on ∂H d+1 admits a countable basis, Ψ is almost surely dense in ∂H d+1 .

The proof of (v) is done by exploiting the planarity in the bi-dimensional case (Proposition 2.2). Let us associate to any z ∞ ∈ Ψ a couple of vertices

P (z ∞ ) = (z 1 , z 2 ) ∈ N 2 with z 1 = z 2 such that z ∞ ∈ Ψ z1 ∩ Ψ z2 .
By planarity, such an application P must be injective. Indeed, if z ∞ = z ∞ are such that P (z ∞ ) = P (z ∞ ) = (z 1 , z 2 ), then there exists four distinct backward infinite paths joining z ∞ to z 1 , z ∞ to z 2 , z ∞ to z 1 and z ∞ to z 2 . This implies that two paths among them intersect each other, even if the representation of edges are replaced by geodesics, which contradicts planarity. Therefore Ψ is a.s. countable in the case d = 1.

The uniqueness part: proof of (iii)

The strategy is to exploit the link between the hyperbolic RST and the hyperbolic DSF. Let us consider the (d + 1)-dimensional half-space model (H, ds 2 ):

H = {(x 1 , ..., x d , y) ∈ R d+1 , y > 0}, ds 2 = dx 2 1 + ... + dx 2 d + dy 2 y 2 .
In the following, we will identify the point (x 1 , ..., x d+1 ) ∈ H with the couple

(x, y) ∈ R d × R * + with x := (x 1 , ..., x d ), y := x d+1 . (3.1)
The coordinate x is referred as the abscissa and y as the ordinate. Let us remind that, in the halfspace representation, the boundary set ∂H d+1 is identifyied as the boundary hyper-plane R d × {0}, plus an additinal point at infinity denoted by ∞, obtained by compactifying the closed half-space R d × R + . Let us define I ∞ as the boundary point represented by (0, 0) in H.

Let N be a PPP inside (H, ds 2 ), and, for any h ≥ 0, let us define RST(h) as the Radial Spanning Tree of N with origin O(h) := (0, e h ), and let us define DSF as the Directed Spanning Forest of N with direction ∞ (defined in [START_REF] Flammant | The directed spanning forest in the hyperbolic space[END_REF]). Given z ∈ N let A DSF (z) be the parent of z in DSF, and let A RST(h) (z) be the parent of z in RST(h). We will also consider, for any given h ≥ 0, the direction toward I ∞ defined as u := (0, ..., -1) ∈ U T O(h) H d+1 S d . The proof is based on the two following propositions. The next one asserts that the RST(h) and the DSF coincide in a given compact set when h is large enough. Proposition 3.1 (Coupling between RST and DSF). Let K ⊂ H d+1 be some compact set. Then

lim h→∞ P[∀z ∈ N ∩ K, A RST(h) (z) = A DSF (z)] = 1.
For A, a, h ≥ 0, let us define:

Vois(A, h) := Cone O(h) (I ∞ , Ae -h )\B(O(h), h), Vois (A, a, h) := (B(O(h), h + a) ∩ Cone O(h) (I ∞ , Ae -h ))\B(O(h), h), Vois (A, a, h) := Cone O(h) (I ∞ , Ae -h-a )\B(O(h), h + a),
where Cone z0 (z, θ) denotes the cone with apex z 0 , direction z and aperture θ. Let us also define: For A, h ≥ 0, let us define the event E(A, h) saying that every infinite backward path converging to I ∞ in RST(h) restricted to the annulus H d+1 \S(O(h), h) are contained in Vois(A, h):

Cyl(A) := B R d (0, A) × 0, 3 2 ⊂ H, Cyl (A, a) := B R d (0, A) × 1 2 e -a , 3 2 ⊂ H, Cyl (A, a) := B R d 0, Ae -a × 0, 3 2
E(A, h) := {∀z ∈ RST(h) ∩ (H d+1 \S(O(h), h)), I ∞ ∈ D RST(h) (z) =⇒ z ∈ Vois(A, h)},
where D RST(h) (z) denotes the descendents subtree of z in RST(h). Let us also define, for A, h, a > 0, the event E (A, h, a) saying that every infinite backward path converging to I ∞ in RST(h) restricted to the annulus H d+1 \S(O(h), h + a) are contained in Vois (A, a, h):

E (A, a, h) := {∀z ∈ RST(h) ∩ (H d+1 \S(O(h), h + a)), I ∞ ∈ D RST(h) (z) =⇒ z ∈ Vois (A, a, h)}.
The following proposition asserts that, uniformly in h, the events E(A, h) and E(A, a, h) occur with high probability when A is large.

P[E(A, h)] > 1 -q/4, P[E (A, a, h)] > 1 -q/4.
Then, by Proposition 3.1 applied to the compact set K := Cyl (A, a) and Lemma 3.2, h can be chosen large enough such that inclusions (3.2) hold and such that

P[CO(A, a, h)] > 1 -q/4, P[E(A, h)] ≥ 1 -q/4 and P[E (A, a, h)] ≥ 1 -q/4.
Let us define the event Z(a) as

Z(a) := U c ∩ E(A, h) ∩ E (A, a, h) ∩ CO(A, a, h),
and define

Z := a0>0 a≥a0 Z(a).
For the choices of q, h done before, P[Z(a)] ≥ q/4 and so P[Z] ≥ q/4 > 0. On the event Z(a), and because inclusion (3.2) holds, there exists two infinite backward paths in RST(h) whose restrictions to R d ×(0, 3/2] are contained in Cyl(A, a) and converging to I ∞ , and intersecting Cyl (A, a). These two infinite backward paths coincide with those of DSF inside Cyl (A, a). Thus, in DSF, there exists two infinite backward paths contained in Cyl(A, a) and intersecting Cyl (A, a). Therefore, since this is true for all a > 0, on the event Z, it is possible to construct two infinite backward paths converging to I ∞ in DSF using the fact that DSF is locally finite (it is true by [START_REF] Flammant | The directed spanning forest in the hyperbolic space[END_REF]Proposition 9]). However, by [START_REF] Flammant | The directed spanning forest in the hyperbolic space[END_REF]Theorem 3], there almost surely a unique infinite backward path converging to I ∞ in DSF. This leads to a contradiction, which achieves the proof.

Figure 4: Representation of the sets Vois(A, h), Vois (A, a, h), Vois (A, a, h) and Cyl(A), Cyl (A, a), Cyl (A, a). The backward paths of RST(h) converging to 0 (in blue) are all contained in Vois (A, a, h) up to level h + a and contained in Vois(A, a, h) up to level h. In the dashed area (Cyl (A, a)), the DSF and RST(h) coincide.

0 x ∈ R d y ∈ R O(h) Cyl (A, a) Cyl (A, a) Cyl(A)
Vois (A, a, h)

Vois(A, h)

Vois (A, a, h)

S(h) S(h + a)

Proof of Proposition 3.1. Let us define, for z = (x, y) ∈ N and h ≥ 0:

B + DSF (z) := B(z, d(z, A(z))) ∩ (R d × (y, ∞)), B + RST(h) (z) := B(z, d(z, A(z))) ∩ (y, ∞).
Let K ⊂ H d+1 be some compact set. For any given z

∈ N ∩ K, h ≥ 0, A DSF (z) = A RST(h) (z) if and only if N ∩ (B + DSF (z)∆B + RST(h) (z)) = ∅.
For any z ∈ H d+1 , Vol(B + DSF (z)∆B + RST(h) (z)) → 0 as h → ∞ (recall that Vol is the hyperbolic volume). Campbell formula [START_REF] Chiu | Stochastic geometry and its applications[END_REF] gives,

E #{z ∈ N ∩ B(z, d(z, A(z))), N ∩ (B + DSF (z)∆B + RST(h) (z )) = ∅} = λ B(z,d(z,A(z))) P N ∩ (B + DSF (z)∆B + RST(h) (z )) = ∅ dVol(z) = λ B(z,d(z,A(z))) 1 -exp(-λVol(B + DSF (z)∆B + RST(h) (z))) dVol(z) → 0 as h → ∞
by dominated convergence. Proposition 3.1 follows.

Proof of Proposition 3.3.

For n ∈ N and h ≥ 0, let us define the event

F n (h) := {∃z ∈ B S(h) (u, 2 n+1 e -h ), MBD ∞ h (z) > 2 n e -h }.
We now show that for n ∈ N and h ≥ 0, E(2 n , h) c ⊂ m≥n F n (h). If E(2 n , h) c does not occur, then there exists some z ∈ S(h)\B S(h) (u, 2 n e -h ) such that I ∞ ∈ D RST(h) (z), so, for the value

of m ∈ N ≥n such that 2 m ≤ z0I ∞ < 2 m+1 , z ∈ B S(h) (u, 2 m+1 e -h ) and MBD ∞ h (z) > 2 n e -h ≥ z0I ∞ ≥ 2 m , thus F m (h) occurs.
Therefore

P[E(2 n , h)] ≥ 1 - m≥n P[F n (h)].
We now upper-bound P[F n (h)]. On F n (h), the following occurs:

z∈B S(h) (u,2 n+1 e -h )∩RST MBD ∞ h (z) p > 2 np e -ph , (3.3)
thus, by Markov inequality,

P [F n (h)] ≤ 2 -np e ph E   z∈B S(h) (u,2 n+1 e -h )∩RST MBD ∞ h (z) p   P rop.2.6 ≤ C fl 2 n(d-p)
for some C fl > 0 depending only on p. Combining this with (3.3) for some p > d leads to

P[E(2 n , h)] ≥ 1 -C fl 2 m(d-p) 1 -2 d-p ,
this proves the first part of Proposition 3.3. The second part be deduced from the first part by applying the dilation (x, y) → (e a x, e a y) (which is an isometry of (H, ds 2 )).

