V Konakov

S Menozzi

LIMIT THEOREMS FOR RANDOM WALKS IN THE HYPERBOLIC SPACE

Keywords: Hyperbolic space, Random Walks and Brownian motion, Local Limit Theorems. MSC2020: 60F05, 43A85

We prove central and local limit theorems for random walks on the Poincaré hyperbolic space of dimension n ě 2. To this end we use the ball model and describe the walk therein through the Möbius addition and multiplication. This also allows to derive a corresponding law of large numbers.

Introduction

Let H n (n ě 2q denote the real hyperbolic space of dimension n ě 2. This is the complete and simply connected Riemannian manifold with constant negative sectional curvature equal to -1.

We will be interested in establishing limit theorems (central, local, law of large numbers) for a corresponding underlying random walk. In the case n " 2, 3, the central limit theorem was first obtained for the models of the Poincaré-Lobachevski ball and plane in the seminal paper by Karpelevich et al. [START_REF] Karpelevich | Limit theorems for the compositions of distributions in the Lobachevsky plane and space[END_REF]. It was shown therein that the suitably renormalized sum converges in law to the normal law, corresponding to the fundamental solution of the heat equation involving the Laplace-Beltrami operator, at a certain time that could be seen as a variance in this setting. We can as well refer to the monograph of Terras [START_REF] Terras | Harmonic Analysis on Symmetric Spaces[END_REF], Section 3.2 for additional details.

In [START_REF] Karpelevich | Limit theorems for the compositions of distributions in the Lobachevsky plane and space[END_REF] the Möbius addition already appeared in order to provide a non-Euclidean analogue of the sum of random variables. In a more general setting, the connection with Harmonic analysis on the underlying appropriate structure, like e.g. Lie groups, provided a natural way to define a corresponding random walk and to investigate the associated limit theorems. Indeed, the harmonic analysis somehow provides natural extensions of the characteristic functions, variances. We again refer to [START_REF] Terras | Harmonic Analysis on Symmetric Spaces[END_REF] and [START_REF] Terras | Harmonic Analysis on Symmetric Spaces[END_REF] for results in that direction.

The above results concentrate on the behavior of a renormalized sum. Concerning a more functional approach we can mention the works by Pinsky (see [START_REF] Pinsky | Lectures on Random Evolution[END_REF] and [START_REF] Pinsky | Isotropic Transport Process on a Riemannian Manifold[END_REF]) which established how the so-called isotropic transport process 1 actually converges in law, when suitably normalized, to the Brownian motion on the manifold when the Ricci curvature is bounded from below. The case of a Poincaré plane when the geodesic ball is also discretized, thus leading to a space-time discrete approximation, is discussed in Gruet [START_REF] Gruet | Hyperbolic random walks[END_REF].

We will here focus on the central limit theorem, the local limit (we will as well establish as a by-product of our analysis a law of large numbers) for any dimension n of the hyperbolic space. Whereas local limit theorems were previously established for some nilpotent groups, see e.g. Breuillard [Bre05] for the Heisenberg group, this seems to be, to the best of our knowledge, new in the current setting. To this end we will rely on the Poincaré ball model and use related harmonic analysis tools.

The paper is organized as follows. We describe the random walk and the related convergence results in Section 2. We provide in Section 3 the main tools from harmonic analysis needed to perform the analysis. Eventually, Section 4 is dedicated to the proof of the main results.

Main Results

2.1. Setting. Let H n (n ě 2q denote the real hyperbolic space of dimension n ě 2. This is the complete and simply connected Riemannian manifold with constant negative sectional curvature equal to -1. We will consider throughout the document the Poincaré ball model for the hyperbolic space H n (n ě 2q. Namely, we introduce: B n :" tx P R n : }x} ă 1u, where } ¨} stands for the Euclidean norm, which we endow with the metric

ds 2 " 4 `dx 2 1 `¨¨¨dx 2 n p1 ´}x} 2 q 2
and the corresponding Riemannian volume

(2.1) µ B n pdxq " 2 n p1 ´}x} 2 q ´ndx.

We recall that the hyperbolic distance to the origin writes for z P B n :

η " 2 atanhp}z}q " log ´1 `}z}

1 ´}z} ¯.,
where we use the notation η for coherence with [START_REF] Karpelevich | Limit theorems for the compositions of distributions in the Lobachevsky plane and space[END_REF].

Hence, for z P B n , z " rΘ " tanhp η 2 qΘ, Θ P S d´1 .

The coordinates pr, Θq P r0, 1q ˆSd´1 are the usual polar coordinates and pη, Θq P r0, `8q ˆSd´1 the geodesic ones. In the geodesic coordinates, we have:

(2.2) µ B n pdxq " sinhpηq n´1 dηΛpdΘq,

where Λ stands for the Lebesgue measure of the sphere S n´1 .

To define a random walk on B n we will introduce the Möbius addition px, yq P B n ˆBn Þ Ñ B n which writes in Euclidean coordinates:

(2.3)

x ' y :" p1 `2xx, yy `}y} 2 qx `p1 ´}x} 2 qy 1 `2xx, yy `}x} 2 }y} 2 , where x¨, ¨y stand for the usual Euclidean scalar product on R n . We will also use the left inverse, i.e. for all x P B n , (2.3) directly gives that ax :" ´x satisfies ax ' x " 0. We refer to [START_REF] Ahlfors | Möbius transformations in several dimensions[END_REF] for a thorough presentation of related operations and properties (see also Section 3 below). Let us also mention that the operation (2.3) can also be written through the formalism of Clifford algebras, which then allows to extend the expression of the Möbius transform/addition in the complex plane to the current setting (see e.g. [START_REF] Ferreira | Harmonic analysis on the Möbius gyrogroup[END_REF]).

We also introduce the Möbius multiplication, i.e. for all γ P R, z P B n , (2.4) γ b z :" tanhpγ atanhp}z}qq z }z} , and refer to [START_REF] Barabanov | Binary operations in the unit ball: A differential geometry approach[END_REF] for a thorough discussion about this operation and other operations on the unit ball connected with differential geometry.

2.2. The random walk and associated convergence results. Let Z be a B n -valued random variable defined on some probability space pΩ, A, Pq. We will further assume that:

[R] Z has radial density f Z P C 8 0 pB n q w.r.t. the Riemannian volume. Let now pZ j q jě1 be a sequence of i.i.d random variables which have the same law as Z. Define then for N P N,

SN :" ' N j"1 1 N b Z j , S N :" ' N j"1 1 ? N b Z j . (2.5)
Then the following results hold:

Theorem 2.1 (Law of large numbers). Under [R], SN P ÝÑ N 0.
It will be shown in Section 4 that S N has a density f S N , which can be expressed as the non Euclidean convolution of the densities of the pZ j q jPrr1,N ss . We quantify below the asymptotic behavior of that density.2 Theorem 2.2 (Central limit theorem). Under [R], it holds that for measurable sets A in B n , ż

B n I A pxqf S N pxqµ B n pdxq ÝÑ N ż B n I A pxqΨpt, xqµ B n pdxq, with t :" 1 n ş `8 0 η2 µ Z,R
pdηq, , where µ Z,R stands for the measure induced by the law of Z in geodesic coordinates. Also, Ψpt, ¨q stands for the hyperbolic heat kernel in the model B n for H n . Namely it denotes the fundamental solution of the equation

1 2 ∆ B n Ψpt, xq " B t Ψpt, xq, Ψp0, ¨q " δp¨q.
The specific expression of Ψpt, ¨q will be given in Section 3.2 below. It plays the same role in the current setting as the normal density in the classical Euclidean central limit theorem.

We are furthermore able to specify the previous result giving a convergence rate.

Theorem 2.3 (Local limit theorem). Under [R], there exists C :" Cpn, µ Z q s.t. for all x P B n and N large enough

(2.6) |f S N pxq ´Ψpt, xq| ď C N .
From now on we will denote by C a generic constant, that may change from line to line and depends on n and the law µ Z , i.e. C " Cpn, µ Z q.

Remark 2.1 (About the radial densities). The radial assumption in [R] is somehow standard in this setting, see [START_REF] Karpelevich | Limit theorems for the compositions of distributions in the Lobachevsky plane and space[END_REF] or Section 3.2.7 in [START_REF] Terras | Harmonic Analysis on Symmetric Spaces[END_REF]. It allows to focus on the asymptotic behavior of the radius to the origin for S N . This isotropy makes the analysis essentially scalar. In particular, the rotation invariance of the underlying law yields that the law of the sum does not depend on the order of the summands. It is indeed shown in [START_REF] Ferreira | Möbius gyrogroups: a Clifford algebra approach[END_REF] that the gyrogroup pB n , 'q is gyro-commutative, i.e. defining for a, b, c P B n the gyration as gyrra, bsc " apa ' bq ' pa ' pb ' cqq then a ' b " gyrra, bspb ' aq. Since the gyration gyr induces a rotation, for i.i.d. random variables pZ j q jě1 with rotationally invariant laws , SN plawq " ' N j"1

1 N b Z ppjq (resp. S N plawq " ' N j"1 1 ? N b Z ppjq )
, where p is any permutation of p1, ¨¨¨, N q. Remark 2.2 (About the local theorem). The convergence rate in Theorem 2.3 is due to the fact that, since we are dealing with radial densities the first moment is already zero. On the other hand one could expect that on the r.h.s. of (2.6) there should as well be an exponential decay term of the form expp´cd 2 pxq{tq, d 2 pxq " 2atanhp}x}q standing for the hyperbolic distance of x to the origin. This would indeed correspond to the Aronson type bound for the hyperbolic heat kernel, see e.g. Molchanov [START_REF] Molchanov | Diffusion processes and Riemannian geometry[END_REF] and would extend the full local limit theorem of Batthacharya and Rao [START_REF] Bhattacharya | Normal approximations and asymptotic expansions[END_REF] to the current setting. One can expect this result to hold but it would require a much more involved analysis than the one in the current work.

