Directed, cylindric and radial Brownian webs
David Coupier Jean-François Marckert Viet Chi Tran
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            #1ℤ/#1ℤℤ/nℤℤ/2nℤ Nb Rk Ext Slice Multinomial Mat Poisson Binomial#1#1 Trace Var Cov sign Pl uniform𝒞#1#2#3#4#5[ #1:  #2   ⟶  #3;      #4   ⟼  #5 ]#1#20pt#1#2#1#1 Proof. #1 Proof of #1 . ℝ/2πℤℚ/2πℤℝ/ℤℚ/ℤ=(d)i.i.d.r.v.æa.e.#1#1#1=0pt#1#1#1#1#2 0pt#1_#2#1#2 0pt#1^#2 DA⟶ x z C K Exp#1I_t_#1  mod `E𝔼`Pℙ`Rℝ`eε`oω#1
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            lemLemma[section]
            pro[lem]Proposition
            theo[lem]Theorem
            cor[lem]Corollary
            conj[lem]Conjecture
            defi[lem]Definition
            OQ[lem]Open question
            noteNoteexm[lem]Examplerem[lem]Remarkmetath[lem]Meta-theorem Directed, cylindric and radial Brownian webs
                David CoupierUniv. Lille, CNRS, UMR 8524 - Laboratoire P. Painlevé, Département GIS - Polytech'Lille, F-59000 Lille, France; E-mail :  Univ. de Valenciennes, CNRS, EA 4015 - LAMAV, F-59313 Valenciennes Cedex 9, France,      Jean-François MarckertCNRS UMR 5800 - LaBRI, Univ. Bordeaux, 351 cours de la Libération, 33405 Talence Cedex-France; E-mail : ,      Viet Chi TranUniv. Lille, CNRS, UMR 8524 - Laboratoire P. Painlevé, UFR de Mathématiques, F-59000 Lille, France; E-mail : 
                Accepted, Received; in original form 
            ===================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
            
            
            The Brownian web (BW) is a collection of coalescing Brownian paths (W_(x,t),(x,t) ∈ `R^2) indexed by the plane. It appears in particular
            as continuous limit of various discrete models of directed forests of coalescing random walks and navigation schemes. Radial counterparts have been considered but global invariance principles are hard to establish. In this paper, we consider cylindrical forests which in some sense interpolate between the directed and radial forests: we keep the topology of the plane while still taking into account the angular component. We define in this way the cylindric Brownian web (CBW), which is locally similar to the planar BW but has several important macroscopic differences. For example, in the CBW, the coalescence time between two paths admits exponential moments and the CBW as its dual contain each a.s. a unique bi-infinite path. This pair of bi-infinite paths is distributed as a pair of reflected Brownian motions on the cylinder. Projecting the CBW on the radial plane, we obtain a radial Brownian web (RBW), i.e. a family of coalescing paths where under a natural parametrization, the angular coordinate of a trajectory is a Brownian motion. Recasting some of the discrete radial forests of the literature on the cylinder, we propose rescalings of these forests that converge to the CBW, and deduce the global convergence of the corresponding rescaled radial forests to the RBW. In particular, a modification of the radial model proposed in Coletti and Valencia is shown to converge to the CBW.
            
            Keywords :  Brownian web, navigation algorithm, random spanning forests, weak convergence of stochastic processes.   
            AMS classification : Primary  60J05, 60G52, 60J65, 60D05  Secondary 60G57; 60E99.
            Acknowledgements : This work has benefitted from the GdR GeoSto 3477. J.F.M. has been partially funded by ANR GRAAL (ANR-14-CE25-0014) and D.C. by ANR PPPP (ANR-16-CE40-0016). D.C. and V.C.T. acknowledge support from Labex CEMPI (ANR-11-LABX-0007-01). 
            
            
            § INTRODUCTION
            
            
            The Brownian web, BW in the sequel, is a much fascinating object introduced in <cit.>. It is formed by a family of coalescing Brownian trajectories (W_x,t,(x,t)∈ℝ^2), roughly speaking starting at each point (x,t) of the plane ℝ^2 (we consider only 2D objects in this paper). For (x,t)∈^2,
            
            (W_x,t(s),s≥ t) x+ (B^(x,t)_s-t, s ≥ t)
            
            where B^(x,t) is a standard Brownian motion (BM) starting at 0 and indexed by (x,t). The trajectories started from two different points of the time-space ^2 are independent Brownian motions until they intersect and coalesce.
            The BW appears as the continuous limit of various discrete models of coalescing random walks and navigation schemes (e.g. <cit.>).
            
            
            Recently, radial (2D) counterparts of these discrete directed forests have been considered and naturally, attempts have been carried to obtain invariance principles for these objects and define a “radial Brownian web” (RBW; <cit.>). Nevertheless, the rescaling needed in the BW case is somehow incompatible with a “nice Brownian limit” in the radial case. For directed forests in the plane, time is accelerated by n^2 say, while space is renormalized by n, for a scaling parameter n→+∞. In the radial case, the “space and time” parameterizations are related by the fact that the circle perimeter is proportional to its radius. This hence prevents a renormalization with different powers of n (2 and 1 for n^2 and 1/n) unless we consider only local limits.
            
             The main idea of this paper is the creation of the cylindric Brownian web (CBW) that allows to involve the angular characteristic of the radial problems, while keeping a geometry close to the plane. The usual BW is indexed by ×, where the first component is the space component. The CBW is an object indexed by the cylinder
            
                =()×
            
            where the first component  is the circle. Topologically,  somehow interpolates between the plane `R× `R and the plane equipped with the polar coordinate system ()× `R_+ suitable to encode a RBW, as we will see.
            
            
             Similarly to (<ref>), we can define the CBW ^=(^_(x,t), (x,t)∈) as the family of coalescing trajectories
            
            (^_x,t(s),s≥ t)ł(x+ B^(x,t)_s-t,s≥ t)̊1
            
            that is, independent Brownian motions taken modulo 1 which coalesce upon intersecting. Note that the time will be flowing upwards in the graphical representations of this paper, and hence the notation with the upward arrow. Later, dual objects will be defined with their inner time running downward. Also, to distinguish between planar and cylindrical objects, cylindrical objects will be denoted with bold letters.
            
            
            In Section <ref>, we recall the topological framework in which the (planar) BW as introduced by Fontes et al. <cit.> is defined. Many convergence results on the plane leading to the BW can be turned into convergence results on the cylinder with the CBW as limit since the map
            
                proj`R`R / ℤxx 1
            
            is quite easy to handle and to understand. We recall some criteria established in the literature that allow to obtain the BW as limit of discrete directed forests. Then, we extend these results to the cylinder for the CBW. We show that the CBW can arise as the limit of a cylindrical lattice web that is the analogous of the coalescing random walks introduced by Arratia <cit.>. We end the section by showing different ways to project the CBW on the radial plane to obtain radial `Brownian' webs. 
            In Section <ref>, the properties of the CBW are investigated. We show in particular that there is almost surely (a.s.) a unique bi-infinite branch in the CBW as well as in its dual, which is a main difference with the planar BW. Starting with a discrete lattice and taking the limit, we can characterize the joint distribution of these two infinite branches as the one of a pair of reflected Brownian motions modulo 1, in the spirit of Soucaliuc et al. <cit.>. We also prove that the coalescence time between two (or more) branches admits exponential moment, when its expectation in the plane is infinite. All these behaviors are closely related to the topology of the cylinder.
             In Sections <ref>and <ref>, we play with the convergences to the BW in the directed plane, to the CBW in the cylinder and to the RBW in the “radial” plane. In the plane, several examples of directed forests in addition to the coalescing random walks of Arratia are known to converge to the Brownian webs, for example <cit.>. Other radial trees such as the one introduced by Coletti and Valencia <cit.> are known to converge locally to Brownian webs. We consider the corresponding cylindrical forests and show that they converge to the CBW with a proper rescaling. For example, in Section <ref>, we propose a radial forest similar to the radial forest of <cit.>, built on a sequence of circles on which a Poisson processes are thrown. When carried to the cylinder, this amounts to throwing Poisson processes with different rates on circles of various heights. We show how the rates and heights can be chosen to have the convergence of the cylindrical forest to the CBW, which is carefully established by adapting well-known criteria (e.g. <cit.>) to the cylinder. The convergence for the latter model has its own interest: as the intensity of points increases with the height in the cylinder, the convergence is obtained for the shifted forests. It is classical in these proofs that the key ingredient for checking the convergence criteria amounts in proving estimates for the tail distribution of the coalescence time between two paths. In our case, this is achieved by using the links between planar and cylindrical models, and thanks to the Skorokhod embedding theorem which connects our problem to available estimates for Brownian motions. However we have to use clever stochastic dominations as well to obtain these estimates. Projecting the cylinder on the (radial) plane then provides a radial discrete forests which converges after normalisation to the radial Brownian web. This convergence is a global convergence, whereas only local convergences are considered in <cit.>.
            
            
            
            
            § CYLINDRIC AND RADIAL BROWNIAN WEB
            
            
            In this Section we introduce the cylindric Brownian web, several natural models of radial Brownian webs together with some related elements of topology, in particular, some convergence criteria. But we start with the definition of the standard BW given in <cit.>.
            
            
            
            
             §.§ The standard Brownian Web
            
            
            
            Following Fontes & al. <cit.> (see also Sun <cit.> and Schertzer et al. <cit.>), we consider the BW as a compact random  subset of the set of continuous trajectories started from every space-time point of ^2=[-∞,∞]^2 equipped with the following distance ρρ((x_1,t_1),(x_2,t_2))=
            A(x_1,t_1)-A(x_2,t_2)_∞,
            where the map A is given by
            A^2[-1,1]^2(x,t)(Φ(x,t),Ψ(t))=ł(tanh(x)/1+|t|,tanh(t))̊.
            
            For t_0∈, C[t_0] denotes the set of functions f from [t_0,+∞] to  such that Φ(f(t),t) is continuous.
            Further, the set of continuous paths started from every space-time points is
            
                Π=⋃_t_0∈ C[t_0]×{t_0}.
            (f,t_0)∈Π represents a path starting at (f(t_0), t_0). For (f,t_0)∈Π, we denote by f̃ the function that coincides with f on [t_0,+∞]  and which is constant equals f(t_0)  on [-∞,t_0). The space Π is equipped with the distance d defined by
            
                d((f_1,t_1),(f_2,t_2))=ł(sup_tł|Φ(f̃_̃1̃(t),t)-Φ(f̃_̃2̃(t),t)|̊)̊∨ |Ψ(t_1)-Ψ(t_2)|.
            
            The distance depends on the starting points of the two elements of Π, as well as their global graphs.
            Further, the set H of compact subsets of (Π,d) is equipped with the d_ H Hausdorff metric (induced by d), and F_ H, the associated  Borel σ-field.
            
            The BW W=(W_x,t, (x,t)∈^2) is a random variable (r.v.) taking its values in ( H, F_ H). It can be seen as a collection of coalescing Brownian trajectories indexed by ^2. Its distribution is characterized by the following theorem due to Fontes & al. <cit.>:
            
            
            There exists an ( H, F_ H)-valued r.v. W whose distribution is uniquely determined by the following three properties.
            
              (o) From any point (x,t)∈`R^2, there is a.s. a unique path W_x,t from (x,t),
            
              (i) For any n≥ 1, any (x_1,t_1),…,(x_n,t_n), the W_x_i,t_i's are distributed as coalescing standard Brownian motions,
            
              (ii) For any (deterministic) dense countable subset D of `R^2, a.s., W is the closure in ( H,d_ H) of (W_x,t,(x,t)∈ D).
            
            
            In the literature, the BW arises as the natural limit for sequences of discrete forests constructed in the plane. Let χ be a family of trajectories in H. For t>0 and t_0,a,b∈ℝ with a<b, let
            
                η_χ(t_0,t; a,b) := { f(t_0+t)  |  (f,s) ∈χ, f(t_0)∈[a,b] }
            
            be the number of distinct points in ℝ×{t_0+t} that are touched by paths in χ which also touch some points in [a,b]×{t_0}. We also consider the number of distinct points in [a,b]×{t_0+t} which are touched by paths of χ born before t_0:
            
                η_χ(t_0,t; a,b) := { f(t_0+t) ∈ [a,b]  |  (f,s) ∈χ, s≤ t_0 } .
            
            Th. 6.5. in <cit.> gives a criterion for the convergence in distribution of sequences of r.v. of ( H, F_ H) to the BW, which are variations of the criteria initially proposed by <cit.>:
            
            
            Let (χ^n)_n≥ 1 be a sequence of ( H, F_ H)-valued r.v. which a.s. consists of non-crossing paths. If (χ^n)_n≥ 1 satisfies conditions (I), and either (B2) or (E) below, then χ^n converges in distribution to the standard BW.
            
              (I) For any dense countable subset 𝒟 of ^2 and for any deterministic y_1,⋯,y_m∈ D, there exists paths χ^n_y_1,…χ^n_y_m of χ^n which converge in distribution as n→ +∞ to coalescing Brownian motions started at y_1,…,y_m.
            
              (B2) For any t>0, as `e→ 0^+,
            
                `e^-1lim sup_n→+∞sup_(a,t_0)∈ `R^2( η_χ^n(t_0,t;a,a+`e) ≥ 3 ) → 0  .
            
              (E) For any limiting value χ of the sequence (χ^n)_n≥ 1, and for any t >0, t_0∈, a<b∈,
            
                ( η_χ(t_0,t ; a,b)) ≤(η_W(t_0,t ; a,b))  ,
            
            where W denotes the BW.
            
            
            In this paper we focus on forests with non-crossing paths. But there also exist in the literature convergence results without this assumption: see Th. 6.2. or 6.3. in <cit.>. For forests with non-crossing paths, the condition (I) implies the tightness of (χ^n)_n≥ 1. The conditions (B2) or (E) somehow ensure that the limit does not contain `more paths' than the BW. In the literature, proofs of (B2) and (E) are both based on an estimate of the coalescence time of two given paths. However, condition (B2) is sometimes more difficult to check. It is often verified by applying FKG positive correlation inequality <cit.>, which turns out to be difficult to verify in some models. When the forest exhibits some Markov properties, it could be easier to check (E) as it is explained in <cit.> or <cit.>, Section 6.1. Let us give some details. The condition (E) mainly follows from
            
                lim sup_n→ +∞( η_χ^n(t_0,ε ; a,b) ) < +∞ ,
            
            for any ε>0, t_0∈ and a<b∈, which can be understood as a coming-down from infinity property. Statement (<ref>) shows that for any limiting value χ,
            the set of points χ(t_0 ; t_0+ε)
             of ×{t_0+ε} that are hit by the paths of χ(t_0)–
             paths of χ born before time t_0–
             constitutes a locally finite set.  Thus, condition (I) combined with the Markov property, implies that the paths of χ starting from χ(t_0 ; t_0+ε) are distributed as coalescing Brownian motions. Hence,
            
                (η_χ(t_0)(t_0,t ; a,b)) ≤(η_W(t_0+ε,t-ε ; a,b))     =    b-a/√(π (t-ε))
                   →   b-a/√(π t) = (η_W(t_0,t ; a,b))
            
            as ε→ 0 and (E) follows. For details about the identity (<ref>) see <cit.>.
            
            
            
            
             §.§ The Cylindric Brownian Web
            
            
            
            
            We propose to define the CBW =(_x,t  , (x,t)∈) on a functional space similar to H so that the characterizations of the distributions and convergences in the cylinder are direct adaptations of their counterparts in the plane (when these counterparts exist! See discussion in Section <ref>). In particular, this will ensure that the convergences in the cylinder and in the plane can be deduced from each other provided some conditions on the  corresponding discrete forests are satisfied.
            
             The closed cylinder is the compact metric space =()×,
            for the metric
            
                ρ_O((x_1,t_1),(x_2,t_2))= d_(x_1,x_2)∨ |Ψ(t_1)-Ψ(t_2)|
            
            where d_(x,y)= min{ |x-y|, 1-|x-y|} is the usual distance in . In the sequel, we use as often as possible the same notation for the CBW as for the planar BW, with an additional index O (as for example ρ and ρ_O).
            
            For t_0∈, the set C_O[t_0] denotes the set of continuous functions f from [t_0,+∞] to , and Π_O  the set ⋃_t_0∈ C_O[t_0]×{t_0}, where (f,t_0)∈Π_O represents a path starting at (f(t_0), t_0). For (f,t_0)∈Π_O, we denote by f̃ the function that coincides with f on [t_0,+∞]  and which equals to f(t_0)  on [-∞,t_0). On Π_O, define a distance d_O by
            
                d_O((f_1,t_1),(f_2,t_2))=ł(sup_t d_(f̃_̃1̃(t),f̃_̃2̃(t)))̊∨ |Ψ(t_1)-Ψ(t_2)|.
            
            Further, H_O, the set of compact subsets of (Π_O,d_O) is equipped with the d_ H_O Hausdorff metric (induced by d_O), and F_ H_O, the associated  Borel σ-field.
            The CBW is a r.v. taking its values in ( H_O, F_ H_O), and is characterized by the following theorem (similar to the Theo. 2.1. in Fontes & al. <cit.> for planar BW).
            
            
            There is an ( H_O, F_ H_O)-valued r.v.  whose distribution is uniquely determined by the following three properties.
            
              (o) From any point (x,t)∈, there is a.s. a unique path _x,t from (x,t),
            
              (i) for any n≥ 1, any (x_1,t_1),…,(x_n,t_n) the joint distribution of  the _x_i,t_i's is that of coalescing standard Brownian motions modulo 1,
            
              (ii) for any (deterministic) dense countable subset D of , a.s.,  is the closure in ( H_O,d_ H_O) of (W^_x,t,(x,t)∈ D).
            