Proof of Lemma 3.2. Let A, a, h ≥ 0, and let z = (x, y) ∈ Vois(A, h). Considering the totally geodesic plane containing I ∞ , z and O(h) (represented by a half-plane in H), it is possible to suppose d = 1 without loss of generality. We apply the distance and angle formulas in (H, ds 2 ) (Propositions 7.1 and 7.2). Let z = (x, y) ∈ Vois(A, h). On the one hand, zO(h)I ∞ ≤ Ae -h , so, taking h large enough such that Ae -h < π/2, arctan 2xe h e 2h -x 2 -y 2 ≤ Ae -h , thus, for h large enough,

|x|e -h ≤ arctan |2xe -h | ≤ arctan 2xe h e 2h -x 2 -y 2 ≤ Ae -h , so |x| ≤ A.
On the other hand, d(O(h), z) ≥ h, so

2 tanh -1 1 - 4ye h A 2 + (y + e h ) 2 |x|≤A ≥ 2 tanh -1 1 - 4ye -h (xe -h ) 2 + (ye -h + 1) 2 = d(O(h), z) ≥ h, so e h ≤ 1 + 1 - 4ye h A 2 +(y+e h ) 2 1 -1 - 4ye h A 2 +(y+e h ) 2 = 2 1 -1 - 4ye h A 2 +(y+e h ) 2 -1 = e h y + o(e h ) when h → ∞,
for h large enough this implies y ≤ 3/2. The two other inclusions are shown by similar computations.

Proof of Proposition 2.5

We use a bloc control argument similar to [13, Section 6.3]. Let δ > 0 small and A > 0 large that will be chosen later. For r 0 ≥ 2 and z ∈ S(r 0 ), let us define

Ψ 1 (r 0 , z) := Cone(z, 3Ae -r0 ) ∩ (B(r 0 + δ)\B(r 0 )), Ψ 2 (r 0 , z) := Cone(z, Ae -r0 ) ∩ (B(r 0 )\B(r 0 -1)). (4.1)
A point z ∈ S(r 0 ) is said to be good if the following event G(r 0 , z) occurs:

G(r 0 , z) := {N ∩ Ψ 1 (r 0 , z) = ∅ and N ∩ Ψ 2 (r 0 , z) = ∅} . (4.2)
A good point is represented in Figure 5.

Let us define the random subsets χ(r 0 ) ⊂ S(r 0 ) and χ(r 0 ) ⊂ H d+1 \{0} as

χ(r 0 ) := z∈S(r0), G(r0,z) occurs Cone(z, Ae -r0 ), χ(r 0 ) := z∈ χ(r0) ]0, z) ⊂ H d+1 \{0}. (4.3)
The region χ(r 0 ) is the controlled region, where the cumulative forward deviations in the annulus S(r 0 + δ)\S(r 0 ) will be upper-bounded. This control of fluctuations are given by the following lemma:

Lemma 4.1. There exists some deterministic constant C geom > 0 such that for any r 0 ≥ 2, s ∈ [r 0 , r 0 + δ], and z ∈ L s ∩ χ(r 0 ), CFD s r0 (z) ≤ C geom e -r0 .

Lemma 4.1 will be proved in Section 4.4.

For any r 0 ≥ 2, we cover the sphere S(r 0 ) by balls of angular radius e -r0 such that the number of balls overlapping at a given point never exceeds some constant K. To proceed, we use the following lemma: Lemma 4.2. There exists K = K(d, p) ∈ N * such that, for any r 0 ≥ 2, there exist a non-negative integer N (r 0 ) ≥ 0 and a family of points z 1 , ..., z N (r0) ∈ S(r 0 ) such that:

0 ∈ N ∅ ∅ ∅ z S(r 0 -1) S(r 0 ) S(r 0 + δ) Ae -r0 Ψ 1 (r 0 , z) Ψ 2 (r 0 , z)
Figure 5: The point z is a good point; the fluctuations of trajectories crossing B S(r0) (z, Ae -r0 ) (in blue) are well controlled.

• 1≤i≤N (r0) B S(r0) (z i , e -r0 ) = S(r 0 ),

• ∀z ∈ S(r 0 ), #{1 ≤ i ≤ N (r 0 ), z ∈ B S(r0) (z i , e -r0 )} ≤ K.
Moreover, there exists C ball = C ball (K, d) > 0 such that, for any r 0 ≥ 2, z ∈ S(r 0 ) and A ≥ 1, the number of balls intersecting B S(r0) (z, Ae -r0 ) is upper-bounded by C ball A d :

∃C ball > 0, ∀r 0 ≥ 2, ∀z ∈ S(r 0 ), ∀A ≥ 1, #{1 ≤ i ≤ N (r 0 ), B S(r0) (z i , e -r0 ) ∩ B S(r0) (z, Ae -r0 ) = ∅} ≤ C ball A d .
We refer to Section 4.4 for the proof of Lemma 4.2.

For 1 ≤ i ≤ N (r 0 ), z i is said to be inhibited if the ball B S(r0) (z i , e -r0 ) intersects S(r 0 )\ χ, and the corresponding event is denoted by Inhib(i). Let Ψ(r 0 ) ⊂ 1, N (r 0 ) be the union of all inhibited balls:

Ψ(r 0 ) := 1≤i≤N (r0), B S(r 0 ) (zi,e -r 0 )∩(S(r0)\ χ) =∅ B S(r0) (z i , e -r0 ).
The region Ψ is the augmented uncontrolled region, that contains H d+1 \{0} \χ. For z ∈ S(r 0 ), let Ĉl(z) be the cluster of z in Ψ(r 0 ) and let us also define (recall that ]0, z) is the semi-geodesic starting at 0 and containing z, without 0):

Cl(z) = z∈ Ĉl(z) ]0, z) ⊂ H d+1 \{0}.
We define the angular radius of Cl(z) as Rad(z) := sup z ∈Cl(z) z 0z.

The next lemma asserts that the connected components of the augmented uncontrolled region Ψ are small (the radius admits exponential tail decay): Lemma 4.3. There exists δ > 0 small enough, A > 0 large enough and some constant c dec > 0 such that, for any B large enough, r 0 ≥ 2 and z ∈ S(r 0 ),

P [e r0 Rad(z) > B] ≤ e -c dec B . Lemma 4.3 is proved in Section 4.4.
In addition, we need a control of the number of points in a given region of a sphere S(r 0 ), which is given by the next lemma: Lemma 4.4. For any p ≥ 1, there exists a constant C = C(d, p) > 0 such that, for any r 0 ≥ 0 and any direction z ∈ B S(r0) ,

E # L r0 ∩ B S(r0) (z, e -r0 ) p ≤ C.
We refer to Section 4.4 for the proof of Lemma 4.4. Let us choose A, δ > 0 as in Lemma 4.3 and C geom as in Lemma 4.1.

4.1

Step 1: a deterministic upper-bound of MBD r 0 +δ r 0 (•)

For z ∈ S(r 0 ), let us define

M (z) := #N ∩ Cl(z) ∩ (S(r 0 + δ)\S(r 0 )). (4.4)
This step is devoted to the proof of the following upper-bound: almost surely, for any z ∈ S(r 0 ),

MBD r0+δ r0 (z) p ≤ 2 p-1 2 p Rad(z) p (M (z) + 1) p + C p geom e -pr0 . (4.5)
The quantity MBD r0+δ r0 (z) takes into account backward paths from z that ends (in the backward direction) before level r 0 + δ and backward paths reaching level r 0 + δ. For z ∈ S(r 0 ), we define Stop(z) as the set of ending points of backward paths from z stopping before level r 0 + δ (recall that D(z) := r≥r0 D r r0 (z)):

Stop(z) := {z = (r ; u ) ∈ N ∩ D(z), r 0 ≤ r < r 0 + δ, A (-1) (z ) = ∅}.
For any z ∈ D r0+δ r0 (z) ∪ Stop(z), one the two following cases occur. Either the branch from z to z stays inside Cl(z), or it crosses χ(r 0 ). Let us define C (resp. C ) as the set of couples (z, z ) such that the branch from z to z crosses χ(r 0 ) (resp. does not cross χ(r 0 )):

C := {(z, z ), z ∈ S(r 0 ), z ∈ D r0+δ r0 (z) ∪ Stop(z), ∃s ∈ [r 0 , r 0 + δ], A r0+δ s (z ) ∈ χ}, C := {(z, z ), z ∈ S(r 0 ), z ∈ D r0+δ r0 (z) ∪ Stop(z), ∀s ∈ [r 0 , r 0 + δ], A r0+δ s (z ) / ∈ χ}.
Moreover, for any (z, z ) ∈ C, we define hit(z, z ) as the highest level where the branch from z to z hits χ(r 0 ):

hit(z, z ) := max{s ∈ [r 0 , r 0 + δ], A r0+δ s (z ) ∈ χ},
where the max is over a set which is finite a.s.

Let

1 ≤ i ≤ N (r 0 ). Let z ∈ B S(r0) (z i , e -r0 ), z = (r ; u ) ∈ D r0+δ r0 (z) ∪ Stop(z)
, we now upperbound CFD r r0 (z ). Let us first consider the case where (z, z ) ∈ C (the branch between z and z does not cross χ(r 0 )). Then this branch stays inside Cl(z). Thus it crosses at most M (z) points of N , therefore CFD r r0 (z ) ≤ 2Rad(z)(M (z) + 1). In the other case, (z, z ) ∈ C, let z = (r ; u ) := hit(z, z ). Let p ≥ 1. Then, by Jensen inequality,

CFD r r0 (z ) p ≤ CFD r r0 (z ) + CFD r0+δ r (z ) p ≤ 2 p-1 CFD r r0 (z ) p + CFD r0+δ r (z ) p

Step 2: a control of the tail decay of M (z)

Recall that, for z ∈ S(r 0 ), M (z) is defined in (4.4). In this step, it is shown that, for any z ∈ S(r 0 ):

E M (z) 4p ≤ C (4.7)
for some C = C(p) > 0.

The quantity M (z) is the number of Poisson points inside a random part of the thin annulus S(r 0 + δ)\S(r 0 ), whose diameter admits exponential tail decay (Lemma 4.3).

Let z ∈ S(r 0 ). For given R ≥ 0, let us define Reg(R) := Cone(z i , Re -r0 ) ∩ S(r 0 + δ)\S(r 0 ) .

