published or not. The documents may come

CARING FOR YOUR DATA

By Konrad Hinsen

D

ata is at the heart of science. A scientist is expected to be able to back up all published conclusions with data. Data management should thus be a priority in science. Scientists can't afford to lose data, be uncertain of what it means, or not know where it came from. In the experimental sciences, there's a long tradition of writing down all experimental setups, parameters, and results meticulously in a lab notebook. Unfortunately, computational science is much less rigorous about data handling, although there are clear signs of improvement. Here, we'll consider what you can do to better prepare and manage your data.

Overview of the Problem

Most scientific computing follows a simple pattern: the computer runs a program that reads in data, manipulates it in some way, and produces output data. Input and output data are typically stored in files, but programs written for interactive use also take input directly from the user and provide output in a visual form (plots, animations, and so on).

For scientific research, the data matters more than the programs. In processing experimental results, the observations are the input data, and some quantities more directly related to the problem at hand (or simply a more familiar quantity to experts in the field) are the output data. In the computational evaluation of theoretical models, the input data consists of parameters and the output data is the theoretical prediction. It's the input and output data that's shown and discussed in scientific publications.

The program that produces output data from input data is of little direct scientific interest. What matters is the mathematical equations that it applies to do its work. As scientific problems and computational resources evolve, the programs are often adapted for technical reasons such as performance, but they still implement the same mathematical equations. A typical scientific publication discusses only the equations, not the other aspects of the program. There's an increasing awareness in the computational science community of the need to publish the full programs as well, in order to permit verification of the computations by other scientists. However, this doesn't change the fact that the central items in computational science are data and equations.

Despite the importance of data for science, the storage and processing of data has traditionally been given little priority in scientific programming. The typical scientific programmer concentrates on the mathematical equations and on the algorithms that implement them. The in-memory representation of data in the program is usually chosen for convenience of implementation, as is (in many cases) the file format used to store the data more permanently. To make it worse, the file formats of many scientific programs aren't documented by anything other than the program's source code.

An unfortunate consequence of this priority of code over data is that computational scientists spend a significant amount of time converting data between the formats used by different programs. In addition to requiring a lot of effort, file format conversion is an error-prone process, in particular when the file formats aren't properly documented. A less-visible consequence of the absence of data design is that scientific programs sometimes get trapped by a legacy data representation that can no longer be adapted to evolving requirements.

Data Models and File Formats

An important distinction to keep in mind is the one between data models and file formats. A data model defines how a specific collection of information is expressed in terms of basic data items such as integers, floating-point numbers, or characters, and simple aggregate data structures such as lists, arrays, sets, or associative arrays. A file format defines how a data model is represented by a sequence of bytes. A data model covers in-memory storage as well as file storage.

A given data model can be represented by several file formats. This is often useful because different stages of data processing have different requirements. For example, it might be advantageous to have a compact and platform-independent file format for archiving, a file format optimized for I/O performance for high-throughput computations, and a text-based file

In the long run, your data matters more than your code. It's worth investing some effort to keep your data in good shape for years to come.

format for manual inspection and editing of data. As long as these three file formats implement the same data model, conversion between them is straightforward and lossless.

As a simple example, consider an address book. A simple data model would be as follows:

1. An address book is a list of address book entries. 2. An address book entry consists of a fixed set of fields, whose values are character strings. Let's say the fields are "first name," "last name," "phone number," and "e-mail address."

A simple file format representing this data model can be defined by writing the value of each field in an address book entry as one line of a text file. Each entry would then take up four lines. Because the entries have a fixed length, they can simply follow each other in the file. An alternative file format could use one line per entry, with the fields separated by commas. Conversion between these two formats would be straightforward and not lose any information either way.

In practice, you would want to define a more elaborate data model for an address book. Many people have several phone numbers and e-mail addresses. You could give each of them a label, such as "home" or "work." The value of the "phone number" field is then an associative array whose keys and values are character strings. Another generalization would be to make the whole entry an associative array as well, allowing any number of fields such as "Twitter account," "nickname," or "postal address."

However, this generalization creates a new problem: What if different users of your data model pick different field names or the same information? One user could choose "postal address" and another one just "address." Interoperability between programs would be much more difficult. A partial solution would be a set of mandatory field names. This example illustrates that it's not necessarily a good idea to make a data model overly general.

A more subtle problem with our address book data model is that it defines an address book as a list of entries. A list is a sequential data structure, meaning that the order of address book entries is significant. Two address books with the same entries arranged differently would be considered different. A more appropriate choice for the top-level data structure would be a set. On the other hand, a set is more difficult to implement in software, and you lose the convenience of a list in which each element has a simple and unique handle: its index in the list. Data model design is thus often a matter of choosing the right compromise.

Designing Data Representations

How can we design a good data model? Unfortunately, this question has no simple answer. Data model design, like software design, is an art rather than a science. Nevertheless, the following guidelines are a good starting point.

Look at your mathematical model.

Most scientific computing is applied mathematics, and in that case your mathematical model is usually the best starting point for your data model. Write down a precise and detailed definition of it. Of course, input from the application domain is useful as well.

Inspect your algorithms. Write down their inputs and outputs in particular. Group data items that belong together.

Think in terms of plain English. Describe your data as you would explain it to a fellow scientist, not as you would implement it in your programs. If you are a Fortran or C programmer, you might automatically translate "a list of points in 3D space" into "an N × 3 array of floats." However, other programming languages encourage different choices.

Avoid redundancy. Any information that can be deduced from other data shouldn't be part of your data model. Redundant information opens the door to inconsistent datasets. You of course might want to have redundant data in your implementation, usually for efficiency reasons, but it shouldn't leak out.

Design for extension. It's hardly ever a good idea in software development to design beyond clearly identified needs. Such over-engineering costs time and effort and can actually make future extensions more difficult if they don't turn out to be exactly the ones envisaged initially. However, imagining various possible extensions of your data model can help to make it more future-proof.

Always keep in mind that a data model is defined at a more abstract

Any information that can be deduced from other data shouldn't be part of your data model.

level than any concrete implementation. The role of data models in software design is similar to the role of pseudocode. Both should contain the essential features of the data structures and algorithms used in a program, but none of the technical implementation details.

As a concrete example that will also serve to illustrate file format design, I propose a minimal data model for the chemical structure of molecules. The term molecule is taken to mean a set of atoms linked by chemical bonds. As a useful generalization of standard chemical terminology, commonly made in computational chemistry, a molecule may contain atoms that don't participate in any bond, and in fact a set of atoms without any bonds is accepted as an extreme case of a molecule. Here are the rules that define the data model:

• A molecule is defined by a name, a set of atoms, and a set of bonds. • The name of a molecule is a character string. • Each atom is identified by a unique name (a character string) consisting of its chemical element symbol optionally followed by an integer. • Each bond is defined by a bond order (an integer) and a set of exactly two distinct atoms, each of which must also be an element of the atom set of the molecule.

A water molecule would thus be described as name "water" atoms {"O", "H1", "H2"} bonds {(1, {"O", "H1"}), (1, {"O", "H2"})}.

Note the use of sets, rather than lists or arrays, for atoms and bonds, which expresses the fact that there's no natural order to these constituents. This is a design choice that favors physical realism over simplicity of implementation. Many data models in actual use in computational chemistry do impose an order on the atom set, because this turns out to be quite convenient in practice. However, such convenience, once it becomes a habit, can lead to bad design decisions, even at the level of mathematical models. For example, in the widely used Amber force field (http://ambermd.org/#ff ), the potential energy of a molecule depends on the order of the atoms, even though that order is arbitrary and completely unrelated to any physical property of the molecule. Data models can help to avoid such mistakes.

Designing File Formats

Once you have your data model, you can think about file formats that implement it for archiving and for exchange with other programs. The ideal file format would have the following characteristics:

• Independence of specific operating systems and compilers-the specification should make it possible to write portable programs that read and write the file format anywhere. • Compactness-even with today's terabyte-sized disks, wasting space is never a good idea.

• I/O efficiency-reading and writing

data should be as fast as possible on all common computing platforms. • Simplicity for programmers-if existing I/O libraries can be reused, your file format will be much more popular.

• Convenient handling by scientists-

looking inside a file, or changing some value, should be doable with simple, powerful, and well-known tools.

In practice it's rarely possible to satisfy all of these conditions. As with other aspects of programming, the criteria of convenience and simplicity on one side and efficiency and compactness on the other often lead to contradicting requirements.

The most fundamental choice to make in designing a file format is the choice between a text-based and a binary format. Text formats ultimately represent all data as a sequence of ASCII or Unicode characters. The main advantage is that any text editor on any computer can be used to inspect and modify the data. On the other hand, text formats are neither compact nor efficient to read and write. Binary formats, which represent data by bit patterns identical or very similar to those used by the computer's CPU, have exactly the opposite characteristics: they're compact and I/O efficient, but require format-specific and sometimes platform-specific tools for any inspection or manipulation. It's often useful to define both a text and a binary format for the same data model.

Because data storage in files is a common need in all areas of computing, several generic file formats have been developed that can be adapted to a wide range of applications. These formats specify the representation of basic data items (numbers and text strings) and common data structures (lists, arrays, tables, associative arrays, and so on), which are exactly the basic units in terms of which data models are defined. Ready-to-use libraries for many popular programming languages make them convenient for the programmer, and the existence of generic tools that deal with these formats makes the users' life easier.

The historically first generic text representation of complex data structures was created in the 1950s together with the Lisp programming language. Known as s-expressions (for symbolic expressions), it's based on the list as the central data structure, with list elements being symbols, numbers, character strings, and other lists. Nested lists that also allow non-list elements are a way of encoding a tree structure, with list-type elements representing nodes and non-list elements representing leaves. A tree structure is a good, flexible choice, because most relevant data structures can easily be mapped to a tree.

Today's best-known generic text format is XML (www.w3.org/TR/ rec-xml), defined as a simplified subset of the Standard Generalized Markup Language (SGML) for use in Web technology. XML and SGML were designed for applications where the majority of the data consists of textbasically character strings-but they can be used for different kinds of data at the cost of being rather verbose. The strong point of XML is a rich ecosystem of tools for creating, reading, validating, and transforming XML files.

Similar to s-expressions, XML is based on the concept of a tree structure. The items in the tree are called elements in XML terminology. Each element is delimited by a start tag and an end tag, and can have content (consisting of text and other elements) and attributes (an associative array). The label in the start and end tags identifies the element type. Defining a specific format based on XML comes down to choosing the tags for various element types and specifying constraints on the elements' contents and attributes. Examples of scientific data formats based on XML are Mathematics Markup Language (MathML; www.w3.org/ Math) for mathematical formulas, Extensible Scientific Interchange Language (XSIL; http://resolver.caltech. edu/CaltechCACR:CACR-1999-171) for array and table data, Chemistry Markup Language (CML; www.xmlcml.org) for chemistry, and Systems Biology Markup Language (SBML; http://sbml.org) for systems biology.

Although XML has gained a lot of support in recent years, it isn't always the best choice for a text-based format. Its main disadvantage is its often verbose style, which increases file sizes and tends to make the data more difficult to understand to the human reader. XML is also rather complex, meaning that implementing XML I/O can be a significant effort, even when using XML libraries. Two other formats worth looking at are YAML (www.yaml.org), which stands for "YAML Ain't Markup Language," to emphasize its differences from XML, and JavaScript Object Notation (JSON; http://json.org), which despite its name is supported by more languages than just JavaScript. The two have much in common, with JSON scoring on ease of handling in programs and YAML having the edge in human readability.