            
            As in the planar case, the CBW  admits a dual counterpart, denoted by  and called the dual Cylindric Brownian Web. For details (in the planar case) the reader may refer to Section 2.3 in <cit.>. For any t_0∈, identifying each continuous functions f∈ C_O[t_0] with its graph as a subset of , f:=-f={(-x,-t) : (x,t)∈ f} defines a continuous path running backward in time and starting at time -t_0. Following the notations used in the forward context, let us define the set Π_O of such backward continuous paths (with all possible starting time), equipped with the metric d_O (the same as d_O but on Π_O). Further, H_O denotes the set of compact subsets of (Π_O,d_O) equipped with the Hausdorff metric induced by d_O. Theorem 2.4 of <cit.> admits the following cylindric version.
            
            
            
            There exists an H_O× H_O valued r.v. (,) called the double Cylindric Brownian Web, whose distribution is uniquely determined by the two following properties:
            
              (a) and - are both distributed as the CBW.
            
              (b) A.s. no path of  crosses any path of .
            
            Moreover, the dual CBW  is a.s. determined by  (and vice versa) since for any point (x,t)∈,  a.s. contains a single (backward) path starting from (x,t) which is the unique path in Π_O that does not cross any path in .
            
            For all -∞≤ t≤ t'<+∞, let us denote by F^_t,t' the σ-algebra generated by the CBW  between time t and t':
            
                ℱ^_t,t' = σ( {{^_(x,s)(s') ,   t<s'≤ t' } ,   x∈ ,   t<s≤ t' })  .
            We write ℱ^_t' instead of ℱ^_-∞,t'. The CBW is Markov with respect to the filtration (ℱ^_t)_t∈`R and satisfies the strong Markov property, meaning that for any stopping time T a.s. finite, the process
            
                {{^_(x,T+t)(T+s) ,   s≥ t } ,   x∈ ,   t≥ 0 }
            
            is still a CBW restricted to the semi-cylinder ^+:=()× `R^+ which is independent of ∩_t>Tℱ^_t. In the same way, we can also define the σ-algebra F^_t,t', where t≥ t', with respect to the dual CBW .
            
            The convergence criteria <cit.> or Theorem <ref> above has hence a natural counterpart on the cylinder. For a,b ∈  denote by [a→ b] the interval from a to b when turning around the circle counterclockwise, and by  |a→ b| its Lebesgue measure (formally: for a<b, [a→ b] = [a,b] and if a>b, [a→ b]=[a,1]∪[0,b]). For X a r.v. in H_O, denote by
            
                η_X^O(t_0,t;[a→ b]) := { f(t_0+t) |  (f,s)∈ X, f(t_0) ∈ [a→ b] }
            
            be the number of distinct points in ×{t_0+t} that are touched by paths in X which also touch some points in [a→ b]×{t_0}. We also set
            
                η_X^O(t_0,t ; [a→ b]) := { f(t_0+t) ∈ [a→ b]  | (f,s)∈ X, s≤ t_0 } .
            
            Here is the counterpart of Theorem <ref>  in the cylinder:
            
            Let (χ^n)_n≥ 1 be a sequence of ( H_O, F_ H_O)-valued r.v. which a.s. consist of non-crossing paths. If (χ^n)_n≥ 1 satisfies conditions (IO), and either (B2O) or (EO), then χ^n converges in distribution to the CBW .
            
              (IO) For any dense countable subset 𝒟 any deterministic y_1,⋯,y_m∈ D, there exists for every n≥ 1, paths χ^n_y_1…χ^n_y_m in χ^n such that χ^n_y_1…χ^n_y_m converge in distribution as n→ +∞ to coalescing Brownian motions modulo 1 started at y_1,…,y_m.
            
              (B2O) For any t>0, as `e→ 0^+,
            
                `e^-1lim sup_n→+∞sup_(a,t_0)∈( η^O_χ^n(t_0,t;[a→ a+`e 1]) ≥ 3 ) → 0  .
            
              (EO) For any limiting value χ of the sequence (χ^n)_n≥ 1, and for any t >0, t_0∈ and a,b∈,
            
                ( η^O_χ(t_0,t ; [a→ b]) ) ≤( η^O_(t_0,t ; [a → b]) )  .
            This section ends with a summary of the relationships between η_W, η_W, η^O_ and η^O_ where W denotes the planar BW. First, in the plane, as noticed in <cit.> Section 2, η_W(t_0,t ; a,b) and η_W(t_0,t ; a,b)+1 are identically distributed. This can be shown using duality arguments. In the cylinder the situation is a little bit different: it is not difficult to show that, for t,t_0>0 and a,b∈,
            
                η^O_(t_0,t ; [a → b]) η^O_(t_0,t ; [a → b]) + _ ,
            
            where the event  means that the cylindric BMs starting from (a,t_0) and (b,t_0) are allowed to coalesce before time t_0+t but not from the side [b → a] (more precisely, |_(a,t_0)(s)→_(b,t_0)(s)| stays in [0,1) for s∈[t_0,t_0+t]).
            
            Moreover, for any t,t_0>0 and a,b∈ with |a → b|<1, we will prove at the end of the current section that
            
                η^O_(t_0,t ; [a → b]) ≤_Sη_W(t_0,t ; a,b)  ,
            
            where ≤_S stands for the stochastic domination. Statement (<ref>) traduces the following natural principle: trajectories merge easier in the cylinder than in the plane. However there is no stochastic comparison between η^O_ and η_W. Indeed, the expectation of η_W(t_0,t ; a,b) tends to 0 as t→∞ thanks to identity (<ref>) whereas this does not hold in the cylinder. Theorem <ref> (below) states the a.s. existence in  of a bi-infinite path. So, for any t,t_0, η^O_(t_0,t ; [0 → 1]) is larger than 1 and, by rotational invariance,
            
                (η^O_(t_0,t ; [a → b])) = |a→ b|  (η^O_(t_0,t ; [0 → 1])) ≥ |a→ b|  .
            
            It then remains to prove eta^O<eta. Let us focus on the planar BW W restricted to the strip ×[t_0,t_0+t]. First, by continuity of trajectories, with probability at least 1-ε, there exists δ>0 such that sup_0≤ d≤δ|W_a,t_0(t_0+d)- W_b,t_0(t_0+d)|<1 (since |a-b|<1) where W_x,t denotes the BM starting at (x,t). The coming-down from infinity property satisfied by the BW ensures that the number of remaining BMs at level ×{t_0+δ} and starting from [a,b]×{t_0} is a.s. finite. Let κ be this (random) number. When defining a realization of the BW, we need to decide, in case of coalescence of two trajectories, which one survives. In order to compute η_W(t_0,t ; a,b) we label these remaining BMs by 1,…,κ from left to right and when two of them merge, the BM having the lower label is conserved while the other one is stopped. This stopping rule allows us to determine the set of labels of remaining BMs at level ×{t_0+t}, say ℒ, whose cardinality is η_W(t_0,t ; a,b). Now, let us complete the previous stopping rule as follows: if the BM with label 2≤ j≤κ meets the path 1+W_a,t_0 between times t_0+δ and t_0+t then it stops. Although 1+W_a,t_0 does not correspond to any trajectory in the planar BW W– and then appears as artificial –, it coincides with W_a,t_0 in the cylinder and then has label 1. According to this completed rule, we obtain a new set of labels of remaining BMs at level ×{t_0+t}. It is included in ℒ and its cardinality has the same distribution than η^O_(t_0,t ; [a → b]). In conclusion the previous construction allows us to bound from above η^O_(t_0,t ; [a → b]) by η_W(t_0,t ; a,b) on an event of probability at least 1-ε, for any ε>0.
            
            
            
            
             §.§ The Cylindric Lattice Web
            
            
            
            As for the BW, the CBW can be constructed as the limit of a sequence of discrete directed forests on the cylinder.
            For any integer n≥ 1, define the “cylindric lattice” as :
            
                = {(x,t),  x ∈ℤ/2nℤ,  t ∈ℤ,  x-t 2 = 0 },
            
            and consider (ξ(w), w ∈) a collection of i.i.d. Rademacher r.v. associated with the vertices of . The cylindric lattice web (CLW) is the collection of random walks
            
                =ł(_w,  w ∈)̊
            
            indexed by the vertices of , where for w=(x,t),
            lcl_(x,t)(t)   =   x
            _(x,t)(s)   =   _(x,t)(s-1) + ξ(_(x,t)(s-1),s-1)  2n,   s >t.
            
            The sequence of paths (_w,w∈) is equivalent to that introduced by Arratia <cit.> in the planar case. The union of the random paths ((_(x,t)(s),s),s≥ t) for (x,t)∈, coincides with the set of edges {(w,w+(ξ(w),1)), w ∈} (see Figure <ref>).
            
            The dual  of  is a reversed time CLW (and shifted by 1) defined on the “dual” of  :
            
                = {(x,t), x ∈, t ∈ℤ, x-t  2 = 1 }.
             is the collection of random walks =ł(_w, w ∈)̊ indexed by the vertices of  such that for w=(x,t)∈, and using the same family (ξ(w), w ∈) as before:
            lcl_(x,t)(t)   =   x,
            _(x,t)(s)   =   _(x,t)(s+1)-ξ(_(x,t)(s+1)-(0,1)),s),        for  s ≤ t.
            
            We define, for any h∈ℤ, for any direction D∈{,}, the horizontal slice by
            
                _2n^D(h)= ^D_2n∩ł( ×{h})̊,
            
            so that the random walks (_w^2n,D,w ∈_2n^D(h)) start from the points of _2n^D(h).
            
            
            The normalized CLW and its dual are defined as follows. For D∈{ , } and for any (x,t) in _2n^D, set
            ^(2n),D_(x/2n,t/n^2)(s):= 1/2n_(x,t)^2n, Dł( 4n^2 s )̊   for s≥t/n^2D=,s≤t/n^2D=.
            Since ^(2n),D_(x,t)(4 n^2 s ) takes its values in , 2n is the right space normalization, which implies the time normalization as usual.
            
            
            
            The pair of renormalized CLW (^(2n),,^(2n),) converges in distribution to the pair of CBW (,).
            
            Let us first prove the convergence of the marginals. Since  and  have the same distribution (up to a reversal of time and a shift by 1).
            
            To do it, we mainly refer to the proof of the convergence towards the (planar) BW of the sequence of lattice webs (W^(2n))_n≥ 1, obtained from normalizing the random walks on the grid ={(x,t)∈^2,  x-t  2 =0 } similarly to (<ref>): see <cit.> for further details. As for (I) the proof of (IO) is a basic consequence of the Donsker invariance principle and is omitted here. The same coupling argument used to prove eta^O<eta leads to the following stochastic domination: for n≥ 1, t_0∈, t>0, a∈ [0,2π] and ε>0,
            
                η^O_^(2n),(t_0,t ; [a→  a+ε]) ≤_S η_W^(2n)(t_0,t ; a, a+ε)  .
            
            Hence condition (B2) satisfied by the rescaled (planar) lattice web W^(2n) (see Section 6 in <cit.>) implies condition (B2O) for ^(2n),. Then Theorem <ref> applies and gives the convergence of (^(2n),)_n≥ 1 to .The convergence of the marginals implies that the distributions of {(,)}_n≥ 1 form a tight sequence in the set of measures on ℋ_O×ℋ_O. It then suffices to prove that any limiting value of this sequence, say (𝒳^,𝒳^), is distributed as the double CBW (,). To do it, we check the criteria of Theorem <ref>. Item (a) has already been proved. To check (b), let us assume by contradiction that with positive probability there exists a path π_z∈𝒳^ which crosses a path π̂_ẑ∈𝒳^.
            
            By definition of (𝒳^,𝒳^), this would lead to the existence, for n large enough and with positive probability, of a path of  crossing a path of .
            This is forbidden since the lattice webs have non crossing paths.
            
             §.§ Radial Brownian Webs
            
            
              §.§.§ The standard Radial Brownian Web and its dual
            
            
            
            Our goal is now to define a family of coalescing paths, indexed by the distances of their starting points to the origin in ℝ^2, that we will call  radial Brownian web. Let us start with some topological considerations.
            Our strategy consists in sending the semi-cylinder ^+:=()× `R^+ onto the plane equipped with the polar coordinate system ()× `R^+ by using the
            map
            
                φ_⋆× `R^+() × `R^+(x, f_⋆(t))(2π x, t) ,
            
            where f_⋆(t):=t/(4π^2). The presence of factor 1/(4π^2) will be discussed below. Let
            
                (h)={ (x,h), x ∈},
            
            be the horizontal slice at height h of . For any t>0, φ_⋆ projects (f_⋆(t)) on (0,t):=×{t}. It also identifies (0) with the origin.
            
            The map φ_⋆ induces the metric ρ_∙ on the radial plane × `R^+  by
            
                ρ_∙((x_1,t_1),(x_2,t_2)) := ρ_O(φ_⋆^-1(x_1,t_1),φ_⋆^-1(x_2,t_2)),
            
            for any elements (x_1,t_1),(x_2,t_2)∈ ()× `R^+. Following the beginning of Section <ref>, we can construct a measurable space ( H_∙, F_ H_∙) equipped with the distance ρ_∙.
            Of course, the map φ_⋆ is continuous for the induced topology, so that the image of a (weak) converging sequence by φ_⋆ is a (weak) converging sequence.
            We call  standard in-radial Brownian web, and denote by , the image under φ_⋆ of the dual CBW  restricted to ^+. In particular, φ_⋆ sends the trajectory _x,f_⋆(t)(s) for s going from f_⋆(t) to 0 on the path _2π x,t(s) for s going from t to 0 where
            
                _2π x,t(s) := s expł( 2iπ_x,f_⋆(t)(f_⋆(s)) )̊ .
            
            Notice that the natural time of the trajectory _2π x,t is given by
            the distance to the origin, since the radius satisfies:
            
                | _2π x,t(s) | = s  .
            
            The families of paths (_x,f_⋆(t), (x,f_⋆(t)) ∈^+) that coalesce on the cylinder when t evolves from +∞ to 0, are then sent on radial paths  (_x,t,(t exp(i x) ∈ℂ)) that coalesce when they are approaching the origin 0.  This is the reason why  is said in-radial, and the notation →∙ evokes the direction of the paths, “coalescing towards the origin”.
            
            Moreover, for any 1<s≤ t, φ_⋆ sends the part of cylinder delimited by times f_⋆(s-1)=(s-1)/(4π^2) and f_⋆(s)=s/(4π^2) (i.e. with height 1/(4π^2)) to the ring centered at the origin and delimited by radii s-1 and s (i.e. with width 1). Then, on the unit time interval [s-1;s], the increment of the argument of _2π x,t, i.e.
            
                2π_x,f_⋆(t)(f_⋆(s-1)) - 2π_x,f_⋆(t)(f_⋆(s))  2π
            
            is distributed according to the standard BM at time 1 taken modulo 2π. This is the reason why  is said to be standard. As a consequence, the trajectory _x,t turns  a.s. a finite number of times around the origin.
            
            As the standard BW, the CBW and the in-radial Brownian web admit special points from which may start more than one trajectory and whose set a.s. has zero Lebesgue measure. See Section 2.5 in <cit.> for details. Except from these special points, the in-radial Brownian web  can be seen as a tree made up of all the paths {_x,t(s), 0≤ s ≤ t}, (x,t)∈× `R^+, and rooted at the origin. Its vertex set is the whole plane. Th. <ref> in the sequel also ensures that this tree contains only one semi-infinite branch with probability 1.
            
            Let us denote by  the image under  φ_⋆ of the CBW  restricted to ^+. We call it the  standard out-radial Brownian web. The map φ_⋆ sends the trajectory {_x,f_⋆(t)(s), s ≥ f_⋆(t)} of the cylindric BM _x,f_⋆(t) starting at (x,f_⋆(t))∈^+ on the out-radial (continuous) path {_2π x,t(s), t≥ s} where
            
                _2π x,t(s) := s expł( 2iπ_x,f_⋆(t)(f_⋆(s)) )̊ .
            
            Unlike the in-radial path _x,t, _x,t is a semi-infinite path which moves away from the origin. Finally, the out-radial Brownian web  appears as the dual of the in-radial Brownian web . Indeed, the CBWs  and  are dual in the sense that no trajectory of  crosses a trajectory of  with probability 1 (see the proof of Prop. <ref>). Clearly, the map φ_⋆ preserves this non-crossing property which then holds for  and . 
             
            
            
            Let us recall that ^(2n), and ^(2n), denote the normalized cylindric lattice webs obtained from ^2n, and ^2n,: see (<ref>). Let us respectively denote by _2π x,t and _2π x,t the radial lattice webs obtained as
            images under φ_⋆ of ^(2n), and ^(2n), restricted to ^+. Using the continuity of φ_⋆, it is possible to transfer the convergence result of Prop. <ref> from the cylinder to the plane. Then, the convergence result below is a direct consequence of Prop. <ref>.
            
            
            The pair (,) converges in distribution to the pair of standard radial Brownian webs (,).
            
            
              §.§.§ Other Radial Brownian Webs
            
            
            
            In this section we explore different radial projections of the cylindric Brownian web (,) into the plane. Let us first describe the general setting. Let f be an increasing continuous function, defining a one-to-one correspondence from an interval I⊂ `R^+ onto an interval J⊂ `R. Define the bijective map φ_f by:
            
            
                φ_f× J× I(θ, f(t))(2πθ, t) .
            
            As previously, × I represents a subset of `R^2 (actually a ring) parametrized by polar coordinates. The map φ_f sends the restriction of the CBW  to the part of cylinder × J on a radial object defined on the ring × I, denoted by f- and also called radial Brownian web. In this construction, the function f is a winding parameter. For instance, if 1,2∈ I, the argument variation (in ) around the origin of the f-_x,2 between radii 1 and 2 (where x∈[0,2π] is the initial argument) is a centered Gaussian r.v.  with variance 4π^2(f(2)-f(1)). The standard radial Brownian web introduced in the previous section corresponds to the particular case I=`R^+, J=`R^+ and f(t)=t/(4π^2), for which the argument variation of a trajectory on a ring with width c is simply a Gaussian 𝒩(0,c).
            