For any R, m ≥ 0, for some constant C > 0 independent of r 0 , R, since Vol(S(r 0 + δ)) = O e dr0 by (2.6). Thus #(N ∩ Reg(R)) st P(CR d ). Thus Chernoff bound for the Poisson distribution [START_REF]Probability and computing: randomized algorithms and probabilistic analysis[END_REF] leads to:

{M (z) > m} ⊂ {Rad(z) > Re -r0 } ∪ {# (N ∩ Reg(R)) > m} thus P [M (z) > m] ≤ P Rad(z) > Re -r0 + P [# (N ∩ Reg(R)) > m] . ( 4
P [#(N ∩ Reg(R)) ≥ m] ≤ e -CR d (CeR d ) m m m (4.10)
for any m ≥ CR d . Let us chose R = (m/(2eC)) 1/d (thus m = 2eCR d ). It leads to:

P [#(N ∩ Reg(R)) ≥ m] ≤ e -1/(2e) 2 m ≤ 1 2 m . (4.11)
Finally, we combine (4.8), Lemma 4.3 and (4.11) to obtain:

P [# (N ∩ Cl(z) ∩ (S(r 0 + δ)\S(r 0 ))) > m] ≤ e -c dec R + 1 2 m = exp -c dec (m/2eC) 1/d + 1 2 m ≤ exp(-cm 1/d ) (4.12)
for some C > 0. Therefore:

E # (N ∩ Cl(z) ∩ (S(r 0 + δ)\S(r 0 ))) 4p = ∞ 0 P # (N ∩ Cl(z) ∩ (S(r 0 + δ)\S(r 0 ))) > m 1/(4p) dm (4.12) ≤ ∞ 0 exp(-cm 1/(4dp) ) dm < ∞,
which proves (4.7).

Step 3: conclusion

By (4.5), for any

1 ≤ i ≤ N (r 0 ), z∈B S(r 0 ) (zi,e -r 0 )∩RST MBD r0+δ r0 (z) p ≤ 2 p-1 #{RST ∩ B S(r0) (z i , e -r0 )} 2 p Rad(z) p (M (z) + 1) p + C p geom e -pr0 .
It follows by Cauchy-Schwartz and Minkowski that

E   z∈B S(r 0 ) (zi,e -r 0 )∩RST MBD r0+δ r0 (z) p   ≤ CE #{RST ∩ B S(r0) (z i , e -r0 )} 2 1/2 E Rad(z i ) p M (z) p + e -pr0 2 1/2 ≤ CE #{RST ∩ B S(r0) (z i , e -r0 )} 2 1/2 E Rad(z i ) 2p M (z) 2p 1/2 + e -pr0 ≤ CE #{RST ∩ B S(r0) (z i , e -r0 )} 2 1/2 × E Rad(z i ) 4p 1/4 E M (z) 4p 1/4 + e -pr0 (4.13)
for some C = C(p) > 0. By Lemma 4.4 applied to p = 2,

E #{RST ∩ B S(r0) (z i , e -r0 )} 2 ≤ C, (4.14)
for C independent of r 0 , z i . By Lemma 4.3,

E Rad(z i ) 4p = e -4pr0 ∞ 0 P e 4pr0 Rad(z i ) 4p > B dB = e -pr0 ∞ 0 P e 4r0 Rad(z i ) > B 1 4p dB ≤ e -4pr0 ∞ 0 e -c dec B 1 4p dB < ∞. (4.15)
Then, by combining (4.13), (4.14), (4.15) and (4.7),

E   z∈B S(r 0 ) (zi,e -r 0 )∩RST MBD r0+δ r0 (z) p   ≤ Ce -pr0 , (4.16)
for some C = C(p) > 0.

The final step is to sum over all i such that B S(r0) (z i , e -r0 ) intersects B S(r0) (u, θ) for any given u ∈ S d and θ > 0. Let θ > 0. By Lemma 4.2 it can be assumed that B S(r0

) (z i , e -r0 ) intersects B S(r0) (z i , θ) for at most C ball e dr0 θ d values of i ∈ 1, N (r 0 ) . Therefore E   z∈B S(r 0 ) (u,θ)∩RST MBD r0+δ r0 (z) p   = 1≤i≤N (r0), B S(r 0 ) (zi,e -r 0 ) ∩B S(r 0 ) (u,θ) =∅ E   z∈B S(r 0 ) (zi,e -r 0 )∩RST MBD r0+δ r0 (z) p   (4.16) ≤ Ce -pr0 #{1 ≤ i ≤ N (r 0 ), B S(r0) (z i , e -r0 ) ∩ B S(r0) (u, θ) = ∅} ≤ Ce (d-p)r0 θ d ,
which achieves the proof of Proposition 2.5. Lemmas 4.1,4.2,4.3,4.4 Proof of Lemma 4.1. Let r 0 ≤ s ≤ r 0 + δ and z 1 ∈ χ(r 0 ) ∩ L(s). Since z 1 ∈ χ(r 0 ), G(z 1 ) occurs. Let z 0 = (r 0 ; u 0 ) := z 1↓ , z 2 = (r 2 , u 2 ) := A r0 s (z 1 ), z 4 = (r 3 , u 3 ) := z 1↑ and z 3 ∈ [z 0 , z 4 ] * is such that z 1 0z 4 = 3Ae -r0 . Since G(z 1 ) occurs, there exists some z = (r ; u ) ∈ Ψ 2 (r 0 , z 1 ). Let us suppose that z 1 0z 2 ≥ 3Ae -r0 . It will be shown that

Proof of

d(z 0 , z ) ≤ d(z 0 , z 3 ). (4.17)
It implies by construction of [z 0 , z 4 ] * that d(z 0 , z ) ≤ d(z 0 , z 4 ), which contradicts the fact that z 4 = A(z 0 ). It follows that z 1 z z 2 ≤ 3Ae -r0 , and since Ψ 1 (r 0 , z 1 ) ∩ N = ∅, it implies that CFD r0 s (z 1 ) ≤ 3Ae -r0 , so Lemma 4.1 holds with C geom = 3A. We move on to show (4.17). By the hyperbolic law of cosines (2.7),

cosh d(z 0 , z ) = cosh(r 0 ) cosh(r ) -cos( z 0 0z ) sinh(r 0 ) sinh(r ) = 1 + (1 -cos( z 0 0z )) sinh(r 0 ) sinh(r ) ≤ 1 + 1 -cos z 0 0z 1 + Ae -r0 sinh(r 0 ) sinh(r ) ≤ 1 + 1 -cos z 0 0z 1 + Ae -r0 sinh(r 0 ) 2 and cosh d(z 0 , z 3 ) = cosh(r 0 ) cosh(r 3 ) -cos( z 0 0z 3 ) sinh(r 0 ) sinh(r -3) = 1 + (1 -cos( z 0 0z 3 )) sinh(r 0 ) sinh(r 3 ) = 1 + 1 -cos z 0 0z 1 + z 1 0z 3 sinh(r 0 ) sinh(r 3 ) ≥ 1 + 1 -cos z 0 0z 1 + 3Ae -r0 sinh(r 0 ) sinh(r 3 ) ≥ 1 + 1 -cos z 0 0z 1 + Ae -r0 sinh(r 0 ) 2 .
This proves (4.17), which achieves the proof of Lemma 4.1.

Proof of Lemma 4.2. Proving the first part of Lemma 4.2 is equivalent to show that there exists some K ∈ N such that, for any ε > 0, the Euclidean unit sphere S d can be covered by balls of radius ε such that the number of balls overlapping some given point x ∈ S d is bounded by K, which is a standard fact.

We move on to show the second part, i.e. the existence of C ball > 0 such that, for any r 0 ≥ 2, z ∈ S(r 0 ) and A ≥ 1, the number of balls intersecting B S(r0) (z, Ae -r0 ) is upper-bounded by C ball A d . Let u ∈ S d be the direction of z and let A ≥ 1.

For i ∈ 1, N (r 0 ) , the ball B S(r0) (z i , e -r0 ) intersects B S(r0) (z, Ae -r0 ) if and only if z i 0z ≤ (A + 1)e -r0 . Thus         1≤i≤N (r0), B S(r 0 ) (zi,e -r 0 )∩ B S(r 0 ) (z,Ae -r 0 ) =∅ B S(r0) (z i , e -r0 )         ⊂ B S(r0) (z, (A + 2)e -r0 ).
Recall that ν denotes the d-dimensional volume measure on S d . There exists C > 0 such that, for any r 0 ≥ 2, ν{u, uu 0 ≤ e -r0 } ≥ Ce -r0d . Moreover, ν{u, uu 0 ≤ (A + 2)e -r0 } ≤ (A + 2)e -r0 d , thus the number of balls intersecting B S(r0) (z, Ae -r0 ) is upper-bounded by:

K ν{u, uu 0 ≤ (A + 2)e -r0 } ν{u, uu 0 ≤ e -r0 } ≤ K ((A + 2)e -r0 ) d ce -r0d A≥1 ≤ 3 d K c A d ,
the conclusion follows.

Proof of Lemma 4.3. Recall that, for

1 ≤ i ≤ N (r 0 ), B S(r0) (z i , e -r0
) is said to be inhibited if B S(r0) (z i , e -r0 ) ∩ (S(r 0 )\χ) = ∅. Let us first estimate the probability that a given z i is inhibited. Let 1 ≤ i ≤ N (r 0 ). Let us consider the following events:

E(i) := {N ∩ Cone(z i , (3A + 1)e -r0 ) ∩ (B(r 0 + δ)\B(r 0 )) = ∅}, E (i) := {N ∩ Cone(z i , (A -1)e -r0 ) ∩ (B(r 0 )\B(r 0 -1)) = ∅}.
We now show that

E(i) ∩ E (i) ⊂ Inhib(i) c . Let z ∈ B S(r0) (z i , e -r0
). By triangular inequality, Ψ 1 (r 0 , z) ⊂ Cone(z i , (3A + 1)e -r0 ) ∩ (B(r 0 + δ)\B(r 0 )) and Cone(z i , (A -1)e -r0 ) ∩ (B(r 0 + δ)\B(r 0 )) ⊂ Ψ 2 (r 0 , z).