Figure 1 shows how the aforementioned chemical structure data model can be represented in these four file formats. YAML is the only format that has a way to represent sets. In the (molecule "water" (atoms "O" "H1" "H2") (bonds (1 "O" "H1") (1 "O" "H2"))) (a)

{"type": "molecule", "name": "water", "atoms": ["O", "H1", "H2"], "bonds": [{"order": 1, "atoms": ["O", "H1"]}, {"order": 1, "atoms": ["O", "H2"]}]} (d) other formats, the sets are replaced by lists. The same choice could be made in YAML to yield more compact files. In the s-expression example, a bond is represented by a three-element list, which is more compact than the attribute-style representation I used in the other formats. Again, other choices can be made as a function of the design priorities.

For binary formats, the two main contenders are the Hierarchical Data Format (HDF5; www.hdfgroup.org/ HDF5) 1 and the Network Common Data Form (NetCDF; www.unidata. ucar.edu/software/netcdf). Of the two, HDF5 is more flexible but also more difficult to use. Both HDF5 and NetCDF are based on multidimensional arrays as their basic data structure. They permit the storage of arrays that are too big to fit into the computer's working memory, providing efficient access to subarrays. Both libraries come with a set of generic tools for data management.

Data Models inside Your Programs

Most scientists are easily convinced of the utility of well-defined data models and file formats for archiving data and for exchanging it between programs and research groups. After all, we've all wasted a lot of time in guessing at file formats and writing conversion programs. The advantage of defining data models for the in-memory data structures of scientific software is less evident, but quite important as well. In fact, there are two main advantages.

This first one should be evident: the more similar the in-memory data model of your software is to the data model you use for file storage, the easier it is to read and write files. It's easier to program, but also more efficient for the computer, because there's less conversion to be done.

The second advantage is that welldefined data models make software more modular. What you pass around between functions are of course the data structures that your programming language proposes. But if your software is designed around data models, you'll probably pass around combinations of data structures that you identified in your data model as useful units. Without a data model, you're likely to pass around just the data that every individual function needs. You then have to think about each function's interface separately, whereas with a data model you see at a glance what your function parameters mean. For example, you might pass a point object to a function instead of three coordinates if your data model makes use of geometrical points.

Just as implementing a data model for disk storage requires defining a file format, implementing an in-memory representation of a data model requires a translation into the concrete data structures available in your chosen programming language. Because programming languages vary enormously in their data definition features, implementing a data model can take highly different forms. At one end of the spectrum, there are languages such as Fortran 77 that don't allow any user-defined data constructs. The only way to implement a data model is thus to provide a list of variables (scalars and arrays) and describe the relations between them by comments. At the other end, there are modern object-oriented or functional languages in which each data item in the data model can be implemented as one problem-specific data type.

U nfortunately, the art of data modeling in scientific computing is still in its infancy, with few tutorials or textbooks available to serve as guides. A good starting point is to look at existing scientific data models. Try an Internet search with some keywords from your field of research followed by "XML" or "HDF5." You might discover that there are already good data models around that you can simply adopt, or at least use as a starting point for your own development.

Figure 1 .

 1 Figure 1. Possible representations of the chemical structure data model. The representations are shown in (a) s-expression, (b) XML, (c) "YAML Ain't Markup Language" (YAML), and (d) JavaScript Object Notation (JSON) formats.

Without a data model, you're likely to pass around just the data that every individual function needs.


David Coupier
email: david.coupier@imt-nord-europe.fr

Lucas Flammant

Viet Chi Tran

Thick trace at infinity for the Hyperbolic Radial Spanning Tree

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

, it is known that in straight radial rooted trees, with probability 1, infinite paths all have an asymptotic direction and each asymptotic direction is reached by (at least) an infinite path. Moreover, there exists a set of 'exceptionnal' directions reached by (at least) two infinite paths which is random, dense and only countable in dimension 2. Howard & Newman's method says nothing about (random) directions reached by more than two infinite paths and, in particular, if such 'very exceptionnal' directions exist in dimension 2. In this paper, we prove that the answer is no for the hyperbolic Radial Spanning Tree (RST): in dimension 2, this tree does not contain 3 infinite paths with the same (random) asymptotic direction with probability one. Turned in another way, this means that there is no infinite but thin subtree in the hyperbolic RST, i.e. whose infinite paths would all have the same asymptotic direction. We actually prove a stronger result in dimension d + 1, d ≥ 1, stating that any infinite subtree of the hyperbolic RST a.s. generates a thick trace at infinity, i.e. the set of asymptotic directions reached by its infinite paths has a positive measure.

Introduction

Unlike combinatorial random graphs whose Erdös-Rényi model is certainly the most famous specimen (see [START_REF] Erdös | On the evolution of random graphs[END_REF][START_REF] Van Der Hofstad | Random Graphs and Complex Networks[END_REF]), the structure of a geometric random graph depends on the locations of its vertices (embedded in a metric space) and on some geometrical rules. This feature makes geometric random graphs suitable to model real phenomena (in biology, material science, image analysis or telecommunication networks) and this is the reason why they have been intensively studied in the last decades. See the book of Penrose [START_REF] Penrose | Random geometric graphs[END_REF] for a general reference on the topic.

For geometric random graphs, macroscopic properties trigger challenging questions, such as studying the number of the topological ends of these structures. Alexander solves this question in [START_REF] Alexander | Percolation and Minimal Spanning Forests in Infinite Graphs[END_REF] for Minimal Spanning Forests on infinite graphs. In the case of Euclidean trees, i.e. trees whose vertices are points of R d+1 with d ≥ 1, a fundamental step has been taken by Howard & Newman for straight rooted trees, i.e. whose subtrees are all the thinner as their roots are far away from that of the entire tree [START_REF] Howard | Geodesics and spanning trees for Euclidean first-passage percolation[END_REF]Section 2.3]. They develop an efficient method [START_REF] Howard | Geodesics and spanning trees for Euclidean first-passage percolation[END_REF]Proposition 2.8] ensuring that any straight tree T satisfies the two following properties almost surely (a.s.):

(A) Every infinite path (z n ) n≥0 of T -a sequence of different vertices (z n ) n≥0 such that {z n , z n+1 } is an edge of the tree for every n ≥ 0 -admits an asymptotic direction u in the unit sphere S d of R d+1 , i.e. lim n→+∞ z n |z n | = u .

(1.1) (B) Every direction u ∈ S d is asymptotically reached by (at least) one infinite path of the tree T .

We will also say that the direction u satisfying (1.1) is asymptotically reached or targeted by the path (z n ) n≥0 . In the whole paper, the dimension of the ambiant space is denoted by d + 1.

Howard and Newman applied their method to the case of first-passage percolation trees built on a Poisson Point Process (PPP). They also proved that, for any deterministic u ∈ S d , the tree T a.s. contains exactly one infinite path with u as asymptotic direction. However there exists a.s. a set of random directions, thought as 'exceptional', targeted by at least two infinite paths which is dense in S d , and only countable in dimension 2. See Fig. 1 and Remark 3.7 for details about these 'exceptional' directions.

Howard & Newman's method then motivated many works on geometric random trees (in particular in dimension 2). Various authors stated the straightness of their favorite tree so as to obtain Properties (A) and (B) for its infinite paths. Here are some examples. The Directed Last-Passage Percolation (LPP) Tree is obtained by a last-passage procedure from i.i.d. weights associated to the vertices of the grid N d+1 . In the case of exponential weights and d + 1 = 2, Ferrari & Pimentel [START_REF] Ferrari | Competition interfaces and second class particles[END_REF] established the straightness of the directed LPP tree. From asymptotic directions of infinite paths of the LPP tree, they deduced the existence of an asymptotic direction for a competition interface. Other works focused on geometric trees directed towards a distinguished point 0, with edges defined by local geometric rules. A first example is the Radial Spanning Tree (RST) introduced by Baccelli & Bordenave in the Euclidean plane to model communication networks [START_REF] Baccelli | The radial spanning tree of a Poisson point process[END_REF]. Each vertex of this tree has an outgoing edge towards the nearest vertex among those closer to 0. The hyperbolic RST studied in this paper is an extension of their RST. Using a directed forest (namely the Directed Spanning Forest) approximating locally, in distribution and far from the root the RST, they proved the straightness of the RST. A second example is the Navigation Tree defined by Bonichon & Marckert in [START_REF] Bonichon | Asymptotics of geometrical navigation on a random set of points in the plane[END_REF]. Each vertex has also outdegree one and the corresponding edge links it to the closet vertex in a given cone directed towards 0. To describe the asymptotic geometry of the navigations paths, the authors stated the straightness of the Navigation Tree.

Since in dimension 2, only a countable number of random directions are targeted by at least two infinite paths, it is natural to ask whether some of them are 'very exceptional', i.e. targeted by more than two infinite paths. Assuming the non-crossing path property (satisfied by all the previously mentioned trees), the existence of such a 'very exceptional' direction would mean that of an infinite subtree-containing the middle infinite pathwhich would be very thin because trapped between both extreme infinite paths having the same asymptotic direction. The existence of such a subtree is suspicious since it is involved in a spatial competition that it neither wins (it fails to occupy a macroscopic part of the space) nor loses (it is unbounded). This heuristic sustains the following refinement of the Howard & Newman's results: Conjecture 1.1. For most of straight bi-dimensional geometric random trees having the non-crossing path property which are studied in the literature (including those cited above), there is no (random) direction targeted with more than two infinite paths with probability 1.

To our knowledge, this conjecture has been established in only one case: for the Directed LPP Tree with exponential weights in N 2 by Coupier [START_REF] Coupier | Multiple geodesics with the same direction[END_REF]. His proof is based on a surprising coupling-exhibited for the first time by Rost [START_REF] Rost | Nonequilibrium behaviour of a many particle process: density profile and local equilibria[END_REF] -between the LPP model and the Totally Asymmetric Simple Exclusion Process (abbreviated in the literature to TASEP) and on results on some special particles (called second class particles) of this particle system [START_REF] Amir | The TASEP speed process[END_REF].

In this paper, we present a second example satisfying this conjecture, namely the hyperbolic Radial Spanning Tree introduced in [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF]. See Fig. 1. Let us emphasize that the nature of this new example is completely different from the first one since it lies in a continuum hyperbolic space instead of N 2 . The construction of the hyperbolic RST is the same as for the bi-dimensional Euclidean RST of Baccelli & Bordenave [START_REF] Baccelli | The radial spanning tree of a Poisson point process[END_REF], whatever the dimension d + 1 ≥ 2. The set of vertices is given by a homogeneous Poisson Point Process (PPP) N of intensity λ > 0. 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 procedure defines a random radial tree rooted at 0 which is straight in H d+1 ; see [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF]Proposition 2.7] or Property 3.2 below. This key property is available here thanks to the fact that the hyperbolic metric guarantees that angular deviations of RST paths decay exponentially fast with the distance to the origin. This feature entails in fact much more than the straightness of the hyperbolic RST. It also implies that there is a positive proportion of RST edges at level r giving rise to infinite paths (Proposition 3.8 below). This later statement does not occur in the Euclidean case, where the probability that a given edge at level r belongs to an infinite path tends to 0 as r → ∞ [START_REF] Baccelli | Semi-infinite paths of the 2d-radial spanning tree[END_REF].