            
            Our second example of maps f allows to project the complete pair (,) parametrized by  to the plane. Let us consider the bijection from I=(0,+∞) onto J=`R defined by f(t):=ln t (or any other map f sending (0,+∞) onto ℝ). Then, the radial Brownian web f-– image of  by φ_f–presents an accumulation phenomenon in the neighborhood of the origin. Indeed, the argument variation around the origin between radii ε and 1 has distribution 𝒩(0,4π^2|ln(ε)|), and thus goes to +∞ in the neighborhood of 0 (ε→ 0^+) when it stays bounded in any other bounded ring far away from 0.
            
            
            Our third example of map f provides a tree – given by the trajectories of f-– having many semi-infinite branches with asymptotic directions.
            A semi-infinite branch ζ (if it exists) of the tree f- is said to admit an asymptotic direction θ∈ whenever (z_k)→θ, for any subsequence (z_k)⊂ζ such that |z_k|→∞. To show that f- admits many semi-infinite branches, let us consider the bijection f from I=`R^+ onto J=[0;1) defined by f(t):=2/πarctan t. For a small ε∈(0,1), the map φ_f projects the thin cylinder × [1-ε;1) on the unbounded set × [tan(π(1-ε)/2);+∞). On the (small) time interval [1-ε;1), the CBMs have small fluctuations, and then the tree f- admits semi-infinite branches with asymptotic directions. The next result proposes a complete description of the semi-infinite branches of f-.
            
            
            
            The standard radial Brownian web could appear a bit impetuous to the reader: the fluctuation of the argument along a trajectory parametrized by the modulus, being a BM mod 2π, the trajectories may have important fluctuations far from the origin. The choice f(t)=ln t of Example 2 provides a radial forest where the paths look like coalescing BMs locally and far from O: between radii r and r+1, the fluctuations are of variance 1/r. This model is invariant by inversion.
            
            Consider the f-RBW for a bijection f from I=_+ into an interval J with compact closure such that f can be extended continuously to (J). With the above notations, the following statements hold.
            
              *  A.s. any semi-infinite branch of f- admits an asymptotic direction.
            
              *  A.s. for any θ∈, the tree f- contains (at least) one semi-infinite branch with asymptotic direction θ.
            
              *  For any (deterministic) θ∈, a.s. the tree f- contains only one semi-infinite branch with asymptotic direction θ.
            
              *  A.s. there exists a countable dense set 𝒟⊂ such that, for any θ∈𝒟, the tree f- contains two semi-infinite branches with asymptotic direction θ.
            
              *  A.s. the tree f- does not contain three semi-infinite branches with the same asymptotic direction.
            
            The first two items generally derive from the straight property of the considered tree: see Howard & Newman <cit.>. However, in the present context, it is not necessary to use such heavy method and we will prove them directly. For the sake of simplicity, we can assume that J=[0,1). Let us first consider a semi-infinite branch ζ of f-.
            
            By construction of the f-RBW, there exists a path γ of the CBW on × J such that ζ=φ_f(γ).
            The path γ of the CBW on × J is a Brownian motion that can be extended by continuity to × [0,1] by (θ̅,1), say, implying that the first coordinate of ζ converges to θ̅/(2π) when the radius tends to infinity. This means that the semi-infinite branch ζ admits θ̅ as asymptotic direction. The proof of the second item is in the same spirit.
            
            The key argument for the three last statements of Th. <ref> is the following. With probability 1, for any θ∈ and x:=θ/2π, the number of CBMs of  starting at (x,1) is equal to the number of semi-infinite branches of f- having 2π x as asymptotic direction. With Th. <ref>(o), it then follows that the number of semi-infinite branches of f- having the deterministic asymptotic direction θ∈ is a.s. equal to 1. This key argument also makes a bridge between the (random) directions in which f- admits several semi-infinite branches and the special points of . Given t∈`R, Th. 3.14 of <cit.> describes the sets of points on the real line `R×{t} from which start respectively 2 and 3 BMs. The first one is dense and countable whereas the second one is empty, with probability 1. These local results also hold for the CBW  (but we do not provide proofs).
            
            Cylinders may also be sent easily on spheres, by sending the horizontal slices h∈ (a,b) of the cylinder to the horizontal slice g(h) of the sphere {(x,y,h) ∈ℝ^3: x^2+y^2+g(h)^2=1}, where -∞≤ a < b ≤ +∞, and g is an increasing and bijective function from (a,b) to (-1,1). Somehow, sending cylinders onto the plane allows to contract one slice (or one end) of the cylinder, and sending it on the sphere amounts to contracting two slices (or the two ends) of the cylinder.
            Again, this point of view will provide a suitable definition for the spherical Brownian web and its dual.
            
            
            § ELEMENTS ON CYLINDRIC LATTICE AND BROWNIAN WEBS
            
            
            In this section, two differences between the CBW and its plane analogous are put forward. Firstly, each of CBW  and  contains a.s. exactly one bi-infinite branch; this is Th. <ref>, the main result of this section. This property is an important difference with the planar BW which admits a.s. no bi-infinite path (see e.g. <cit.> in the discrete case).
            The distributions of these bi-infinite paths are identified by taking the limit of their discrete counterparts on the cylindric lattice web.
            
            Secondly, the coalescence time of all the Brownian motions starting at a given slice admits exponential moments (Prop. <ref>). This is also an important difference with the planar case, where the expectation of the coalescence time of two independent Brownian motions is infinite, which comes from the fact that the hitting time τ_1 of 0 by a Brownian motion starting at 1 is known to have distribution (τ_1∈ dt)=e^-1/(2t)/√(2π t^3).
            
            
            
             §.§ The bi-infinite branch of the CBW
            
            
            
            For any x,x'∈^D, t∈ℝ, denote by
            
                T^(x,x',t) =     infł{ s>t :  ^_(x,t)(s) = ^_(x',t)(s) }̊
                
                T^(x,x',t) =     supł{ s<t :  _(x,t)(s) = _(x',t)(s) }̊
            
            the coalescence times of the cylindric Brownian motions ^_(x,t) and ^_(x',t) one the one hand, and of ^_(x,t) and ^_(x',t) on the other hand. Set for D∈{, },
            
                T^D(t)= maxł{T^D(x,x',t), (x,t),(x',t)∈(t)}̊,
            
            the coalescence time of all the Brownian motions (going upward if D= and downward if D=) starting at (t).
            Consider a continuous function γ : ↦.  We say that γ, or rather, its graph {(γ_t,t), t ∈} is a bi-infinite path of the CBW , if there exists an increasing sequence (t_k, k ∈ℤ) such that lim_k→ -∞ t_k=-∞,  lim_k→ +∞ t_k=+∞, and a sequence (x_k, k ∈ℤ) such that for any k∈ℤ,  ^_(x_k,t_k)(t_k+1)=x_k+1, and ^_(x_k,t_k)(s)=γ_s for s∈[t_k,t_k+1].
            Similarly, we say that  {(γ_t,t), t ∈} is a bi-infinite path of the CBW ^, if
            there exists an decreasing sequence (t_k, k ∈ℤ) such that lim_k→ -∞ t_k=+∞,  lim_k→ +∞ t_k=-∞ and a sequence (x_k, k ∈ℤ) such that for any k∈ℤ,
            _(x_k,t_k)(t_k+1)=x_k+1, and _(x_k,t_k)(s)=γ_s for s∈[t_k+1,t_k].
            
            
            With probability 1, any two branches of the CBW  eventually coalesce. Furthermore, with probability 1, the CBW  contains exactly one bi-infinite branch (denoted ).
            
            A notion of semi-infinite branch is inherited from the cylinder via the map φ_⋆:
            The standard out-radial Brownian web  possesses a unique semi-infinite branch.
             The first statement is a consequence of the recurrence of the linear BM.
            
            Let us introduce some stopping times for the filtration F^.
            First let τ^,1=T^(0) (the coalescing time of the CBW ^ coming from (0) in the dual), and successively, going back in the past,  τ^,k=T^(τ^,k-1). Since the primal and dual paths do not cross a.s., it may be checked that all primal Brownian motion ^_(x,τ^,k) for x ∈(τ^,k) have a common abscissa, say x'_k-1 at time τ^,k-1, that is in (τ^,k-1). In other words, they merge before time τ^,k-1.
            A simple picture shows that at x'_k-1, the dual ^ has two outgoing paths, and thus the primal ^_(x'_k-1,τ^,k-1) is a.s. a single path (see e.g. <cit.>, and use the fact that the special points of the CBW are clearly the same as those of the BW).
            
            
            We have treated the negative part of the bi-infinite path. The positive path is easier, since a bi-infinite path must coincide with the trajectory ^_(x'_0,0) for its part indexed by positive numbers.
            As a consequence, the sequence defined by :
            – for k≥ 0 by t_-k=τ^,k, x_-k=x'_k,
            – for k≥ 1 by t_k=k, x_k=W^_(x_k-1,t_k-1)(t_k)
            
             does the job if we prove that τ^,ks are finite times that go to -∞ a.s. But this is a consequence of the strong law of large numbers, since τ^,k is a sum of i.i.d. r.v. distributed as  τ^,1 a.s. finite and positive (by continuity of the BM and comparison with the planar BW).
             
            
            Similarly, it can be proved that any two branches of  eventually coalesce and that  contains a.s. a unique bi-infinite path that we denote 𝐂^.
            
            
            
             §.§ The bi-infinite branch of the CLW
            
            
            As we saw in Prop. <ref>, the CBW can be obtained as a limit of a CLW when the renormalization parameter n in the CLW tends to +∞. We first show that the CLW also has a bi-infinite path and use the explicit transition kernels for the trajectories of the CLW to obtain, by a limit theorem, the distribution of (𝐂^,𝐂^). The latter are two reflected Brownian motions, as described by <cit.>.
            
            
            The coalescence times of the random walks starting at height h∈ are respectively :
            ccl
            T_n^(h)   =   infł{t≥ h : _w(t)=_w'(t), ∀ w,w'∈^_2n(h)}̊,
            
            T_n^(h)   =   supł{t≤ h : _w(t)=_w'(t), ∀ w,w'∈^_2n(h)}̊.
            
            Since for any two points w,w'∈, _w and _w' eventually coalesce a.s., we have a.s., for any h,
            
                T^_n(h)<+∞,      T^_n(h)>-∞.
            
            
            For D∈{, }, a bi-infinite path of ^D_2n is a sequence (x_i,i)_i∈, such that for all i∈, x_i-x_i-1 2n ∈{1,2n-1}.  We say that ^2n,D contains a bi-infinite path ^,D if there is a bi-infinite path ^,D of ^D_2n whose edges are included in the set of edges of ^2n,D.
            
             A.s.,  and  each contains a unique bi-infinite path.
            
            Take the slice h and consider T^_n(h). Since the paths from  do not cross those of , the paths in  started from h+T^_n(h) all meet before slice h. Let (h) be their common position at height h.
            Let us consider the sequence (τ^,k_n, k∈) defined similarly to the one introduced in the proof of Th. <ref>: τ^,0_n=0 and for k≥ 1, τ^,k_n=T_n^(τ^,k-1_n). This sequence converges to +∞ a.s. since τ^,k_n is the sum of k independent r.v. distributed as τ^,1_n=T_n^(0).
            The sequence of paths
            γ_k=_(C^2n,(τ_n^,k),τ_n^,k) is increasing for inclusion and defines a bi-infinite path 𝐂^=lim_k→ +∞γ_k that is unique by the property (<ref>). The construction of the bi-infinite path for  follow the same lines.
            
            
            Let us describe more precisely the distribution of (,). Let h_1≤ h_2 be two heights. We show that (,) is distributed on a time interval [h_1,h_2], as a Markov chain with explicit transitions.
            For any process X=(X_i,i ∈ℤ) indexed by ℤ, and h_1 ≤ h_2, let us denote
            
                X[h_1,h_2]:= (X_h_1, X_h_1+1, ⋯, X_h_2), X[h_2,h_1]:= (X_h_2, X_h_2-1,⋯,X_h_1).
             For h_1≤ h_2, we have
            (i)(h_1) and (h_2) are independent r.v. respectively uniformly distributed in _2n(h_1) and _2n(h_2),
            (ii) For any (x_1,x_2)∈_2n(h_1) ×_2n(h_2), conditionally on ((h_1),(h_2))=(x_1,x_2),
            ł([h_1,h_2],[h_2,h_1])̊ł(_(x_1,h_1)[h_1,h_2],_(x_2,h_2)[h_2,h_1])̊
            If Pair^,(x_1,x_2,h_1,h_2) denotes the support of (_(x_1,h_1)[h_1,h_2], _(x_2,h_2)[h_1,h_2]),
            then for any (C_1,C_2)∈ Pair_n^,(x_1,x_2,h_1,h_2)
            `Pł(( _(x_1,h_1)[h_1,h_2],_(x_2,h_2)[h_1,h_2])=(C_1,C_2))̊=2^-2(h_2-h_1)+(C_1,C_2)
            where (C_1,C_2) is the “number of contacts" between C_1 and C_2 :
            (C_1,C_2)=#{i ∈ [h_1,h_2-1] : C_1(i)= C_2(i+1)}.
             The family ((h), h<h_1) (resp. ((h), h≥ h_2)) is a function of the Rademacher r.v. placed on ∪_h<h_1_2n(h) (resp. on ∪_h≥ h_2_2n(h)). Hence, ((h), h<h_1) and ((h), h≥ h_2) are independent, and independent of [h_1, h_2-1]. Clearly, (h_1) and (h_2) have invariant distributions by rotation, so they are uniform, and (<ref>) holds.
            
            Using the Rademacher r.v. ξ's defined at the beginning of Section <ref>, we have
            
                {(_(x_1,h_1)[h_1,h_2],_(x_2,h_2)[h_1,h_2])=(C_1,C_2)}= 
                {∀ h_1≤ i < h_2,   ξ(C_1(i),i)=C_1(i+1)-C_1(i) 2n 
                  ξ(C_2(i+1),i)=C_2(i+1)-C_2(i) 2n},
            since the edges of the dual are determined by the edges of the primal. The number of Rademacher (ξ(w),w ∈) contributing to the above event is 2(h_2-h_1)-(C_1,C_2), hence the result. (C_1,C_2)=#{i : (C_1(i),i)= (C_2(i+1),i)} is the number of edges (ξ(u),u ∈) contributing to the definition of both (C_1,C_2). Apart these edges, each increment of  and of C^ are determined by some different Rademacher r.v. Hence 2(h_2-h_1)-(C_1,C_2) edges determine the event
            { (_(x_1,h_1)^[h_1,h_2],_(x_2,h_2)^[h_1,h_2])=(C_1,C_2)}.
            From the above Lemma, it is possible to give a representation of the vectors [h_1,h_2] and [h_2,h_1] with a Markov chain whose components both go in the same direction ↑.
             We have
            
                ł([h_1,h_2],[h_1,h_2])̊ (M_1[h_1,h_2],M_2[h_1,h_2])
            
            where M=(M_1,M_2) is a Markov chain whose initial distribution is uniform on _2n(h_1)×_2n(h_1), and whose transition kernel K is defined as follows:
            
            if d_ℤ/2nℤ(a,a')>1,
            
            K((a,a'),(a +  2n, a' + '  2n)=1/4,    for any (,')∈{-1,1}^2
            if d_ℤ/2nℤ(a,a')=1,
            {[ K((a,a+1 ),(a+1  ,a+2 ))                        =                     1/2,; K((a,a+1 ),(a-1  ,a+2 ))                        =                     1/4,;   K((a,a+1 ),(a-1  ,a ))                        =                     1/4,;   K((a+1,a ),(a  ,a-1 ))                        =                     1/2,;  K((a+1,a ),(a+2, a-1 ))                        =                     1/4,; K((a+1,a ),(a+2  ,a+1 ))                        =                     1/4, ].
            where a,a-1,a+1,a+2 are considered modulo 2n.
            
            Notice that the starting points of M is a pair of uniform points at time h_1, while for [h_1,h_2] and [h_1,h_2] the starting points were on two different slices (see Lemma <ref>).
            
             First, both distributions have same support, which is
            
                ⋃_(x_1,x_2)∈^_2n(h_1)×^_2n(h_2) Pair^,(x_1,x_2,h_1,h_2),
            
            the set of pairs of non-crossing paths living on ×. By Lemma <ref>, we see that for any pair (C_1,C_2) in this support we have `P( (_h_1[h_1,h_2],_h_2([h_1,h_2])= (C_1,C_2))= 2^-2(h_2-h_1)+(C_1,C_2). The Markov kernel has been designed to satisfy the same formula.
            
            
             §.§ Distribution of (^, ^)
            
            
            In the sequel, we consider the sequence (,)_n∈ correctly renormalized and interpolated as a sequence of continuous functions. We will prove its convergence in distribution on every compact set [h_1,h_2] (with h_1<h_2) to (^,^), a pair of reflected Brownian motions modulo 1 (see Figure <ref>). This result is similar to that of  Soucaliuc et al. <cit.> introduced in the next paragraph.
            
            Let F:ℝ→[0,1] be the even, 2-periodic function defined over [0,1] by F(x)=x.
            
            Let us consider U_1 and U_2 two i.i.d uniform r.v. on [0,1], and B and B' two i.i.d. BM starting at 0 at time h_1 and independent of U_1 and U_2.
            Let (Y^,Y^) be the following continuous process defined for t∈ [h_1,h_2] and taking its values in ^2
                (Y^, Y^)(t)=ł( U_1+B'_t/√(2)-H(t) 1, U_1+B'_t/√(2)+H(t) 1 )̊,
            
            where H(t)  represents half the “distance”|Y^(t)→ Y^(t)| :
            
                H(t)=Fł(U_2+√(2)B_t)̊/2.
            