The Poincaré ball: Fourier inversion and heat kernel

To prove our results, as for any (local) limit theorem see e.g. [START_REF] Bhattacharya | Normal approximations and asymptotic expansions[END_REF] in the Euclidean case, we aim at comparing the characteristic functions of the underlying normal law and the one of the corresponding approximating walk.

In the current non-Euclidean setting, the characteristic function will be expressed in terms of the Fourier-Helgason transform, see [START_REF] Helgason | Groups and geometric analysis[END_REF].

3.1. The Fourier-Helgason transform and its inverse. In the following for a generic radial function f : B n Ñ R we will write with a slight abuse of notation f pzq " f prq, r " }z} P r0, 1q. For a radial function f P C 8 0 pB n , Rq, and λ P R, its Fourier-Helgason transform is given by the expression

(3.1) f pλq " ż B n f pxqϕ λ pxqµ B n pdxq " Ω n´1 ż `8 0 psinh ηq n´1 f ptanhp η 2 qqϕ λ ptanhp η 2 qqdη,
where Ω n´1 " 2π n 2

Γp n 2 q denotes the area of the unit sphere S n´1 and the radial functions ϕ λ are eigen-functions of the Laplace-Betrami operator in B n expressed in radial coordinates. Namely, for a generic smooth enough φ : B n Ñ R,

∆ B n φpzq " ´1 ´}z} 2 2 ¯2 n ÿ j"1 B 2 zj φpzq `pn ´2q 1 ´}z} 2 2 n ÿ j"1 z j B zj φpzq.
When φ is radial, this operator rewrites as ∆ B n φprq " B 2 r φprq `pn ´1q cothprqB r φprq, and the function ϕ λ in (3.1) solves the differential equation:

(3.2) # ∆ B n ϕ λ prq `pλ 2 `ρ2 qϕ λ prq " 0, ρ " n´1 2 , ϕ λ p0q " 1.
This equation is solved using the functions e λ,ω pxq " p1 ´}x} 2 q 1 2 pn´1`iλq }x ´ω} n´1`iλ , (3.3) which are eigenfunctions of ∆ B n associated with the eigenvalue ´pλ 2 `ρ2 q and therefore. Those functions actually play in the current context a similar role to the complex exponential exppix ¨yq in the usual Fourier analysis on the Euclidean space R n .

We then define the corresponding elementary spherical function ϕ λ setting for all x P B n ,

ϕ λ pxq " 1 Ω n´1 ż S n´1
e λ,ω pxqΛpdωq, (3.4) where Λ stands for the Lebesgue measure on the unit sphere S n´1 . It is then clear that ϕ λ is also an eigenfunction of ∆ B n with eigenvalue ´pλ 2 `ρ2 q and that ϕ λ p0q " 1. Hence, the above ϕ λ indeed solves (3.2).

Note that its expression also writes in geodesic polar coordinates (see e.g. [START_REF] Anker | An introduction to Dunkl theory and its analytic aspects[END_REF]):

ϕ λ ptanhp η 2 qq " Cn sinhpηq 2´n ż η 0 dspcoshpηq ´coshpsqq n´3 2 cospλsq, Cn " 2 n´1 2 Γ `n 2 ?πΓ `n´1 2 ˘. (3.5)
For computational purposes we also make the following link with the Legendre special function, see Gradstein and Ryzhik [START_REF] Gradshteyn | Table of integrals, series and products[END_REF], p. 1016, formula 8.715(1),

P 1´n 2 ´1 2 `iλ pcoshpηqq " c 2 π sinhpηq 1´n 2 Γpρq ż η 0 dspcoshpηq ´coshpsqq n´3 2 cospλsq, ϕ λ ptanhp η 2 qq " 2 ρ´1 2 Γpρ `1 2 q sinhpηq 1 2 ´ρP 1 2 ´ρ ´1 2 `iλ pcoshpηqq. (3.6)
The inversion formula associated with (3.1), for a radial function f P C 8 0 pB n , Rq, reads for all }z} " r P r0, 1q,

(3.7) f pzq " f prq " C n ż `8 0 dλ|cpλq| ´2 f pλqϕ λ prq,
where cpλq is the generalized Harish-Chandra function: andC n " 1 2 n´3 πΩn´1 . Formulas (3.1) and (3.7) can be derived from Liu and Peng [START_REF] Liu | Generalized Helgason-Fourier transforms associated to variants of the Laplace-Beltrami operators on the unit ball in R n[END_REF] (taking therein ϑ " 2 ´n) or Ferreira [START_REF] Ferreira | Harmonic analysis on the Möbius gyrogroup[END_REF] (taking therein σ " 2 ´n, t " 1).

(3.8) cpλq " 2 3´n´iλ Γp n 2 qΓpiλq Γ `n´1`iλ 2 ˘Γ `1`iλ 2 ˘,
3.2. The heat kernel on B n . The heat kernel will provide the limit law which is somehow the analogue in the current non-Euclidean setting of the normal law. The normal density of parameter t ą 0 in B n is defined as the solution to 1 2 ∆ B n Ψpt, xq " B t Ψpt, xq, Ψp0, ¨q " δp¨q.

In the literature the usual heat equation considered is ∆ B n ψpt, xq " B t ψpt, xq, ψp0, ¨q " δp¨q.

It can actually be solved through the Fourier-Helgason transform (assuming the solution is radial). Namely, from (3.1), one derives that for all λ P R:

(3.9) ψpt, λq " expp´pλ 2 `ρ2 qtq, ρ " n ´1 2 .

Observe that the expression in (3.9) in particular coincides when n " 2, 3 with the one obtained in [START_REF] Karpelevich | Limit theorems for the compositions of distributions in the Lobachevsky plane and space[END_REF].

With the terminology of this work the expression in (3.9) corresponds to the so-called characteristic functions of the first kind. In order to match the Euclidean probabilistic set-up we will consider the characteristic functions of the second kind, (3.10) ψ2 pt, λq :" ψpt, λq ψpt, 0q " expp´λ 2 tq, so that in particular ψ2 pt, 0q " 1. By inversion it follows that for m P N:

(3.11) ψpt, xq " ψpt, atanhp η 2 qq " $ ' & ' % expp´m 2 t 2 q p2πq m ? 2πt ´´1 sinh η B η ¯m exp ´´η 2 2t ¯, n " 2m `1, expp´p m´1 2 q 2 t 2 q p2πq m ? πt ş `8 η ds ? coshpsq´coshpηq p´B s q ´´1 sinh s B s ¯m´1 exp ´´s 2 2t ¯, n " 2m.
Namely, the heat kernel on H n is expressed in term of the hyperbolic distance to the origin. We can refer to [START_REF] Anker | An introduction to Dunkl theory and its analytic aspects[END_REF] for the derivation of (3.11) through the Abel transform and its inverse or to [START_REF] Grigor'yan | The heat kernel on hyperbolic space[END_REF] in which the authors derive the heat kernel through the fundamental solution of the wave equation.

From the previous definitions it is clear that for Ψ as in Theorem 2.2, Ψpt, xq " ψp t 2 , xq.

3.3. Some additional tools from harmonic analysis on B n . To establish our main theorems we will need some additional tools. Namely, we need to define the convolution on B n . To this end, following Ahlfors [START_REF] Ahlfors | Möbius transformations in several dimensions[END_REF] we define for fixed a P B n the translation operator,

T a : x P B n Þ Ñ T a pxq " ´a ' x P B n .
The previous mapping is bijective, it is easily checked that T ´1 a " T ´a, and has the next important properties.

Proposition 3.1 (Some properties of T a , cf. [Ahl81], pp. 18-30). For a fixed a and x P B n it holds that:

1.

1 ´}T a pxq} 2 " p1 ´}a} 2 qp1 ´}x} 2 q 1 ´2xx, ay `}x} 2 }a} 2 . 2. detpD x T a pxqq " ˜1 ´}a} 2 1 ´2xx, ay `}x} 2 }a} 2 ¸n, ~Dx T a pxq~" 1 ´}a} 2 1 ´2xx, ay `}x} 2 }a} 2 ,
where ~¨~denotes the spectral norm. 3. T a preserves the Riemann measure. Namely,

~Dx T a pxq1 ´}T a pxq} 2 " 1 1 ´}x} 2 .

T a pxq " ´DxTapxq

~DxTapxq~T x paq. In particular, for a radial function f :

B n Ñ R, it holds that f pT a pxqq " f pT x paqq.
The translation operator allows to define in a quite natural way the convolution. Namely, for f, g P C 8 0 pB n , Rq, we set:

(3.12) @x P B n , f ˚gpxq "

ż B n f p´y ' xqgpyqµ B n pdyq " ż B n f pT y pxqqgpyqµ B n pdyq.
This definition enlarges the one in [START_REF] Karpelevich | Limit theorems for the compositions of distributions in the Lobachevsky plane and space[END_REF] to the current multi-dimensional setting. We now state some very useful properties of the convolution in order to prove limit theorems Proposition 3.2 (Some properties of the convolution). Let f, g P C 8 0 pB n , Rq and be radial functions. The following properties hold: -Commutativity: for the definition in (3.12) it holds that f ˚gpxq " g ˚f pxq.