Our main result (Theorem 3.5) holds in any dimension. Heuristically, we will show that the set of directions u ∈ S d that are asymptotically reached by infinite paths stemming from an arbitrary Poisson point z, called the trace of z at infinity, is either empty or has positive measure. In the first case, the subtree rooted at z is finite while in the second case, it is infinite and generates a thick trace at infinity. In dimension 2, thanks to planarity and the non-crossing path property, it is easily deduced from Theorem 3.5 that a.s. there is no direction reached by more than two infinite paths, i.e. Conjecture 1.1. See Corollary 3.6, Item (iii).

The proof of Theorem 3.5 can be summarized by the geometric construction depicted in Fig. 2 and mainly relies on the fact that the probability with which a given Poisson point z at level r generates a thick trace at infinity (plus extra conditions as in particular a stabilization criteria) is bounded away from 0 uniformly on r (Lemma 4.5). This technical result is specific to the hyperbolic metric: this explains why we are able to prove Conjecture 1.1 for the hyperbolic RST and not for its Euclidean counterpart (although the Euclidean RST contains in proportion fewer infinite paths than the hyperbolic RST). Lemma 4.5 also uses a control of path fluctuations developed in [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF] (recalled Lemma 5.2).

In Section 2, we recall shortly some results on the hyperbolic geometry that will be useful in the paper. In Section 3, we enounce our main result: Theorem 3.5 for the general dimension d + 1, d ≥ 1, and its Corollary 3.6 in dimension 2, that answers the Conjecture 1.1 for the hyperbolic Radial Spanning Tree, providing the first example in continuum space where this question can be answered. Section 4 presents the proofs of Theorem 3.5 and of the technical Lemma 4.5 introduced in the previous paragraph. Section 5 is devoted to the proof of this later result. Finally, in Section 6, we prove that there is a positive proportion of points of the tree at level r that generates thick tracks at infinity.

Hyperbolic geometry and notations

Generalities on hyperbolic geometry. We refer to [START_REF] Cannon | Hyperbolic geometry[END_REF][START_REF] Chavel | Riemannian Geometry: A Modern Introduction, Second Edition[END_REF][START_REF] Paupert | Introduction to hyperbolic geometry[END_REF][START_REF] Ratcliffe | Foundations of Hyperbolic Manifolds[END_REF] for a complete introduction to hyperbolic geometry. For d ∈ N * = {1, 2, . . . }, 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 the open-ball model D (or Poincaré disc model in dimension 2) consisting in the unit open-ball

D := {(x 1 , . . . , x d+1 ) ∈ R d+1 : x 2 1 + . . . + x 2 d+1 < 1}
endowed with the metric:

ds 2 D := 4 dx 2 1 + . . . + dx 2 n+1 (1 -x 2 1 -. . . -x 2 d+1 ) 2 . The volume measure on (D, ds 2 D ) is then given by 2 d+1 dx 1 . . . dx d+1 /(1-x 2 1 -. . .-x 2 d+1 ) d+1 .
A convenient way to represent points in the open-ball model is to use polar coordinates (w.r.t. the origin 0). Any z ∈ D can be written as z = (r; u) where r = d(z, 0) is its distance to 0 and u ∈ S d is its direction. Let σ be the spherical probability measure on S d (in particular σ(S d ) = 1). In polar coordinates, the rescaled volume measure Vol, corresponding to the rescaled probability measure σ, becomes dVol(r; u) = sinh(r) d dr dσ(u) .

(2.1)

The choice of rescaling of the volume measure is adopted to avoid writing the constant 2π (d+1)/2 /Γ((d + 1)/2) that would appear a lot in the paper otherwise (this constant is the non-rescaled measure of S d , see [START_REF] Chavel | Riemannian Geometry: A Modern Introduction, Second Edition[END_REF](III.3.10) p.125]).

The hyperbolic space H d+1 is also naturally equipped with a set of points at infinity denoted by ∂H d+1 . In the open-ball model, this set is identified with the unit sphere S d . The distances are distorted in comparison with the Euclidean distance and become 'smaller' when we approach the boundary ∂D = S d , which is at infinite hyperbolic distance from the center 0. We also set H d+1 := H d+1 ∪ ∂H d+1 endowed with the topology given by the closed ball.

In (D, ds 2 D ), geodesics are of two types: either diameters of D or arcs perpendicular to the boundary ∂D. Also, this model is conformal in the sense that the hyperbolic angle between two geodesics is equal to the Euclidean angle between them, in the open ball representation. Another important fact about hyperbolic geometry is that all points and all directions play the same role: H d+1 is homogeneous and isotropic.

Notations. We denote by d(•, •) the hyperbolic distance in H d+1 . For z, z ′ ∈ H d+1 , let [z, z ′ ] be the geodesic between z and z ′ . We will denote by (z, z ′ ) the geodesic without the extremities z and z ′ .

For z ∈ H d+1 and r > 0, we respectively denote by B(z, r) := {z ′ ∈ H d+1 : d(z, z ′ ) < r} and S(z, r) := {z ′ ∈ H d+1 : d(z, z ′ ) = r} the hyperbolic open ball and hyperbolic sphere centered at z with radius r > 0. We will write B(r) := B(0, r) and S(r) := S(0, r) for short. Writting Vol(B(r)) = r 0 sinh d (ρ)dρ (see e.g. [22, p.79]), it is easy to prove that there exists c = c(d) ∈ (0, 1) such that for any r ≥ 1,

ce dr ≤ Vol(B(r)) ≤ νe dr . (2.2)
Let us also denote by C(z, r, R) := {z ′ ∈ H d+1 : r < d(z, z ′ ) < R} the annulus with radii r < R and C(r, R) := C(0, r, R).

For z, z ′ , z ′′ ∈ H d+1 , zz ′ z ′′ is the measure of the corresponding (non-oriented) hyperbolic angle. For any z ∈ H d+1 and θ > 0, Cone(z, θ) := {z ′ ∈ H d+1 : z0z ′ ≤ θ} is defined as the cone of apex 0, axis z and aperture θ. In addition, for z ∈ H d+1 and r > 0, we will use the spherical cap B S(r) (z, θ) := Cone(z, θ) ∩ S(r) .

(2.3)

For its surface, note that there exists a constant C = C(d) > 0 such that:

σ B S(r) (z, θ) ≤ Cθ d .
(2.4)

Main results

In the sequel, we pick some arbitrary origin point 0 ∈ H d+1 (thought as the center of D in the open-ball representation) to be the root of the hyperbolic RST that we now define.

The hyperbolic RST. Let N be a homogeneous PPP of intensity λ > 0 in H d+1 . The definition of the hyperbolic RST is similar to the Euclidean case. This is a directed graph whose vertex set is N ∪ {0} and in which each vertex z ∈ N is connected to the closest Poisson point among (N ∪ {0}) ∩ B(r), with r := d(0, z), for the hyperbolic distance.

Definition 3.1 (Radial Spanning Tree in H d+1 ). The ancestor of z ∈ N is defined as

A(z) := argmin z ′ ∈(N ∪{0})∩B(d(0,z)) d(z ′ , z) . (3.1)
The Radial Spanning Tree (RST) in H d+1 is the directed graph (V, E) where V := N ∪ {0} and E := {(z, A(z)) : z ∈ N }.

Since N ∪ {0} contains no isosceles triangles with probability 1, the ancestor A(z) of any Poisson point z is a.s. well-defined. In other words, any Poisson point admits only one outgoing edge, but possibly several ingoing edges. In any case, these ingoing edges are finitely many: Property 3.2 (Proposition 2.2 of [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF]). A.s. the RST is a tree rooted at 0 where all vertices have finite degrees. In the bi-dimensional case (d = 1), the union of geodesics [z, A(z)], z ∈ N , is planar, i.e. whatever z = z ′ ∈ N , the geodesics [z, A(z)] and [z ′ , A(z ′ )] may only overlap on their endpoints.

For z ∈ N and r > 0, we also define B + (z, r) := B(z, r) ∩ B(d(0, z)) and B + (z) := B + (z, d(z, A(z))) .

(3.2)

By construction of the ancestor, the random set B + (z) avoids the PPP N . This fact is responsible for many difficulties when studying the RST. Indeed, when restarting from A(z) = (r ′ ; u ′ ) and constructing the path forward (towards 0), with probability 1, B(r ′ ) ∩ B + (z) is non-empty. This means that the geometric information used to determine A(z) is still involved for next steps of the process, generating statistical dependencies.

A path of the RST is a sequence (finite or not) of different vertices (z 0 , z 1 , z 2 . . .) in N ∪ {0} such that A(z n+1 ) = z n for any n ≥ 0. By (3.1), we have that for all z ∈ N ,

d(0, A(z)) < d(0, z). Applying this to z = z n , n ≥ 1, we obtain that d(0, z n ) < d(0, z n+1 ).
The path (z 0 , z 1 , z 2 . . .) can thus be viewed as an exploration of the RST starting at z 0 and moving away from 0. We will say that the forward direction is towards 0 and the backward direction is towards infinity. Every path can be started at z 0 = 0, but this is not an obligation.

The set of descendants D(z) of a given vertex z is made up of all vertices that can be reached by a backward finite path starting at z (including z itself by convention). Because a vertex can be the ancestor of no other vertex, all sets of descendants are not infinite.

As mentioned in the Introduction, a very important feature of the hyperbolic RST is its straightness: Property 3.3 (Proposition 2.7 of [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF]). A.s. for any ε > 0 there exists some r 0 > 0 such that, for any radius r ≥ r 0 and for any vertex z of the hyperbolic RST with d(0, z) ≥ r, the set of descendants D(z) is contained in a cone of apex 0 and aperture e -(1-ε)r , i.e. for any z

′ , z ′′ ∈ D(z), z ′ 0z ′′ ≤ e -(1-ε)r .
Trace on the boundary ∂H d+1 . Let us consider a point z ∈ N and its set of descendants D(z). When it is infinite, it contains (at least) an infinite path (z n ) n≥0 by the finite degree property (Property 3.2). Theorem 1.1 of [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF] then asserts that the path (z n ) n≥0 admits an asymptotic direction u ∈ ∂H d+1 . Thus, let us define by D ∞ (z) the set of asymptotic directions in ∂H d+1 that can be reached by the infinite paths of D(z). Roughly speaking, D ∞ (z) is the trace left by the set of descendants D(z) on the boundary ∂H d+1 (see the traces at infinity left by the children of 0 in Fig. 1). We just have proved that:

Property 3.4. A.s. for any vertex z ∈ N , #D(z) = +∞ ⇐⇒ D ∞ (z) = ∅ . (3.3)
Our main result (Theorem 3.5) establishes a stronger statement: infinitely many descendants in D(z) actually implies that D(z) leaves a trace with positive volume at infinity, i.e. σ(D ∞ (z)) > 0. In this case, we will say that z generates a thick trace at infinity. Theorem 3.5. Consider the hyperbolic RST in H d+1 , for any dimension d ≥ 1. A.s. for any vertex z ∈ N , either z admits finitely many descendants or σ(D ∞ (z)) > 0.

Consequence in H 2 . Theorem 3.5 holds in any dimension. In dimension 2 (i.e. with d = 1), combined to the non-crossing path property (Property 3.2), it leads to Conjecture 1.1: a.s. the RST does not contain three infinite paths with the same (random) asymptotic direction. This statement-Item (iii) of Corollary 3.6 -completes the description of infinite paths and their asymptotic directions of the hyperbolic RST started in [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF]. The first three items below are given by [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF]Theorem 1.1]. Their proofs are based on the strategy developed by Howard and Newman in [START_REF] Howard | Geodesics and spanning trees for Euclidean first-passage percolation[END_REF] and on the straightness of the hyperbolic RST.