            Since F is bounded by 1, Y^ and Y^ never cross.
            
            We have the following convergences in distribution:
            
            (i) Let h_1 <h_2. Let U^_n and U^_n be two independent uniform r.v. on _2n(h_1) and _2n(h_2) respectively. Then in ([h_1,h_2],()^2):
            ł(_U^_n (4n^2   .) /2n, _U^_n(4n^2   .)/2n)̊ (Y^,Y^).
            
            (ii) In (,()^2):
            
                ł((4n^2 .)/2n, (4n^2 .)/2n)̊ (,)
            
            and (,)(Y^,Y^).
            
            Notice that for t=h_1, 
                (Y^(h_1),Y^(h_1))=(U_1-F(U_2)/2 1,U_1+F(U_2)/2 1)
            
            which is indeed a pair of i.i.d. uniform r.v. on [0,1] as expected in view of Lemma <ref>(i).
            
            
            
            The remaining of this section is devoted to the proof of Theorem <ref>, which is separated into several steps. Let us start with point (i). 
            Step 1: Tightness of ł(_U^_n(4n^2   .)/(2n), _U^_n(4n^2  .)/(2n))̊
            
            
            By translation invariance, we may suppose that h_1=0 and set h_2=T.
            The tightness of the family of distributions of ł(_U^_n(4n^2   .)/(2n), _U^_n(4n^2  .)/(2n))̊ in C([0,T],()^2) follows from the tightness of its marginals that are simple well rescaled random walks on the circle. Now, our aim is to identify the limiting distribution. For that purpose, and in view of Lemmas <ref> (ii) and <ref>, we study more carefully the Markov chain (M_1,M_2).
            Step 2: Angle process between M_1 and M_2
            
            
            Let us extend the notation [a→ b] and |a→ b| for a and b in /2n. For the Markov chain M defined in Lemma <ref>, the angle process between the two components is
            
                A(i)= |M_1(i) → M_2(i)|,     i≥ 0.
            
            Of course, for any i, (M_1(i),M_2(i))= (M_1(i),M_1(i)+A(i)  2n). We will focus on the asymptotics of ((M_1(i), A(i)), i≥ 0).
            
            
            
            Recall that M_1 and M_2 are simple non-independent random walks with Rademacher increments. Let us write:
            
                ccl
                M_1(i)     =     M_1(0)+∑_j=1^i R_2j-1,
                
                M_2(i)     =     M_2(0)+∑_j=1^i R_2j
            
            where (R_2i,i≥ 1) and (R_2i-1,i≥ 1) are two families of i.i.d. Rademacher r.v., the two families being possibly dependent from each other.
            The process A takes its values in the set of odd integers in [0,2n], and its are sums of 2 Rademacher r.v.
            
            
            Now, let us consider the simple random walk
            
                Z(i) = A(0) + ∑_j=1^i (-1)^jR_j=M_2(i)-M_1(i)
            
            starting from A(0). If M_1 and M_2 were allowed to cross, then A(i) would be equal to Z(2i). We have to account for the non-crossing property of the paths of .
            
            
            A random walk (Z_i,i≥ 0) is said to be the simple random walk reflected at 0 and 2n, and starting at some  b∈0,2n if (Z_i,i≥ 0) is a Markov chain such that
            
                rcl(Z_i+1=1  |  Z_i=0)    =   (Z_i+1=2n-1  |  Z_i=2n)=1
                (Z_i+1=a± 1 |  Z_i=a)   =    1/2, for any  a ∈1,2n-1.
            
            For any discrete time process X, denote by
            
                Δ X_i:=X_i-X_i-1,
             the ith increment of X.
            We have
            
            The distribution of the process ((A(i),M_1(i)),i≥ 0) starting at (A(0),M_1(0)) where A(0)∈1,2n-1 is odd, and M_1(0) ∈0,2n-1 is even is characterized as follows: 
            
            
            For (Z_i,i≥ 0) a simple random walk reflected at 0 and 2n, and starting from A(0), we have:
            
            (A(i),i≥ 0) (Z_2i,i≥ 0)  F_2n(Z_2i,i≥ 0),
            
            where F_2n:ℤ→0,2n, the even 4n-periodic function, defined on [0,2n] by F_2n(x):= x.
            
            
            The random walk M_1 starting at M_1(0) admits as increments the sequence
            ł(Δ M_1(i),i≥ 1)̊=ł(-Δ Z_2i-1,i ≥ 0)̊,
             that is the opposite of the increments with odd indices of Z.
            
            
            Notice that the second identity in eq:tehez holds in distribution only: as defined, the reflection only modifies the increments that follow the hitting times of 0 and 2n, whereas the map F_2n turns over large part of the trajectory (Z_i,i≥ 0). Denoting by w_2n(ℓ)=⌊ℓ/(2n)⌋ the discrete “winding number” of ℓ, according to Lemma <ref>, the increments of the process M_1 under this representations are
            
                Δ M_1(ℓ) = (-1)^w(Z _2ℓ-1)ΔZ _2ℓ-1.
            
            The distance |M_1(i)→ M_2(i)| decreases when Δ M_1(i)=R_2i-1=1 increases and increases when Δ M_2(i)=R_2i=1, so that Z(2i) would be equal to A(i) if the two walks were not constrained to not cross. We would also have
            
                Δ M_1(i) =  -(Z_2i-1-Z_2i-2)=-ΔZ_2i-1.
            
            Since 0 and 2n are even, and since Z(0)=A(0) is odd, the random walk can hit 0 and 2n only after an odd number of steps. In other words, the reflection will concern only the steps with even indices. Therefore, let (Z_i,i≥ 0) be the random walk Z reflected at 0 and 2n: the odd increments of Z and of Z are the same, and the even increments correspond except when Z_2i-1∈{0,2n}, in which case the reflection implies that Z_2i=_Z_2i-1=0+(2n-1)_Z_2i-1=2n. It is easy to check from (<ref>)-(<ref>) that (Z_2i,  i≥ 0) has the same distribution as the angle process (A(i), i≥ 0) started from A(0) one the one hand, and as (F_2n(Z_2i), i≥ 0) on the other hand.
            
            Finally, notice that because the odd increments are the same, (<ref>) also holds for Z.
            Step 3: Identification of the limit
            
            Let U_1, U_2 are two uniform r.v. on [0,1], let B and B' be two BMs, all being independent.
            We have in (_+,) that
            ł(M_1(4n^2.)/2n,A(4n^2.)/2n)̊ł( U_1+B'_.-H(.) 1, 2H(.))̊
            where H(t):=F(U_2 + √(2) B_t)/2, has been defined in (<ref>).Let us first consider the angle component. Since the discrete process A(4n^2t)/(2n) is the difference between two (dependent) suitably rescaled random walks which are both tight under this rescaling, the process A(4n^2t)/2n is tight in (_+,). To characterize the limiting process, write
            
                A(4n^2t)/2n= F_2n(Z_8n^2t)/2n
                                   = F ł( Z_8n^2t/2n)̊
            
            since for every x and every n, F_2n( 2n x ) = 2n F( x ). The central limit theorem implies the convergence  Z_8n^2t/2n U_2+ N(0, 2t) for a fixed t≥ 0. Since, the mapping g↦ (t↦ F(g(t)) is continuous on (_+,)s, the independence and stationarity of the increments of Z provide the finite dimensional convergence of the angle process in (<ref>).
            
            
            For the first component, we know that M_1(4n^2.)/2n converges in distribution to a BM modulo 1, but that is not independent from the limit H of A(4n^2.)/2n. The result is a consequence of the following lemma, proved in the sequel. Let B and B' be two independent BM, and let X=B+B' be the sum process. For any (b_0,x_0)∈^2, conditionally on {(X_t=x_t,t∈[0,T]) ,B_0=b_0,B'_0=x_0-b_0}, we have
            
            (B.,B'.)_[0,T]      ł(b_0-x_0/2+x_./2 +B”./√(2), -b_0+x_./2-B”./√(2))̊_[0,T]
            for an independent BM B”.
             Step 4: Proof of Theorem <ref>(ii).
            
            
            Consider two levels h_1≤ h_2.
            First, remark that the restriction of (^, ^) to the compact interval [h_1,h_2] has same distribution as (_U^,_U^) on [h_1,h_2], where U^ and U^ are independent and uniformly distributed on (h_1) and (h_2) (indeed, (h_1) depends only on what happens below the level h_1 and (h_2) depends only on what happens above the level h_2.
            
            
            From (<ref>), it  remains to prove that (_U^,_U^) on [h_1,h_2] is distributed as (Y^,Y^).
            
            For any (x,h)∈, the map Π_(x,h) :  F∈ (ℋ_O,d_ℋ_O) ↦ W_(x,h)∈([h,+∞),∖) that associates to a forest the path started at (x,h) is continuous. From Prop. <ref>, we thus deduce that the Markov chain M of Lemma <ref> correctly renormalized converges on [h_1,h_2], when n→ +∞, to (_u_1-F(u_2)/2(t),_u_1+F(u_2)/2(t))_t∈ [h_1,h_2]
             the paths of  and its dual. We deduce that (Y^,Y^) has the same distribution.
            This concludes the proof of Theorem <ref>.
            
            
            Since we are dealing with Markov processes with stationary increments and simple scaling properties, it suffices to show that for X_1,X_2,N 3 i.d.d. N(0,1) r.v. , we have that conditionally on S=X_1+X_2,
            
                (X_1,X_2) (d)=(S/2+N/√(2),S/2-N/√(2)).
            
            This is a consequence of Cochran theorem, which gives that (X_1-S/2, X_2-S/2) is a Gaussian vector independent from S. Since X_1-S/2=(X_1-X_2)/2=-(X_2-S/2), introducing N/√(2)=X_1-S/2 finishes the proof.
            
            
             §.§ The coalescence times have exponential moments
            
            
            Th. <ref> states that the coalescence times T^(x,x',t) and T^(t) are finite a.s. Due to the compactness of the space , we can prove in fact that they admit exponential moments.
            
            (i) There exist b>0, M<+∞ such that for any x,x'∈ and any t∈`R,
            
                `E [ e^b (T^(x,x',t)-t)] < M.
            (ii) For any t∈ `R, the coalescence time T^(t) admits exponential moments :
            
                ∃ a>0 ,   `E [ e^a (T^(t)-t)] < ∞ .
            For both assertions, by the time translation invariance of the CBW, it suffices to consider only the case t=0 . 
            (i) We can assume that 0≤ x≤ x'< 1.
            We have before crossing time
            
                (^_(x',0)(t)-^_(x,0)(t), 0≤ t≤ T(x,x',0))  (x'-x+ √(2)B(t) 1, 0≤ t≤ T(x,x',0)),
            
            where B is a standard usual Brownian motion.
            Hence  T(x,x',0) has same distribution as the exit time of a linear BM B from the segment [-(x'-x),1-(x'-x)].
            This exit time is known to admit exponential moments (see e.g. Revuz & Yor <cit.>).
            (ii) We will use a very rough estimate to prove this fact.
            
            Let for k≥ 0, A_k be the following independent events :
            
                A_k={T^(2k)≤ 2k+2}
            
            meaning that all trajectories born at height 2k have coalesce before time 2k+2. If we show that
            p:=P(A_k)> 0, then T(0) ≤min{ k, A_k  holds} is bounded by twice  a geometric r.v. p, and then has some exponential moments. So let us establish this fact.
            
            For this, we use a single argument twice. Consider Z the hitting time
            of two BM starting at distance 1/2 on . Clearly
            q:=ℙ(Z≤ 1)>0.
            Let us now bound P(A_0). For this consider “half of the dual CBW”(_(x,1), 0≤ x≤ 1/2) starting at (1). With probability q these trajectories merge before (0). Conditionally to this event, all primal trajectories (^_(x,0), x ∈(0)) starting at time 0 a.s. avoid the dual trajectories, and satisfy
            ^_(x,0)(1) ∈ (1/2,1),
            meaning that, with probability q at least, they will be in the half interval (1/2,1). But now, the two trajectories ^_(1/2,1) and ^_(1,1), will merge before time 2 with probability q. Conditionally to this second event, with probability ≥ 1/2, the merging time of (^_(x,1), x∈ (1/2,1)) is smaller than 1. Indeed on , by symmetry, when ^_(1/2,1) and ^_(1,1) merge, they “capture” all the trajectories starting in [1/2,1] (which will merge with them) or they capture all the trajectories starting in  [0,1/2]. Since both may happen, the probability of each of this event are larger than 1/2. Hence p ≥ q^2/2 and the proof is complete.
            
            
             §.§ Toward explicit computations for the coalescence time distribution
            
            Notice that an approach with Karlin-McGregor type formulas can lead to explicit (but not very tractable) formulas for the distribution of the coalescing time of several Brownian motions.
            Let us consider 0<x_1<x_2<… <x_k<1, and denote by T_k the time of global coalescence of the k Brownian motions W_(x_1,0)^,… W_(x_k,0)^.
            
            Taking to the limit formulas obtained by Fulmek <cit.>, we can describe the distribution of the first coalescence time T^k→ k-1 between two of these paths :
            
                T^k→ k-1(x_i,1≤ i ≤ j)=min{T^(x_i,x_i+1,0), 1≤ i≤ k}
            
            with the convention that x_k+1=x_1, and where T^(x_i,x_i+1,0) is the time of coalescence of ^_(x_i,0) and ^_(x_i+1,0) as defined in (<ref>). We will omit the arguments (x_i,1≤ i ≤ j) in the notation T^k→ k-1_(x_i,1≤ i ≤ j) unless necessary. For t>0,
            
                ℙ(T^j→ j-1>t)=∫ dy_1…∫ dy_j _0<y_1<y_2<… <y_j<1[∑_i=0^j-1(σ^i) ∏_ℓ=1^j Φ_t(y_ℓ-x_σ^i(ℓ))]
            
            where σ^i denotes the rotation σ^i(ℓ)= ℓ+i  j  and where
            
                Φ_t(x)=1/√(2π t)∑_m ∈exp(-(x-m)^2/2t).
            
            Explicit formulas for the Laplace transform of T^j→ j-1 are not established in general cases to our knowledge, except for the following special case when k=2 and θ<0 (see e.g. Revuz & Yor <cit.>):
            
                (e^θ T^2→ 1)=cosh(√(|θ|)1+2x_1-2 x_2/2)/cosh(√(|θ|)/2).
             Using that (e^θ T^j+1→ j)=1+∫_0^+∞θ e^θ t(T^j+1→ j>t)dt and the Markov property, we can finally link (<ref>) and T_k:
            
                (e^θ T_k)=∏_j=1^k-1((e^θ T^j+1→ j(_1(T_j),…_k(T_j)) | _1(T_j),…_k(T_j))),
            
            where T_j is the time of the k-jth coalescence (at which there are j Brownian motions left) and (_1(T_j),…_k(T_j)) are the values of the k coalescing Brownian motions at that time (and hence only j of these values are different).
            It is however difficult to work out explicit expressions from these formula.
            
            
            
            § DIRECTED AND CYLINDRIC POISSON TREES
            
            
            Apart from the (planar) lattice web W^2n, defined as the collection of random walks on the grid ={(x,t)∈^2,  x-t  2 =0 } (see <cit.> or Figure <ref>), several discrete forests are known to converge to the planar BW; in particular the two-dimensional Poisson Tree studied by Ferrari & al. in <cit.>. In Section <ref>, a cylindric version of this forest is introduced and we state the convergence of this (continuous space) discrete forest to the CBW. See Th. <ref> below.
            Our proof consists in taking advantage of the local character of the assumptions (B2O) and (B2). Indeed, the cylinder locally looks like the plane and we can couple (on a small window) the directed and cylindrical Poisson trees in order to deduce (B2O) from (B2).
            
            Finally, in Section <ref>, we discuss under which assumptions, conditions (B2) and (B2O) can be deduced from each other.
            
            
            
             §.§ Convergence to the CBW
            
            
            Let n≥ 1 be an integer, and  r>0 be a real-valued parameter. Consider a homogeneous Poisson point process (PPP in the sequel) 𝒩_λ with intensity λ >0 on the cylinder  defined in (<ref>).
            
            Let us define a directed graph with out-degree 1 having 𝒩_λ as vertex set as follows: from each vertex X=(x,t)∈𝒩_λ add an edge towards the vertex Y=(x',t')∈𝒩_λ which has the smallest time coordinate t'>t among the points of 𝒩_λ in the strip {(x”,t”) ∈ |  d_`R/ℤ(t,t”) ≤ r} where  d_`R/ℤ(x,x”):=min{|x-x”|, |1+x-x”|}. Let us set α(X):=Y the out-neighbor of X. Notice that even if X does not belong to 𝒩 the ancestor α(X)∈𝒩 of this point can be defined in the same way.
            For any element X∈, define α^0(X):=X and, by induction, α^m+1(X):=α(α^m(X)), for any m≥ 0. Hence, (α^m(X))_m≥ 0 represents the semi-infinite path starting at X. We define by _X^λ,r, the continuous function from [t;+∞) to `R/ℤ which linearly interpolates the semi-infinite path (α^m(X))_m≥ 0.
            
            The collection ^λ,r,:={_X^λ,r,, X∈𝒩_λ} is called the Cylindric Poisson Tree (CPT). This is the analogue on  of the two-dimensional Poisson Tree introduced by Ferrari et al. in <cit.>. Also, ^λ,r, can be understood as a directed graph with edge set {(X,α(X)) : X ∈𝒩_λ}. Its topological structure is the same as the CBW (see Th. <ref>) or as the CLW (see Prop. <ref>). The CPT a.s. contains only one connected component, which justifies its name: it is a tree and admits only one bi-infinite path (with probability 1).
            
            Let us choose λ=n and rescale ^λ,r, into ^(n),r, defined as
            
                ^(n),r, := {^n,r/n,_(x,t)(n^2 s) ;   (x,t)∈𝒩_n, s≥t/n^2} .
            