-Stability of the radial property through convolution : f ˚g is a radial function.

-Fourier-Helgason transform of the convolution: it holds with the definition in (3.1) that for all λ P R,

z f ˚gpλq " f pλqĝpλq.
Proof. Since f is radial, we have from point 4 of Proposition 3.1 that f pT y pxqq " f pT x pyqq. Setting z " T x pyq " ´x ' y ðñ x ' z " T ´xpzq " y and using point 3 of Proposition 3.1, we derive:

f ˚gpxq " ż B n f pT y pxqqgpyq dy p1 ´}y} 2 q n " ż B n f pzqgpT ´xpzqq dz p1 ´}z} 2 q n " x'z"´rp´xq'p´zqs ż B n f pzqgpT x p´zqq dz p1 ´}z} 2 q n " z"´z ż B n f pzqgpT x pzqq dz p1 ´}z} 2 q n " ż B n f pzqgpT z pxqq dz p1 ´}z} 2 q n " g ˚f pxq,
where we also used that f, g are radial (and therefore even), for the equality in the second and third lines. This proves the first point.

Let us now turn to the second point. It suffices to prove that for any orthogonal matrix A P Opnq and x P B n , it actually holds that f ˚gpAxq " f pxq.

Observe first from (2.3) that for all a, b P B n ApA T a ' bq " a ' Ab.

(3.13) Indeed,

ApA T a ' bq "A p1 `2xA T a, by `}b} 2 qA T a `p1 ´}A T a} 2 qb 1 `2xA T a, by `}A T a} 2 }b} 2 " p1 `2xa, Aby `}Ab} 2 qa `p1 ´}a} 2 qAb 1 `2xa, Aby `}a} 2 }Ab} 2 "a ' Ab. Write now, f ˚gpAxq " ż B n f pyqgp´y ' Axqµ B n pdyq " (3.13) ż B n f pyqgpApA T p´yq ' xqqµ B n pdyq " ż B n f pyqgp´pA T yq ' xqqµ B n pdyq " ż B n f pyqgpT A T y pxqqµ B n pdyq " ż B n f pyqgpT x pA T yqqµ B n pdyq " ż B n f pAyqgpT x pyqqµ B n pdyq " ż B n f pyqgpT y pxqqµ B n pdyq " f ˚gpxq,
where we used that f, g are radial and property 3 of Proposition 3.1. This proves the second point.

Let us eventually prove the property concerning the Fourier-Helgason transform of the convolution. From (3.1) write for all λ P R:

z f ˚gpλq " ż B n ´żB n f pT y pxqqgpyqµ B n pdyq ¯ϕ´λ pxqµ B n pdxq " ż B n gpyq ˜żB n f pT y pxqqϕ ´λpxqµ B n pdxq ¸µB n pdyq " ż B n gpyq ˜żB n f pzqϕ ´λpT ´y pzqqµ B n pdzq ¸µB n pdyq.
Going back to the definition (3.4) and (3.3) we derive:

ϕ ´λpT ´y pzqq " 1 Ω n´1 ż S n´1
e ´λ,ω pT ´y pzqqΛpdωq.

Now, it holds from Lemma 4.3 in [LP09] that

e ´λ,ω pT ´y pzqq " e ´λ,ω p´yqe ´λ,T´ypωq pzq.

We point out that for ω P S n´1 it indeed holds from Property 1. of Proposition 3.1 that T ´y pωq P S n´1 . Thus,

z f ˚gpλq " ż B n gpyq ˜żB n f pzq 1 Ω n´1 ż S n´1 e ´λ,ω p´yqe ´λ,T´ypωq pzqΛpdωqµ B n pdzq ¸µB n pdyq " 1 Ω n´1 ż S n´1 ż B n gpyqe ´λ,ω p´yq ˆżB n f pzqe ´λ,T´ypωq pzqµ B n pdzq ˙µB n pdyqΛpdωq. (3.14)
We now recall that for possibly non-radial functions, h P C 8 0 pB n q the general Fourier-Helgason transform has two arguments (the polar coordinates of the phase variable) and writes for λ P R, ζ P S n´1 : ĥpλ, ζq "

ż B n hpxqe ´λ,ζ pxqµ B n pdxq.
When h is radial we derive passing to polar coordinates that ĥpλ, ζq " ĥpλq so that i.e. the Fourier transform does not depend on the considered point on the sphere S n´1 . Taking h " f , we therefore deduce from (3.14) that

Ω n´1 ĥpλq " ż B n hpxq ż S n´1
z f ˚gpλq " 1 Ω n´1 ż S n´1 ż B n
gpyqe ´λ,ω p´yq f pλqµ B n pdyqΛpdωq " ĝpλq f pλq, which gives the third point and concludes the proof of the proposition.

3.4. Non Euclidean mean, variance ans scaling. We define, coherently with the Euclidean case, the mean and variance associated with a B n -valued random variable Z defined on some probability space pΩ, F, Pq satisfying assumption [R].

The analogue of the characteristic function (of the second kind with the terminology of [START_REF] Karpelevich | Limit theorems for the compositions of distributions in the Lobachevsky plane and space[END_REF]) writes:

(3.15) @λ P R, Φ Z pλq " fZ pλq fZ p0q .

Again the normalization guarantees that Φ Z p0q " 1 as in the Euclidean case. We also emphasize that, since we assumed the density to be radial, it follows from (3.1) and (3.5) that for all m " 2j `1, j P N,

(3.16) B m λ Φ Z pλq| λ"0 " 0.
In other terms, the odd moments of the random variable are 0.

From the above definition we then define the analogue of the variance as

(3.17) V Z " ´B2 λ Φ Z pλq| λ"0 .
Proposition 3.3 (Variance of the sum). Let Z 1 and Z 2 be two B n -valued independent random variables with radial densities f Z1 , f Z2 P C 8 0 pB n , Rq w.r.t. the Riemannian volume (2.1) of B n . It then holds that

V Z1'Z2 " V Z1 `VZ2 .
Proof. It is clear from (3.12) and Proposition 3.2 that Z 1 ' Z 2 has a radial density f Z1'Z2 w.r.t. µ B n and that

f Z1'Z2 " f Z1 ˚fZ2 , fZ1'Z2 " fZ1 fZ2 .
Hence, from (3.16)-(3.17),

V Z1'Z2 " ´B2 λ ´f Z1 pλq ¯|λ"0 fZ2 p0q ´2´B λ fZ1 pλqB λ fZ2 pλq ¯|λ"0 ´f Z1 p0qB 2 λ ´f Z2 pλq ¯|λ"0 fZ1 p0q fZ2 p0q " ´B2 λ ´f Z1 pλq ¯λ"0 fZ1 p0q ´B2 λ ´f Z2 pλq ¯λ"0 fZ2 p0q " V Z1 `VZ2 .
In order to make a connection with the usual normalizing rate in the CLT, in [START_REF] Karpelevich | Limit theorems for the compositions of distributions in the Lobachevsky plane and space[END_REF] the authors introduce the following invariance property on the radial measures. Let Z be a B n -valued random variable with density f Z P C 8 0 pB n q and denote again by µ Z,R the induced measure in geodesic coordinates. For a parameter ε ą 0, the authors consider scaled variables Z ε (associated with Z) for which they assume that the corresponding measure µ Zε,R in radial geodesic coordinates satisfies for all τ ą 0:

µ Zε,R ´!η P R `: η ď τ ε )¯" µ Z,R ´!η P R `: η ď τ )¯. (3.18)
Proposition 3.4 (Scaled Variables). In order to fulfill property (3.18), the density f Zε must be chosen such that, in geodesic polar coordinates:

(3.19) @η P p0, `8q, f Zε ptanhp η 2 qq " 1 ε f Z ptanhp η 2ε qq ´sinhp η ε q sinhpηq ¯n´1 .
Observe in particular that if supppf Z q P Bp0, Rq then supppf Zε q P Bp0, Rεq. Furthermore, for a bounded measurable function g (not necessarily radial):

ErgpZ ε qs " E " g ´tanhpεatanhp}Z}qq Z }Z} ¯ı. (3.20)
Importantly, from the definition of the Möbius multiplication in (2.4), it holds that the property (3.18) holds if and only if:

Z ε plawq " ε b Z.
Proof. Since we assumed Z to have a smooth compacted support density in B n , the above identity (3.19) gives that this is also the case for Z ε . Denoting the corresponding density w.r.t. µ B n by f Zε , we write:

µ Zε,R ´!η P R `: η ď τ ε )¯" ż τ ε 0 µ Zε,R pdηq " Ω n´1 ż τ ε 0 f Zε ptanhp η 2 qqpsinh ηq n´1 dη " η" η ε Ω n´1 ż τ 0 f Zε ptanhp εη 2 qqpsinhpεηqq n´1 εdη " Ω n´1 ż τ 0 " εf Zε ptanhp εη 2 qq ´sinhpεηq sinhpηq ¯n´1 ı sinhpηq n´1 dη.
The scaling identity (3.18) then holds if and only if

" εf Zε ptanhp εη 2 qq ´sinhpεηq sinhpηq ¯n´1 ı " f Z ptanhp η 2 qq ðñ f Zε ptanhp η 2 qq " 1 ε f Z ptanhp η 2ε qq ´sinhp η ε q sinhpηq ¯n´1 .
Importantly, since supppf Z q Ă Bp0, Rq for some R P p0, 1q it then holds that supppf Zε q P p0, tanhpεatanhpRqqq.