Corollary 3.6. The following properties concern the hyperbolic RST in H 2 .

(o) A.s. any infinite path admits an asymptotic direction and, for any u ∈ ∂H 2 , the RST contains an infinite path with asymptotic direction u.

(i) For any (deterministic) u ∈ ∂H 2 , the RST a.s. contains a unique infinite path with asymptotic direction u.

(ii) A.s. the subset of ∂H 2 of asymptotic directions reached by at least two infinite paths is dense and countable in ∂H 2 .

(iii) A.s. no (random) asymptotic direction of ∂H 2 is reached by more than two infinite paths.

Item (o) says that any u in ∂H 2 is reached by at least one infinite path and exactly one when u is deterministic (Item (i)). There exist by Item (ii) asymptotic directions which are reached by several infinite paths but these directions are random and few (only countable). Finally, Theorem 3.5 specifies that there is no random asymptotic direction reached by more than two infinite paths.

Proof of Corollary 3.6, Item (iii). Let us assume that the hyperbolic RST contains three different infinite paths, say (x n ) n≥0 , (y n ) n≥0 and (z n ) n≥0 having the same asymptotic direction in ∂H 2 . Without loss of generality, we can also assume that these three paths have no vertices in common. By the non-crossing path property and planarity, one of these three infinite paths, say (y n ) n≥0 , is trapped between the two other paths which by hypothesis have the same asymptotic direction. This forces σ(D ∞ (y 0 )) = 0 while the set of descendants D(y 0 ) is infinite. This occurs with null probability thanks to Theorem 3.5. Remark 3.7. Let us show that the asymptotic direction separating the green and red traces at infinity on Fig. 1, is reached by two infinite paths (a green one and a red one). In the green (resp. red) infinite subtree, pick the leftmost (resp. rightmost) infinite path in the trigonometric sense. Such construction makes sense in dimension 2 thanks to the noncrossing path property. These two paths π green and π red admit asymptotic directions, say u green and u red in ∂H 2 by Corollary 3.6, Item (o). If u green and u red were different, any asymptotic direction u ′ located (strictly) between them shoud be reached by an infinite path π thanks to Corollary 3.6, Item (o). By construction of π green and π red , the path π could not be neither green or red, leading to a contradiction.

Positive density with σ(D ∞ (•)) > 0. Although the previous results concern the combinatorial structure of the RST, it will be useful in the proofs to represent the graph RST as a subset of H d+1 , denoted by RST, in which each edge (z, A(z)) is represented by the arc |[z, A(z)]| (defined in Appendix A) and not by the (hyperbolic) geodesic [z, A(z)]: This section ends with a density result. Proposition 3.8 heuristically says that in expectation a macroscopic proportion of elements of L r generates a thick trace at infinity: Proposition 3.8. There exists c = c(d) > 0 such that for any r > 0,

RST := z∈N |[z, A(z)]| . ( 3
E #{z ∈ L r : σ(D ∞ (z)) > 0} ≥ c E #L r .
Proposition 3.8 contrasts with several Euclidean bi-dimensional trees [START_REF] Ahlberg | The number of geodesics in planar first-passage percolation grows sublinearly[END_REF][START_REF] Coupier | Sublinearity of the number of semi-infinite branches for geometric random trees[END_REF], including the Euclidean RST [START_REF] Baccelli | Semi-infinite paths of the 2d-radial spanning tree[END_REF], where the proportion of edges at level r belonging to infinite paths is negligible. Proposition 3.8 is an immediate consequence of some intermediate results leading to Theorem 3.5, and its proof is done in Section 6. It could be improved in several directions (an almost sure result rather than in expectation, a positive limit of the ratio rather than a positive lim inf etc.) but this is not the goal of the current work.

4 Proof of Theorem 3. Consider the event E(r 0 ; u 0 ) on which the set of points at infinity D ∞ (z 0 ) reached by descendants of z 0 is nonempty but with null volume:

E(r 0 ; u 0 ) := D ∞ (z 0 ) = ∅ and σ D ∞ (z 0 ) = 0 . (4.1)
Our purpose is to prove that P(E(r 0 ; u 0 )) = 0.

For δ > 0, let us consider the set G(δ) of descendants z of z 0 whose ancestor A(z) is not too close to z:

G(δ) := z ∈ D(z 0 ) : d(z, A(z)) ≥ δ .
Let H(δ) be the set of corresponding radii:

H(δ) := r > 0 : ∃u ∈ S d , (r; u) ∈ G(δ) .
The next result states that, for δ small enough, if z 0 admits infinitely many descendants (or in an equivalent way D ∞ (z 0 ) = ∅) then infinitely many of them are in G(δ):

Lemma 4.1. For any δ > 0 small enough, D ∞ (z 0 ) = ∅ a.s. implies that H(δ) is un- bounded.
For any radius r > 0, let N B(r) := N ∩ B(r) be the PPP N restricted to the ball B(r). The following Lemma 4.2 says that a.s. on D ∞ (z 0 ) = ∅ the random variable

P(σ(D ∞ (z 0 )) > 0 | N B(r)
) is bounded away from 0 for all radii r ∈ H(δ) and uniformly on those radii. Lemma 4.2. For any δ > 0 small enough, there exists ε > 0 such that, a.s. on The second lemma says that for this choice of δ and ε, we have a.s. on {D ∞ (z 0 ) = ∅} and for any r ∈ H(δ),

D ∞ (z 0 ) = ∅, ∀r ∈ H(δ), P σ(D ∞ (z 0 )) > 0 | N B(r) ≥ ε .
P E(r 0 ; u 0 ) | N B(r) ≤ P σ(D ∞ (z 0 )) = 0 | N B(r) ≤ 1 -ε .
Since ε is uniform on r ∈ H(δ) and since we work on {D ∞ (z 0 ) = ∅}, the set H(δ) is a.s. unbounded. We can take the lim inf: a.s. on {D ∞ (z 0 ) = ∅},

lim inf r→∞ P E(r 0 ; u 0 ) | N B(r) ≤ 1 -ε . (4.3)
But the martingale convergence theorem asserts that:

lim r→∞ P E(r 0 ; u 0 ) | N B(r) = 1 E(r 0 ;u 0 ) . (4.4)
So, on E(r 0 ; u 0 ) ⊂ {D ∞ (z 0 ) = ∅}, the limit in (4.4) equals 1. Thus, both statements (4.3) and (4.4) are compatible only if P(E(r 0 ; u 0 )) = 0. Hence, the union of the events E(r 0 ; u 0 ) with rational radius r 0 and rational direction u 0 has null probability too. Since any Poisson point is the closest one to some rational element (r 0 ; u 0 ) of H d+1 , we can conclude that with probability 1, any vertex z ∈ N satisfies either D ∞ (z) = ∅ (or D(z) is finite in an equivalent way) or σ(D ∞ (z)) is (strictly) positive.

Proof of Lemma 4.1

The proof of Lemma 4.1 is based on a percolation argument. To set it up, we first need a covering of the hyperbolic space H d+1 (except in the vicinity of the origin) with a uniform control of overlappings. This will be done using the Covering Lemma (below). This classical result will be used several times in this paper and is stated in [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF]Lemma 4.2].

Lemma 4.3 (Covering Lemma

). There exists a covering constant K = K(d) > 0 such that, for any r > 0, there exists a collection of N (r) points z 1 , ..., z N (r) ∈ S(r) such that:

(a) 1≤i≤N (r) B S(r) (z i , e -r ) = S(r), (b) ∀z ∈ S(r), #{1 ≤ i ≤ N (r) : z ∈ B S(r) (z i , e -r )} ≤ K.
Moreover, there exists C = C(K, d) > 0 such that, for any r > 0, z ∈ S(r) and A ≥ 1, the number of caps overlapping B S(r) (z, Ae -r ) is bounded by CA d :

# 1 ≤ i ≤ N (r) : B S(r) (z i , e -r ) ∩ B S(r) (z, Ae -r ) = ∅ ≤ CA d .
In particular (taking A := πe r in the previous inequality), N (r) ≤ Ce dr .

For any integer radius r > 0 and any index

1 ≤ m ≤ N (r), let us set B r,m := C(r, r + 1) ∩ Cone(z r,m , e -r )
where {z r,1 , . . . , z r,N (r) } is the collection of points of the sphere S(r) given by Lemma 4.3 (in this proof we stress the dependence on r of the z m 's by adding the index r in z r,m ). Each block B r,m is based on the spherical cap B S(r) (z r,m , e -r ) = S(r) ∩ Cone(z r,m , e -r ) and has thickness 1. Two blocks B r,m and B r ′ ,m ′ are said to be adjacent, which is denoted by B r,m ∼ B r ′ ,m ′ , if they are at distance less than δ from each other. Lemma 4.4 asserts that the volumes of B r,m 's are uniformly bounded and the degrees in the graph generated by the adjacency relation are bounded. Lemma 4.4. There exist two positive constants C 1 and C 2 only depending on d such that for all integer radius r > 0 and m ∈ {1, . . . , N (r)}, the following holds :

Vol(B r,m ) ≤ C 1 ,
and

# (r ′ , m ′ ) : B r ′ ,m ′ ∼ B r,m ≤ C 2 . (4.5)
Proof. First, using (2.4),

Vol(B r,m ) = r+1 r sinh d (ρ) × σ Cone(z ρ,m , e -ρ ) ∩ S d dρ ≤ C r+1 r e dρ dρ × e -dr = C d (e d -1) =: C 1 .
For the second inequality of (4. The block B r,m is said δ-bad if it contains a Poisson point z such that d(z, A(z)) < δ. Let us first prove that it is possible to choose δ > 0 small enough so that the set of δ-bad blocks denoted as Ψ δ is a.s. subcritical w.r.t. the adjacency relation, i.e. Ψ δ only admits finite connected components. To do it, we first use the Mecke's formula [START_REF] Daley | An introduction to the theory of point processes[END_REF]Prop. 13.1.IV] to bound the probability for a block to be δ-bad: for any (r, m),

P B r,m is δ-bad ≤ E [#{z ∈ B r,m ∩ N : d(z, A(z)) < δ}] = E #{z ∈ B r,m ∩ N : B + (z, δ) ∩ N = ∅ = λ Br,m P B + (z, δ) ∩ N = ∅ dz ≤ λVol(B r,m ) 1 -e -λVol(B(δ)) ≤ λC 1 1 -e -λVol(B(δ)) =: p(δ) (4.6)
thanks to Lemma 4.4. Hence, P(B r,m is δ-bad) tends to 0 as δ → 0 uniformly on the couple (r, m).

In a second step, we adapt the Peierls argument to our context to establish the subcriticality of Ψ δ for δ small enough. Given (r, m), let P r,m (k) be the set of paths with length k of adjacent blocks starting at B r,m . By Lemma 4.4, #P r,m (k) ≤ (C 2 ) k . Such a path π = (B 1 , . . . , B k ) is said δ-bad if all the blocks B i it contains are δ-bad. Henceforth the probability for B r,m to belong to an infinite connected component of δ-bad blocks is upperbounded by lim sup k→∞ π∈Pr,m(k)

P π is δ-bad .