            For r=1/2, the normalized CPT ^(n),r, converges in distribution to the CBW as n→+∞.
            
            As noticed in Section <ref>, only criteria (IO) and (B2O) of Th. <ref> have to be checked. The proof of (IO) is very similar to the one of (I) for the two-dimensional Poisson Tree (see Section 2.1 of <cit.>) and is omitted. The suitable value r=1/2 ensures that the limiting trajectories are coalescing standard Brownian motions.
            
            Let us now prove (B2O). By stationarity of the CPT, it suffices to prove that for all t>0,
            
                lim_`e → 0^+1/`elim sup_n→+∞(η^O_^(n),1/2,(0,t;[0→ε]) ≥ 3 ) = 0.
            
            Recall that among all the trajectories in ^(n),1/2, that intersect the arc [0→ `e] at time 0, η^O_^(n),1/2,(0,t;[0→ `e]) counts the number of distinct positions these paths occupy at time t.
            
            A first way to obtain (<ref>) consists in comparing η^O_^(n),1/2,(0,t;[0→ `e]) and η_W^(n)(0,t;0,`e), where W^(n) denotes the normalized two-dimensional Poisson tree– whose distribution converges to the usual BW, see <cit.>–by using stochastic dominations similar to (<ref>) traducing that it is easier to coalesce on the cylinder than in the plane. Since W^(n) satisfies (B2) (see Section 2.2 of <cit.>), η^O_^(n),(0,t;[0→ `e]) ≤_S η_W^(n)(0,t;0,`e) implies that ^(n),1/2, satisfies (<ref>) which achieves the proof of Th. <ref>.
            
            A second strategy is to investigate the local character of the assumptions (B2) and (B2O). Indeed, the map t↦η^O_^(n),1/2,(0,t;[0→ `e]) is a.s. non-increasing. It is then enough to prove (<ref>) for (small) 0<t≪ 1 in order to get it for any t>0. The same holds when replacing ^(n),1/2, with W^(n). Now, when t and `e are both small, the (normalized) CPT ^(n),1/2, restricted to a small window containing [0→ `e]×[0;t] behaves like the (normalized) two-dimensional Poisson tree W^(n) restricted to a window containing [0;`e]×[0;t] with high probability. As a consequence, ^(n),1/2, and W^(n) should simultaneously satisfy (B2O) and (B2).
            
            Let us write this in details. We use a coupling of the environment (the PPP) on some larger window since the trajectories of the discrete trees on a window are also determined by the environment around. Using some control of the deviations of the paths issued respectively from the intervals I^_ε=[0 →ε ]×{0} and I_ε=[0;ε]×{0}, we determine larger windows _ε^ and  _ε which will determine the trajectories started from this sets to a certain time t_ε up to a negligible probability p_ε. Using the constants that emerge from this study, we thereafter design a coupling between the PPP on the cylinder and on the plane that coincides on _ε^ and  _ε (up to a canonical identification). This will allow us to deduce (B2) or (B2O) from the other.
            
            To design the windows that contains all paths crossing I^_ε (or I_ε) up to time t,  it suffices to follow the trajectories starting at (0,0) and (ε,0). Consider the path (X_k=(x_k,y_k), k≥ 0) started from (0,0) and consider the successive i.i.d. increments of this path
            denoted by (ξ^x_k,ξ^y_k)=Δ X_k. Before normalisation, (ξ^x_1, ξ^y_1) consists of two independent r.v., where ξ^x_1 is uniform on [-r,+r] with r=1/2, and ξ^y_1 has exponential distribution with parameter λ=1, since
            
                P(ξ^y_1 ≥ y)=`P(𝒩∩([-1/2,1/2]× [0,y])  = ∅) = e^- y .
            
            
            Now, starting at 0, the renormalized trajectory on ^(n),1/2, is a random walk whose increments (ξ^(n),x_k, ξ^(n),y_k, k≥ 0) are i.i.d. such that nξ^(n),x_k ξ^x_1, and n^2ξ^(n),y_kξ^y_1. Let us define the number of steps for the rescaled path to hit ordinate t by
            
                τ^n_t := infł{ j≥ 1  | ∑_k=1^j ξ^(n),y_k ≥ t }̊infł{j≥ 1  | ∑_k=1^j ξ^y_k ≥ n^2t}̊ .
            
            The points {∑_k=1^j ξ^y_k, j≥ 1} form a PPP Θ on the line with intensity 1, so that τ^n_t=1+#(Θ∩[0,n^2t]) a.s. Therefore
            
                p_c,t,n := `P(τ^n_t ≥ cn^2) = `P(1+P(n^2t)≥ n^2c)
            
            where P(x) is a Poisson r.v. with parameter x.
            For c=2t  this probability p_2t,t,n is exponentially small in n and the event A_t,n:={τ_n^t≤ 2n^2 t} has probability exponentially close to 1. Now, on the event A_t,n, we can control the angular fluctuations of ^(n),1/2,:
            
            q_t,n    :=     `Pł(sup_j≤τ^n_tł|∑_k=1^j ξ^(n),x_k|̊≥ c √(t))̊
               ≤    `P(A^c_2t,n)+`Pł(sup_j≤ 2n^2 tł|∑_k=1^jξ^(n),x_k|̊≥ c √(t))̊ .
            
            Thus, consider the process defined by
            
                s_n(j/n^2) := ∑_k=1^jξ^(n),x_k   1/n∑_k=1^j ξ^x_k  ,   j≥ 0,
            
            and interpolated in between. A simple use of Donsker theorem shows that
            
                (s_n(a))_a≥ 0ł(1/√(12)B(a))̊_a≥ 0
            
            in (_+,) where B is a Brownian motion. Since for every t, on ([0,2t],), the functional g↦max|g| is continuous, one sees that
            
            q_t,n   =    `P(A^c_2t,t,n)+`Pł(sup_a≤ 2 tł|s_n(a)|̊≥ c √(t))̊
                   `Pł(sup_a≤ 2 t |B(a)| ≥ c√(12 t))̊= `Pł(sup_a≤ 1 |B(a)| ≥ c√(6 ))̊.
            
            Take ε>0. Choose c large enough such that `Pł(sup_a≤ 1 |B(a)| ≥ c√(6))̊≤ε^2/2, and n large enough so that q_t,n≤ε^2, and t small enough so that c√(t)<1/4. We have proved that with probability larger than 1-O(ε^2), the walk hits  ordinate t before its abscissa exits the window [-c√(t),c√(t)]. Since the decision sector for each step of the walker has width 2r/n, with probability more than 1-O(ε^2), the union of the decision sectors of the walk before time t are included in
            [-c√(t)-2r/n,c√(t)+2r/n]⊂[-1/3,1/3] for n large enough.
            It is now possible to produce a coupling between the PPP on the cylinder and the plane that coincides on a strip with width 2/3 : take the same PPP on the two strips (up to a canonical identification of these domains), and take an independent PPP with intensity 1 on the remaining of the cylinder or of the plane. Henceforth, any computation that depends only of such a strip in the cylinder and in the plane will give the same result. Here, we then have here, for any event  that depends on the trajectories passing through I_ε^ or I_ε up to time t (for the constant satisfying what is said just above)
            
            `P_n^() = `P_n()+ O(ε^2),
            
            so that the inheritance of (B2) from the plane to the cylinder is guaranteed, as well as the converse.
            
            
             §.§ From the plane to the cylinder, and vice-versa: principles
            
            
            
            When a convergence result of some sequence of coalescing processes defined on the plane to the BW has been shown, it is quite natural to think that the similar convergence holds on the cylinder too, and that the limit should be the CBW. The converse, also, should hold intuitively.
            
            The main problem one encounters when one wants to turn this intuition into a theorem, is that, in most cases the constructions we are thinking of are trees that are defined on random environments (RE) as a PPP or as lattices equipped with Rademacher r.v.. Both these models exist on the cylinder and on the plane, leading to clear local couplings of these models. But, more general RE and more general random processes exist, and it is not possible to define a “natural” model on the cylinder inherited from that of the plane. We need to concentrate on the cases where such a natural correspondence exists.
            
            A similar restriction should be done for the algorithms that build the trajectories using the RE. In the cases studied in the paper,
            the trajectories are made by edges, constructed by using a navigation algorithm, which decides which points to go to depending on a “decision domain” which may depend on the RE. For example, in the cylindric lattice web, the walker at position (x,t) just needs to know the Rademacher variable attached to this point, so that its decision domain is the point (x,t) itself. In the generalization of  Ferrari & al. <cit.> treated at the beginning of Section <ref>, the decision domain is a rectangle  [x-r,x+r]×(t,t+h] where h is smallest positive real number for which this rectangle contains a point of the point process (many examples of such navigation processes have been defined in the literature, see <cit.>).
            We may call such model of coalescing trajectories as coming from “navigation algorithms, with local decision domains”.
            
            There exist models of coalescing random processes of different forms, or that are not local (such as minimal spanning trees). Again, it is not likely that one may design a general theorem aiming at comparing the convergence on the cylinder with that on the plane.
            
            “For a model defined on the cylinder and on the plane on a RE” as explained in  the proof of Theorem <ref>, when a local coupling between windows (or strip) of the cylinders and of the plane exists, (B2) and (B2O)“are morally equivalent”.
            Informally, the 4 conditions are:  
            
            1) the models are invariant by translations on respectively, the cylinder and the plane;
             2) there exists a coupling between both probabilistic models which allows to compare  ^(n), and W^(n) at the macroscopic level: on a window :=[0,A]× [0,B] for some (small) A,B>0, the environments on which are defined  ^(n), and W^(n) can be coupled, and, under these coupling, these RE  coincide a.s.;
             3) the restriction of the trajectories from ^(n), and W^(n) on [0,ε]×[0,t_ε] are measurable with respect to the environment in  with probability 1-O(ε^1+a) for some a>0;
             4) the largest decision domain before hitting ordinate n^2t is included in a rectangle [a_n,b_n] with probability 1-O(ε^1+a) where a_n=o(1) and b_n=o(1) (for the rescaled version).
            
            
            
            § DISCRETE CYLINDRIC AND RADIAL POISSON TREE
            
            
            
            Coletti and Valencia introduce in <cit.> a family of coalescing random paths with radial behavior called the Discrete Radial Poisson Web. Precisely, a Poisson point process Θ with rate 1 on the union of circles of radius k∈∖{0}, centered at the origin, is considered. Each point of Θ in the circle of radius k is linked to the closest point in Θ in the circle of radius k-1, if any (if not, to the closest point of Θ in the first circle of radius smaller than k-1 which contains a point of Θ). They show in <cit.> that under a diffusive scaling and restricting to a very thin cone (so that the radial nature of paths disappears), this web converges to some mapping of the (standard) BW. A similar result is established in Fontes et al. <cit.> for another radial web.
            
            Our goal in this section is to establish a convergence result for an analogous of the Discrete Radial Poisson Web of Coletti and Valencia <cit.> but which holds in the whole plane. Our strategy consists in considering a cylindrical counterpart to the Discrete Radial Poisson Web and to prove its convergence to the CBW (Theorem <ref>). Thenceforth, it suffices to map the cylinder on the (radial) plane with φ_⋆ defined in (<ref>) to obtain a global convergence result for the corresponding planar radial forest.
            
            
            We modify a bit the model of <cit.> to make the involved normalizations more transparent and to reduce as much as possible the technical issues, while keeping at the same time the main complexity features. Consider an increasing sequence of non-negative numbers (h_k, k ∈), with h_0=0, and the associated slices of the cylinder:
            
                ' = ×{h_k,  k∈}=⋃_k∈(h_k)  .
            
            Consider the following Poisson point process on ',
            
                Ξ=⋃_k≥ 0Ξ_k  ,
            
            where  Ξ_k is a PPP on (h_k) with intensity n_k>0. The sequences (h_k)_k≥ 1 and (n_k)_k ≥ 1 are the parameters of the model. Remark that the choice of n_k=n (a constant) is treated in previous sections. Here we are interested in the case where n_k, h_k→ +∞.
            
            
            Given Ξ, let us define the ancestor α(Z) of a point Z=(x,h_k)∈(h_k) as the closest point of Ξ_k+1 if the latter is not empty and the point (x,h_k+1) otherwise. This second alternative means that instead of moving to the closest point of the first non-empty slice with rank k'>k (as in <cit.>), one just moves vertically to the next slice.
            
            The ancestor line AL_Z of Z=(x,h_k) is the sequence (Z_j=(x_j,h_j),j≥ k) such that Z_k=Z and for j> k, Z_j+1=α(Z_j). Upon Ξ we define the Discrete Cylindric Poisson Tree𝒯 as the union of the ancestor lines of the elements of Ξ:
            
                𝒯 := ⋃_(x,h) ∈Ξ AL_(x,h) .
            Notice that when (x,h)∈Ξ, AL_(x,h)=𝒯_(x,h) is the path of 𝒯 started at (x,h). The notation AL_(x,h) allows to consider ancestor lines started from any points Z∈'.
            
            Contrary to Section <ref>, we do not consider a sequence of point processes parametrized by n which goes to infinity, but rather we shift the cylinder which also implies that we see more and more points. Precisely, for any k≥ j≥ 1 and any (x,h_k)∈Ξ, let AL^(j)_(x,h_k) be the ancestor line AL_(x,h_k) translated by the vector -(0,h_j). We can then associate to 𝒯, the sequence of shifted forests (𝒯^(j))_j≥ 1 by
            
                𝒯^(j) := ⋃_(x,h) ∈∪_k≥ j Ξ_k AL^(j)_(x,h) .
            
            
            Our purpose is to prove that:
            
            Let us consider two sequences (n_k)_k ≥ 1 and (f_k)_k ≥ 1 of positive real numbers such that
            lim_k→∞ f_k = 0   ,  ∑_k e^-n_kf_k < +∞  ∑_k1/n_k^2 = +∞ .
            
            Then there exists a sequence (h_k)_k ≥ 1 tending to infinity such that the sequence of shifted forests (𝒯^(j))_j≥ 1 converges in distribution to the CBW restricted to the half cylinder ^+=×_+.
            
            The map φ_⋆, as defined in def:phi, sends the  half cylinder ^+:=()× `R^+ onto the radial plane ()× `R^+. The image of the PPP Ξ is a PPP Ξ' on the plane, which is the superposition of the Poisson point processes Ξ'_k of intensities n_k/(8π^3h_k) on the circles with radii 4π^2h_k. The image of the tree we built on the cylinder is a tree on “the radial plane”, which can in fact be directly built by adapting the navigation used in the cylinder in the plane (go to the closest point in the next circle if any, and otherwise to the point with same argument). See Figure <ref>. To get a convergence result on the radial plane, with the same flavour as that obtained in the plane, we need to discard the neighborhood of zero by a shift. The most economic way to state our results is as an immediate corollary of the previous result:
            
            Under the hypotheses of Theorem <ref>, when j→+∞, the sequence (φ_⋆(𝒯^(j)))_j≥ 1 converges in distribution to the .
            
            
            Let us comment on the hypothesis eq:newcond. First, it implies that n_k→∞. A consequence of Borel-Cantelli's lemma is that whenever ∑_k≥ 0e^-n_k is finite, there exists a.s. a random rank from which the Ξ_k's are non-empty. Hypothesis eq:newcond is actually slightly more demanding: the condition ∑_k 1/n_k^2=+∞ and the link between sequences (n_k)_k ≥ 1 and (f_k)_k ≥ 1 will appear in Sections <ref> and <ref>. 
            
            
            To prove Theorem <ref>, we check the criteria of Theorem <ref>, namely (IO) and (EO). The convergence of an ancestor line of 𝒯^(j) to a BM modulo 1, when j→ +∞, is first stated in Section <ref>. In the proof, we see that the condition ∑_k 1/n_k^2=+∞ of (<ref>) is necessary. We then deduce (IO) in Section <ref>, and use at some point the second item of (<ref>). The proof of (EO) is devoted in Section <ref> and is based on a coalescence time estimate (Proposition <ref> in Section <ref>) whose proof uses the links between the cylindric and planar forests highlighted in Section <ref>.
            
            
            
             §.§ Convergence of a path to a Brownian motion
            
            
            
            Let us consider the ancestor line 𝒯_(X_j,h_j)=((X_k,h_k), k≥ j) started at a point (X_j,h_j)∈Ξ_j for j≥ 1. For k>j, this path goes to infinity by jumping from (h_k-1) to (h_k).
            The random increments (Δ X_k:=X_k-X_k-1,k >j) are independent.
            The distribution of Δ X_k is characterized for any measurable bounded map f:[-1/2,1/2]→ by,
            
                (f(Δ X_k))=e^-n_kf(0)+∫_-1/2^1/2 f(x) n_k e^-2n_k |x| dx.
            
            In other words, conditionally on the (h_k) being not empty, Δ X_k is a Laplace r.v. conditioned on having absolute value smaller than 1/2.
            Hence,
            {[                                                 (Δ X_k)                                                       =                                                       0;                                                 (Δ X_k)                                                       =           1/2n_k^2-e^-n_k(n_k^2+2n_k+2)/4n_k^2 =: σ_k^2;                                             ((Δ X_k)^4)                                                       = 3/2n_k^4-e^-n_k(n_k^4+4n_k^3+12n_k^2+24n_k+24)/16n_k^4. ].
            
            As n_k→∞, the variance σ_k^2 is equivalent to 1/(2n_k^2). For the sequel, let us denote the variance of X_k-X_0 by:
            
                V_k := (X_k-X_0) = σ_1^2+⋯+ σ_k^2.
            
            The variance V_k is hence related to n_k by (<ref>).
            