Let us specify what is the transform of the initial random variable Z which has density f Zε . For a bounded measurable function g and from (2.1)-(2.2) write:

ErgpZ ε qs " ż B n gpzqf Zε pzqµ B n pdzq " ż S d´1 ΛpdΘq ż `8 0 gptanhp η 2 qΘqf Zε ptanhp η 2 qq sinhpηq n´1 dη " ż S d´1 ΛpdΘq ż `8 0 gptanhp η 2 qΘq 1 ε f Z ptanhp η 2ε qq ´sinhp η ε q sinhpηq ¯n´1 sinhpηq n´1 dη " η" η ε ż S d´1 ΛpdΘq ż `8 0 gptanhp εη 2 qΘqf Z ptanhp η 2 qq sinhpηq n´1 dη " ż S d´1 ΛpdΘq ż `8 0 g ´tanhp ε2atanhptanhp η 2 qq 2 qΘ ¯fZ ptanhp η 2 qq sinhpηq n´1 dη " ż B n g ´tanhpεatanhp}z}qq z }z} ¯fZ pzqµ B n pdzq " E " g ´tanhpεatanhp}Z}qq Z }Z} ¯ı.
Hence, to verify the condition (3.18) we have to take Z ε plawq " tanhpεatanhp}Z}q Z }Z} q" ε b Z ‰ εZ, that would somehow be the natural scaling in the Euclidean setting. (3.21)

V Zε ď Cε 2 .
In particular, choosing ε " 1 ? N , the above control can be specified to derive that with the notation of (2.5):

(3.22) V S N " V ' N j"1 1 ? N bZ j ÝÑ N t :" 1 n ż `8 0 η2 µ Z,R pdηq,
which is precisely the asymptotic variance appearing in Theorems 2.2 and 2.3. Furthermore, there exists C ě 1 s.t.

(3.23)

|V S N ´t| ď C N , and
(3.24) V SN " V ' N j"1 1 N bZ j ÝÑ N 0,
which is also coherent with the statement of Theorem 2.1.

Proof. For Z ε , we again derive from (3.17), (3.5) that

V Zε " ´B2 λ fZε pλq fZε p0q ˇˇλ "0 " ´B2 λ Ω n´1 fZε p0q ż `8 0 f Zε ptanhp η 2 qqpϕ λ ptanhp η 2 qqq sinhpηq n´1 dη ˇˇλ "0 " ´B2 λ Ω n´1 fZε p0q ż `8 0 f Z ptanhp η 2 qqϕ λ ptanhp εη 2 qq sinhpηq n´1 dη ˇˇλ "0 " 1 fZε p0q ż `8 0 ´C n sinhpηq 2´n ż η 0 dspcoshpηq ´coshpsqq n´3 2 s 2 cospλsq ¯ˇˇη "εη µ Z,R pdηq ˇˇλ "0 (3.25) " 1 fZε p0q ż `8 0 ´C n sinhpηq 2´n ż η 0 dspcoshpηq ´coshpsqq n´3 2 s 2 ¯ˇˇη "εη µ Z,R pdηq ď 1 fZε p0q ż `8 0 ´C n η2 sinhpηq 2´n ż η 0 dspcoshpηq ´coshpsqq n´3 2 ¯ˇˇη "εη µ Z,R pdηq ď ε 2 fZε p0q ż `8 0 η2 ϕ 0 ptanhpε η 2 qqµ Z,R pdηq ď C ε 2 f Zε p0q ż 8 0 φ0 ptanhpε η 2 qqµ Z,R pdηq ď Cε 2 ,
using that µ Z,R has compact support in R `for the last but one inequality and recalling that ş 8 0 φ0 ptanhpε η 2 qqµ Z,R pdηq " fZε p0q for the last one.

In particular, considering now S N " ' n j"1 1 ?

N b Z i where the pZ i q iě1 are independent with the same law as Z, there exists C ě 1 s.t. for all N ě 1,

V S N " V ' N j"1 1 ? N bZ i " N V 1 ? N bZ ď C.
Let us now establish (3.22). To this end we must precisely expand the inner integral in (3.25). To this end we introduce the quantity:

I ε pηq :" sinhpεηq 2´n ż εη 0 dspcoshpεηq ´coshpsqq n´3 2 s 2 "pεηq 3 sinhpεηq 2´n ż 1 0 ds `2 sinhp εη 2 p1 `sqq sinhp εη 2 p1 ´sqq ˘n´3 2 s2
"pεηq 3 pεη `pεηq 3 `opε

ˆż 1 0 dss 2 `p1 `sqp1 ´sq ˘n´3 2 `p1 `p εη 2 q 2 p1 `sq 2 `opε 2 qqp1 `p εη 2 p1 ´sqq 2 `opε 2 qq ˘n´3 2 .
It therefore follows that there exists a constant c :" cpn, f Z q, s.t. for all η P p0, 2atanhpRqq

|I ε pηqpεηq ´2 ´cn | ď cε 2 , c :" cpn, f Z q.
Observe now that, similarly to the previous computations:

fZε p0q " Cn ż `8 0 J ε pηqµ Z,R pdηq, (3.26) J ε pηq " sinhpεηq 2´n ż εη 0 dspcoshpεηq ´coshpsqq n´3 2 "p1 `pεηq 2 `opε 2 qq 2´n 2 3´n 2 ˆż 1 0 ds `p1 `sqp1 ´sq ˘n´3 2 `p1 `p εη 2 q 2 p1 `sq 2 `opε 2 qqp1 `p εη 2 p1 ´sqq 2 `opε 2 qq ˘n´3 2 .
We now rewrite from (3.25) and (3.26),

V Zε " ş `8 0 Cn I ε pηqµ Z,R pdηq ş `8 0 Cn J ε pηqµ Z,R pdηq " ş 1 0 dss 2 `p1 `sqp1 ´sq ˘n´3 2 ş 1 0 ds `p1 `sqp1 ´sq ˘n´3 2 ε 2 ş `8 0 η2 p1 `Oppεηq 2 qqµ Z,R pdηq ş `8 0 p1 `Oppεηq 2 qqµ Z,R pdηq " ş 1 0 dss 2 `1 ´s 2 ˘n´3 2 ş 1 0 ds `1 ´s 2 ˘n´3 2 ε 2 ´ż `8 0 η2 µ Z,R pdηq `Opε 2 q ¯.
Setting for m P N, Ipmq :" ş π 0 sinpθq m dθ and changing variables setting s " cospθq, it is easily seen that:

ş 1 0 dss 2 `1 ´s 2 ˘n´3 2 ş 1 0 ds `1 ´s 2 ˘n´3 2 " ş π 0 cos 2 pθq sinpθq n´2 dθ ş π 0 sinpθq n´2 dθ " Ipn ´2q ´Ipnq Ipn ´2q " 1 ´Ipnq Ipn ´2q " 1 ´n ´1 n " 1 n ,
where the last but one inequality follows from a direct integration by part. We have thus established that:

V Zε " ε 2 n ´ż `8 0 η2 µ Z,R pdηq `Opε 2 q ¯.
In particular, for ε " 1 ? N this precisely gives

|V ' N j"1 1 ? N bZ j ´t| ď C N ,
which is precisely (3.23) and readily gives as well (3.22). . Equation (3.24) is derived similarly using the corresponding specific scaling.

Proof of the main results

For a given N P N, let us consider the variables SN and S N defined in (2.5). From (3.12) and Proposition 3.2 we derive that they both possess a (radial) density, f SN and f S N respectively, and:

f SN " f 1 N bZ 1 ˚¨¨¨˚f 1 N bZ N , f S N " f 1 ? N bZ 1 ˚¨¨¨˚f 1 ? N bZ N . (4.1)
We will denote by

(4.2) µ εbZ,R pdηq " Ω n´1 f εbZ ptanhp η 2 qqpsinh ηq n´1 dη, η ě 0, ε P t 1 ? N , 1 N u,
the radial density of ε b Z in geodesic polar coordinates. This corresponds for ε " 1

?

N to what in [KTS59] is denoted by μN,1 . 4.1. Proof of the Theorem 2.1: Law of large numbers.

4.1.1. Proof of the main estimate. It suffices to establish the pointwise convergence of the characteristic function of SN as defined in (3.15) towards the constant 1. From (4.1) and Proposition 3.2, we have for all λ P R:

Φ SN pλq " f SN pλq f SN p0q " ˜f 1 N bZ pλq f 1 N bZ p0q ¸N " exp ˜N log ˜f 1 N bZ pλq f 1 N bZ p0q
¸¸" exppN logpΦ 1 N bZ pλqqq. We insist that the last equality is a definition and purely notational. Theorem 4.7 in [START_REF] Sturm | Probability measures on metric spaces of nonpositive curvature[END_REF] establishes that the walk asymptotically converges towards the mean in law and probability (weak law of large numbers). The convergence is strong provided Z n is bounded. The mean is intended here as the barycenter of the underlying law µ Z . If Z P L 2 pΩ, N q it corresponds to the unique minimizer of the mapping z Þ Ñ Erd 2 pz, Xqs, where d denotes here the distance between two points on N . We refer to Section 4 of [START_REF] Sturm | Probability measures on metric spaces of nonpositive curvature[END_REF] for additional details.

f p2q 1 N bZ pδλq f 1 N bZ p0q ˇˇď 1 2 V 1 N bZ ď (3.21) C N 2 ,
Let us first describe in the current specific case of H n how the previous construction can be related to the Möbius addition and multiplication. We will actually prove that (4.5)

Sn " xp

1 n q " Sn´1 ' 1 n b pa Sn´1 ' Z n q,
where for a P B n , aa is the left inverse, i.e. aa ' a " 0. Actually, it can be observed from (2.3) that aa " ´a.