For any path π = (B 1 , . . . , B k ) in P r,m (k), we can choose a subset of blocks {B i 1 , . . . , B i ℓ } included in {B 1 , . . . , B k } such that the B i j 's are two by two non adjacent and ℓ ≥ k/C 2 (here we use that each block has at most C 2 adjacent blocks). Besides, the adjacency relation has been defined so that the events {B i j is δ-bad}'s are mutually independent. So

P π is δ-bad ≤ P B i 1 , . . . , B i ℓ are δ-bad ≤ p(δ) k/C 2 ,
where p(δ) is defined in (4.6). It follows

π∈Pr,m(k) P π is δ-bad ≤ C 2 p(δ) 1/C 2 k .
Choosing δ small enough so that C 2 p(δ) 1/C 2 < 1, we then obtain that a.s. the block B r,m cannot belong to an unbounded connected component of δ-bad blocks. Consequently, for such parameter δ, the set Ψ δ is subcritical with probability 1.

To conclude the proof of Lemma 4.1, let us pick δ small enough such that the set of δ-bad blocks does not percolate. Assume also that D ∞ (z 0 ) is non empty, i.e. the subtree of the RST rooted at z 0 admits (at least) one infinite path (z n ) n≥0 of Poisson points. The PPP N being locally finite, the path (z n ) n≥0 cannot be stuck, from some index, inside a δ-bad connected component of blocks which is bounded by choice of δ. It eventually comes out of each δ-bad connected component. Two cases must be distinguished. Either (z n ) n≥0 visits infinitely many δ-good blocks where of course a block is said δ-good if it is not δ-bad. Hence, infinitely many of the z n 's satisfy d(z n , A(z n )) ≥ δ. These vertices are in G(δ) which means that H(δ) is unbounded.

Or, the infinite path (z n ) n≥0 jumps infinitely many times from a δ-bad connected component to another one. But, by construction, two different bad connected components are at distance at least δ from each other. So these jumps provide as many z n 's in G(δ): H(δ) is unbounded in this case too.

Proof of Lemma 4.2

In what follows, we will modify configurations locally. This is why we emphasize the dependence of any random set A (think about D(z), D ∞ (z) or H(δ)) on the current configuration η, a realization of the PPP N , by writting A[η].

Given r 0 > 0 and u 0 ∈ S d , recall that z 0 is the closest Poisson point to (r 0 ; u 0 ). For this section, let us consider a configuration η satisfying D ∞ (z 0 ) = ∅. Let δ > 0 and r > r 0 be an element of the set H(δ)[η]by Lemma 4.1, this latter set is unbounded provided δ is small enough. Let us set z 1 := (r; u) ∈ G(δ) for some u ∈ S d (z 1 is in N ). In the sequel, we will work conditionally on N B(r) = η B(r) , the configuration of the PPP N restricted to the ball B(r).

Let 0 < δ ′ < δ and h > 0 (thought as large). Let also z 2 := (r + h; u) (with the same direction as z 1 ). The event Stab(r, h, δ ′ ) encodes the fact that the set of descendants of any Poisson point in B(z 2 , δ ′ ) is not sensitive to what happens inside B(r):

Stab(r, h, δ ′ ) := η ′ : ∀z ∈ N ∩ B(z 2 , δ ′ ), ∀η ′′ , D(z)[η ′′ B(r) ∪ η ′ B(r) ∁ ] = D(z)[η ′ ] . (4.7)
It will be proved in Lemma 5.4 that Stab(r, h, δ ′ ) has a probability tending to 1 as h → ∞ uniformly on r.

Let us now introduce the event F (r, h, δ ′ ) on which there exists z ∈ N ∩ B(z 2 , δ ′ ) such that the four following items hold:

(i) N ∩ B(z 2 , δ ′ ) = {z}, (ii) σ(D ∞ (z)) > 0, (iii) Stab(r, h, δ ′ ) occurs (iv) and D(z) ⊂ B(r + h + δ ′ ) ∁ .
On the event F (r, h, δ ′ ), there is a unique point of the PPP N in the ball B(z 2 , δ ′ ) that has a thick trace at infinity. This subtree is outside the ball B(r + h + δ ′ ) and does not depend on the points of N in the ball B(r). The distance h is the separation gap ensuring that the subtree rooted at z 2 = (r + h; u) remains independent from the Poisson points inside the ball B(r).

Lemma 4.5 states that the event F (r, h, δ ′ ) has a probability larger than some positive ε which is uniform on r. Getting such uniformity of ε (or A) on r will significantly complicate the proof of Lemma 4.5, postponed to Section 5.

Lemma 4.5.

There exists A > 0 large enough such that, for any δ ′ > 0 small enough, there exists ε = ε(A, d, λ, δ ′ ) > 0 such that for any h 0 > 0 large enough and r > 0, there exists h ∈ [h 0 , h 0 + A] such that P(F (r, h, δ ′ )) ≥ ε. Note also that among the previous parameters, only h may depend on r.

For θ > 0, let us consider the subset U = U (r, h, δ ′ , θ) of H d+1 defined as

U := B(r + h + δ ′ ) \ (B(r) ∪ B(z 2 , δ ′ )) ∩ Cone(0, θ) . (4.8) See Figure 2. Let η ′ ∈ F (r, h, δ ′ ) such that η ′ B(r) = η B(r)
. When all the Poisson points of the configuration η ′ inside the set U are removed, the ancestor of z-the only Poisson point in B(z 2 , δ ′ ) according to the event F (r, h, δ ′ ) -becomes z 1 which itself is a descendant of z 0 . This leads to σ(D ∞ (z 0 ))[η ′

U ∁ ] > 0. This construction requires the hypothesis r ∈ H(δ)[η], i.e. there is no Poisson points in B + (z 1 , δ), and this is the only place in the proof of Lemma 4.2 where it is needed. Lemma 4.6. For any r ∈ H(δ) with r ≥ r 0 , for any h and δ ′ < δ/3, there exists θ large enough such that the following statement holds. Almost every configuration η ′ with η

′ B(r) = η B(r) and η ′ ∈ F (r, h, δ ′ ) satisfies σ(D ∞ (z 0 ))[η ′ U ∁ ] > 0. 0 z 0 z 1 z B(z 2 , δ ′ ) r r + h + δ ′ ∞ r 0
(z)[η ′ U ∁ ] = z 1 which means that σ(D ∞ (z 0 ))[η ′ U ∁ ] > 0.
Edges in dotted lines have been modified after deleting η ′ U .

We are now ready to prove Lemma 4.2.

Proof of Lemma 4.2. Parameters A, δ ′ , ε and h 0 are chosen according to Lemma 4.5. Then for any r > 0 there exists h ∈ [h 0 , h 0 + A] such that P(F (r, h, δ ′ )) ≥ ε. Because it requires Stab(r, h, δ ′ ), the event F (r, h, δ ′ ) does not depend on the configuration inside the ball B(r). So,

P F (r, h, δ ′ ) | N B(r) = η B(r) = P F (r, h, δ ′ ) ≥ ε .
Now r can be chosen in H(δ) with r > r 0 and 0 < δ ′ < δ/3 so that Lemma 4.6 applies:

ε ≤ P η ′ ∈ F (r, h, δ ′ ) | N B(r) = η B(r) ≤ P σ(D ∞ (z 0 ))[η ′ U ∁ ] > 0 | N B(r) = η B(r) = P σ(D ∞ (z 0 ))[η ′ ] > 0, η ′ U = ∅ | N B(r) = η B(r) ≤ P σ(D ∞ (z 0 )) > 0 | N B(r) = η B(r) .
Note that the lower bound ε = ε(A, d, λ, δ ′ ) does not depend on the parameter r.

This section ends with the proof of Lemma 4.6.

Proof of Lemma 4.6. Consider a configuration η ′ equal to η inside the ball B(r) and belonging to the event F (r, h, δ ′ ). Let us first assume that, for the configuration η ′ U ∁ , the ancestor of the Poisson point z (whose existence is given by F (r, h, δ ′ )) is z 1 , i.e.

A(z)[η ′ U ∁ ] = z 1 . (4.9) Let us prove Vol(D ∞ (z 0 ))[η ′ U ∁ ] > 0 from (4.9). Since η ′ ∈ F (r, h, δ ′ ), the set of descendants D(z)[η ′ ] is included in B(r + h + δ ′ ) ∁ . So, removing Poisson points of η ′
U modifies no edges (z ′ , A(z ′ )) of the RST as long as A(z ′ ) ∈ D(z). In fact, removing η ′ U may only add new descendants to the vertex z. Hence,

D(z)[η ′ ] is included in D(z)[η ′ U ∁ ] which leads to σ(D ∞ (z))[η ′ U ∁ ] ≥ σ(D ∞ (z))[η ′ ] > 0. Finally, using A(z)[η ′ U ∁ ] = z 1 which is itself a descendant of z 0 , we get σ(D ∞ (z 0 ))[η ′ U ∁ ] > 0.
It then remains to show (4.9). Let R be the hyperbolic distance between z and z 1 . It is sufficient to prove that

B + (z, R) is included in U ∪ B + (z 1 , δ) since,
R = d(z, z 1 ) ≤ d(z, z 2 ) + d(z 2 , z 1 ) ≤ δ ′ + h ≤ h + 1 with δ ′ < 1. So, taking θ large enough such that B + (z, R) ⊂ Cone(0, θ), it then remains to prove that V := B + (z, R) ∩ B(r) ⊂ B + (z 1 , δ). (4.10)
To do so, let us pick v 1 , v 2 , v 3 on the geodesic γ between 0 and z as follows: v 1 , v 2 and v 3 are the intersection points between γ and resp. the sphere S(0, r + h), S(0, r) and S(z, R).

Let us denote by w the symmetric of z 1 w.r.t. the line (0z). Henceforth, it is sufficient to show that w, z 1 and v 3 belong to B + (z 1 , δ). First, we have

d(v 2 , z 1 ) ≤ d(v 1 , z 2 ) ≤ δ ′ .
Thus, by symmetry, d(w, z 1 ) ≤ d(w, v 2 ) + d(v 2 , z 1 ) = 2d(v 2 , z 1 ) ≤ 2δ ′ . Since v 3 , v 2 and z are on the same geodesic, we can write:

d(v 3 , v 2 ) = d(v 3 , z) -d(z, v 2 ) ≤ R -h + δ ′ . Since R ≤ h + δ ′ , we get d(v 3 , v 2 ) ≤ 2δ ′ and then d(v 3 , z 1 ) ≤ d(v 3 , v 2 ) + d(v 2 , z 1 ) ≤ 3δ ′ .
Whenever δ ′ < δ/3, the three points w, z 1 and v 3 are inside B + (z 1 , δ). This proves (4.10) and concludes the proof.

Proof of Lemma 4.5: uniformity in r

This section is devoted to the proof of Lemma 4.5, i.e. P(F (r, h, δ ′ )) ≥ ε where the lower bound ε does not depend on r, h. Recall the notation of Section 4.3 and of Figure 2. We first prove in Section 5.2 that a positive proportion of Poisson points of B(z 2 , δ ′ ) satisfies σ(D ∞ (•)) > 0-Item (ii) in the definition of F (r, h, δ ′ ) -where z 2 = (r + h; u). Then we prove that the properties described by the other three items occur with high probability.

In particular we state in Section 5.3 that Stab(r, h, δ ′ ) has a probability tending to 1 as h → ∞ uniformly on r. But first, in Section 5.1, we recall some properties of the Maximal Backward angular Deviations (MBD) which is the key tool here to control the path fluctuations in the hyperbolic RST.