Therefore, on the event E(i) ∩ E (i), z is good (i.e. G(r 0 , z) occurs). Thus, Inhib(i) c occurs, which shows that

E(i) ∩ E (i) ⊂ Inhib(i) c . It follows that P[Inhib(i)] ≤ P[E(i) c ] + P[E (i) c ]. Since Vol(Cone(z i , (3A + 1)e -r0 ) ∩ (B(r 0 + δ)\B(r 0 ))) = r0+δ r0 ν{u , u, u ≤ (3A + 1)e -r0 } sinh(r) d dr ≤ CδA d
for some C > 0 independent of A, r 0 , δ. Thus

P [E(i) c ] ≤ 1 -e -λcδA d
An analogous computation for E (i) leads to:

P [E (i) c ] ≤ e -λcA d
for some c > 0 independent of A, r 0 , δ.

We move on to show that A and δ can be chosen such that P[Inhib(i)] ≤ 2C -1 ball (6A + 4) -d /3 for any 1 ≤ i ≤ N (r 0 ). Indeed, let us first chose A such that e -λcA d C ball (6A + 4) d ≤ 1/3. Then let us chose δ such that 1 -e -λcδA d ≤ C -1 ball (6A + 4) -d /3. Hence

P[Inhib(z i )] ≤ P [E(i) c ] + P [E (i) c ] ≤ 1 -e -λcδA d + e -λcA d ≤ 2 3 C -1 ball (6A + 4) -d .
We finally show that, for this choice of A, δ, there exists c dec > 0 such that for any B large enough, r 0 ≥ 2 and z ∈ S(r 0 ),

P [e r0 Rad(z) > B] ≤ e -c dec B .
Fix r 0 ≥ 2 and z ∈ S(r 0 ). For given k, let us denote by P (k) the set of sequences

z i0 , • • • , z i k among the {z i , 1 ≤ i ≤ N (r 0 )} such that: 1. d(z i0 , z) ≤ e -r0 , 2. for any 0 ≤ j ≤ k -1, d(z ij , z ij+1 ) ≤ (6A + 4)e -r0 , 3. for any 0 ≤ j, j ≤ k, d(z ij , z i j ) ≥ (6A + 2)e -r0 .
Let us also denote by P (k) ⊂ P (k) the set of sequences z i0 , • • • , z i k verifying 1., 2., 3. and such that:

4. z i0 , • • • , z i k are inhibited.
It can be noticed that {Rad(z) ≥ k(6A + 4)} ⊂ { P (k) = ∅} for any k ∈ N, thus it is enough to upper-bound { P (k) = ∅} for any k ∈ N. Let (z i0 , • • • , z i k ) ∈ P (k). For 0 ≤ j ≤ k, the event Inhib(z ij ) only depends on the Poisson process N inside Cone(z ij , (3A + 1)e -r0 ), therefore, by 3. the events Inhib(z ij ) are mutually independent. Thus

P[(z i0 , • • • z i k ) ∈ P (k)] = P[Inhib(z i0 )] k+1 ≤ 2 3 k+1 C -(k+1) ball (6A + 4) -d(k+1) .
By Lemma 4.2, for any 1 ≤ i ≤ N (r 0 ), the number of balls intersecting B S(r0) (z, θ) is upperbounded by C ball e dr0 θ d . Thus k+1) .

#P (k) ≤ C k+1 ball (6A + 4) d(
It follows that Proof of Lemma 4.4. Let r 0 , M > 0 and z ∈ S(r 0 ). Let h ≥ 0 that will be fixed later. We divide the set L = {z ∈ N , [z , A(z )] * ∈ B S(r0) (z, e -r0 )} into two subsets L ≤h and L >h according to the length of [z , A(z )]:

P[Rad(z) ≥ k(6A + 4)] ≤ P[ P (k) = ∅] ≤ E[# P (k)] ≤ C k+1
L ≤h := {z ∈ L, d(z , z) ≤ h}, L >h := {z ∈ L, d(z , z) > h}. (4.18)
Thus L = L ≤h ∪ L >h , and

P [#L ≥ M ] ≤ P [#L ≤h ≥ M ] + P [L >h = ∅] . (4.19)
We first upper-bound P [#L ≤h ≥ M ]. Since L ≤h ⊂ B(z, h),

P [#L ≤h ≥ M ] ≤ P [# (N ∩ B(z, h)) ≥ M ] .
By (2.6), Vol(B(z, h) ≤ Ce dh , for some C > 0 independent of r 0 . So the random variable # (N ∩ B(z, h)) is stochastically dominated by a Poisson law with parameter Cλe dh , thus, by the Chernoff bound for the Poisson distribution [START_REF]Probability and computing: randomized algorithms and probabilistic analysis[END_REF], ≤ arcosh cosh(r 0 ) 2 -cos(e -r0 ) sinh(r 0 ) 2 = arcosh 1 + 1 -cos(e -r0 ) sinh(r 0 ) 2 ≤ C dis for some C dis > 0 independent of r 0 . Thus d(z , A(z )) ≥ d(z , z) -C dis , so B + (z, (d(z , z) -C dis ) + ) is empty of points. Therefore, by Campbell formula [START_REF] Chiu | Stochastic geometry and its applications[END_REF], 5 Proof of Propositions 2.6 and 2.7

P [#L ≤h ≥ M ] ≤ e -Cλe dh (Cλe dh ) M M M . ( 4
P [L >h = ∅] ≤ E [#L >h ] = λ B(r0)\B(h) P [z ∈ L >h ] dz ≤ λ B(r0)\B(h) P B + (z, (d(z , z) -C dis ) + ) ∩ N = ∅ dz = λ B(r0)\B(h) P exp(-λVol(B + (z, (d(z , z) -C dis ) + )) dz (6.1) ≤ λ B(r0)\B(h) P exp(-λce -d(d(z ,z)-C dis )/2 ) dz (2.5) = λS(d)
We first prove Proposition 2.6.

Step 1: Let us fix p > 3d/2. For any r 0 > 0, n ∈ N, let us define

S n (A, r 0 ) := z∈B S(r 0 ) (u,Ae -r 0 )∩RST MBD r0+nδ r0 (z) 2p .
The strategy of the proof is to construct a family of non-negative random variables

(Y M n (A, r 0 )) r0,A,M ≥0,n∈N and (Y M (A, r 0 )) r0,A,M ≥0 such that (1) almost surely, Y M n (A, r 0 ) ↑ Y M (A, r 0 ) when n → ∞ for any M, A, r 0 ≥ 0; (2) sup A,r0 P Y M (A, r 0 ) ≥ M = O M -2/3 when M → ∞;
(3) the following implication holds almost surely:

S n (A, r 0 ) ≤ (M ∧ Y M n (A, r 0 ))A d e -2r0p =⇒ S n+1 (A, r 0 ) ≤ Y M n+1 (A, r 0 )A d e -2r0p
.

Let us suppose for the moment that such random variables Y M n (A, r 0 ) and Y M (A, r 0 ) exist. Let A, r 0 ≥ 0 and M ≥ 0. On the event {Y M (A, r 0 ) ≤ M }, it can be shown by induction that S n (A, r 0 ) ≤ M A d e -2r0p for any n ≥ 0. Indeed, S 0 = 0, and if S n (A, r 0 ) ≤ Y M n (A, r 0 )A d e -2r0p , then, using that

M ∧ Y M n (A, r 0 ) (1)
≤ M ∧ Y M (A, r 0 ) ≤ Y M (A, r 0 ) since we are on the event {Y M (A, r 0 ) ≤ M }.

So S n+1 (A, r 0 ) ≤ Y M n+1 (A, r 0 )A d e -2r0p by (3), which achieves the induction. Thus, for any A, r 0 , M ≥ 0,

P[S n (A, r 0 ) ≥ M A d e -2r0p ] ≤ P[Y M (A, r 0 ) ≥ M ] ≤ CM -2/3 by (2).
(5.1)

for M large enough and some constant C > 0 independent of A, r 0 , M . It follows that

C : = sup A,r0 E S n (A, r 0 ) 1/2 A -d/2 e r0p = sup A,r0 ∞ 0 P[S n (A, r 0 ) 1/2 A -d/2 e r0p ≥ M ] dM = sup A,r0 ∞ 0 P S n (A, r 0 ) ≥ M 2 A d e -2r0p
(5.1)

≤ ∞ 0 CM -4/3 dM < ∞. (5.2) Let K := # N ∩ B S(r0) (u, Ae -r0
) . Let us apply Cauchy-Schwartz with the inner product defined by Since r → MBD r r0 (z) is non-decreasing for any z ∈ S(r),

X, Y = E [ i X i Y i ], E   z∈B S(r 0 ) (u,Ae -r 0 )∩RST MBD r0+nδ r0 (z) p   ≤ E   z∈B S(r 0 ) (u,Ae -r 0 )∩RST MBD r0+nδ r0 (z) 2p   1/2 E #B S(r0) (u, Ae -r0 ) 1/2 = E S n (A, r 0 ) 1/2 E #B S(r0) (u, Ae -r0 ) 1/2 (5.2) ≤ C A d/2 e -r0p E #B S(r0) (u, Ae -r0 ) ∩ RST 1/2 . ( 5
z∈B S(r 0 ) (u,Ae -r 0 )∩RST MBD ∞ r0 (z) p = lim n→∞ ↑ z∈B S(r 0 ) (u,Ae -r 0 )∩RST MBD r0+nδ r0 (z) p .
(5.5) Proposition 2.6 follows by (5.4) and by monotone convergence theorem.

Step 2: we build the random variables Y M n (A, r 0 ) and Y M (A, r 0 ). Let A, r 0 ≤ 0, let n ∈ N. The strategy is to upper-bound S n+1 in function of S n . Fix z ∈ B S(r0) (u, Ae -r0 ). The quantity MBD r0+nδ r0 takes into account finite backwards paths that stop before level r 0 + nδ and those (potentially infinite) that continue after level r 0 + nδ. Let us define the random set Stop(z) as the set of ending points (in the backward direction) of finite paths from z stopping before level r 0 + nδ: (z ) .