To prove (4.5) we will use the following result whose proof can be found in Section 4.1 of [START_REF] Barabanov | Binary operations in the unit ball: A differential geometry approach[END_REF]. xp0q " a and xp1q " a ' b corresponds to the initial velocity 9 xp0q " p1 ´}a} 2 q atanhp}b}q }b} b. Furthermore, for t P r0, 1s it holds that xptq " a ' t b b.

To establish (4.5), which relates the geodesic walk Sn in [START_REF] Sturm | Probability measures on metric spaces of nonpositive curvature[END_REF] to the Möbius addition and multiplication, it then suffices to apply Proposition 4.1 with a " Sn´1 and b " a Sn´1 ' Z n3 , for which one indeed derives xp1q " Z n and to eventually take t " 1 n . The corresponding limit mean in the previous sense is clearly 0 (center of the ball). Now, if the Möbius multiplication were distributive w.r.t. the Möbius addition, i.e.

(D) 1 n b ´a Sn´1 ' Z n ¯" 1 n b a Sn´1 ' 1 n b Z n " ´1 n b Sn´1 ' 1 n b Z n ,
we would then obtain from (4.5), using as well that for x P B n , λ, µ P R, λ b x ' µ b x " pλ `µq b x (see [START_REF] Barabanov | Binary operations in the unit ball: A differential geometry approach[END_REF]):

Sn " p1 ´1 n q b Sn´1 ' 1 n b Z n .
A direct induction would then give Sn " ' n j"1 1 n b Z j and the geodesic random walk would then correspond to the one we considered.

However, in order to have the distributivity property (D) for the Möbius multiplication, it is necessary and sufficient (within the setting of gyrogroups) see e.g. [START_REF] Kim | Distributivity on the gyrovector spaces[END_REF], that gyrrx, ysz :" ´px ' yq ' px ' py ' zqq " z, i.e the gyration operator must be the identity. This property clearly fails in the current setting due to the non associativity of the Möbius addition.

Hence, the walks in [START_REF] Sturm | Probability measures on metric spaces of nonpositive curvature[END_REF] and the ones considered here are different objects (even though they converge to the same limit for the corresponding law of large numbers).

Proof of Theorem 2.2.

Proof. From (3.7) it suffices to establish the convergence of the characteristic function that we normalize as in (3.15). From (4.1) and Proposition 3.2, write for all λ P R:

Φ S N pλq " fS N pλq fS N p0q " ¨f 1 ? N bZ pλq f 1 ? N bZ p0q 'N " exp ¨N log ¨f 1 ? N bZ pλq f 1 ? N bZ p0q '' " exppN logpΦ 1 ? N bZ pλqqq. (4.6)
From (3.16), (3.17) we derive that

Φ 1 ? N bZ pλq " 1 ´λ2 2 V 1 ? N bZ `λ4 6
ż 1 0 dδp1 ´δq 3 f p4q 1 ? N bZ pδλq f 1 ? N bZ p0q ": 1 ´λ2 2 V 1 ? N bZ `λ4 6 R N , (4.7)
3 to obtain the appropriate b one has to solve Sn ' b " Z n , which according to Proposition 3 in [START_REF] Ferreira | Möbius gyrogroups: a Clifford algebra approach[END_REF] admits the unique solution given by the indicated b.

where f p4q 1 ? N bZ stands for the fourth derivative of the Fourier-Helgason transform. The point is now, somehow as in a usual limit theorem to control the remainder. We have:

ˇˇˇˇˇˇf p4q 1 ? N bZ pδλq f 1 ? N bZ p0q ˇˇˇˇˇˇď ş `8 0 ˇˇB 4 ν ϕ ν ptanhp η 2 qq ν"δλ ˇˇµ 1 ? N bZ,R pdηq ş `8 0 ϕ 0 ptanhp η 2 qqµ 1 ? N bZ,R pdηq " ş τ 0 ˇˇB 4 ν ϕ ν ptanhp η 2 qq ν"δλ ˇˇµ 1 ? N bZ,R pdηq ş `8 0 ϕ 0 ptanhp η 2 qqµ 1 ? N bZ,R pdηq `ş`8 τ ˇˇB 4 ν ϕ ν ptanhp η 2 qq ν"δλ ˇˇµ 1 ? N bZ,R pdηq ş `8 0 ϕ 0 ptanhp η 2 qqµ 1 ? N
bZ,R pdηq ": I 1 pτ q `I2 pτ q, (4.8) for some τ ą 0 to be specified. Observing from (3.5) that, for η P r0, τ s, (4.9)

ˇˇB 4 ν ϕ ν ptanhp η 2 qq ν"δλ ˇˇď ´τ 2 B 2 ν ϕ ν ptanhp η 2 qq ν"0 ,
write:

I 1 pτ q " V 1 ? N bZ ş τ 0 ˇˇB 4 ν ϕ ν ptanhp η 2 qq ν"δλ ˇˇµ 1 ? N bZ,R pdηq ´ş8 0 B 2 ν ϕ ν ptanhp η 2 qq ν"0 µ 1 ? N bZ,R pdηq ď V 1 ? N bZ τ 2 .
Similarly to (4.9) we derive that for η P rτ, `8q:

ˇˇB 4 ν ϕ ν ptanhp η 2 qq ν"δλ ˇˇď ´η2 B 2 ν ϕ ν ptanhp η 2 qq ν"0 .
From the definition of µ 1 ? N bZ,R introduced in (4.2) and Proposition 3.4 we derivea tail control of the Fourier-Helgason transform. Namely, for any τ ą 0 it holds that (4.10)

ż 8 τ η 2 B 2 ν ϕ ν ptanhp η 2 qq| ν"0 µ 1 ? N bZ,R pdηq ÝÑ N 0.
We then get from (4.10):

I 2 pτ q ď V 1 ? N bZ ş 8 τ η 2 B 2 ν ϕ ν ptanhp η 2 qq ν"0 µ 1 ? N bZ,R pdηq ş 8 0 B 2 ν ϕ ν ptanhp η 2 qq ν"0 µ 1 ? N bZ,R pdηq ": V 1 ? N bZ R N pτ q, R N pτ q ÝÑ N 0, τ ą 0.
Plugging the above controls for I 1 pτ q, I 2 pτ q into (4.8) we derive that the remainder term R N in (4.7) enjoys the upper-bound:

R N ď V 1 ? N bZ 4
pτ 2 `RN pτ qq.

Expanding then the logarithm in (4.6), we thus get:

Φ S N pλq " exp ˆ´λ 2 2 V 1 ? N bZ N `λ4 6 N R N `op1q " exp ˜´λ 2 2 t `λ4 6 t R N V 1 ? N bZ `op1q ¸.
Taking τ small enough and then N large enough we deduce that:

Φ S N pλq ÝÑ N expp´λ 2 2
tq " Ψpt, λq, using (3.10), i.e. we recognize that the limit is the characteristic function of the second kind of the normal law on B n . The statement then follows from the inversion formula (3.7) and domination arguments.

4.3. Proof of the Local limit Theorem 2.3. The point is now to go further than the convergence in law for the random walk established in Theorem 2.2 and to establish under the previous assumptions a (global) pointwise control for the difference of the densities. From (3.7) the difference writes:

f S N ptanhp η 2 qq ´Ψpt, tanhp η 2 qq "C n ż `8 0 r fS N pλq ´ψpt, λqsϕ λ ptanhp η 2 qq|cpλq| ´2dλ "C n ż `8 0 r N ź j"1 f 1 ? N bZ j pλq ´expp´p ρ 2 `λ2 qt 2 qsϕ λ ptanhp η 2 qq|cpλq| ´2dλ "C n ż `8 0 pI λďD N `IλąD N qrp f 1 ? N bZ pλqq N ´expp´p ρ 2 `λ2 qt 2 qsϕ λ ptanhp η 2 qq|cpλq| ´2dλ ": pB N `TN qpηq, (4.11)
where D N is a cutting level to be specified which will allow to balance the contribution for the terms B N pηq and T N pηq corresponding respectively to the bulk and tails of the Fourier-Helgasson integral.

-Contribution of the bulk. We first concentrate on the difference of the two Fourier transforms in the bulk, i.e. for λ ď D N . Namely, write:

ˇˇp f 1 ? N bZ pλqq N ´expp´p ρ 2 `λ2 qt 2 q ˇ" expp´p ρ 2 `λ2 qt 2 q ˆˇˇe xp ˆN ln ´f 1 ? N bZ pλq ¯`pλ 2 `ρ2 qt 2 ˙´1 ˇď expp´p ρ 2 `λ2 qt 2 q ˇˇN ln ´f 1 ? N bZ pλq ¯`pλ 2 `ρ2 qt 2 ˇˇexp ´ˇˇN ln ´f 1 ? N bZ pλq ¯`pλ 2 `ρ2 qt 2
ˇˇ¯, (4.12) using that for all x P C, |e x ´1| ď |x| expp|x|q for the last inequality.