Maximal backward angular deviations

A crucial ingredient in this work is the control of Maximal Backward angular Deviations (MBD) which has been done in [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF] (see also [START_REF] Flammant | The directed spanning forest in the hyperbolic space[END_REF] where these notions are introduced). Let us recall here the main definitions and results. Given r > 0 and z ∈ L r , recall that z ↓ denotes the Poisson point whose arc |[z ↓ , A(z ↓ )]| crosses S(r) at z and set z ↑ := A(z ↓ ). Let also A (k) (z) := A • . . . • A(z) (k times) be the k-th ancestor of z for any k ∈ N (by convention, we set A(0) := 0).

Consider 0 < r ≤ r ′ and z ′ ∈ L r ′ . Let us denote by z ∈ L r the intersection point between the path of RST joining z ′ to the origin and the sphere S(r). Let us define CFD r ′ r (z ′ ) as the Cumulative Forward angular Deviations between levels r and r ′ as

CFD r ′ r (z ′ ) :=        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 ↓ . We also set CFD r ′ r (z ′ ) := 0 when z ′ / ∈ L r ′ .
Definition 5.1 (Maximal Backward angular Deviations). For 0 < r ≤ r ′ , let us define the Maximal Backward angular Deviations between levels r and r ′ as

MBD r ′ r (z) := sup ρ∈[r,r ′ ] max y∈D ρ r (z) CFD ρ r (y) (5.1)
if z ∈ L r and MBD r ′ r (z) := 0 if z / ∈ L r , where D ρ r (z) is the set of points y ∈ L ρ whose path of RST from y to 0 cuts S(r) at z (or, roughly speaking, the set of descendants of z at level ρ).

Since r ′ → MBD r ′ r (z) is non-decreasing, the MBD can be naturally extend to r ′ = ∞ by setting:

MBD ∞ r (z) := lim r ′ →∞ MBD r ′ r (z) .
The next lemma provides a control of the moments of MBD ∞ r (z), which is crucial for proving the positive density. The idea is that because the paths do not fluctuate too much, there is room for a positive number of points at a given level r to have a thick trace at infinity. Lemma 5.2 (Proposition 2.6 of [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF]). For any p ≥ 3d/2, there exists a constant C = C(d) > 0 such that, for any r > 2, A > 0 and any direction u ∈ S d ,

E z∈B S(r) (u,Ae -r )∩RST MBD ∞ r (z) p ≤ CA d e -rp . (5.2)
5.2 Positive density of σ(D ∞ (•)) > 0 Lemma 5.3. There exists A > 0 large and c 0 = c 0 (A, d, λ) > 0 such that for any 0 < δ ′ < 1, h 0 > 0 and r > 0, there exists h ∈ [h 0 , h 0 + A] such that for any

z 2 = (r + h; u) (u ∈ S d ) E # z ∈ N ∩ B(z 2 , δ ′ ) : σ(D ∞ (z)) > 0 ≥ c 0 Vol B(δ ′ ) . (5.3)
Note also that among the previous parameters, only h = h(r) may depend on r.

Proof of Lemma 5.3. The proof is splitted into three steps. We first start with proving an estimate close to (5.3), but for z ∈ L r (Step 1). Then, we extend the result to portion of annuli and to balls.

Step 1. Let us first prove that there exists c = c(d) > 0 such that for any r > 0,

E # z ∈ L r : σ(D ∞ (z)) > 0 ≥ ce dr . (5.4)
Let us first use the Cauchy-Schwarz inequality with the inner product X, Y :

= E[ i X i Y i ]: E z∈Lr σ(D ∞ (z)) 2 ≤ E z∈Lr 1 σ(D ∞ (z))>0 E z∈Lr σ(D ∞ (z)) 2 = E # z ∈ L r : σ(D ∞ (z)) > 0 E z∈Lr σ(D ∞ (z)) 2 . (5.5)
Thus, the left hand side of (5.4) is lower bounded by

E # z ∈ L r : σ(D ∞ (z)) > 0 ≥ E z∈Lr σ(D ∞ (z)) 2 E z∈Lr σ(D ∞ (z)) 2 .
Recall that in the open-ball model, the set of points at infinity ∂H d+1 is identified with the unit d-dimensional sphere S d whose spherical probability measure σ(S d ) is (normalized to) 1. By [12, Theorem 1.1 (i)], with probability 1, any point at infinity is the asymptotic direction of (at least) one infinite path of the hyperbolic RST. So, a.s. 1 = σ(S d ) ≤ z∈Lr σ(D ∞ (z)) .

(5.6)

Given r > 0 and z ∈ L r , let us denote by Ang(z) := sup{ z0I :

I ∈ D ∞ (z)} if D ∞ (z) =
∅ and Ang(z) := 0 otherwise. Then, the set D ∞ (z) is included in the spherical cap Cone(z, Ang(z)) ∩ ∂H d+1 , which means σ(D ∞ (z)) ≤ CAng(z) d . Moreover, the quantity Ang(z) is bounded by the Maximal Backward angular Deviation MBD ∞ r (z): see Definition 5.1. We then get

E z∈Lr σ(D ∞ (z)) 2 ≤ E z∈Lr MBD ∞ r (z) 2d ≤ Ce -dr (5.7)
by Lemma 5.2 (applied with A = πe r and p = 2d), where the constant C = C(d) > 0 does not depend on r > 0.

Finally, (5.4) follows from combining (5.5), (5.6) and (5.7).

Inequality (5.4) refers to the elements of L r while (5.3) concerns Poisson points. Passing from ones to others while preserving the uniformity of A in r requires some technical considerations. We now extend (5.4) to annuli and then to balls.

Step 2. For any A large enough there exists c 0 = c 0 (d, A) > 0 such that for any r > 0 and 0 < δ ′ < 1,

E # z ∈ N ∩ C(r, r + A -2δ ′ ) : σ(D ∞ (z)) > 0 ≥ c 0 Vol C(r, r + A -2δ ′ ) , (5.8)
where we recall that C(r, R) has been defined in the notations of Section 2.

For z ∈ L r , recall that z ↓ denotes the Poisson point whose arc

|[z ↓ , A(z ↓ )]| crosses S(r) in z. In particular, σ(D ∞ (z)) and σ(D ∞ (z ↓ )) are equal. Henceforth, E # z ∈ N ∩ C(r, r + A -2δ ′ ) : σ(D ∞ (z)) > 0 ≥ E # z ∈ L r : z ↓ ∈ B(r + A -2δ ′ ) and σ(D ∞ (z)) > 0 ≥ c 1 E # z ∈ L r : σ(D ∞ (z)) > 0 .
(5.9)

The inequality (5.9) makes the object of Lemma 5.5, and its proof is postponed to Section 5.5. This choice is done as the proof uses arguments similar to the ones developed in the next Section in a more difficult case. The constants c 1 = c 1 (d) > 0 and A above are chosen large enough according to Lemma 5.5. It then remains to use Step 1 and the inequalities

Vol C(r, r + A -2δ ′ ) ≤ Vol B(r + A) ≤ Ce d(r+A) ,
to finally get

E # z ∈ N ∩ C(r, r + A -2δ ′ ) : σ(D ∞ (z)) > 0 ≥c 1 c e dr ≥c 1 × c × (Ce dA ) -1 Vol C(r, r + A -2δ ′ ) .
Step 3. For short, let us set f (z

′ ) := P(σ(D ∞ (z ′ )) > 0 | z ′ ∈ N )
. The Mecke's formula and Fubini's theorem allow us to write:

C(r,r+A) E # z ′ ∈ N ∩ B(z, δ ′ ) : σ(D ∞ (z ′ )) > 0 dz = λ z∈C(r,r+A) z ′ ∈B(z,δ ′ ) f (z ′ ) dz ′ dz = λ z ′ ∈C(r-δ ′ ,r+A+δ ′ ) f (z ′ )Vol B(z ′ , δ ′ ) ∩ C(r, r + A) dz ′ ≥ λVol B(δ ′ ) z ′ ∈C(r+δ ′ ,r+A-δ ′ ) f (z ′ ) dz ′ ≥ λVol B(δ ′ ) E # z ′ ∈ N ∩ C(r + δ ′ , r + A -δ ′ ) : σ(D ∞ (z ′ )) > 0 ≥ λVol B(δ ′ ) c 0 Vol C(r + δ ′ , r + A -δ ′ ) (5.10)
by Step 2. Now, using Inequalities (2.2), it is not difficult to choose c = c(d) > 0 (small) and A = A(d) > 0 large enough and uniform on r > 0 and δ ′ < 1 such that Vol(C(r + δ ′ , r + Aδ ′ )) is bigger than cVol(C(r, r + A)). Combining with (5.10), we get

C(r,r+A) E # z ′ ∈ N ∩ B(z, δ ′ ) : σ(D ∞ (z ′ )) > 0 dz ≥ c 2 Vol B(δ ′ ) Vol C(r, r + A) .
with c 2 := λc 0 c. This forces the existence of some (r + h; u) ∈ C(r, r + A), with h ∈ [0, A] and u ∈ S d , satisfying

E # z ′ ∈ N ∩ B((r + h; u), δ ′ ) : σ(D ∞ (z ′ )) > 0 ≥ c 2 Vol B(δ ′ ) .
(5.11)

To conclude, let us first specify that (5.11) holds for any direction u ∈ S d by isotropy of the model. Moreover, for any given h 0 > 0, the radius r + h with r > h 0 can be written as r ′ + h ′ where r ′ > 0 and h ′ ∈ [h 0 , h 0 + A]. Hence, we have proved that there exists A large and c 2 = c 2 (A, d, λ) > 0 such that for any 0 < δ ′ < 1, h 0 > 0 and r > 0, there exists h ∈ [h 0 , h 0 + A] such that (5.11) holds for any u ∈ S d . This is Lemma 5.3. This last change on quantifiers, i.e. r > h 0 and h ∈ [0, A] replaced with r > 0 and h ∈ [h 0 , h 0 + A], will allow us to take simultaneously h in the good interval [h 0 , h 0 + A] and also large enough. See Section 5.4.

A stabilization result for subtrees of the RST

For 0 < δ ′ < δ and r, h > 0, recall that z 2 = (r + h; u) and recall the definition of Stab(r, h, δ ′ ) in (4.7). In this section, we prove a stabilization result: subtrees of the RST rooted at Poisson points in B(z 2 , δ ′ ) do not depend on what happens inside B(r) w.h.p. as h → ∞. We are going to prove that there exist c, C > 0 (not depending on r, h and M ) such that ∀r, h > 0, P(I) ≤ C M d/2 + e -cM and lim h→∞ sup r>0 P(II) = 0 .

(5.12)

The previous statement holding whatever the constant M > 0, Lemma 5.4 then follows.

Let us deal with P(II). By Mecke's formula and δ ′ < 1,

P(II) ≤ E # z ′ = (r ′ ; u ′ ) ∈ N ∩ Cone(z 2 , M e -r-h ) : d(z ′ , A(z ′ )) ≥ r ′ -r ≤ n≥h-1 λ Vn P d(z ′ , A(z ′ )) ≥ n | z ′ ∈ N dz ′
where V n := Cone(z 2 , M e -r-h ) ∩ C(r + n, r + n + 1). On the one hand, for z ′ ∈ V n ,

P d(z ′ , A(z ′ )) ≥ n | z ′ ∈ N ≤ P N ∩ B + (z ′ , n) = ∅ = e -λVol(B + (z ′ ,n)) .
Here we need a lower bound of B + ((r ′ ; •), ρ) and use the one obtained in [12, eq. (A1)]: there exists c = c(d) > 0 such that, for any radii r ′ , ρ ≥ 1,

Vol B + ((r ′ ; •), ρ) ≥ ce d(r ′ ∧ρ)/2 .
(5.13) So we use (5.13) to get

P(d(z ′ , A(z ′ )) ≥ n | z ′ ∈ N ) ≤ e -λce dn/2 .
On the other hand, we upperbound the volume of V n : -h) .