Stop(z) := {z = (r ; u ) ∈ N ∩ D(z), r 0 ≤ r ≤ r 0 + nδ, A -1 (z ) = ∅} ⊂ N .

By definition of MBD

For any p ≥ 1, a, b ≥ 0 and t ∈ [0, 1], Jensen inequality gives,

(a + b) p = t a t + (1 -t) b 1 -t p ≤ t a t p + (1 -t) a 1 -t p = t 1-p a p + (1 -t) 1-p b p . (5.8)
Applying (5.8) with t = 1/n 2 leads to:

MBD r0+(n+1)δ r0 (z) 2p (5.7) = max z =(r ;u )∈Stop(z) CFD r r0 (z ) 2p ∨ max z ∈D r 0 +nδ r 0 (z) CFD r0+nδ r0 (z ) + MBD r0+(n+1)δ r0+nδ (z ) 2p ≤ max z =(r ;u )∈Stop(z) CFD r r0 (z ) 2p ∨ max z ∈D r 0 +nδ r 0 (z) 1 - 1 n 2 1-2p CFD r0+nδ r0 (z ) 2p + n 4p-2 MBD r0+(n+1)δ r0+nδ (z ) 2p ≤ 1 - 1 n 2 1-2p max z =(r ;u )∈Stop(z) CFD r r0 (z ) 2p ∨ max z ∈D r 0 +nδ r 0 (z) CFD r0+nδ r0 (z ) 2p +n 4p-2 max z ∈D r 0 +nδ r 0 (z) MBD r0+(n+1)δ r0+nδ (z ) 2p (5.6) = 1 - 1 n 2 1-2p MBD r0+nδ r0 (z) 2p + n 4p-2 max z ∈D r 0 +nδ r 0 (z) MBD r0+(n+1)δ r0+nδ (z ) 2p . (5.9)
Summing (5.9) over all z ∈ B S(r0) (u, Ae -r0 ) leads to:

S n+1 ≤ 1 - 1 n 2 1-2p S n + n 4p-2       z∈B S(r 0 ) (u,Ae -r 0 ) z ∈D r 0 +nδ r 0 (z) MBD r0+(n+1)δ r0+nδ (z ) 2p       . (5.10)
Let us place on the event {S n ≤ M A d e -2r0p }. Then, for any z ∈ B S(r0) (u, Ae -r0 ),

MBD r0+nδ r0 (z) 2p ≤ S n ≤ M A d e -2r0p , so, for any z ∈ D r0+nδ r0 (z), z 0z ≤ M 1/2p A d/(2p) e -r0 , so z 0I ∞ ≤ z 0z + z0I ∞ ≤ M d 2p A d 2p e -r0 + Ae -r0 since z ∈ B S(r0) (u, Ae -r0 ) ≤ Ae -r0 M d 2p + 1 since d 2p ≤ 1.
Therefore, for any z ∈ B S(r0) (u, Ae -r0 ), Define R 0 such that, for any r 0 ≥ R 0 , e (1-ε)r0 max z∈Lr 0 MBD ∞ r0 (z) ≤ 1/2. For any r 0 ≥ R 0 , z ∈ L r0 , z 1 , z 2 ∈ D(z), defining r 1 := d(0, z 1 ) and r 2 := d(0, z 2 ), z 1 0z 2 ≤ z 1 0z + z0z 2 ≤ CFD r1 r0 (z 1 ) + CFD r2 r0 (z 2 ) ≤ MBD r1 r0 (z) + MBD r2 r0 (z) ≤ 2MBD ∞ r0 (z) ≤ 2 max z ∈Lr 0 MBD ∞ r0 (z ) ≤ e -(1-ε)r0 . (5.22) This achieves the proof of Proposition 2.7.

D r0+nδ r0 (z) ⊂ B S(r0+nδ) u, Ae -r0 M d 2p + 1 . ( 5
6 Appendix A: proof of Proposition 2.2

We first show that the RST is a tree. If the RST contains some loop z 0 , • • • , z n , then the furthest vertex to the origin in the loop, say z i , must have two parents, which contradicts the definition of the RST. Moreover, for some given vertex z ∈ N , the sequence d A (k) (z), 0 k is decreasing. In addition, since N ∩ B(r) is finite for any r ≥ 0, there is no infinite decreasing sequence d A (k) (z), 0 k . Thus A (k) (z) = 0 for some finite k ≥ 0. Therefore, the RST is a connected graph, so it is a tree.

We move on to show that the RST is locally finite. Let z 0 = (r 0 ; u 0 ) ∈ N ∪ {0}. Let us assume for the moment that, for any z = (r; u) ∈ H d+1 and r > 0, Vol(B + (z, r )) ≥ ce d(r ∧r)/2 . (6.1)

for some c independent of z, r. For z 0 = (r 0 ; u 0 ), z = (r; u) ∈ H d , let us define a(z, z 0 ) = 1 r>r0 1 B + (z,d(z,z0))∩N =∅ .

Thus, for any z 0 ∈ N , z 0 = A(z) if and only if a(z, z 0 ) = 1. By Campbell formula [START_REF] Chiu | Stochastic geometry and its applications[END_REF],

E #{z 0 ∈ N , #A (-1) (z 0 ) = ∞} = E It remains to show (6.1). Let ρ ≥ 0 and 0 ≤ r ≤ r 0 . Let z ∈ S(r ). Using the hyperbolic law of cosines in the triangle 0zz (2.7), z ∈ B + (z, ρ) ⇐⇒ d(z, z ) ≤ ρ ⇐⇒ cosh(ρ) ≥ cosh(r 0 ) cosh(r) -cos( z0z ) sinh(r 0 ) sinh(r) ⇐⇒ cos( z0z ) ≥ cosh(r 0 ) cosh(r) -cosh(ρ) sinh(r 0 ) sinh(r) .

A study of the function r → cosh(r0) cosh(r)-cosh(ρ) sinh(r0) sinh(r)

shows that, if ρ ≥ 1, then this quantity is non-decreasing when r 0 -1 ≤ r ≤ r 0 . Thus, for ρ ≥ 1 and 1 ≤ r 0 -1 ≤ r ≤ r 0 , cosh(r 0 ) cosh(r) -cosh(ρ) sinh(r 0 ) sinh(r) ≤ cosh(r 0 ) 2 -cosh(ρ) sinh(r 0 ) 2 = 1 -cosh(ρ) -1 sinh(r 0 ) 2 ≤ 1 -Ce ρ-2r0 is an isometry sending (0, 1) on (0, 0) [START_REF] Cannon | Hyperbolic geometry[END_REF]. Let us compose it with the dilation of factor e -h to build an isometry from H to I sending O(h) on (0, 0). We obtain the application φ defined as φ (x, y) = 1 (e -h x) 2 + (e -h y + 1) 2 (e -h x) 2 + (e -h y) 2 -1, -2e -h x = 1 x 2 + (y + e h ) 2 x 2 + y 2 -e 2h , -2xe h

The Poincare disc model is conform (i.e. the hyperbolic angles correspond to angles in the disc model), and the geodesics containing the origin (0, 0) are represented by straight lines, thus for any two points z 1 , z 2 ∈ I, the hyperbolic angle z 1 0z 2 coincides with the Euclidean one. For any z = (x, y) ∈ H, zO(h)I ∞ = φ (z)0φ (I ∞ ), where the second angle is taken in the disc model. Since φ (I ∞ ) = (-1, 0), if y < e h then z0(h)I ∞ < π 2 , thus zO(h)I ∞ = arctan 2xe h x 2 + y 2 -e 2h .

Figure 1 :

 1 Figure 1: Simulation of the two-dimensional hyperbolic RST, with λ = 30, in the Poincaré disc model. The edges are represented by geodesics. The different connected components of the RST (appart from the root) are represented with different colors.

Figure 2 :

 2 Figure 2: Geodesics in the open ball model

  For z ∈ N and r ≥ 0, let us define B + (z, r) := B(z, r) ∩ B(0, d(0, z)) and B + (z) := B + (z, d(z, A(z))).

Figure 3 :

 3 Figure 3: Representation of levels r and r , the ancestor A r r (•) and the set of descendents D r r (•)

Lemma 3 . 2 .

 32 e -a ⊂ H.The sets Vois(A, h), Vois (A, a, h), Vois (A, a, h) and Cyl(A), Cyl (A, a), Cyl (A, a) are represented in Figure4. We will use the following geometrical fact: For any A, a ≥ 0, h can be chosen large enough such that Vois(A, h) ⊂ Cyl(A), Vois (A, a, h) ⊂ Cyl (A, a), and Vois (A, a, h) ⊂ Cyl (A, a).(3.2)

  (A, a, h)] = 1 for any a ≥ 0.Let us assume Propositions 3.1, 3.3 and Lemma 3.2 for the moment and let us prove part (iii) of Theorem 1.1. For h ≥ 0, let us define the event U (h) := {there is a unique infinite backward path converging to I ∞ in RST(h)} By isometries invariance, P[U (h)] is independent of h, and let us suppose for contradiction that q := P[U (h) c ] > 0. For A, a, h ≥ 0, let us define the event CO(A, a, h) := {∀z ∈ N ∩ Cyl (A, a), A RST(h) (z) = A DSF (z)} By Proposition 3.3, A can be chosen such that lim inf h→∞

  By the same argument as in the previous case, CFD r r0 (z ) ≤ 2Rad(z)(M (z) + 1), and, by Lemma 4.1, CFD r0+δ r (z ) ≤ C geom e -r0 , since, by definition of z , the part of trajectory between z and z is included in Cl(z). Thus,CFD r r0 (z ) p ≤ 2 p-1 2 p Rad(z) p (M (z) + 1) p + C p geom e -pr0 .(4.6)The upper-bound (4.6) holds whatever (z, z ) belongs to C or C . It follows that (4.5) holds for any z ∈ S(r 0 ).

. 8 )

 8 By Lemma 4.3, P [Rad(z) > Re -r0 ] ≤ e -c dec R . The random variable # (N ∩ Reg(R))) is distributed according to the Poisson law with parameter λVol(Reg(R)). Recall that ν the d-dimensional volume measure on S d . Denoting by u the direction of z, λν(Reg(R)) = λν({u , u, u < Re -r0 })Vol((S(r 0 + δ)\S(r 0 )) ≤ CR d .(4.9)

ball 2 3 k+1( 3 k+1.