            
            Let us now consider the time continuous interpolation of the shifted sequence (X_k,h_k-h_j)_k≥ j. For ℓ∈, we set, if h_j+t∈ [h_j+ℓ, h_j+ℓ+1),
            
                X̅_t^(j)= X_j+ ∑_k=1^ℓΔ X_j+k + (h_j+t)-h_j+ℓ/h_j+ℓ+1-h_j+ℓ Δ X_j+ℓ+1 .
            
            In order to prove the convergence of (X̅^(j))_j≥ 0 to a Brownian motion, it is natural to set
            
            h_k = V_k ,  for any  k≥ 0  .
            
            Then, combining eq:displ, def:Vk, eq:hV with eq:newcond, it follows that h_k→∞ and h_k+1-h_k=σ^2_k+1→ 0, i.e. slices are getting closer and closer.
            
            The hypothesis eq:newcond is satisfied for example for n_k=k^α with 0<α<1/2 and f_k=k^α' with 0<α'<α. This entails that h_k=c_1+c_2 k^1-2α where c_1,c_2 are positive constants.
            
            
            Let us introduce, for the sequel,
            
                R(t) := inf{k∈, V_k≥ t}
            
            the integer index such that h_R(t)-1< t ≤ h_R(t). Note that R(h_k)=k.
            
            
            Under the previous notations and eq:newcond, the following convergence holds in distribution in (_+,)
            (X^(j)_t,t≥ 0)  (B_t,t≥ 0)  ,
            
            where (B_t,t≥ 0) is a standard Brownian motion taken modulo 1.
            
            We are not under the classical assumptions of Donsker theorem, since the Δ X_k's are not identically distributed and since the convergence involves a triangular array because of the shift.
            Because the Δ X_k's are independent, centered, with a variance in 1/n_k^2 that tends to 0, we have for all t≥ 0,
            
                lim_j→ +∞(X̅^(j)_t-X̅^(j)_0)=
                lim_j→ +∞∑_ℓ = j+1^R(h_j+t)(Δ X_ℓ)=
                lim_j→ +∞(V_R(h_j+t)-V_j)=t,
            implying that X̅^(j)_t-X̅^(j)_0 converges in distribution to 𝒩(0,t) by Lindeberg theorem (e.g. <cit.>). The convergence of the finite dimensional distributions is easily seen by using the independence of the Δ X_k's.
             The tightness is proved if (see e.g. <cit.>) for every positive ε>0 and η>0, there exists δ∈ (0,1) and j_0∈ such that for every j≥ j_0 and every t∈_+,
            
                1/δ(sup_t≤ s≤ t+δ| X̅^(j)_s-X̅^(j)_t|≥ε)≤η.
            
            For t∈_+ and j∈,
            
                (sup_t≤ s≤ t+δ| X̅^(j)_s-X̅^(j)_t| ≥ε) =    (sup_t≤ s≤ t+δ| X̅^(j)_s-X̅^(j)_t|^4≥ε^4)
                ≤   ( max_R(h_j+t)≤ℓ≤ R(h_j+t+δ)   S_ℓ^4   ≥ε^4 ),
            where
                S_ℓ=h_R(h_j+t)-(h_j+t)/h_R(h_j+t)-h_R(h_j+t)-1 Δ X_R(h_j+t)+ ∑_k=R(h_j+t)+1^ℓΔ X_k.
            
            Since the Δ X_k are centered, S_ℓ defines a martingale (in ℓ≥ R(h_j+t)), and S_ℓ^4 is a submartingale. Using Doob's lemma for submartingales:
            
                ε^4 ( max_R(h_j+t)≤ℓ≤ R(h_j+t+δ) S_ℓ^4 ≥ε^4 ) ≤(S^4_R(h_j+t+δ))
                ≤(∑_k=R(h_j+t)^R(h_j+t+δ) (Δ X_k)^4)+∑_k≠ℓR(h_j+t)≤ k,ℓ≤ R(h_j+t+δ)((Δ X_k)^2)((Δ X_ℓ)^2),
            using that the Δ X_k's are independent and centered. The last sum in the r.h.s. of (<ref>) is upper bounded by
            
                (∑_k=R(h_j+t)^R(h_j+t+δ)((Δ X_k)^2))^2
            
            that converges to δ^2 when j→ +∞. For the first term, there exists from (<ref>) a constant C such that for k large enough, ((Δ X_k)^4)≤ C (Δ X_k)^2 = C σ_k^4. Thus:
            
                ł(∑_k=R(h_j+t)^R(h_j+t+δ) (Δ X_k)^4)̊≤    C M_j,t,δ∑_k=R(h_j+t)^R(h_j+t+δ)σ_k^2 ∼_j→ +∞ C M_j,t,δδ
            
            with M_j,t,δ=sup{σ_k^2,  R(h_j+t)≤ k ≤ R(h_j+t+δ) }→ 0 when j→+∞.
             Gathering these results, we see that up to a certain constant C,
            
                1/δ(sup_t≤ s≤ t+δ| X̅^(j)_s-X̅^(j)_t|≥ε)≤C/ε^4 δ(δ^2 + CM_t_1,t,δδ),
            which converges to zero when δ→ 0 and j→ +∞.
            
            
             §.§ Coalescence time estimate
            
            
            
            In this section, we establish a coalescence time estimate that will be useful for proving (IO) and (EO). Following the lines of Section <ref>, we can introduce a planar model corresponding to our cylindrical tree and ensuring the possibility of couplings between the cylinder and the plane. In the plane, the use of the Skorokhod embedding theorem and the results known for planar Brownian motions make it easier to obtain such estimates.
            We thus first introduce in Section <ref> a planar model corresponding to the forest 𝒯. We establish estimates for the coalescence time of two paths in this planar model. For this, we start with studying how the distance between the two paths evolves. The core of the proof relies on the Skorokhod embedding theorem (as in <cit.>), but with a clever preliminary stochastic domination of the distance variations. In Section <ref>, we return to the original model and deduce from the previous result estimates for the coalescence time of two paths of 𝒯^(j).
            
            
            
            
              §.§.§ Planar analogous
            
             We first define the planar model corresponding to our cylindrical problem. We consider the horizontal lines with ordinate (h_k)_k∈ in the upper half plane. For each k∈∈, we consider on the line L_k:=×{h_k} (or level h_k), an independent PPP Υ_k with intensity n_k. The Poisson point process on the union of the lines is denoted by Υ, similarly to (<ref>). Each point of the level h_k is linked with the closest point of the next level, namely level h_k+1.
            This generates a forest that we denote 𝒲, and which can be seen as the analogous of 𝒯 in the plane.
            
            
            
            For a given point Z∈∪_k∈L_k, denote by α^P(Z) the ancestor of Z for this navigation. This allows us to define, as for the cylinder, the ancestor line  AL^P_Z of any element Z∈×_+.
            
            The aim of this section is to provide an estimate on the tail distribution of the hitting time between the ancestor lines started from two points at a distance d>0 on the line L_0. Without restriction, we consider Z=(0,0) and Z'=(d,0), and denote their ancestor lines by (Z_k, k≥ 0) and (Z'_k, k≥ 0).
            Let us denote by
            
                D_k= Z_k'-Z_k,
            
             the distance between the two paths at level h_k. The result proved in this section is the following:
            Let us define by τ=inf{k∈, D_k= 0}. There exists C>0 such that for K∈∖{0},
            
                (τ>K) ≤C d/√(h_K).
            
            
            The remaining of the section is devoted to the proof of Prop. <ref>.
            In the proof, we will also need the following quantity for k≥ 1:
            
                Δ_k= D_k-D_k-1.
            
            
            The proof is divided into several steps. For the first step, we consider a PPP with an intensity constant and equal 1. In doing so, we introduce a sort of companion model that will help finding estimates for the planar model considered above.
             We will then proceed to the control of the hitting time of two ancestors lines, by using some rescaling properties.
            
            
            
              §.§.§ Step 1: Evolution of the distance in one step, when the intensity is 1 
            
            
            Take two points Z=(0,0) and Z'=Z(d,0) at distance d in L_0.
            Assume that the PPP Υ_1 on L_1 has intensity 1: let (X_1,h_1)=α^P(Z) be the closest point to Z in Υ_1 and by (X_2,h_1)=α^P(Z') the closest point to Z' in Υ_1. Let
            
                D(d)=X_2-X_1=|α^P(Z')-α^P(Z)|
            
            be the “new distance”, and denote by
            
                Δ(d)=D(d)-d
             the variation of the distance between the levels L_0 and L_1.
            
            The distribution μ_d of Δ(d) is the following probability measure on μ_d(du) = (d+1)e^-2dδ_-d(du)+ f_d(u) _[-d,d](u)  du+de^-u-d_[d,+∞)(u)  du ,
            where
             f_d(u)=-e^-2d/2 +e^-2|u|(|u|+1/2).
            
            The atom   m_d= (d+1)e^-2d of μ_d at -d corresponds to case where coalescence occurs, that is α^P(Z)=α^P(Z'). Apart from this atom, μ_d is absolutely continuous with respect to the Lebesgue measure, and
            (Δ(d))   =   0    and   (D(d))=d
            (Δ(d))   =   V(d)=1-e^-2 d+2/3 e^-2 dd^3+e^-2 dd^2.
            
            The proof is postponed to the end of the section.
            
            Notice that the distribution of Δ(d) does not depend on the height h_1 of the level L_1, and that f_d is a symmetric function.
            
            
            
              §.§.§ Step 2: The sequence (D_k,k≥ 0) is a martingale
            
            
            Denote by D^λ(d)=X_2-X_1=α^P(Z')-α^P(Z) the new distance if the PPP Υ_1 has intensity λ>0, and by Δ^λ(d)=D^λ(d)-d. A simple scaling argument allows one to relate the distribution of Δ^λ(d) to that of Δ^1(d)Δ(d) :
            Δ^λ(d)1/λΔ^1(λ   d),  d≥ 0,
            
            Now, since the intensity on L_k is n_k, conditional on D_k-1,
            ccl
            D_k       D^n_k(D_k-1)
            Δ_k       Δ^n_k(D_k-1)1/n_kΔ^1( D_k-1 n_k).
            
            
            Because of (<ref>), (D_k, k≥ 0) defines a martingale. The particular form of  μ_d makes it difficult to control the time at which it hits 0. We will dominate D_k by another martingale that is easier to handle.
            
            
            
              §.§.§ Step 3: Introduction of an auxiliary distribution μ_d
            
            
            We introduce  the following family of distributions indexed by d>0:
            
                μ_d(du):=α_d δ_-(c_d+d)(du)+f_d(u)_[-d,d](u)  du 
                
                + de^-u-d_[d,+∞)(u)  du+ β_d e^-u-d_[d +ν_d,+∞)(u)  du.
            Let Δ(d) be a r.v. with distribution μ_d, and set D(d) to be the r.v. defined by
            
                Δ(d)=D(d)-d.
            
            Our strategy is as follows: we will choose carefully the functions α_d,c_d,r_d,β_d satisfying for any d>0,
            α_d ≤ (d+1)e^-2d,    
            c_d ≥ 0,    ν_d  ≥ 0,    β_d ≥ 0.
            
            in such a way that for any d>0, μ_d is a probability distribution with mean 0, and which will dominate μ_d in the sense of the forthcoming Lemma <ref>. The difference between μ_d and μ_d is that the atom at -d in μ_d is replaced by an atom at -c_d-d<-d with, as a counterpart, a modification of the distribution at the right of d which is replaced by a distribution larger for the stochastic order. 
            Proceeding like this, our idea is to bound stochastically the hitting of 0 by (D_k,k≥ 0) (the coalescing time τ) by the hitting time of (-∞,0) by an auxiliary Markov chain (D_k,k≥ 0).
             The measure μ_d is a density probability with mean  (Δ(d))=0 iff
            
                α_d = e^-2d(1+d)(1+2d+ν_d)/1+2d+c_d+ν_d        β_d = e^ν_d(1+d)c_d/1+2d+c_d+ν_d.
            
            Compute the total mass of μ_d:
             
                ⟨μ_d,1⟩= β_d e^-2d-ν_d+α_d+1-e^-2dd-e^-2d
            
            and if the total mass is 1, the expectation of Δ(d) is :
             
                (Δ(d))
                 =      - α_d  (c_d +d)+ d(d+1)e^-2d + β_d  (d+ν_d+1) e^-2d-ν_d.
            Solving these equations in α_d and β_d provides the announced result.
            
            
            We hence see that we have two degrees of freedom. In the sequel, we will choose:
            
                c_d=1,    ν_d=2,
            independent of d. This implies that:
            
                α_d=e^-2d2d^2+5d+3/2(d+2),    β_d=e^2 d+1/2(d+2).
            From this, we can compute (Δ(d)):
            
                (Δ(d))
                =1+( 4 d^4+26 d^3+66 d^2+75 d+27 )
                e^-2 d/6 d+12.
            
            
            For the measure μ_d which we have now completely constructed, we have:
             Under (<ref>) and (<ref>), we have for d<d',
            
            D(d) _D(d)>0≤ _S D(d') _D(d')>0,
            
            in the sense that for all t>0, (D(d)>t)≤(D(d')>t).
            
            First, for any d', by construction of the measure μ_d',
            
                D(d')_D(d')>0≤ _S D(d')_D(d')>0.
            
            Now, recall that D(d) provides the new distance in the model of Step 1 when the intensity of the PPP on L_1 is 1 and when the starting points Z=(0,0) and Z'(d,0) are at distance d. One can follow a third point Z”(d',0).
            Since these paths do not cross, the distance D_1 D(d) between Z_1 and Z'_1 remains smaller than the distance D'_1=Z”_1-Z_1 D(d'). This implies that D(d)_D(d)>0≤ _SD(d')_D(d')>0 holds. This concludes the proof.
            
            
              §.§.§ Step 4: Introduction of an auxiliary Markov chain
            
            
            To dominate (D_k,k≥ 0) we introduce the Markov chain (D_k, k ≥ 0) whose distribution respects the same scaling eq:efgz as (D_k,k≥ 0): conditionally on D_k-1, we let Δ_k= D_k-D_k-1 have distribution
            
                Δ_k1/n_kΔ( D_k-1 n_k).
            
            Let us define
            
                τ_^-=inf{k ≥ 0 : D_k≤ 0},        τ_^-=inf{k ≥ 0 : D_k≤ 0} .
            
            For any d>0, if D_0=D_0=d, we have
            
                τ_^-≤_S τ_^-.
            
            Just observe that in (<ref>), the r.v. in both sides have atoms at 0 that correspond to the entrance of D_k and D_k in (-∞,0] (in fact the hitting time of {0} for D_k and of (-∞,0) for D_k). The Markov property and (<ref>) allow to conclude.
            
            
              §.§.§ Step 5: Skorokhod embedding
            
            By the Skorokhod embedding theorem (see <cit.>), there exists a BM B started at 0 and a stopping time T_1(d) such that Δ(d)(d)= B_T_1(d). Moreover, it is possible to construct two r.v. U(d)≤ 0 and V(d)≥ 0 such that T_1(d)=inf{t≥ 0, B_t∉ [U(d),V(d)]}. U(d) and V(d) are independent from the BM B, but not independent (in general) one from the other. Since B_T_1(d)=U(d)≤ 0 or B_T_1(d)=V(d)≥ 0, U(d) and V(d) can be constructed from the distribution of B_T_1(d)(d)=Δ(d), i.e. μ_d, as follows (recall (<ref>)):
            
              *  With probability 2p_d, U(d)=-V(d) and V(d) is a r.v. with density _[0,d](v) (e^-2v(v+1/2)-e^-2d/2) /p_d. We denote by A_d this event.
            
              *  With probability 1-2p_d=(1+2d)e^-2d, on A^c_d, we set U(d)=-d-1. For V(d), we have two cases since the right tail of μ_d is the sum of two exponential tails, de^-d-u_[d,+∞)(u) and β_d e^-d-u_[d+ν_d,+∞)(u). Conditionally on A^c_d:
            
              ∙ With probability q_d=2d/(1+2d), V(d) is a r.v. with density v+d+1/2(1+d)e^d-v_[d,+∞)(v) with respect to the Lebesgue measure. We call this event E_d.
            
              ∙ With probability 1-q_d=1/(1+2d), V(d) is a r.v. with density
            v+d+1/3+2d e^d+2-v_[d+2,+∞)(v). This event is E^c_d∩ A^c_d.
            
            Recall from (<ref>) that μ_d admits a symmetric density f_d on [-d,d]. Thus, on the event A_d={|Δ_d|<d}, which has probability
            
                2p_d= ∫_-d^d f_d(u)du=1-e^-2d-2d e^-2d,
            it is sufficient to define U(d)=-V(d) with V(d) a r.v. of density f_d(v)/p_d_[0,d](v). Since the Brownian motion B started at 0 exits the symmetric interval [-V(d),V(d)] through the upper or lower bound with equal probabilities 1/2, the likelihood of B_T_1(d) for this part is as expected:
            
                2p_d (1/2f_d(v)/p_d_[0,d](v)+1/2f_d(-v)/p_d_[-d,0](v))=f_d(v)_[-d,d](v).
            
            Let us now consider A_d^c ∩ E_d. The lower bound is necessarily U(d)=-d-1, since it is the only possible value for Δ_d below -d. As for the density of V(d) conditionally to A_d^c∩ E_d, say g(v), it has to be chosen such that we recover d e^-d-u_[d,+∞)(u) once multiplied by (1-2p_d), q_d and by the probability that B exits through the upper bound V(d) rather than through the lower bound U(d)=-d-1:
            
                (1-2p_d) q_d d+1/v+d+1 g(v)=d e^-d-v_[d,+∞)(v)  ⇒   q_d g(v)= e^2d/1+2dv+d+1/d+1 d e^-d-v_[d,+∞)(v).
            Since g is a probability density, integrating over v gives q_d: q_d=2d/(1+2d). We then deduce the density of V(d) conditionally to A_d^c∩ E_d. We proceed similarly for A_d^c∩ E_d^c.
            