Vol(V n ) = r+n+1 r+n sinh(ρ) d dρ × σ {u ′ ∈ S d : u0u ′ ≤ M e -r-h } ≤ c d e d(r+n) × M d e -d(r+h) = c d M d e d(n
Putting together the previous upperbounds, we get:

P(II) ≤ λc d M d e -dh n≥h-1
e dn-λce dn/2 , (

which tends to 0 as h → ∞ uniformly on r.

Let us now consider the term P(I). Let us denote by π z the path of RST starting from the Poisson point z until the origin. We define Rad(z 2 ) as the maximal angular deviation generated by a path π z starting from some z ∈ B(z 2 , δ ′ ) when it goes through the sphere S(r + hδ ′ ):

Rad(z 2 ) := max u0u ′ : ∃z ∈ N ∩ B(z 2 , δ ′ ) s.t. π z intersects S(r + h -δ ′ ) at (•; u ′ ) .
The maximal angular deviation cannot be too large. Precisely, setting with Rad := {Rad(z 2 ) ≤ M e -(r+h-δ ′ ) }, Lemma 4.3 of [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF] states that P(Rad ∁ ) ≤ e -cM for c = c(d) > 0 and M large enough. Besides,

P(I ∩ Rad) = P ∃z ′ = (r ′ ; •) ∈ N ∩ B(z 2 , δ ′ ) : MBD ∞ r ′ (z ′ ) ≥ M e -(r+h) ∩ Rad ≤ E 1 Rad z ′ ∈N ∩B(z 2 ,δ ′ ) 1 {MBD ∞ r ′ (z ′ ) d ≥M d e -d(r+h) } ≤ M -d e d(r+h) E 1 Rad z ′ ∈N ∩B(z 2 ,δ ′ ) MBD ∞ r ′ (z ′ ) d .
Now, in order to apply Lemma 5.2, we have to turn the sum over elements of N ∩ B(z 2 , δ ′ ) into a sum over elements of L r+h-δ ′ . Given z ′ = (r ′ ; •), let us denote by A r+h-δ ′ r ′ (z ′ ) the intersection point between the path π z ′ of RST and S(r

+ h -δ ′ ). It is the 'ancestor' at radius r + h -δ ′ < r ′ of z ′ in RST. A.s. 1 Rad z ′ ∈N ∩B(z 2 ,δ ′ ) MBD ∞ r ′ (z ′ ) d ≤ 1 Rad z ′ ∈N ∩B(z 2 ,δ ′ ) MBD ∞ r+h-δ ′ (A r+h-δ ′ r ′ (z ′ )) d ≤ z ′′ ∈B S(r+h-δ ′ ) (z 2 ,M e -(r+h-δ ′ ) )∩RST # z ′ ∈ N ∩ B(z 2 , δ ′ ) : z ′ ∈ D(z ′′ ) MBD ∞ r+h-δ ′ (z ′′ ) d .
Thus, the Cauchy-Schwarz inequality gives:

E 1 Rad z ′ ∈N ∩B(z 2 ,δ ′ ) MBD ∞ r ′ (z ′ ) d ≤ E z ′′ ∈L r+h-δ ′ # z ′ ∈ N ∩ B(z 2 , δ ′ ) : z ′ ∈ D(z ′′ ) 2 1/2 ×E z ′′ ∈B S(r+h-δ ′ ) (z 2 ,M e -(r+h-δ ′ ) )∩RST MBD ∞ r+h-δ ′ (z ′′ ) 2d 1/2 . (5.15)
On the one hand, let us write

z ′′ ∈L r+h-δ ′ # z ′ ∈ N ∩ B(z 2 , δ ′ ) : z ′ ∈ D(z ′′ ) 2 ≤ z ′′ ∈L r+h-δ ′ # z ′ ∈ N ∩ B(z 2 , δ ′ ) : z ′ ∈ D(z ′′ ) 2 = #N ∩ B(z 2 , δ ′ ) 2
which means that the first term of the upper bound in (5.15) is bounded by

C 1 := E[#(N ∩ B(1)) 2 ] 1/2

Conclusion

Let us prove Lemma 4.5. We pay a special attention to dependencies between parameters.

Let A > 0 and c 0 = c 0 (A, d, λ) > 0 given by Lemma 5.3. It is well known that

E #N ∩ B(z 2 , δ ′ ) 1 #N ∩B(z 2 ,δ ′ )≥2 (5.17)
which does not depend on r, h by stationarity of the PPP, is negligible w.r.t. Vol(B(δ ′ )) as δ ′ → 0. Hence we choose δ ′ > 0 small enough and uniformly on r, h > h 1 = h 1 (d) so that the expectation (5.17) and

P ∃z ∈ N ∩ B(z 2 , δ ′ ), D(z)\{z} ⊂ B(r + h + δ ′ ) ∁
are both smaller than c 0 4 Vol(B(δ ′ )). This second upper-bound and the constant h 1 = h 1 (d) are proved in Lemma 5.6, whose proof is postponed in Section 5.5. At this stage, parameters r > 0 and h > h 1 are still free. Now we choose h 0 ≥ h 1 large enough so that for any h ≥ h 0 and uniformly on r > 0, the following holds by Lemma 5.4:

P Stab(r, h, δ ′ ) ∁ ≤ c 0 4 Vol(B(δ ′ )) .
Finally, for any given radius r > 0, we choose h (possibly depending on r) in [h 0 , h 0

+ A] such that E # z ∈ N ∩ B(z 2 , δ ′ ) : σ(D ∞ (z)) > 0 ≥ c 0 Vol B(δ ′ )
by Lemma 5.3.

For these fixed parameters A, c 0 , δ ′ , h 0 , r and h,

c 0 Vol(B(δ ′ )) ≤ E # z ∈ N ∩ B(z 2 , δ ′ ) : σ(D ∞ (z)) > 0 ≤ P ∃!z ∈ N ∩ B(z 2 , δ ′ ), σ(D ∞ (z)) > 0 +E #{N ∩ B(z 2 , δ ′ )} 1 #N ∩B(z 2 ,δ ′ )≥2 ≤ P ∃!z ∈ N ∩ B(z 2 , δ ′ ), σ(D ∞ (z)) > 0 + c 0 4 Vol(B(δ ′ )) from which we get P(∃!z ∈ N ∩ B(z 2 , δ ′ ), σ(D ∞ (z)) > 0) ≥ 3c 0 4 Vol(B(δ ′ )
). Thus we conclude with

P F (r, h, δ ′ ) ≥ P ∃!z ∈ N ∩ B(z 2 , δ ′ ), σ(D ∞ (z)) > 0 -P Stab(r, h, δ ′ ) ∁ -P ∃z ∈ N ∩ B(z 2 , δ ′ ), D(z)\{z} ⊂ B(r + h + δ ′ ) ∁ ≥ c 0 4 Vol(B(δ ′ )) .
So ε := c 0 4 Vol(B(δ ′ )) depending on A, d, λ and δ ′ , is suitable, and Lemma 4.5 is proved.

Technical lemmas

To complete the proof of Lemma 4.5, it remains to state the two following technical lemmas.

Lemma 5.5. There exists c 1 = c 1 (d) > 0 such that for any A large enough, r > 0 and 0

< δ ′ < 1, E # z ∈ L r : z ↓ ∈ B(r + A -2δ ′ ) and σ(D ∞ (z)) > 0 ≥ c 1 E # z ∈ L r : σ(D ∞ (z)) > 0 .
Proof. For the proof, we will consider first a portion of the sphere of radius r, and then extend the result to the whole L r using the (Covering) Lemma 4.3.

Step 1: Given z 0 ∈ S(r), let us consider the event

L(A, r) := ∃z ∈ B S(r) (z 0 , e -r ) ∩ L r : z ↓ / ∈ B(r + A -2δ ′ ) .
For short, let us set X 0,r := # z ∈ B S(r) (z 0 , e -r ) ∩ L r : σ(D ∞ (z)) > 0 . We will prove that

E X 0,r 1 L(A,r) ∁ ≥ 1 2 E[X 0,r ] . (5.18) With X 0,r 1 L(A,r) ∁ = # z ∈ B S(r) (z 0 , e -r ) ∩ L r : z ↓ ∈ B(r + A -2δ ′ ) and σ(D ∞ (z)) > 0 .
Lemma 5.5 then immediatly follows from the above inequality with c 1 := 1/(2K) using the (Covering) Lemma 4.3, where the covering constant K = K(d) is given in that lemma.

The proof of (5.18) relies on the two following inequalities (5.19) and (5.20) which will be proved in a second time. The first inequality states that the probability of L(A, r) tends to 0 when A → ∞, uniformly on r, δ ′ : lim The second inequality ensures that there exists c = c(d) > 0 such that for any r > 0, E X 0,r ≥ c .

(5.20)

Then, the Cauchy-Schwarz inequality gives

E X 0,r 1 L(A,r) ≤ C 1/2 P(L(A, r)) 1/2
where C = C(d) > 0 upperbounds the expectation of # B S(r) (z 0 , e -r ) ∩ L r 2 thanks to Lemma 6.1. Thus, combining (5.19) and (5.20), we can choose A large enough uniformly on r ≥ 2 and δ ′ < 1 such that

E X 0,r 1 L(A,r) ≤ 1 2 E X 0,r .
This proves (5.18).

Step 2: proof of (5.19). The proof is very close to that of Lemma 5.4 with fewer technical difficulties. By analogy, we write L(A, r) ⊂ I ∪ II where

I := ∃z ∈ B S(r) (z 0 , e -r ) ∩ L r : D(z) ⊂ Cone(z 0 , Ae -r )
and

II := ∃z ∈ B S(r) (z 0 , e -r ) ∩ L r : z ↓ ∈ Cone(z 0 , Ae -r ) ∩ B(r + A -2δ ′ ) ∁ .
We upperbound P(I) more easily than in the proof of Lemma 5.4 since the points z concerned by the event I are already in L r :

P(I) ≤ P ∃z ∈ B S(r) (z 0 , e -r ) ∩ L r : MBD ∞ r (z) ≥ Ae -r ≤ A -2d e 2dr E z∈B S(r) (z 0 ,e -r )∩Lr MBD ∞ r (z) 2d ≤ CA -2d
(by Lemma 5.2) which tends to 0 uniformly on r, δ ′ . The same holds for P(II) as in the proof of Lemma 5.4. For this reason, we omit the details.

Step 3: proof of (5.20). This inequality is a consequence of isotropy of the model, (Covering) Lemma 4.3 and Step 1 in the proof of Lemma 5.3:

E # z ∈ B S(r) (z 0 , e -r ) ∩ L r : σ(D ∞ (z)) > 0 = 1 N (r) N (r) i=1 E # z ∈ B S(r) (z i , e -r ) ∩ L r : σ(D ∞ (z)) > 0 ≥ 1 N (r) E # z ∈ L r : σ(D ∞ (z)) > 0 ≥ c C -1
with N (r) ≤ Ce dr . This concludes the proof of the Lemma.