 33 6A + 4) d(k+1) C -(k+1) ball (6A + 4) -d(k+1) ≤2Lemma 4.3 follows.

. 20 )

 20 The second step is to upper-boundP [#L >h = ∅]. Recall that, for r ≥ 0, B + (z, r) := B(z, r) ∩ B(0, d(0, z)).For any z ∈ L >h , by triangular inequality, denoting by z * the meeting point of [z , A(z )] * and S(r 0 ), d(z , A(z )) ≥ d(z , z * ) ≥ d(z , z) -d(z * , z). The hyperbolic law of cosines (2.7) gives, d(z * , z) = arcosh cosh(r 0 ) 2 -cos( z * 0z) sinh(r 0 ) 2

  λce -rd/2 ) sinh(r) d dr, where S(d) denotes the surface area of the Euclidean unit ball S d . When r → ∞, exp(-λce -rd/2 ) sinh(r) d = o e -rd/3 e e -rd/3 thus λS(d) ∞ h exp(-λce -rd/2 ) sinh(r) d dr = o e -e -rd/3 , thus, for h large enough, and for any r 0 ≥ 0, P [L >h = ∅] ≤ e -e -hd/3 . (4.21) Finally, combining (4.19), (4.20) and (4.21) with h = -ln(M/(2Cλ))/d leads to Lemma 4.4.

. 3 )

 3 Let us show that E #B S(r0) (u, Ae -r0 ) ∩ RST ≤ CA d for some C > 0 independent of A, r 0 . We use the covering of S(r 0 ) by balls of radius e -r0 introduced by Lemma 4.2 in Section 4. For any 1 ≤ i ≤ N (r 0 ), by Proposition 4.4 applied with p = 1, E[#RST ∩ B S(r0 (z i , 1)] ≤ C for C independent of r 0 , z i . By Lemma 4.2, the number of balls intersecting B S(r0 (z i , 1) is bound byC ball A d . It follows that E #B S(r0) (u, Ae -r0 ) ∩ RST ≤ CA d .Thus, by (5.3), E   z∈B S(r 0 ) (u,Ae -r 0 )∩RST

  r0+nδ

  r0 (z) (resp. MBD r0+(n+1)δ r0 (z)) (Definition 2.4), MBD r0+nδ r0 (z) = max z =(r ;u )∈Stop(z) CFD r r0 (z ) ∨ max z ∈D

. 11 ) 1 dPn∈N

 111 Let us defineZ n (A, r 0 ) := z ∈B S(r 0 +(n+1)δ) (u,Ae -r 0 (M 1/(2p) +1))∩RST MBD r0+(n+1)δ r0+nδ (z ) 2p .(5.12) assertion is clear for n = 0 and, for n ≥ 0,E[Y M n+1 (A, r 0 )] = 1n (A, r 0 ) + n 4p-2 A -d e 2r0p E [Z n (A, r 0 )] = p(n)E[Y M n (A, r 0 )] + A -d e 2r0p-n(d-2p)δ q(n)E [Z n (A, r 0 )] n)P (n)Q(n) + q(n)] by induction hypothesis (n + 1)Q(n + 1) since P (n + 1) ≤ 1,(5.19)which achieves the induction. Thus, there exists some constant C > 0 such that, for any M, A, r 0 ≥ 0, for anyn ∈ N, E Y M n (A, r 0 ) ≤ C M 1/(2p) + 1 d . By monotone convergence, E Y M (A, r 0 ) ≤ C M 1/(2p) + 1 d .Thus,for any M, A, r 0 ≥ 0, Markov inequality gives,P Y M (A, r 0 ) ≥ M ≤ C M 1/(2p) + 1 d M = O(M -2/3 ) since 2p > 3d.Thus the family of random variables Y M (A, r 0 ) verifies (2). This achieves the proof.Proof of Proposition 2.7. This is a direct consequence of Proposition 2.6. Let ε > 0 and let us choose p such that d/p < ε. Applying Proposition 2.6 with A = πe r0 gives that, for any r 0 ≥ 0,Ce n((d-p)+(1-ε)p) < ∞ since ε < d/p. Therefore, a.s.,lim n→∞ e (1-ε)n max z∈Ln MBD ∞ n (z) → 0 as n → ∞.Moreover, r 0 → max z∈Lr 0 MBD ∞ r0 is non-increasing, so for any n ≤ r 0 < n + 1,e (1-ε)r0 max z∈Lr 0 MBD ∞ r0 (z) ≤ max z∈Ln MBD ∞ n (z)e (1-ε)r0 ≤ e 1-ε max z∈Ln MBD ∞ n (z)e(1-ε)n , e (1-ε)r0 → 0 as r 0 → ∞.

z0∈N 1

 1 z∈N a(z,z0)=∞ + P z∈N a(z, 0) = ∞ = P z∈N a(z, 0) = ∞ + λ H d+1 P z∈N a(z, z 0 ) = ∞ dz 0 , thus it suffices to show that P z∈N a(z, z 0 ) = ∞ = 0 for any z 0 ∈ H d+1 . Let z 0 = (r 0 ; u 0 ) ∈ H d+1 . Note that, if d(z, z 0 ) ≥ r 0 , then 0 ∈ B + (z, d(z, z 0 )) so a(z, z 0 ) = 0. Thus, Campbell formula gives, E z∈N a(z, z 0 ) = λ H d+1 E [a(z, z 0 )] dz ≤ λ H d+1 1 d(z,z0)<r0 P[B + (z, d(z, z 0 )) ∩ N = ∅] dz = λ H d+1 1 d(z,z0)<r0 P exp(-λVol(B + (z, d(z, z 0 ))) dz(6.1)≤ λH d+1 P exp(-λce -d/2 d(z,z0) ) dz (2.5) = λS(d) ∞ 0 exp(-λce -rd/2 ) sinh(r) d dr, < ∞,where S(d) denotes the surface area of the Euclidean unit ball S d . Thus P z∈N a(z, z 0 ) = ∞ = 0.

Acknowledgments. This work has been supervised by David Coupier and Viet Chi Tran, who helped a lot to write this paper. It has been supported by the LAMAV (Université Polytechnique des Hauts de France) and the Laboratoire P. Painlevé (Université de Lille). It has also benefitted from the GdR GeoSto 3477, the Labex CEMPI (ANR-11-LABX-0007-01) and the ANR PPPP (ANR-16-CE40-0016).

By (5.11), z∈B S(r 0 ) (u,Ae -r 0 ) z ∈D r 0 +nδ r 0 (z) MBD r0+(n+1)δ r0+nδ

(z ) 2p ≤ Z n (A, r 0 ), (5.13) thus, combining (5.10) and (5.13), on the event {S n ≤ M A d e -2r0p },

S n + n 4p-2 Z n . (5.14) This upper-bound of S n+1 suggests the following definition of the random variables Y M n (A, r 0 ). We set Y M 0 (A, r 0 ) := 0, and for any n ≥ 0,

Let us also define

Thus the random variables Y M n (A, r 0 ) verify (3). We move on to show that (Y M n (A, r 0 )) n,M,A,r0 and (Y M (A, r 0 )) M,A,r0 also verify [START_REF] Baccelli | Semi-infinite paths of the two-dimensional radial spanning tree[END_REF]. To proceed, we upper-bound E[Y M n (A, r 0 )] by induction on n.

For any M, A, r 0 , n, Proposition 2.5 applied for θ = Ae -r0 M 1/(2p) + 1 gives,

Let us define, for any n ∈ N,

and

(5.17)

with the convention P (0) = 1 and Q(0) = 0. It can be noticed that

Let us show by induction on

The for some C > 0 independent of r 0 , r, ρ. Thus, there exists C > 0 independent of r 0 , r, ρ such that, if ρ ≥ 1 and 1 ≤ r 0 -1 ≤ r ≤ r 0 , if z0z ≤ Ce ρ/2-r0 then z ∈ B + (z, ρ). Therefore, by (2.5), if ρ ≥ 1 then (recall that ν is the d-dimensional volume measure on S d and u is the direction of z),

Therefore, the RST has finite degree a.s.

It remains to show that the geodesics [z, A(z)] for z ∈ N do not cross a.s. in the bi-dimensional case (d = 1). Let us suppose that there are no two points z 1 , z 2 with d(0, z 1 ) = d(0, z 2 ) (this happens with probability 0). Let z 1 = (r 1 ; u 1 ), z 2 = (r 2 ; u 2 ) ∈ N and let us set A(z 1 ) := (r 1 ; u 1 ), A(z 2 ) := (r 2 ; u 2 ). Suppose that [z 1 , A(z 1 )] and [z 2 , A(z 2 )] meet at some point P hyp := (r hyp ; u hyp ). We have r 1 < r hyp < r 2 , thus by definition of the parent,

so d(P hyp , A(z 2 )) < d(P hyp , A(z 1 )). On the other hand, interchanging z 1 and z 2 in the previous calculation leads to d(P hyp , A(z 1 )) < d(P hyp , A(z 2 )). This is a contradiction. Therefore [z 1 , A(z 1 )]∩ [z 2 , A(z 2 )] = ∅. This achieves the proof of Proposition 2.2.

7 Appendix B: computing distances and angles in the half-space model

We refer to [13, Proposition 5] for a proof.