Write now:

ˇˇN ln ´f 1 ? N bZ pλq ¯`pλ 2 `ρ2 qt 2 ˇˇ" ˇˇN ln ´f 1 ? N bZ p0q ´1 ´p1 ´f 1 ? N bZ pλq f 1 ? N bZ p0q q ¯¯`p λ 2 `ρ2 qt 2 ˇď |N ln ´f 1 ? N bZ p0q ¯`ρ 2 2 t| `λ2 2 |t ´N V 1 ? N bZ | `N |pln ´1 ´p1 ´f 1 ? N bZ pλq f 1 ? N bZ p0q q ¯`λ 2 2 V 1 ? N bZ | ď|N ln ´f 1 ? N bZ p0q ¯`ρ 2 2 t| `λ2 2 |t ´VS N | `N ´8 ÿ r"2 r ´1|1 ´f 1 ? N bZ pλq f 1 ? N bZ p0q | r `| ´1 `f 1 ? N bZ pλq f 1 ? N bZ p0q `λ2 2 V 1 ? N bZ ˇˇ" :R N,1 `R2,N `R3,N . (4.13)
We can write similarly to (4.7), expanding again up to order 4, that

f 1 ? N bZ pλq f 1 ? N bZ p0q " 1 ´λ2 2 V 1 ? N bZ `λ4 6
ż 1 0 dδp1 ´δq 3 f p4q 1 ? N bZ pδλq f 1 ? N bZ p0q ": 1 ´λ2 2 V 1 ? N bZ `λ4 6 R N . (4.14) From (3.5) it follows that max δPr0,1s |ϕ p4q δλ ptanhp η 2 qq| ď η 2 2 |ϕ p2q 0 ptanhp η 2 qq| and therefore (4.15) |R N | ď Er|Z 1 ? N ,R | 2 ϕ p2q 0 `tanhp Z 1 ? N ,R 2 q f 1 ? N bZ p0q s,
where Z 1 ? N

,R is a random variable with law µ 1 ? N bZ,R introduced in (4.2) corresponding to the geodesic polar coordinates of 1 ? N b Z. It follows that:

Er|Z 1 ? N ,R | 2 |ϕ p2q 0 `tanhp Z 1 ? N ,R 2 q ˘| f 1 ? N bZ p0q s "Ω n´1 ż `8 0 η 2 |ϕ p2q 0 ptanhp η 2 qq| f 1 ? N bZ p0q f 1 ? N bZ ptanhp η 2 qq sinhpηq n´1 dη " (3.19) Ω n´1 ż `8 0 η 2 |ϕ p2q 0 ptanhp η 2 qq| f 1 ? N bZ p0q ? N f Z ptanhp ? N η 2 qq ´sinhp ? N ηq sinhpηq ¯n´1 sinhpηq n´1 dη "Ω n´1 1 f 1 ? N bZ p0q ż `8 0 η2 N |ϕ p2q 0 ptanhp η 2N 1 2 qq|f Z ptanhp η 2 qq sinhpηq n´1 dη ďCΩ n´1 1 f 1 ? N bZ p0q ż `8 0 η4 N 2 |ϕ 0 ptanhp η 2N 1 2 qq|f Z ptanhp η 2 qq sinhpηq n´1 dη ď C N 2 . (4.16)
It therefore holds that there exists C s.t.

(4.17)

|R N | ď C N 2 . Set now, A N :" pV 1 ? N bZ q ´1 4 , B N :" ´N E " |Z 1 ? N ,R | 2 |ϕ p2q 0 `tanhp Z 1 ? N ,R 2 q ˘| f 1 ? N bZ p0q ı¯´1 4 , C N :" |t ´VS N | ´1 2 .
In the previous setting, from (3.21), (

A ´2 N `B´2 N `C´1 N ď CN ´1 2 . 4.16), (3.23), (4.18)
With these notations, it is clear from the definitions in (4.13) and (4.18) that

(4.19) R 2,N ď λ 2 2 C ´2 N ď C λ 2 N .
On the other hand,

R 3,N ď N ˜|1 ´f 1 ? N bZ pλq f 1 ? N bZ p0q | ÿ r"2 r ´1|1 ´f 1 ? N bZ pλq f 1 ? N bZ p0q | r´1 `λ4 6 E » -Z 2 1 ? N ,R ˇˇϕ p2q 0 `tanhp Z 1 ? N ,R 2 q f 1 ? N bZ p0q ˇˇfi fl ¸": R p1q 3,N `Rp2q 3,N . Assume now that |λ| ď D N " minp A N ? 3 , B N , C N ?
3 q. It readily follows from (4.16) that:

R p2q 3,N ď λ 4 6 B ´4 N ď C λ 4 N .
We also have from (4.14), (4.15) that:

| f 1 ? N bZ pλq f 1 ? N bZ p0q ´1| " λ 2 2 V 1 ? N bZ `λ4 6 R N ď 1 6 `1 6 " 1 3 .
We point out that this is a very coarse estimate which is only given to ensure the convergence of the series appearing for the term R p1q 3,N . Then,

R p1q 3,N ď N |1 ´f 1 ? N bZ pλq f 1 ? N bZ p0q | 2 ÿ r"2 r ´1|1 ´f 1 ? N bZ pλq f 1 ? N bZ p0q | r´2 ď CN pλ 4 V 2 1 ? N bZ `λ8 R 2 N q ÿ rPN p2 `rq ´1p 1 3 q r ď C ´λ4 N A ´8 N `λ4 N 2 ˆλ4 B ´4 N ¯ď λ 4 N ´1,
exploiting (4.18), which gives that N A ´4 N ď C and A ´4 N ď C{N , as well as λ 4 B ´4 N ď 1 for the last inequality. We carefully mention that it is the term involving the variance which is responsible for the above rate.

We

f 1 ? N bZ p0q ´1 " Ω n´1 ż `8 0 f Z ptanh η 2 q `ϕ0 ptanhp η 2 ?
N qq ´1˘s inhpηq n´1 dη.

Setting now, gpζq " ϕ 0 ptanhp ζ 2 qq, we perform a fourth order Taylor expansion which yields:

f 1 ? N bZ p0q ´1 "Ω n´1 ż `8 0 f Z ptanh η 2 q ´gp1q p0q η N 1 2 `1 2 g p2q p0q η2 N `1 6 g p3q p0q η3 N 3 2 `1 6 ż 1 0 g p4q pγ ηqp1 ´γq 3 dγ η4 N 2 ¯sinhpηq n´1 dη,
where g piq denotes the i th derivative of g. It now remains to compute the derivatives of g. We will actually prove that, the derivatives of odd index are zero at zero and we aim at retrieving for the second order derivative a contribution, which once summed in (4.21) will precisely exactly give the contribution ´ρ2 2 t. To compute the derivative of g let us recall from (3.5) that: Hence,

f 1 ? N bZ p0q ´1 " k n p´ρpρ `1qIpnq `ρ2 Ipn ´2qq 2N ż `8 0 η2 µ Z,R pdηq `Op 1 N 2 q.
Having in mind the explicit expression of t in (3.22), to investigate the difference of the first l.h.s term in (4.21) it remains to specify properly the constants involved (in both the definition of t and the above expression).

From [START_REF] Gradshteyn | Table of integrals, series and products[END_REF], formula 3.631 (8), p. 386, it holds that for ν P C, pνq ą 0,

ż π 0 sinpθq ν´1 dθ " π 2 ν´1 νBp ν`1 2 , ν`1 2 q ,
where Bp¨, ¨q denotes the β-function. Therefore taking respectively ν " n `1 and ν " n ´1, recalling as well that Bpx, yq " ΓpxqΓpyq Γpx`yq , one derives

Ipnq " πΓpn `2q 2 n pn `1qΓp n`2 2 q 2 " πn! 2 n Γp n`2 2 q 2 , Ipn ´2q " πpn ´2q! 2 n´2 Γp n 2 q 2 . Recalling from (3.9) that ρ " n´1 2 , this in turn gives:

´kn ρpρ `1qIpnq " ´Γp n 2 q ? πΓp n´1 2 q n ´1 2 n `1 2 πn! 2 n `n 2 ˘2 Γp n 2 q 2 " ´n2 ´1 2 n n 2 Γp n´1 2 qΓp n 2 q ? πn! " ´n2 ´1 4Γpn ´1q n! n 2 " ´pn 2 ´1q 4 n ´1 n "
´ρ2 pn `1q n using the Legendre duplication formula for the third equality. We also similarly derive,

k n ρ 2 Ipn ´2q " Γp n 2 q ? πΓp n´1 2 q ´n ´1 2 ¯2 πpn ´2q! 2 n´2 Γp n 2 q 2 " pn ´2q! Γpn ´1q ´n ´1 2 ¯2 " ρ 2 .
We have thus established:

1 ´f 1 ? N bZ p0q " ρ 2 n 1 2N ż `8 0 η2 µ Z,R pdηq `Op 1 N 2 q,
which eventually gives in (4.21)

R 1,N " ´ρ2 2 1 n ż `8 0 η2 µ Z,R pdηq `Op 1 N q `ρ2 2 t.
This eventually gives from the very definition of t (see e.g. (3.22)):

R 1,N " Op 1 N q. (4.22) Therefore, plugging (4.22), (4.20), (4.19) into (4.13) and (4.12) we thus derive, choosing D N " N 1 4 so that in particular λ 4 {N ď 1:

|B N pηq| ďC ż 0ăλďD N expp´p ρ 2 `λ2 qt 2 q 1 N `1 `λ4 ˘exp ´1 N `1 `λ4 ˘¯|ϕ λ ptanhp η 2 qq||cpλq| ´2dλ ď C t 2 N ż 0ăλďD N expp´p ρ 2 `λ2 qt 4 q|ϕ λ ptanhp η 2 qq||cpλq| ´2dλ ď C t 2 N ż R `expp´p ρ 2 `λ2 qt 4 q|ϕ λ ptanhp η 2 qq||cpλq| ´2dλ. (4.23)
From (3.5) and the definition of the Harish-Chandra function in (3.8), see e.g. Remark 2.7 in [START_REF] Anker | An introduction to Dunkl theory and its analytic aspects[END_REF] for its asymptotic behavior, we get (4.24)

|ϕ λ ptanhp η 2 qq| ď ϕ 0 ptanhp η 2 qq ď C, |cpλq| ´2 ď Cpλ 2 I λďC `λn´1 I λąC q,
from which we eventually derive:

|B N pηq| ď C t 2 N ´1 t 1 2 ^1 t n 2 ¯. (4.25)
-Contribution of the tails, general case. For λ ą N 1{4 , we write:

T N pηq ď ˇˇż `8 N 1{4 ´N ź j"1 f 1 ? N bZ j pλq ¯ϕλ ptanhp η 2 qq|cpλq| ´2dλ ˇˇ`ˇˇż `8 N 1{4 expp´p ρ 2 `λ2 qt 2 qϕ λ ptanhp η 2 qq|cpλq| ´2dλ ˇ" :pT 1 N `T 2 N qpηq. (4.26)
Let us first consider the term T 2 N which can be handled globally. We get again from (4.24) (similarly to (4.25)):

T 2 N pηq ď expp´N 1 2 t 4 q ż `8 0 expp´p ρ 2 `λ2 qt 4 q|ϕ λ ptanhp η 2 qq||cpλq| ´2dλ ď C t 2 N ´1 t 1 2 ^1 t n 2 ¯, (4.27)
which gives an upper bound similar to the one obtained for the bulk in (4.25).

Let us now turn to T 1 N pηq, and split as follows:

T 1 N pηq ď ˇˇż c0N 1 2 N 1 4 ´f 1 ? N bZ pλq ¯N ϕ λ ptanhp η 2 qq|cpλq| ´2dλ ˇż 8 c0N 1 2 ´f 1 ? N bZ pλq ¯N ϕ λ ptanhp η 2 qq|cpλq| ´2dλ ˇˇ": pT 11 N `T 12 N qpηq, (4.28)
for some small enough constant c 0 to be specified.

Let us now recall that from Lemma 2.1 of [START_REF] Liu | Boundary regularity for the Dirichlet problem for the invariant Laplacians ∆ on the unit real ball[END_REF], for x P B n and λ P C

ż S n´1 σpdζq |x ´ζ| 2λ " 2 F 1 pλ, λ ´n 2 `1, n 2 , }x} 2 q,
where 2 F 1 stands for the hypergeometric function and σ " Λ{Ω n´1 for the normalized surface measure on S n´1 (i.e. σpS n´1 q " 1). We now recall, see e.g. formula 15.1.1. in [START_REF] Gradshteyn | Table of integrals, series and products[END_REF], that for a, b, c, z, where c is not a negative integer and |z| ă 1, (4.29) 2 F 1 pa, b, c, zq " `8 ÿ q"0 paq q pbq q pcq q z q q! , denoting for d P ta, b, cu, pd q q " # 1, q " 0, ś q´1 j"0 pd `jq, q ‰ 0.

The series gives an analytic function if |z| ă 1 and can be continued on the complex plane (cut on r1, `8qq. From (3.3), (3.4), we thus get:

ϕ λ pxq " p1 ´}x} 2 q ρ`i λ 2 ż S n´1 σpdζq }x ´ζ} 2pρ`i λ 2 q " p1 ´}x} 2 q ρ`i λ 2 2 F 1 pρ `i λ 2 , 1 2 `i λ 2 , n 2
, }x} 2 q.

Recalling the Pfaffian identity (see e.g. (2.15) in [START_REF] Liu | Generalized Helgason-Fourier transforms associated to variants of the Laplace-Beltrami operators on the unit ball in R n[END_REF])

2 F 1 pa, b, c, zq " p1 ´zq ´a2 F 1 pa, c ´b, c, z z ´1 q, we get for z " }x} 2 " tanhp η 2 q 2 , a " ρ `i λ 2 , b " 1 2 `i λ 2 , c " n 2 , ϕ λ ptanhp η 2 qq " 2 F 1 pρ `i λ 2 , ρ ´i λ 2 , n 2 , ´sinh 2 p η 2 qq.
We therefore get from (4.29):

ϕ λ ptanhp η 2 qq "1 `8 ÿ q"1 ś q´1 j"0 `pρ `i λ 2 `jqpρ ´i λ 2 `jq śq´1 j"0 p n 2 `jq p´1q q psinhp η 2 qq 2q q! "1 `8 ÿ q"1 ś q´1 j"0 `pρ `jq 2 `λ2 4 q śq j"1 pm `jq p´1q q psinhp η 2 qq 2q q! , m " ρ ´1 2 " n 2 ´1, "1 `8 ÿ q"1 ś q´1 j"0 `p2 ρ`j λ q 2 `1q 4q ś q j"1 pm `jq p´1q q pλ sinhp η 2 qq 2q q! "1 `8 ÿ q"1 p´1q q pλ sinhp η 2 qq 2q « 1 `´n´1 λ ¯2ff ˆ¨¨¨ˆ«1 `´n`2q´3 λ ¯2ff 4 q q!pm `1qpm `2q ¨¨¨pm `qq (4.30) "1 `8 ÿ q"1 p´1q q pλ sinhp η 2 qq 2q « 1 `´2m`1 λ ¯2ff ˆ¨¨¨ˆ«1 `´2m`2q´1 λ ¯2ff
4 q q!pm `1qpm `2q ¨¨¨pm `qq .

We emphasize that the representation (4.30) will be actually used to investigate the behavior of the quantity ´f 1 ? N bZ pλq ¯N in (4.28) for the term T 11 N pηq. From (3.1), (3.19), and since on the considered integration interval λ P rN 1 4 , c 0 N 1 2 s, this actually means that we will consider (4.30) for a small spatial argument. Namely,

f 1 ? N bZ pλq " Ω n´1 ż 8 0 ϕ λ ptanhp η 2 ? N qqf ptanhp η 2 qq sinhpηq n´1 dη " ż 8 0 ϕ λ `tanhp η 2 ? N q ˘µZ,R pdηq. (4.31)
Let us turn to the estimation of the r.h.s. in (4.30). We will actually establish that, for sufficiently large q: (4.32)

« 1 `´2m`1 λ ¯2ff ˆ¨¨¨ˆ«1 `´2m`2q´1 λ ¯2ff 4 q q!pm `1qpm `2q ¨¨¨pm `qq " m! « 1 `´2m`1 λ ¯2ff ˆ¨¨¨ˆ«1 `´2m`2q´1 λ ¯2ff
4 q q!pm `qq! ď Cpmq4 ´q .

To establish the above inequality we first observe that

« 1 `´2m `1 λ ¯2ff ˆ¨¨¨ˆ«1 `´2m `2q ´1 λ ¯2ff
ď exp ˜q ln ´1 `´2m `2q ´1 λ ¯2¯ḑ exp ˜q ln ´q2 p 1 q 2 `Cpmq λ 2 q ¯¸ď exppq lnpr q e s 2 qq " expp´2qq expp2q ln qqq, (4.33) where the last inequality holds for q ě 3 and N large enough (recall λ P rN 1{4 , c 0 N 1{2 s). On the other hand, from Stirling's formula we get: q!pm `qq! " 2πpqpm `qqq 1 2 e q ln q e pm`qq lnpm`qq e ´pm`2qq e 1 12q pθ1q`θ2qq , θ iq P p0, 1q, i P t1, 2u, ą 2πe ´me 2q ln q e ´2q , (4.34) for sufficiently large q. From (4.34) and (4.33), we derive (4.32).

On the other hand, on the considered range for λ, i.e. λ P rN 1 4 , c 0 N 1 2 s we have:

λ sinhp η 2 ? N q ď λ η ? N ď c 0 R ă 1,
choosing c 0 small enough and exploiting that µ Z in (4.31) is compactly supported on r0, Rs. We then derive that on the considered range: η 2 µ Z,R pdηqq " expp´cλ 2 tq, up to a modification of c for this very last inequality. We then eventually get from (4.28), analogously to (4.27), that: N pηq in (4.28). To this end we exploit, see e.g. Lemma 5.1 formula (5.3) in [START_REF] Petrova | Approximation on a hyperboloid in the L 2 metric[END_REF], that (4.37) Dc P p0, 1q, @λ, r P R 2 `, λr ą 1, |ϕ λ ptanhprqq| ă 1 ´c.

T 11 N pηq ď C expp´c p´ρ 2 `N 1 2 qt 2 q ż c0N 1 2 N 1 4 expp´c pρ 2 `λ2 qt 2 q|ϕ λ ptanhp η 2 qq||cpλq| ´2dλ ď C t 2 N ´1 t 1 2 ^1 t

Write then,

T 12 N pηq " ˇˇż We can now use the Plancherel equality, see e.g. Theorem 6.14 in [START_REF] Liu | Boundary regularity for the Dirichlet problem for the invariant Laplacians ∆ on the unit real ball[END_REF] with ϑ " 2 ´n, to derive: ¯`1 ¯.