Lemma 5.6. There exists h 1 = h 1 (d) > 0 such that the following limit is uniform on r > 0 and h > h 1 : For z in B(z 2 , δ ′ ),

lim δ ′ →0 1
P ∃z ′ ∈ N ∩ B(r + h + δ ′ ) s.t. A(z ′ ) = z | z ∈ N ≤ n≥0 E # z ′ ∈ N ∩ V δ ′ ,n : A(z ′ ) = z | z ∈ N
where we set V δ ′ ,n := C(r + hδ ′ , r + h + δ ′ ) ∩ Cone(u, ne -r-h , (n + 1)e -r-h ) and Cone(u, ne -r-h , (n + 1)e -r-h ) is the set of directions u ′ ∈ S d such that ne -r-h ≤ u0u ′ ≤ (n + 1)e -r-h . A second use of the Mecke's formula gives:

E # z ′ ∈ N ∩ V δ ′ ,n : A(z ′ ) = z | z ∈ N = λ V δ ′ ,n P A(z ′ ) = z | z, z ′ ∈ N dz ′ .
Given z ′ ∈ V δ ′ ,n , we have A Arcs of the RST as a subset of H d+1

Given z 1 , z 2 ∈ H d+1 , the arc |[z 1 , z 2 ]| is precisely defined in Section 2.2 of [START_REF] Coupier | Hyperbolic radial spanning tree[END_REF] but for convenience we recall the main lines. Let us write z i = (r i ; u i ) with polar cooordinates, i ∈ {1, 2}. Whenever u 1 and u 2 are not antipodal (and it will be a.s. the case when z 2 is the ancestor of z 1 ), we consider the unique geodesic γ Hence, such a point z will be counted with multiplicity k in L r . Also, to identify without ambiguity this point z, we should formally represent it as a couple made up with its location in H d+1 and one of the arcs generating it, say |[z i , A(z i )]|. In this case the vertex z ↓ ∈ N is defined as z ↓ := z i . In this article, we will commit the following abuse of notations: we will count elements of L r with multiplicity without specifying the arcs distinguishing them.

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 (apart from the root) are represented with different colors. This coloring allows to distinguish some random directions reached by two infinite paths, namely the three directions separating the three colored traces at infinity: see Remark 3.7 for details.

. 4 )

 4 This choice (somewhat unnatural) is motivated by the fact that the (hyperbolic) distance to the origin 0 is monotonous along the arc |[z, A(z)]| which fails for the geodesic [z, A(z)] (or for the Euclidean segment between z and its ancestor A(z)). In particular each arc |[z, A(z)]| crosses any given sphere at most once. Thus, we define the RST at level r > 0 as the following random set:L r := RST ∩ S(r) . (3.5) Elements of L r are a.s. not Poisson points on S(r) with probability 1. However we extend to elements of L r the notations D(•) and D ∞ (•) as follows. For z ∈ L r , we denote by z ↓ the Poisson point whose arc |[z ↓ , A(z ↓ )]| crosses S(r) at z: with a slight abuse of notations, z ↓ can be defined without ambiguity (see Appendix). Then we set D(z) = D(z ↓ ) and D ∞ (z) = D ∞ (z ↓ ).

5

 5

4. 1

 1 Global strategyLet r 0 ∈ (0, 1), u 0 ∈ S d and select the closest Poisson point to (r 0 ; u 0 ) (with polar coordinates) denoted by z 0 := argmin z∈N d(z, (r 0 ; u 0 )) .

(4. 2 )

 2 As shown below, Lemmas 4.1 and 4.2 together imply P(E(r 0 ; u 0 )) = 0 from which Theorem 3.5 immediatly follows. The proofs of Lemmas 4.1 and 4.2 are respectively postponed to Sections 4.2 and 4.3. Proof of Theorem 3.5. Choose parameters δ and ε such that both Lemmas 4.1 and 4.2 hold.

  [START_REF] Baccelli | Semi-infinite paths of the 2d-radial spanning tree[END_REF], two blocks B r ′ ,m ′ and B r,m are adjacent whenever B r ′ ,m ′ overlaps {z ∈ H d+1 : d(z, B r,m ) ≤ δ}. Since δ < 1 this forces r ′ to be in {r -1, r, r + 1} and B r ′ ,m ′ to overlap Cone(z r,m , (1 + δ)e -r ). Now, Lemma 4.3 asserts that for each layer r ′ ∈ {r -1, r, r + 1}, there are at mostCA d = C(1 + δ) d e d(r ′ -r) ≤ C2 d e d blocks B r ′ ,m ′overlapping Cone(z r,m , (1+δ)e -r ). Consequently, B r,m has at most C 2 := 3C2 d e d adjacent blocks.

Figure 2 :

 2 Figure 2: Illustration of Lemma 4.6. Black dots are Poisson points. This picture represents a configuration η ′ U ∁ (i.e. Poisson points of η ′ U have been removed) where η ′ B(r) = η B(r) and η ′ ∈ F (r, h, δ ′ ). The ball B(z 2 , δ ′ ) whose (deterministic) center z 2 is marked by a grey square, contains only one Poisson point, namely z. Its set of descendants D(z) is represented by the hatched region and satisfies σ(D ∞ (z))[η ′ ] > 0 (the bold black curve on the unit sphere S d ). When the set U is emptying of Poisson points then A(z)[η ′ U ∁ ] = z 1 which means that σ(D ∞ (z 0 ))[η ′ U ∁ ] > 0.Edges in dotted lines have been modified after deleting η ′ U .

Lemma 5 . 4 .

 54 For any 0 < δ ′ < δ,lim h→∞ sup r>0 P(Stab(r, h, δ ′ )) = 1 .Proof. Let us denote by χ the union of descendant sets D(z) with z in N ∩ B(z 2 , δ ′ ). So, for the configuration in B(r) to alter χ, it must exist a vertexz ′ = (r ′ ; u ′ ) ∈ χ whose B + (z ′ , d(z ′ , A(z ′ ))) overlaps B(r), which means d(z ′ , A(z ′ )) ≥ r ′r.Roughly speaking, this would imply the occurrence of a large ball empty of Poisson points with radius r ′r ≥ hδ ′ which is very unlikely as h → ∞. Hence, Stab(r, h, δ ′ ) ∁ is included in I ∪ II where, for a positive constant M > 0,I := χ ⊂ Cone(z 2 , M e -r-h )and II := ∃z ′ = (r ′ ; u ′ ) ∈ N ∩ Cone(z 2 , M e -r-h ) : d(z ′ , A(z ′ )) ≥ r ′r .

  A→∞ sup r>0, δ ′ <1 P(L(A, r)) = 0 .(5.[START_REF] Liggett | Domination by product measures[END_REF]

  Vol(B(δ ′ )) P ∃z ∈ N ∩ B(z 2 , δ ′ ), D(z)\{z} ⊂ B(r + h + δ ′ ) ∁ = 0 .Proof. The Mecke's formula allows to write:P ∃z ∈ N ∩ B(z 2 , δ ′ ), D(z)\{z} ⊂ B(r + h + δ ′ ) ∁ ≤ P ∃z ∈ N ∩ B(z 2 , δ ′ ), ∃z ′ ∈ N ∩ B(r + h + δ ′ ) s.t. A(z ′ ) = z ≤ E # z ∈ N ∩ B(z 2 , δ ′ ) : ∃z ′ ∈ N ∩ B(r + h + δ ′ ) s.t. A(z ′ ) = z = λ B(z 2 ,δ ′ ) P ∃z ′ ∈ N ∩ B(r + h + δ ′ ) s.t. A(z ′ ) = z | z ∈ N dz .(5.21)

P 1 P-λce d 4 cLemma 6 . 1 (

 1461 A(z ′ ) = z | z, z ′ ∈ N = P B + (z ′ , d(z, z ′ )) ∩ N = ∅ = e -λVol(B + (z ′ ,d(z,z ′ ))) .MoreoverVol(B + (z ′ , d(z, z ′ ))) ≥ ced 2 d(0,z ′ )∧d(z,z ′ ) by (5.13) ≥ ce d 4 d(z,z ′ ) since d(z, z ′ ) ≤ 2d(0, z ′ ) ≥ ce d 4 c ′ n because z ′ ∈ V δ ′ ,n and r + h is larger than some h 1 = h 1 (d) > 0.Let us now compute the volume of V δ ′ ,n :Vol(V δ ′ ,n ) = r+h+δ ′ r+h-δ ′ sinh(ρ) d dρ × σ {u ′ ∈ S d : ne -r-h ≤ u0u ′ ≤ (n + 1)e -r-h } . (5.22)The first term of the r.h.s. of (5.22) is bounded by c 1 δ ′ e d(r+h) while the second one is bounded by c 2 n d e -d(r+h) using (2.2). The previous constants c i , i ∈ {1, 2}, are positive and only depend on d. Hence, the volume ofV δ ′ ,n is smaller than c 1 c 2 n d δ ′ .Combining what precedees, we finally get for any r > 0 and h > h ∃z ′ ∈ N ∩ B(r+ h + δ ′ ) s.t. A(z ′ ) = z | z ∈ N ≤ n≥0 λe ′ n c 1 c 2 n d δ ′whose upperbound can be expressed as Cδ ′ with C = C(λ, d, h 1 ) is a positive, finite constant. It then remains to plug this bound in (5.21) to conclude.6 Proof of Proposition 3.8Let us first recall an upperbound for the number of elements of L r in a cap B S(r) (•, e -r ). Lemma 4.4 of[START_REF] Coupier | Hyperbolic radial spanning tree[END_REF]). For any p ≥ 1, there exists a constant C = C(d, p) > 0 such that, for any r ≥ 0 and any direction z ∈ S(r),E # RST ∩ B S(r) (z, e -r ) p ≤ C .Thanks to the Covering Lemma (Lemma 4.3), it is now easy to prove that L r = RST ∩ S(r) admits in expectation about e dr elements:E #L r ≤ N (r) i=1 E #(B S(r) (z i , e -r ) ∩ L r ) ≤ Ce dr . (6.1)Finally, combining the previous inequality and (5.4) proved in Step 1 of Lemma 5.3, we get Proposition 3.8.

u 1 ,u 2 : 1 u 1

 1211 [0, 1] → S d on the sphere connecting u 1 to u 2 . Then, the arc|[z 1 , z 2 ]| is the path t ∈ [0, 1] → (1t)r 1 + tr 2 ; γ u 1 ,u 2 (φ z 1 ,z 2 (t)) ∈ H d+1 where φ z 1 ,z 2 : [0, 1] → [0, 1] is defined as φ z 1 ,z 2 (t) := 0u 2 arccos (1t) sinh(r 1 ) + t cos( u 1 0u 2 ) sinh(r 2 ) sinh((1t)r 1 + tr 2 ) .This construction of the arc |[z 1 , z 2 ]| ensures that the distance to the origin (as well as the distance to z 1 ) are monotonous along the path|[z 1 , z 2 ]|.By Property 3.2, we know that the geodesics [z, A(z)] and [z ′ , A(z ′ )], for z, z ′ ∈ N , can overlap only at their extremities. This is not the case any more when we use the arcs|[z, A(z)]| and |[z ′ , A(z ′ )]|.For any r > 0, it may exist some points z ∈ L r belonging to several arcs, say |[z 1 , A(z 1 )]|, . . . , |[z k , A(z k )]|.

  after removing Poisson points in U , z 1 would become the closest Poisson point to z. Here we use that r ∈ H(δ), i.e. the ball B + (z 1 , δ) contains no other Poisson points except z 1 . Let us first remark that

Acknowledgments. This work has been supported by the RT GeoSto 3477, the Labex Bézout (ANR-10-LABX-58), the ANR PPPP (ANR-16-CE40-0016) and the ANR GrHyDy (ANR-20-CE40-0002). V.C.T. thanks the CRM Montréal (IRL CNRS 3457) where part of this work was completed.