            
            
            
            By recursion, we can define for k≥ 1 the time T_k by
            
                T_k=inf{t≥ T_k-1, B_t-B_T_k-1∉ [U_k(D_k-1),V_k(D_k-1)]}
            where U_k(D_k-1) and V_k(D_k-1) are independent r.v. conditionally on D_k-1, such that for any D>0,
            
                U_k(D)(d)=1/n_k U(n_k D),     V_k(D)(d)=1/n_k V(n_k D),
            where U(d) and V(d) have the law described above in the representation of T_1(d) for d>0.
            With this construction, we have that for k≥ 1, B_T_k(d)=D_k-d.
            
            
            
              §.§.§ Step 6: Laplace transforms of T_1(d) and T_k(d):
             For λ>0, there exists c_0(λ)∈ (0,1) independent of d such that
            
                0≤φ_d(λ)=  (e^-λ T_1(d))≤ c_0(λ)<1.
            Moreover, for λ small, there exists a constant C>0 such that
            c_0(λ)≤ e^-Cλ.
            Using the Skorokhod embedding described above,
            
                φ_d(λ) =    (e^-λ T_1(d) | A_d ) 2p_d + (e^-λ T_1(d) | A^c_d , E_d)  (1-2p_d)q_d 
                   + (e^-λ T_1(d) | A^c_d , E_d^c)  (1-2p_d)(1-q_d).
             Our purpose is to bound φ_d(λ) uniformly in d by a constant strictly smaller than 1. On the events, A_d and A^c_d ∩ E_d, the interval [U(d),V(d)] which defines T_1(d) has at least one extremity that gets closer and closer to zero when d tends to zero. So upperbounding the expectations in the first and second terms of the r.h.s. of (<ref>) by a constant strictly less than 1 uniformly in d is difficult. For the third term of (<ref>) however, because U(d)<-c_d=-1 and V(d)>ν_d=2, we have that
            
                (e^-λ T_1(d) | A^c_d , E_d^c)≤(e^-λ T')<1
            
            where T'=inf{t≥ 0, B_t ∉ [-1,2]}.
            Additionally, since
            (1-2p_d)(1-q_d)=e^-2d→_d→ 0 1, this shows (<ref>) with
            
                c_0(λ)=    _0(e^-λ T')
                =  cosh(√(λ/2))/cosh(3√(λ/2))<1.
            When λ→ 0, c_0(λ)=  1-2λ+o(λ) ≤ e^-2λ which shows the second assertion with C=2.
            
            From this by using (<ref>) and the self-similarity of the standard BM started at 0,
            
                T_k-T_k-1(d)=   inf{t≥ 0, B_t ∉[1/n_kU(n_k D_k-1),1/n_kV(n_k D_k-1)]}
                (d)=   inf{t≥ 0, 1/n_k B_n_k^2 t∉[1/n_kU(n_k D_k-1),1/n_kV(n_k D_k-1)]}
                (d)=   1/n_k^2 T_1(n_k D_k-1).
            Hence it follows that
            
                (e^-λ (T_k-T_k-1) | ℱ_T_k-1)=φ_n_k D_k-1(λ/n_k^2).
            
            
              §.§.§ Step 7: Estimate  for the tail distribution of the coalescing time
            
            With the ingredients developed above, we can now follow ideas developed in <cit.> for instance.
            Recall that τ__-=inf{k∈, D_k≤ 0} and define θ=inf{t≥ 0, B_t=-d}. Let us consider ζ>0. Then for K∈∖{0}:
            
                (τ__->K)=    (θ > T_K)
                ≤   (θ > ζ h_K)+(θ>T_K, T_K < ζ h_K)
                ≤   Cd/√(ζ h_K)+ e^λζ h_K(e^-λ∑_k=1^K (T_k-T_k-1)).
            
            For the Laplace transform in the last term, using (<ref>):
            
                (e^-λ∑_k=1^K (T_k-T_k-1)) =    (_d(e^-λ∑_k=1^K (T_k-T_k-1) | ℱ_T_K-1))
                
                =    (e^-λ∑_k=1^K-1 (T_k-T_k-1)(e^-λ (T_K-T_K-1) | ℱ_T_K-1))
                
                 =
                   _d(e^-λ∑_k=1^K-1 (T_k-T_k-1)φ_n_K D_K-1(λ/n_K^2))
                ≤    c_0 (λ/n_K^2)_d(e^-λ∑_k=1^K-1 (T_k-T_k-1))≤∏_k=0^K-1c_0 (λ/n_k+1^2)
                ≤   exp(-2 λ∑_k=1^K1/n_k^2).
            
            Recall from (<ref>) that h_K=V_K∼∑_k=1^K 1/2n_k^2. Thus, from (<ref>) and (<ref>):
            
                (τ__->K)≤Cd/√(ζ h_K)+ C'exp(λ h_K(ζ-4)).
            Because h_K→ +∞, and because the term in the exponential is negative for ζ sufficiently small, there exists λ_0>0 and ζ_0>0 such that the r.h.s. of (<ref>) is smaller than C” d/√(ζ_0 h_K) for K large enough.
            
            This together with Proposition <ref> allow to conclude the proof of Proposition <ref>. Starting from two points Z and Z' of L_0 at distance d and denoting by τ the index of the level at which they coalesce, we have for any K∈∖{0},
            
                (τ>K)≤(τ__->K)≤Cd/√(h_K).
            
            
            Let us finish this subsection with the proof of Proposition <ref> that had been postponed.
            
            
            First, notice that X_1 has density e^-2|x|_x ∈. Then, we can compute the distribution of X_2 conditionally on X_1. In what follows, all r.v. are independent, R is a Rademacher r.v., (k) denotes an exponential r.v. with expectation 1/k.
            
              – Conditional on X_1=x_1>0, with x_1<d: 
            ∙X_2=-(d-x_1) (merge) with probability e^-2(d-x_1),
            ∙ with probability1-e^-2(d-x_1), X_2∼ L(R  (2) | (2) <d-x_1).
            
              – Conditional on X_1=-x_1 <0, with x_1<d
            ∙X_2=-(d+x_1) (merge) with probability e^-2(d-x_1)-2x_1
            ∙X_2∼ L(R  (2) | (2) <d-x_1) with probability  1-e^-2(d-x_1)
            ∙X_2∼ L((1)+d-x_1 | (1) <2x_1) with probability  e^-2(d-x_1)(1-e^-2x_1)
              – Conditional on X_1=x_1>0, with x_1>d: 
            ∙  merge with probability1
            
              – Conditional on X_1=-x_1 <0, with x_1>d
            ∙X_2=-(d+x_1) (merge) with probability e^-2d
            ∙X_2∼ L(x_1-d+ (1) | (1) <2d) with probability 1-e^-2d
            This yields the announced result.
            In particular, the two trajectories started at (0,0) and (d,0) merge at ordinate 1 with probability:
            
            `P(D(d)=0)   =   ∫_x=0^d e^-2x (e^-2(d-x)+e^-2(d-x)-2x)dx+∫_x=d^+∞e^-2x (1+e^-2d)dx,
             which is (d+1)e^-2d, as announced.
            
            
              §.§.§ Extension to the shifted cylinder
            
            
            We now conclude the section with a corollary establishing an estimate for the coalescence time in 𝒲^(j), which is the forest 𝒲 shifted by (0,-h_j) similarly to 𝒯^(j). Then, we enounce an estimate for the shifted cylindrical forest 𝒯^(j).
            
            Let d>0 and j∈. 
            
            (i) Let us consider the paths in 𝒲^(j) started at (0,0) and (d,0) (if (0,h_j) and (d,h_j), these points are connected at the level j+1 to the closest point of 𝒲). Define their coalescing time as τ=inf{k≥ j, D_k=0}. There exists a constant C>0 such that for any K> j,
            
                (τ>K)≤Cd/√(h_K-h_j).
            
            This can be translated, for any t_0>0 as:
            
                (𝒲^(j)_(0,0)(t_0)≠𝒲^(j)_(d,0)(t_0))≤C d/√(h_R(h_j+t_0)-1-h_j)→_j→ +∞C d/√(t_0).
            (ii) Let us consider the paths of 𝒯^(j) started at (0,0) and (d,0) for d∈(0,1/2] (d=1/2 is the maximal distance in the cylinder). Then, there exists C>0 such that for any t_0>0:
            
                (𝒯^(j)_(0,0)(t_0)≠𝒯^(j)_(d,0)(t_0))≤C d/√(h_R(h_j+t_0)-1-h_j)→_j→ +∞C d/√(t_0).
            
            The proof of (i) is an adaptation of the proof Step 7 of Prop. <ref> by summing between levels L_j and L_K. 
            
            
            Let us now consider (ii). Intuitively, the coalescence time in the cylinder is stochastically dominated by the coalescence time in the plane. But since some slices in the cylinder may contain no points of the PPP (when no line L_k in the plane is empty), and since the increments of the distance between the two paths are non standard when this distance is close to 0 and 1 (when only the case 0 matters in the plane), an additional argument is needed in the discrete case to establish the domination rigorously.
            
            Recall the model introduced in Section <ref>. We consider the Markov chain (D_k,k≥ 0) and denote by τ_d be stopping time at which the Markov chain started from d hits 0. We also introduce similarly the distance process (D̅_k,k≥ 0) in the cylinder.
            
            Now, let us define another Markov chain (D'_k,k≥ 0) with the following transitions: 
             L(D'_k+1 |  D'_k=d)=  L( min(D_k+1,|1-D_k+1|)  |  D_k=d ).
            The distance D'_k somehow mimics the distance on the cylinder by considering the minimum distance between two points of the same level in the clockwise and counter clockwise senses. Let us define by τ'_d the stopping time at which (D'_k,k≥ 0) started from d hits 0. Since min(D_k+1,|1-D_k+1|)≤ D_k+1, and since
            τ_d ≤_S τ_d' when  d<d',
            
            by using the same argument as in the proof of Lemma <ref>, we obtain by using iteratively eq:tjdktu that
            
                τ'_d ≤_S  τ_d.
            
            To conclude, it remains to show that (D'_k,k≥ 0) coincides with (D̅_k,k≥ 0), up to a probability going to 0 in j. Since we may produce a local coupling between D'_k and D̅_k as long as D_k possesses small fluctuations, it suffices to prove that all the increments of the paths (Z̅_k, k≥ 0) and (Z̅'_k,k≥ 0) that define (D'_k,k≥ 0) in the cylinder are not 0 and smaller than 1/6 after the slice j with probability going to 1 when j→ +∞. This indeed guarantees that the cylinder effects do not prevent the coupling: no jumps “0” occur and “decision domains” do not see that the environment is a cylinder.
            The probability that there is no point within distance ± 1/6 for a walk is e^-n_k/3, and by Borel-Cantelli's lemma, with probabilty 1 the two walks  (Z̅_k) and (Z̅'_k) will do a finite number of jumps larger than 1/6. Hence, for any `e>0, for j large enough, the distribution of (D'_k,k≥ 0) and (D̅_k,k≥ 0) coincides with probability at least 1-`e. Thus the coupling works, which allows to conclude.
            
            
            
            We have now the tools to prove the criteria of the convergence Theorem <ref>, (IO) and (EO). Both of these criteria make use of the estimates on coalescing time that we hve just established.
            
            
            
             §.§ Proof of (IO)
            
            
            
            The purpose of this section is to prove the next Proposition which implies (IO).
            
            Assume eq:newcond. Let m∈∖{0} and y_1=(x_1,t_1),…,y_m=(x_m,t_m)∈^+. For j≥ 0 and 1≤ℓ≤ m, let us denote by γ^(j)_y_ℓ the path interpolating linearly the shifted ancestor line AL^(j)_y_ℓ+(0,h_j).
            Then, the sequence (γ^(j)_y_1,…γ^(j)_y_m) converges in distribution, when j→ +∞, to coalescing Brownian motions modulo 1 started at y_1,… y_m.
            
            
            Notice that the path γ^(j)_y_ℓ starts at y_ℓ. We also recall that the ancestral line AL_y_ℓ+(0,h_j) does not necessarily starts from a point of Ξ, but links the starting point y_ℓ+(0,h_j) to the closest point of Ξ in the first non-empty slice of height greater or equal to R(t_ℓ+h_j). For the sequel, let us denote by y_j,ℓ this point.
            
            
            The result for m=1 is due to Lemma <ref> and the fact that y_j,1-(0,h_j) converges a.s. to y_1.
            The proof can be done by recursion, and we focus here on the case m=2 which can be generalized directly by following Arratia <cit.> and Ferrari, Fontes and Wu <cit.>.
            
            Let us first recall a simple fact. Let t≤ t' and a,b ∈. Two BM (^_(a,t)(s),s≥ t) and (^_(b,t')(s),s ≥ t') on the cylinder are said to be coalescing BM if ^_(a,t)(s) for t≤ s ≤ t' is a standard BM taken modulo 1, and if the two trajectories (^_(a,t)(s),s≥ t') and (^_(b,t')(s),s ≥ t') are BM till their hitting time τ. After this time, they coincide with (^_(b,t')(s), s≥τ).
            
            To prove that (γ^(j)_y_1,γ^(j)_y_2) converges to two coalescing BM, a strategy consists in decomposing the trajectories as follows, where we can assume to simplify that y_1 and y_2 are such that t_1=t_2:
            (a) as long as the two paths are far apart, say if
            d_/ł(γ^(j)_y_1(s),γ^(j)_y_2(s))̊>a^j(s) for a good sequence a^j(s)→ 0, then the next steps of these trajectories are likely to be characterized by Ξ∩ I and Ξ∩ I' for two random influence intervals I and I' that will not intersect. By the spatial properties of the PPP, it means that as long as I∩ I'=∅,  the two trajectories behave as if they were constructed on different spaces, and then eventually behave as independent BM before their coalescing time (here, since the intensity is not constant, a dependence in s is needed).
            (b)  when eq:far fails (the two paths are close) then another argument is developed to prove that the two paths will merge with a probability going to 1, within a o(1) delay. This is given by the Corollary <ref>.
            
            It remains to see in details how (a) can be handled. Let us denote by (X̅^(j)_s)_s≥ t (as in (<ref>)) the path γ^(j)_(x,t). The distribution of the increment Δ X_j+k (with the notation of section <ref>) satisfies
            (|Δ X_j+k|≥ r)=e^-2n_k+j r
            so that for r=f_j+k/2, with the f_j+k's appearing in Assumption (<ref>), the event {∀ k≥ 0, |Δ X_j+k|≤ f_j+k/2} will occur a.s. for j large enough thanks to Borel-Cantelli's lemma. We then decree that two walks are close if when they get in the slice with intensity n_k+j, their distance is smaller than f_j+k, i.e. we choose a^j(s)=f_R(h_j+s)-1. This suffices to complete the proof.
            
            
             §.§ Proof of (EO)
            
            
            
            We follow the strategy developed in <cit.>: we first show that the sequence (𝒯^(j))_j≥ 1 satisfies (<ref>), stated below, from which (EO) follows.
            
            
            The sequence (𝒯^(j))_j≥ 1 satisfies for all t_0,t>0, a>0,
            
                lim sup_j→ +∞( η_𝒯^(j)^O(t_0,t ; [0 → a]) ) < +∞ .
            
            This implies that (𝒯^(j))_j≥ 1 satisfies (EO).
            
            
            The rest of this section is devoted to the proof of Proposition <ref>.
            
            
            Let us remark that, by translation invariance and linearity of the expectation, it is enough to prove (<ref>) for small values of a.
            
            
            Let us start with proving that (<ref>) implies (EO). We follow the corresponding proof in the planar context which is recalled at the end of Section <ref> (see also the end of Section 6.1 in <cit.> or Section 6 of <cit.>). Except that contrary to the planar context where explicit computation is possible, namely (η_W(t_0,t ; a,b))=(b-a)/√(π t) where W is the standard BW, another argument is needed to get the limit (<ref>) in the cylinder context. We then have to show:
            
                lim_ε→ 0( η^O_(t_0+ε,t-ε ; [a → b]) ) = ( η^O_(t_0,t ; [a → b]) )  .
            
            
            Let us consider t,t_0>0 and a,b∈. Let us first prove that there exists ε_0∈(0,t) such that
            
                ( η^O_(t_0+ε_0,t-ε_0 ; [a → b]) ) < ∞ .
            
            Without loss of generality, we still write t_0 and t instead of t_0+ε_0 and t-ε_0. The inequality (be careful to the presence or not of the hat η on η ),
            
                η^O_(t_0,t ; [0 → 1])  ≤ η^O_(t_0,t ; [0 → 1/2]) + η^O_(t_0,t ; [1/2 → 1])
            
            implies by rotational invariance
            
                ( η^O_(t_0,t ; [a → b]) ) ≤ 2 |a→ b| ( η^O_(t_0,t ; [0 → 1/2]) )  .
            
            We finally get (<ref>) in combining the fact that η^O_(t_0,t ; [0 → 1/2]) is stochastically dominated by η_W(t_0,t ; 0,1/2)) (already stated in (<ref>)) and
            
                (η_W(t_0,t ; 0,1/2)) = 1+(η_W(t_0,t ; 0,1/2)) = 1+1/2√(π t) .
            With η^O_(t_0+ε,t-ε ; [a → b]) η^O_(t_0,t-ε ; [a → b]), (<ref>) and
            
                lim_ε→ 0η^O_(t_0,t-ε ; [a → b]) = η^O_(t_0,t ; [a → b])  ,
            
            the Lebesgue's dominated convergence theorem applies and leads to the searched limit (<ref>). Mainly because there is no coalescence on the arc {t_0+t}× [a → b] for the trajectories starting before t_0 with probability 1, there exists a random ε>0 such that for any 0≤ε'≤ε, η^O_(t_0,t-ε' ; [a → b]) is equal to the limit η^O_(t_0,t ; [a → b]). This proves (<ref>). 0.3cm
            
            Now, let us show that (𝒯^(j))_j≥ 1 satisfies (<ref>). The strategy to get (<ref>) can be divided into two steps. First, we bound from above the expectation in (<ref>) by twice the expected number of remaining paths γ at time t_0+t which are born before t_0 and such that γ(t_0)∈ [0→ a], i.e. 2 (η^O_𝒯^(j)(t_0,t;[0 → a])). See Lemma <ref>. Thus, using the coalescence time estimate (Corollary <ref>), we obtain an upper bound for this latter expectation when j→ +∞. This is Lemma <ref>.
            The various lemma on which the proof of (<ref>) is based are proved at the end of the present section.
            