Proposition 7.2 (Angle formula). Let z = (x, y) ∈ H and let h ≥ 0. Recall that O(h) = (0, e h ) and I ∞ = (0, 0). If y < e h , then

Proof of Proposition 7.2. The proof is done by considering the isometry that sends the half-space model on the Poincaré disc model [START_REF] Cannon | Hyperbolic geometry[END_REF]. The Poincaré disc model is defined as:

endowed with the following metric:

2)

The application φ : H → I defined as:

(x, y) → 1 x 2 + (y + 1) 2 x 2 + y 2 -1, -2x


Léandre Ponthus

| Michel De Saint Blanquat
email: michel.desaintblanquat@get.omp.eu

Damien Guillaume

Marc Le Romancer

Norman Pearson

Suzanne O'reilly

Michel Grégoire

Michel De

One of the main remaining questions about earth magmatism in both oceanic and continental domains is the role(s) of the crustal Plutonic processes in transitional oceanic plateau crust: Structure, age and emplacement of the South Rallier du Baty laccolith, Kerguelen Islands

The main output of Earth magmatic activity is represented by in-

that it is a laccolith built between 11.6 and 7.9 Ma by successive injections of magma sheets around the crust-mantle boundary, with an average injection rate between 0.8 and 1.4 × 10 -4 km 3 /year. These results establish strong similarities between the SRBIC, the only recorded example of a felsic laccolith in an oceanic intraplate setting, and many continental plutons emplaced in various geodynamic setting. The SRBIC thus has the characteristics of a continental plutonic complex emplaced in an oceanic plateau crust. We postulate the critical parameter relevant to causing such similarities and plutonic magmatism is crustal thickness.

K E Y W O R D S

alkaline magmatism, Kerguelen, oceanic plateau, pluton emplacement, syenite intrusion "filter" (including thickness, composition, rheological stratification and structure) in magma production, differentiation, contamination, transport and final emplacement style and geometry. This question can be explored by characterizing and comparing the structure and detailed construction of plutons in different geodynamic settings.

Although there are numerous existing datasets for continental plutons, there are very few that address a detailed understanding of the structure and detailed geochronology of oceanic plutons. We therefore focused this study on the South Rallier du Baty Intrusive Complex (SRBIC) outcropping in the south-western part of the Kerguelen Islands within a thick oceanic plateau crust. This plutonic complex is the most spectacular known occurrence of felsic oceanic plutonic rocks and, along with the intrusive complexes from Iceland [START_REF] Burchardt | Mode of emplacement of the Slaufrudalur Pluton, Southeast Iceland inferred from three-dimensional GPS mapping and model building[END_REF][START_REF] Padilla | Elucidating the magmatic history of the Austurhorn silicic intrusive complex (southeast Iceland) using zircon elemental and isotopic geochemistry and geochronology[END_REF]), represents the only example of outcropping felsic plutonic rocks in an oceanic contexts as other occurrences are only inferred from xenoliths in the Azores and Ascension islands [START_REF] Kar | Origin of differentiated volcanic and Plutonic rocks from Ascension Island, South Atlantic Ocean[END_REF][START_REF] Widom | Syenite nodules as a long-term record of magmatic activity in Agua de Pao Volcano, Sao Miguel, Azores[END_REF]. Therefore, the SRBIC is an ideal target to obtain precise quantitative data (internal structure, geometry, duration of emplacement) on the construction of a pluton in an oceanic domain, and thus allows a comparison between oceanic and continental plutonic processes. Moreover, previous studies already questioned the oceanic/continental/transitional nature of the northern Kerguelen plateau crust considering its geodynamic setting [START_REF] Grégoire | The meta-igneous granulite xenoliths from Kerguelen Archipelago: Evidence of a continent nucleation in an oceanic setting[END_REF][START_REF] Watkins | Kerguelen: Continental fragment or oceanic island?[END_REF]. We present in this study new constraints for the construction of the SRBIC based on a set of new field structural observations, accompanied by detailed U-Pb geochronological data.

| G EOLOG I C AL CONTE X T AND PRE VIOUS S TUDIE S

The Kerguelen Islands represent the emergent part of the Northern Kerguelen oceanic Plateau (NKP), assumed to be formed by activity of the Kerguelen plume [START_REF] Gautier | Petrology and geochemistry of the Kerguelen Archipelago basalts (South Indian Ocean), evolution of the mantle sources from ridge to intraplate position[END_REF][START_REF] Watkins | Kerguelen: Continental fragment or oceanic island?[END_REF].

These islands primarily consist of tholeiitic to transitional basalt flows emplaced from ~29 to ~24 Ma (Figure S1; Table S1). Several volcano-plutonic complexes emplaced into this basalt sequence are distributed through the islands and range in age from ~24 to ~4.9 Ma (Figure S1; Table S1). With the exception of the oldest tholeiitic-transitional gabbroic complexes, all the younger differentiated intrusive complexes show a bimodal composition with either Si-oversaturated or Si-undersaturated alkaline rock series. The SRBIC is part of a ~350 km 2 plutonic province [START_REF] Nougier | Contribution à l'étude géologique etgéomorphologique des îles Kerguelen: CNFRA[END_REF]; Figure 1; Figure S1). The plutonic rocks of the SRBIC belong to the Si-oversaturated series and range from monzogabbros and monzodiorites to mostly quartzmonzonites and syenites. Rare mafic and intermediate intrusive coarse-grained rocks are only associated with the syenites in the external parts of the SRBIC. The ages obtained for syenites by previous authors with the K-Ar and Rb-Sr whole rock methods range between ~15.5 and ~7.5 Ma from the south to the north of the SRBIC (Table S1). Geophysical studies revealed an abnormally thick crust beneath the NKP with a total crustal thickness estimated to range from 20 to 25 km [START_REF] Charvis | Deep structure of the northern Kerguelen Plateau and hotspot-related activity[END_REF]. Based on the study of oceanic mafic granulite xenoliths, [START_REF] Grégoire | Oceanic mafic granulite xenoliths from the Kerguelen archipelago[END_REF][START_REF] Grégoire | The meta-igneous granulite xenoliths from Kerguelen Archipelago: Evidence of a continent nucleation in an oceanic setting[END_REF] related the deep seismic properties to underplating processes of basaltic melts and associated cumulates equilibrated in the granulites facies, resulting in a transitional and thick crust-mantle boundary. The SRBIC has been considered in early studies as a ring complex and a typical example of emplacement by cauldron subsidence [START_REF] Bonin | The Neogene to Recent Rallier-du-Baty nested ring complex, Kerguelen Archipelago (TAAF, Indian Ocean): Stratigraphy revisited, implications for cauldron subsidence mechanisms[END_REF][START_REF] Giret | Le plutonisme océanique intraplaque: Exemple des Iles Kerguelen: Laboratoire de pétrologie, Université Pierre et Marie Curie[END_REF].

| G EOME TRY OF THE S RB I C

The SRBIC intrudes the plateau lavas and forms a 17 × 11 km elliptical body with a WSW-ENE long axis (Figure 1). The contact between the intrusion and the basalts shows two contrasting geometries (Figure 2 and Figure S2): (a) steeply dipping and cross-cutting or (b) shallowly dipping and concordant. At the scale of the Kerguelen Islands, the orientation of the basaltic lava flows is very regular with a small dip (2-5°) towards the SE, ascribed to recent tilting [START_REF] Nougier | Contribution à l'étude géologique etgéomorphologique des îles Kerguelen: CNFRA[END_REF] Figure S1). This orientation changes towards the intrusive centres, particularly when approaching the SRBIC. Indeed, progressive tilting of the flood basalts is observed all around the NE, SE and SW parts of the SRBIC, reaching a dip of 20-40° away from the intrusion (Figures 1 and2; Figures S1 andS2). Because of recent volcanic activity and ice cover, this contact is not exposed in the NW part of the complex. Nevertheless, measurements of the basalt flows layering along the coast on the western side of the SRBIC dips towards the W. Measurements of primary layering all around the SRBIC confirm that the basalts are tilted away from the intrusive centre with dip increasing towards the contact, a feature already documented by [START_REF] Nougier | Contribution à l'étude géologique etgéomorphologique des îles Kerguelen: CNFRA[END_REF]. The floor and the roof of the SRBIC are not exposed, but the geometry of the surrounding basalt lava flows, the amplitude of their flexure towards the intrusive centre and the location of this flexure, allow us to draw a precise cross-section which indicates that the base of the SRBIC lies at a depth of ~2 km. As the highest syenitic outcrops are at an altitude of ~600 m, we infer that the SRBIC could be modelled as a tabular elliptical magmatic body with a thickness of 2-3 km, a 20 km WSW-ENE long axis and a 16 km NNW-SSE short axis at its base. From these geometrical observations, its volume can be estimated at about ~400 km 3 ± 100 km 3 .

| INTERNAL S TRUC TURE

Three main syenitic units separated by sharp magmatic contacts can be distinguished in the field (Figures 1 and2; Figure S3). Unit 1 occurs along the intrusion margin, Unit 3 at its centre and Unit 2 lies between the two. Field observations indicate that these units are younger towards the centre of the intrusive complex. Each unit consists of several layered intrusions with contrasting textures. All the textures and contacts are magmatic. These different injections of mainly coarse and fine-grained syenites, pegmatites and rare monzodiorites, define a magmatic layering (Figures 1d, 2 and3) which dips consistently outward across the SRBIC, and decreases towards the centre of the intrusion. Moreover, this general magmatic layering is parallel to the external contact with basalts and is also parallel to the main internal contacts that separate the three main syenitic units (Figures 1 and2; Figure S4a). The textures are magmatic throughout the SRBIC; preferred orientation fabrics are poorly developed and rarely visible in the field.

A network of brittle-ductile shear zones is observed throughout the intrusion, except in the youngest Unit 3. They are interpreted to be syn-plutonic as evidenced by mutual cross-cutting relationships between shear zones and dyke intrusions (Figure S5). Their distribution is heterogeneous and shows a denser network at the base of Unit 1 at its contact with the underlying Unit 2 (Figure 2; Figure S4b). The orientation of this network and the observed senses of shear suggest that the shear zones are the result of a shortening perpendicular to the contact between these two units. This defines a ring of more deformed and brecciated syenites (Figure 1), which records flattening of the marginal unit due to injection of the lower units. These features imply an advanced crystallization stage of the older unit before intrusion of the younger one.

| NE W G EO CHRONOLOG I C AL DATA

The methodology, sample location and raw data are available in the Supporting Information (Figures S6 and S7; Tables S2-S4). Most zircons from the SRBIC rocks exhibit a characteristic morphology with (100) prisms and (101) pyramids as well as clear oscillatory and sector zonings (Figure S6). The oldest syenite (13.74 ± 0.2 Ma) was sampled in the southern part of the peninsula and is from one of the first emplaced minor complex clearly disconnected in space and time from the SRBIC (Figures 1 and2; Figure S7). with field observations [START_REF] Ponthus | Origine, évolution et mise en place d'un pluton alcalin récent en contexte intraplaque océanique. Exemple du complexesud de Rallier du Baty[END_REF]; Figures 1 and2).

| D ISCUSS I ON AND CON CLUS I ON

The SRBIC was previously interpreted as a ring complex emplaced by cauldron subsidence that corresponds to a cluster of intersecting ring dykes injected during down-sagging of a crustal block into a deeper magma chamber, and overlain by a caldera [START_REF] Bonin | The Neogene to Recent Rallier-du-Baty nested ring complex, Kerguelen Archipelago (TAAF, Indian Ocean): Stratigraphy revisited, implications for cauldron subsidence mechanisms[END_REF][START_REF] Giret | Le plutonisme océanique intraplaque: Exemple des Iles Kerguelen: Laboratoire de pétrologie, Université Pierre et Marie Curie[END_REF]. Based on our new field and geochronological data, we

propose a new model of construction for the SRBIC based on the following main features:

1. The consistent spatial coincidence between the change in the orientation of layering in the surrounding basalts and the external contacts of the SRBIC indicates a causal relationship, which suggests that the main space-making process for the emplacement of magmas was uplift and up-doming of host basalts. This feature was already recorded by previous authors [START_REF] Nougier | Contribution à l'étude géologique etgéomorphologique des îles Kerguelen: CNFRA[END_REF] and interpreted as being due to the formation of deep lower crustal magma chambers before [START_REF] Giret | Le plutonisme océanique intraplaque: Exemple des Iles Kerguelen: Laboratoire de pétrologie, Université Pierre et Marie Curie[END_REF] or during [START_REF] Bonin | The Neogene to Recent Rallier-du-Baty nested ring complex, Kerguelen Archipelago (TAAF, Indian Ocean): Stratigraphy revisited, implications for cauldron subsidence mechanisms[END_REF] the formation of the SRBIC.

These interpretations are not consistent with field observations, as the uplift produced by deep magma chambers should be more randomly distributed, rather than coinciding exactly with the intrusion geometry. The deflection of the plateau basalts suggests that the SRBIC has a dome shape and is concordant at map scale, which is consistent with emplacement as a laccolith (Figure 4). In summary, our data are compatible with a model where the SRBIC is an incrementally assembled laccolith constructed by the upward inflation (evidenced by basalt uplift) and stacking (evidenced by the internal structure) of initially horizontal thin magma sheets.

Our results establish strong similarities between the SRBIC-a unique example of a felsic laccolith in an oceanic within-plate setting-and continental plutons. First, the 3D geometry, internal structure and emplacement mechanism of the SRBIC are identical to many continental examples. Second, from a dynamic and temporal perspective, it appears that the duration and the average magma flux during SRBIC construction are similar to those proposed for many continental plutons located in contrasting geodynamic settings [START_REF] Rocchi | Two-stage growth of laccoliths at Elba Island, Italy[END_REF]de Saint-Blanquat et al., 2011 and references therein;[START_REF] Barboni | Timing of incremental pluton construction and magmatic activity in a back-arc setting revealed by ID-TIMS U/Pb and Hf isotopes on complex zircon grains[END_REF][START_REF] Barboni | Evaluating the construction and evolution of upper crustal magma reservoir with coupled U/Pb zircon geochronology and thermal modeling: A case study from the Mt. Capanne pluton (Elba, Italy)[END_REF][START_REF] Coleman | The Pace of Plutonism[END_REF][START_REF] Leuthold | Time resolved construction of a bimodal laccolith (Torres del Paine, Patagonia)[END_REF]and references therein).

In analysing similarities between magma emplacement in the Kerguelen plateau oceanic crust and the continental crust, and comparing it with the structure of the standard oceanic crust, we postulate that crustal thickness is the critical parameter that controls the intrusion characteristics. A thick oceanic crust associated with a progressive lithological transition between mantle and crust will favour the formation of a zone where the accumulation and differentiation of magma occur at the crust-mantle boundary, in the manner of MASH/Deep hot zone area of the continental crust [START_REF] Annen | The genesis of intermediate and silicic magmas in deep crustal hot zones[END_REF][START_REF] Hildreth | Crustal contributions to arc magmatism in the Andes of Central Chile[END_REF]. Magma evolution in the "oceanic MASH/ Deep hot zone" from Kerguelen certainly plays the major role, by providing variable volumes of magma at variable timescales, resulting in a final magmatic output forming plutons [START_REF] Ponthus | Origine, évolution et mise en place d'un pluton alcalin récent en contexte intraplaque océanique. Exemple du complexesud de Rallier du Baty[END_REF]. This latter permits to localize injections of magma in close space and time, rather than producing spatially and temporally dispersed network of veins, sills and dykes.

The new ideas we propose on how plutonic processes work in

F

  I G U R E 1 (a) Location of the South Rallier du Baty Intrusive Complex, (b) Structural map of the SRBIC with simplified geology and geochronological data acquired on single zircons (purple circles) and thick sections (yellow circles). (c) Detailed map of a small area with closely spaced U-Pb geochronology sample sites. Equal area, lower hemisphere stereographic projection of poles to magmatic layering (d) and cataclastic shear zones (e); arrows indicate the location of the cross-section in the Figure 2. CRBIC and SRBIC: Central and South Rallier du Baty Intrusive Complexes

  If we interpret U-Pb zircon ages as maximum emplacement ages, the major volume of SRBIC magma was emplaced between 11.57 ± 0.15 and 7.88 ± 0.2 Ma. The zircon ages of the monzodiorites are similar to those of surrounding syenites, suggesting synchronous emplacement of magmas with contrasting degrees of differentiation. U-Pb zircon ages show that the syenites become younger towards the centre of the intrusive complex, from the marginal Unit 1 to the central Unit 3, in agreement

  The discordant and steeply dipping parts of the external contact are irregularities due to local stoping, as suggested by the occurrence of basalt screens within the syenite below those parts of the contact, and/or as extensional F I G U R E 3 Field example of a south-dipping magmatic contact underlined by a dashed white line between two syenitic layers of contrasted magmatic textures. Both layers are cross-cut by basaltic dykes dipping to the N. Central part of the Unit 2F I G U R E 2 SSW-NNE geological cross-section of the SRBIC drawn from field relations. U-Pb results of samples located along or at a distance lower than 2 km of the cross-section are showed; the colour of the small coloured circle corresponds to the sample lithology fractures filled by magma and related to the extensional component of strain in the wall rocks induced by magma infilling.2. The internal structure shows that each of the three main units of magma comprises numerous injections of syenite, with rare monzodiorite in the oldest unit. The magmatic layering is concentric and concordant within and between each unit from external contact to the centre of the intrusive complex. This layering dips towards the outside margin of the complex, decreasing from the periphery towards the centre of the SRBIC. The sharp contacts between the three main syenitic units and the brittle deformation of the two older units demonstrate that in spite of an apparent N-S continuous age distribution, the construction of the SRBIC was discontinuous. The textural difference observed between the coarse and fine-grained syenites was acquired during the emplacement process and we interpret it as the record of the dynamics of magma injection: different volumes of magma injected at variable rates and after variable time intervals between injections will have varying temperature contrasts with previous injections and consequently different cooling rates and thus textures. The two other main facies, the pegmatites and the network of porphyritic syenites, are inferred to have been produced by local mechanical segregation in an older, incompletely crystallized injection during the injection of a younger one. Field relationships and geochronology demonstrate that the SRBIC was constructed by the amalgamation of successive under-accreted magma injections.3. The field relationships and U-Pb geochronology both provide evidence that the magmatic injections become progressively younger towards the centre of the SRBIC. The U-Pb data indicate that the construction of the SRBIC lasted approximately 3.7 Ma, between 11.57 ± 0.15 and 7.88 ± 0.2 Ma. With a total estimated volume of about 300-500 km 3 , this suggests an average injection rate between 0.8 and 1.4 × 10 -4 km 3 /year.

the

  Kerguelen oceanic plateau have implications on the general interpretation of alkaline magmatism in this context, particularly in the light of the comparison made by Grégoire et al. (1998) between the thickened Kerguelen oceanic lithosphere and the thinned F I G U R E 4 Interpreted cross-sections of the general laccolithic geometry of the SRBIC along two sub-perpendicular cross-sections. The representation of the contact between the SRBIC and the CRBIC is conjectural continental lithosphere based on the study of mafic granulitic xenoliths from the Kerguelen Islands.

ACK N OWLED G EM ENTS

We thank the French Polar Institute Paul-Emile Victor (IPEV-TALISKER 1077 program, D. Guillaume; IPEV-DYLIOKER 444 program, B.N. Moine), Y. Le Meur and the IPEV-logistic staff for field assistance and support. We also thank the CNRS-INSU, OMP and the GET for funding allowing data acquisitions. This is contribution 1499 from the ARC Centre of Excellence for Core to Crust Fluid Systems (http://www.ccfs.mq.edu.au) and 1384 from the GEMOC Key Centre (http://www.gemoc.mq.edu.au). The analytical data were obtained using instrumentation funded by DEST Systemic Infrastructure Grants, ARC LIEF, NCRIS/AuScope, industry partners and Macquarie University. W. Powell is thanked for his help during the U-Pb acquisitions. A.R. Cruden, B. Bonin and E.C. Ferré are thanked for their constructive comments on an earlier version of this manuscript. C. Archanjo, an anonymous reviewer, the Scientific Editor K. Mezger and the Associate Editors G. Wörner are warmly thanked for their constructive review, help and encouragement.

DATA AVA I L A B I L I T Y S TAT E M E N T

The data that support the findings of this study are available from the corresponding author upon reasonable request.

S U PP O RTI N G I N FO R M ATI O N

Additional supporting information may be found online in the Supporting Information section. Table S1. Published geochronological data on the Rallier du Baty plutonic province.

Table S3. Zircon U-Pb ages data from the SRBIC, Kerguelen archipelago (T.A.A.F).