            
            For all times t_0,t>0, for all a>0 and any j≥ 1, the following inequality holds:
            
                ł(η^O_𝒯^(j)(t_0,t;[0 → a]))̊≤ 2  ł(η^O_𝒯^(j)(t_0,t;[0 → a]))̊ ,
            from which we deduce that:
            
                lim sup_j→ +∞ł(η^O_𝒯^(j)(t_0,t;[0 → a]))̊≤ 2 lim sup_j→ +∞ł(η_𝒲^(j)(t_0,t;0 , a))̊,
            where 𝒲^(j) is the shifted forest introduced in Section <ref>.
            
            
            In view of (<ref>) of Lemma <ref>, we now focus on showing that
            
                lim sup_j→ +∞ł(η_𝒲^(j)(t_0,t;0 , a))̊<+∞ .
            
            
            Let us choose r∈∖{0} (intended to be large) and m(a,r):=min{m: m≥ ar}. We consider the grid
            
                (t_0,a,r) := ł{k/r, k∈{0,… r_a}}̊×{h_R(h_j+t_0)-1-h_j} .
            
            The height h_R(h_j+t_0)-1-h_j corresponds to the (shifted) largest slice just before height h_j+t_0 (possibly h_j+t_0 itself): see Figure <ref>. Since the sequence h_k-h_k-1=σ^2_k tends to zero by (<ref>), there exists j_0 such that, for any j≥ j_0, there is a slice carrying points between t_0 and t_0+t in 𝒲^(j), i.e. t_0<h_R(h_j+t_0)-h_j<t_0+t.
            
            Let us focus on the paths starting from the points of (t_0,a,r). For 0≤ k≤ m(a,r), denote by γ_k(.) the ancestor line starting at (k/r, h_R(h_j+t_0)-1-h_j). Even if the points (k/r, h_R(h_j+t_0)-1) do not belong to the point process Υ defined in Section <ref>, they connect to the nearest point of Υ∩ L_R(h_j+t_0). So each path γ_k(.) a.s. coincides with a path of 𝒲^(j) after one step. Let us define the event
            
                A_a,j,r := {[ 𝒲^(j)∩([0, a]×{h_R(h_j+t_0)-1-h_j});                           (t_0,a,r) ]} .
            
            The event A_a,j,r is described in Figure <ref>.
            
            We claim that when the mesh 1/r of the grid (t_0,a,r) tends to 0, the probability of A_a,j,r tends to 1:
            
            
            For all times t_0,t>0, for all a>0 and any j≥ j_0,
            
                lim_r→ +∞(A_a,j,r^c) = 0  .
            
            
            The event A_a,j,r has been introduced in order to compare η_𝒲^(j)(t_0,t;0 , a) to the number of remaining paths at height t_0+t, starting from the deterministic points of (t_0,a,r). Then, the coalescence time estimate (Corollary <ref>) leads to the following bound:
            
            
            For all times t_0,t>0, for all a>0, there exists a constant C>0 and an integer j_1=j_1(t_0,t) (which does not depend on r) such that for any j≥ j_1 and any r∈∖{0},
            
                ( η_𝒲^(j)(t_0,t ; 0,a) _A_a,j,r) ≤ 1 + 2 m(a,r) C/r √(t) ,
            
            where C is the universal constant given by Corollary <ref>.
            We can now conclude. Let t_0,t>0 and a>0. First, we bound (η_𝒲^(j)(t_0,t;0, a)) from above by
            
                (η_𝒲^(j)(t_0,t;0,a) _A_a,j,r) + M ( A_a,j,r^c) + ( |Υ_R(h_j+t_0)-1([0,a])| _{|Υ_R(h_j+t_0)-1([0,a])|>M})  ,
            
            for any j,M,r. Let j≥ j_0∨ j_1. Then Lemma <ref> applies and provides a bound for the first term of (<ref>). Then, take M=M_j:=2 n_R(h_j+t_0)-1 (twice the intensity of the PPP). For j large enough and with this choice of M, the third term of (<ref>) is smaller than 1. It then follows:
            
                ł(η_𝒲^(j)(t_0,t;0,a) )̊≤ 2 + 2 m(a,r) C/r √(t) + M_j ( A_a,j,r^c)  .
            
            Let us point out that till now, the parameter r is totally free. So we can choose it large enough so that m(a,r)/r≤ a+1/r≤ 2a and M_j (A_a,j,r^c) ≤ 1 (by Lemma <ref>). In conclusion, for any j large enough, (η_𝒲^(j)(t_0,t;0,a)) is bounded by 3 + 4 a C/√(t). This gives (<ref>), ending the proof of Proposition <ref>.
            
            This section ends with the proofs of Lemmas <ref>, <ref> and <ref>.
            
            Let us first prove (<ref>). We denote, for an interval I of , by η^O_𝒯^(j)(t_0,t ; I ; [0 → a]) the number of paths γ∈𝒯^(j) born before t_0 and such that γ(t_0)∈ I and γ(t_0+t)∈ [0→ a]:
            
                η^O_𝒯^(j) (t_0,t ;I ; [0 → a]) := {γ(t_0+t)∈ [0→ a], γ∈𝒯^(j),  γt_0, γ(t_0)∈ I} .
            
            Then, the following inequality holds almost surely:
            
                η^O_𝒯^(j)(t_0,t ; [0→ a])≤∑_k=0^[1/a]η^O_𝒯^(j)(t_0,t ; [ka → (k+1)a] ;[0→ a]),
            
            where [1/a] is the integer part of 1/a. The inequality (<ref>) is due to the fact that two paths starting from different intervals [ka → (k+1)a] and [ℓ a → (ℓ+1)a] can coalesce and give a single point in the l.h.s. while they are counted twice in the r.h.s. Notice that when a is not the inverse of an integer, the right hand side is a bit larger than what is needed, because the first and last intervals intersect. We conclude using the rotational invariance:
            
                (η^O_𝒯^(j)(t_0,t ; [0→ a]))≤   ∑_k=0^[1/a](η^O_𝒯^(j)(t_0,t ; [ka → (k+1)a] ; [0→ a]))
                
                =    ∑_k=0^[1/a](η^O_𝒯^(j)(t_0,t ; [0→ a] ; [-ka → -(k-1)a]))
                
                =    (η^O_𝒯^(j)(t_0,t ; [0→ a])) +  ( η^O_𝒯^(j)(t_0,t ; [0→ a] ; [1-a[1/a]→ a])) 
                ≤     2(η^O_𝒯^(j)(t_0,t ; [0→ a]))  .
            
            since the arc [1-a[1/a]→ a] is included in [0→ a]. This proves (<ref>).
            
            
            We now turn to the proof of (<ref>). Having proved (<ref>), it is sufficient to show that
            
                lim sup_j→ +∞ł(η^O_𝒯^(j)(t_0,t;[0 → a]))̊≤lim sup_j→ +∞ł(η_𝒲^(j)(t_0,t;0 , a))̊ .
            
            For this, we construct a coupling between 𝒲^(j) and 𝒯^(j). Let us introduce the following event:
            
                E_j := {[ k≥ j, (h_k) ]} .
            
            Notice that (E^c_j)≤∑_k≥ je^-n_k which converges to zero when j→ +∞, and by Borel Cantelli's lemma and (<ref>), there exists a random level J, finite a.s., such that E_J holds. Following the idea in (<ref>), we have:
            
                (η^O_𝒯^(j)(t_0,t;[0 → a]))
                ≤(η^O_𝒯^(j)(t_0,t;[0 → a])_E_R(h_j+t_0)-1)
                 + M ( E_R(h_j+t_0)-1^c) + ( |Ξ_R(h_j+t_0)-1([0,1])| _{|Ξ_R(h_j+t_0)-1([0,1])|>M})  .
            Choosing M=M_j=2n_R(h_j+t_0)-1, we can control the third term as in (<ref>). For this choice of M=M_j, the second term is upper bounded by 2n_R(h_j+t_0)-1∑_k≥ R(h_j+t_0)-1e^-n_k which converges to zero when j→ +∞ by (<ref>).
            Now for the first term in the r.h.s. of (<ref>), let us prove that
            
                (η^O_𝒯^(j)(t_0,t;[0 → a])_E_R(h_j+t_0)-1)≤(η_𝒲^(j)(t_0,t;0 , a)),
            
            in which case, taking the lim sup in (<ref>) when j→ +∞ gives (<ref>).
            
            
            To show (<ref>), we produce a coupling between 𝒲^(j) and 𝒯^(j) ensuring that on E_R(h_j+t_0)-1,
            
                η^O_𝒯^(j)(t_0,t;[0 → a]) ≤η_𝒲^(j)(t_0,t;0 , a)).
            
            Consider the paths of 𝒲^(j) touching [0,a]×{t_0} and that survive until level t_0+t. Let us denote by K the random number of points in [0,a]×{t_0} corresponding to these paths (K≥η_𝒲^(j)(t_0,t;[0 → a])) and let us call 0≤ a_1<… <a_K≤ a the abscissa of these points. Recall that if Z'=α(Z)∈ L_k+1 is the ancestor of Z∈ L_k, then base the isosceles triangle with apex Z and admitting Z' as other vertex contains only one atom of Ξ_k+1: Z'. This triangle is called the influence triangle of Z. Let us denote by θ_L the left border of the influence region of 𝒲^(j)_(a_1,t_0) (i.e. the union of the influence triangles of the vertices constituting 𝒲^(j)_(a_1,t_0)). We consider the point process on the cylinder consisting of the atoms of Ξ in the region
            
                ℛ_j = {(x,t)∈× [h_j+t_0,+∞), θ_L(t)≤ x<θ_L(t)+1 } ,
            
            which is a PPP conditioned on E_R(h_j+t_0)-1. On this PPP, let us construct the corresponding forest 𝒯^(j) and compute the r.v. η^O_𝒯^(j)(t_0,t;[0 → a]).
            
            Let κ∈{1,… K} be the (random) index k of the rightmost path 𝒲^(j)_(a_k,t_0) that does not intersect the border θ_L(.)+1. By construction, all the paths 𝒲^(j)_(a_k,t_0) for k∈{1,…κ} are unchanged on the cylinder (meaning that the successive ancestors of (a_k,t_0) remain the same as in the plane). Hence, among the paths started at (a_1,t_0),…,(a_κ,t_0), the remaining ones at level t_0+t are the same in the plane and in the cylinder: η^O_𝒯^(j)(t_0,t;[0 → a_κ]) and η_𝒲^(j)(t_0,t ; 0,a_κ) are equal. If κ = K then (<ref>) is proved. Else, let us consider 𝒲^(j)_(a_κ+1,t_0) and denote by Z the first vertex of this path (in the plane) such that Z∈ℛ_j and Z'=α(Z)∉ℛ_j. In other words, 𝒲^(j)_(a_κ+1,t_0) intersects θ_L+1 for the first time when going from Z to Z'. In the cylinder, the paths 𝒯^(j)_(a_κ+2,t_0),…,𝒯^(j)_(a_K,t_0) (if they exist) are all trapped between 𝒯^(j)_(a_κ+1,t_0) and 𝒯^(j)_(a_1,t_0). Two cases may be distinguished:
            
              ∙ If 𝒯^(j)_(a_κ+1,t_0) coalesces with 𝒯^(j)_(a_1,t_0) before time t_0+t then the same holds for 𝒯^(j)_(a_κ+2,t_0),…,𝒯^(j)_(a_K,t_0). In this case, the contribution of 𝒯^(j)_(a_κ+1,t_0),…,𝒯^(j)_(a_K,t_0) to η^O_𝒯^(j)(t_0,t;[0 → a]) is null. So,
            
                η^O_𝒯^(j)(t_0,t;[0 → a]) = η^O_𝒯^(j)(t_0,t;[0 → a_κ])     =    η_𝒲^(j)(t_0,t ; 0,a_κ) 
                   ≤   η_𝒲^(j)(t_0,t ; 0,a)  .
            
              ∙ Since the influence triangle of the vertex Z overlaps the influence region of 𝒯^(j)_(a_1,t_0) then all the paths 𝒯^(j)_(a_κ+2,t_0),…,𝒯^(j)_(a_K,t_0) have to coalesce with 𝒯^(j)_(a_κ+1,t_0) or 𝒯^(j)_(a_1,t_0) before time t_0+t. Either 𝒯^(j)_(a_κ+1,t_0) coalesces with 𝒯^(j)_(a_κ,t_0) before time t_0+t and then η^O_𝒯^(j)(t_0,t;[0 → a]) is still smaller than η_𝒲^(j)(t_0,t ; 0,a) (as in the first case). Or, 𝒯^(j)_(a_κ+1,t_0) does not coalesce with 𝒯^(j)_(a_κ,t_0) before time t_0+t and η^O_𝒯^(j)(t_0,t;[0 → a]) = η^O_𝒯^(j)(t_0,t;[0 → a_κ])+1. This also prevents the planar paths 𝒲^(j)_(a_κ+2,t_0),…,𝒲^(j)_(a_K,t_0) to coalesce with 𝒲^(j)_(a_κ+1,t_0). Their contribution to η_𝒲^(j)(t_0,t ; 0,a) is at least 1:
            
                η^O_𝒯^(j)(t_0,t;[0 → a]) = η^O_𝒯^(j)(t_0,t;[0 → a_κ]) + 1 ≤η_𝒲^(j)(t_0,t ; 0,a)  .
            
            This shows (<ref>) and concludes the proof of the Lemma.
             Let j≥ j_0. On the event A_a,j,r^c, there exists a point of 𝒲^(j) at height h_R(h_j+t_0)-h_j, say Z, which is the ancestor of an element of 𝒲^(j)∩([0, a]×{h_R(h_j+t_0)-1-h_j}) but of none of the points of the grid Gr(t_0,a,r). This occurs only if Z belongs to the segment [0, a]×{h_R(h_j+t_0)-h_j} and is surrounded by two other points of 𝒲^(j) on the same level which are very close to it. Precisely, A_a,j,r^c implies the existence of a segment in [-1/r, a+1/r]×{h_R(h_j+t_0)-h_j} with length 2/r and containing at least 3 points of the Poisson point process Υ_R(h_j+t_0) of intensity n_R(h_j+t_0). The number of points of Υ_R(h_j+t_0) in [-1/r, a+1/r]×{h_R(h_j+t_0)-h_j} being a Poisson r.v. with parameter (a+2/r) n_R(h_j+t_0)-1, we can deduce that the minimum distance between two consecutive points of this PPP possesses a density. Thus, the probability of A_a,j,r^c tends to 0 as r→+∞.
             This last proof is based on the proof of Lemma 2.7 in <cit.>. Let us denote by η_r,j(t_0,t;0,a) the number of remaining paths γ_k(.), k=0,…,m(a,r), in 𝒲^(j) at time t_0+t, that started from the grid Gr(t_0,a,r). The event A_a,j,r has been introduced in order to write:
            
                η_𝒲^(j)(t_0,t ; 0,a) _A_a,j,r≤η_r,j(t_0,t;0,a)  .
            
            Besides, the number of paths counted by η_r,j(t_0,t;0,a) is upper bounded by the number m(a,r)+1 of paths γ_k(.)'s starting from the grid Gr(t_0,a,r) minus the number of pairs (k,k+1) that have coalesced before height t_0+t, i.e.
            
                η_r,j(t_0,t;0,a) ≤ (m(a,r)+1) - ∑_k=0^m(a,r)-1_γ_k(t_0+t)=γ_k+1(t_0+t) .
            
            Using (<ref>), we have for sufficiently large j:
            
                ( ∑_k=0^m(a,r)-1_γ_k(t)=γ_k+1(t))     =     m(a,r) ( 1 - (γ_0(t_0+t)≠γ_1(t_0+t)) ) 
                   ≥    m(a,r) - m(a,r) C/r √(h_R(h_j+t_0+t)-h_R(h_j+t_0)-1)
                   ≥    m(a,r) - 2 m(a,r) C/r √(t) ,
            
            since h_R(h_j+t_0+t)-h_R(h_j+t_0)-1 tends to t as j→∞. Finally, (<ref>), (<ref>) and (<ref>) lead to the expected result.
            
            Just above Corollary <ref>, we observed that certain navigations on the cylinder can be sent onto navigations in the radial plane, and that both navigations are very similar in nature. In the whole paper, we often used that some structures can be transported from the plane onto the cylinder, and to the radial plane, provided the transport keeps the crucial features of the models considered. 
            
            With the example adapted from the work of Coletti and Valencia <cit.>, in this Section <ref>, we illustrate how working on the cylinder allows us to state global convergence results for the radial tree correctly renormalized. However, in some cases such as the radial spanning tree (RST) of Baccelli and Bordenave <cit.>, the cylindrical forest can appear very complicated so that is can be easier to stick to the original radial problem, showing the limitations of this method.
            
            In the RST, a homogeneous PPP is given in the plane. A radial tree with vertex set the points of the PPP and rooted at the origin O is constructed. In this tree, the ancestor of a vertex x is the closest point of the PPP with smaller radius. When we send this tree and PPP in the cylinder, the circle of radius ρ is sent on the slice of height h(ρ). The resulting vertex set on the cylinder is not a homogeneous PPP. Additionally, the neighborhood of a given point becomes complicated in the cylinder, as shown in Fig. <ref>.
            
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