T 12 N pηq ď Cδ N ´2 ż R 0 f 2 1 ? N bZ ptanhp η 2 qq sinh n´1 pηqdη ď Cδ N ´2 ż R N 1 2 0 N f 2 Z ptanhp N 1 2 η 2
From the above control and (4.27), (4.26) we thus derive:

T N pηq ď C N ´1 t 2 ´1 t 1 2 ^1 t n 2 ¯`1 ¯,
which together with (4.25) and (4.11) completes the proof of the local limit Theorem 2.3. Email address: stephane.menozzi@univ-evry.fr

e

  ´λ,ζ pxqΛpdζqµ B n pdxq, from which we derive ĥpλq " ż B n hpxqe ´λ,ζ pxqµ B n pdxq, @ζ P S n´1 ,

  Proposition 3.5 (Variance for the random walks). Let Z satisfy [R]. Set for ε ą 0, Z ε :" ε b Z. It then holds that there exists C ě 1 s.t.

  get similarly to the proof of Proposition 3.5, see e.g. (3.25) replacing therein λ " 0 by δλ, that ˇˇż

Proposition 4. 1 (

 1 Geodesic lines and the Möbius addition). Let a, b in B n . The geodesic line `xptq ˘tPr0,1s s.t.

."k n ρ ż π 0 cospθq sinpθq n´2 dθ " 0, g p2q p0q "k n ρpρ `1q ż π 0 pcoshpηq ´sinhpηq cospθqq ´pρ`2q psinhpηq ´coshpηq cospθqq 2 sinpθq 2ρ´1 dθ| η"0 ´kn ρ ż π 0 "k n ρpρ `1q ż π 0 cospθq 2 sinpθq n´2 dθ ´kn ρ ż π 0

 00000 rewrites, see e.g.[START_REF] Anker | An introduction to Dunkl theory and its analytic aspects[END_REF],gpηq " ϕ 0 ptanhp η 2 qq " k n ż π 0 pcoshpηq ´sinhpηq cospθqq ´ρ sinpθq 2ρ´1 dθ, k n :" ΓFrom the above expression we indeed derive:g p1q p0q " ´kn ρ ż π 0pcoshpηq ´sinhpηq cospθqq ´pρ`1q psinhpηq ´coshpηq cospθqq sinpθq 2ρ´1 dθ| η"0 pcoshpηq ´sinhpηq cospθqq ´pρ`1q pcoshpηq ´sinhpηq cospθqq sinpθq 2ρ´1 dθ| η"0 sinpθq n´2 dθ " ´kn ρpρ `1qIpnq `kn ρ 2 Ipn ´2q, @m P N, Ipmq :"

  Z,R pdηqq, | f 1 ? N bZ pλq| N ď expp´cλ 2 ż B n

  now remains to consider the case λ ą c 0 N 1 2 to handle T 12

NN

  for some parameter b ą 0 to be specified. Let us now write from (4.31): R pdηq ": I `II. (4.38)For the term II in (4.38), observe that for η ą 2b ą 1 and by (4.37), II ď 1 ´c.For the term I observe that 2b ? c0 . Since µ Z,R has a density there is in particular no atom at 0 and therefore, since |ϕ λ b ptanhp η 2 ?N qq| ď C on the considered integration range, we then derive ş 2b qq|µ Z,R pdηq ă c{2 for b small enough. Hence, plugging those controls into (4.38), we get: pλq| 2 |ϕ λ ptanhp η 2 qq||cpλq| ´2dλ.

  Connection with the Sturm geodesic walks. In the article[START_REF] Sturm | Probability measures on metric spaces of nonpositive curvature[END_REF], Sturm proposed a very general construction of geodesic random walks in non positive curvature (NPC) metric spaces and established a corresponding weak and strong law of large number. In particular, the Hyperbolic space H n enters this setting. The construction proposed in[START_REF] Sturm | Probability measures on metric spaces of nonpositive curvature[END_REF] is the following. Let N be a NPC metric space. For fixed n P N ˚, let Sn´1 be given (representing the value of the normalized walk, with the scaling of the law of large numbers, at time n ´1). Let Z n be the n th innovation variable (defined on some probabilistic space pΩ, A, Pq and N valued).

	which plugged into (4.4), (4.3) gives Φ SN pλq Ñ N	1 which yields the stated convergence.
	4.1.2. Consider a a geodesic line `xptq ˘tPr0,1s s.t. xp0q " Sn´1 , xp1q " Z n . The walk is then updated setting
	Sn " xp	1 n	q ": ´1	´1 n	¯S n´1	`Zn .

  Cλ 4 N ´1.

		have thus established that
	(4.20) R 3,N ď It remains to handle the contribution R 1,N in (4.13). Write:
		R 1,N " |N ln ´f 1 ? N	bZ p0q	¯`ρ 2 2	t| " |N ln `1 ´p1 ´f 1 ? N	bZ p0qq	˘`ρ 2 2	t|
										8
	(4.21)	" ˇˇ´N p1 ´f 1 ? N	bZ p0qq	`ρ2 2	t ´N p1 ´f 1 ? N	bZ p0qq	ÿ r"2	r ´1p1 ´f 1 ? N	bZ p0qq r´1 ˇˇ.
	Write now from (3.20):

  (3.19) with ε " N ´1 2 for the second inequality. Hence, recalling that δ ă 1 for the last inequality. We have thus established from the above control and (4.36) that,

										qq	´sinhpN sinhpηq 1 2 ηq	¯2pn´1q	sinh n´1 pηqdη
				ď Cδ N ´2N	1 2	ż R 0	f 2 Z ptanhp	η 2	qq sinhpηq 2pn´1q psinhp	η N	2 1	qq ´pn´1q dη
	using as well T 12 N pηq ď CN	n 2 δ N ´2 ż R 0	f 2 Z ptanhp	η 2	qq sinhpηq 2pn´1q η ´pn´1q dη ď CN	n 2 δ N ´2 " CN	n 2 expppN ´2q lnpδqq
	ď	C N	expp´N 2	| lnpδq|q,
								T 1 N pηq ď	C N ´1 t 2 ´1 t 1 2 ^1 t n 2

  Higher School of Economics, National Research University, Pokrovski Boulevard 11, Moscow, Russian Federation. Email address: VKonakov@hse.ru Laboratoire de Modélisation Mathématique d'Evry, (LaMME), UMR CNRS 8071, Université d'Evry Val-d'Essonne, 23 Boulevard de France, 91037 Evry.

From now on we will denote by rr¨, ¨ss integer intervals.


Ph.D V A Montoya-Coronado

MSc L Souillard

MSc R Gaubert

Res. Dir.² P Molle

D Tedoldi
email: damien.tedoldi@insa-lyon.fr

Prof G Lipeme Kouyi

Catchment delineation with different levels of geographical data: which effects on combined sewer overflow modelling?

Keywords: Combined Sewer Overflow, Data scarcity, Hydraulic modelling, Sub-catchment

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Sustainable Urban Drainage Systems (SUDs) are increasingly used to alleviate the sewer system and thus contribute to reduce combined sewer overflows (CSO). However, this approach raises several questions about the minimal number of SUDs, their types, and optimal locations to achieve a significant decreasing of CSO at the catchment scale. Those questions are often appraised through modelling approaches and scenario analyses. However, hydrological and hydraulic (HH) models require a minimal description of the territory and its characteristics, notably the catchment and sub-catchment delineation, and the appropriate level of sewer network plan. Several catchment delimitation methods already exist [START_REF] Li | Methodology of Sub-Catchment Division Considering Land Uses and Flow Directions[END_REF]. A first group of methods are generally applied to a single data source. Delimitation can be made with land use, mostly consisting of urban block divisions according to existing roads [START_REF] Khurelbaatar | Management of Urban Stormwater at Block-Level (MUST-B): A New Approach for Potential Analysis of Decentralized Stormwater Management Systems[END_REF]; network sewer plans [START_REF] Ghosh | Effects of Spatial Resolution in Urban Hydrologic Simulations[END_REF]; or topography. For example, the "D8 algorithm" is used quite frequently, and is based on the Voronoïi geometric division principle. A second group of methods, referred as "hybrid" partitioning methods, sequentially uses the different data sources to identify sub-catchments. The present study investigates the effect of different strategies for catchment and sub-catchment delineation -and when applicable, sewer network identification -based on different spatial data: topography and/or sewer network plans and/or land use data. Four scenarios of data availability are proposed and tested in two different sites, followed by HH modelling to compare the computed flow dynamics. The final objective is to test solutions for CSO mitigation.

Methodology and results

Methodology: Available data were gathered through four scenarios: aerial images only (Scenario 1); aerial images and topography (Scenario 2); aerial images and actual network plans (Scenario 3); all above-mentioned data (Scenario 4). The overall delineation methodology is adapted accordingly. First, a preliminary catchment delimitation is made from aerial views, by excluding the external zones (forests, fields, parks), or by considering natural or artificial frontiers (river, highway). Whenever necessary, we propose to mimic the real sewer network by means of a fictive one following principal roads and driven by gravity. This follows the conclusions of [START_REF] Blumensaat | Sewer model development under minimum data requirements[END_REF] who showed that correlation between the network and roads goes up to 95% depending on the city, and emphasized that older infrastructures are more likely to follow this rule. When digital elevation models (DEM) data is not available (scenarios 1 and 3), each area is considered as contributor to the nearest network branch, so that boundaries between sub-catchments are defined by the equidistance between two pipes. When DEM data is available (scenarios 2 and 4), the extent of this area is determined considering the natural path flow direction (D8 method).

The delineation methodology was applied in two study sites: Ecully (245 ha) and Vernaison (360 ha), France. For those two sites, DEM data with a resolution of 5m x 5m grid is available (RGE ALTI©; IGN, National Geographic Institute). The network plans were provided by local authorities.

Page 2

The outputs of the above-mentioned method were used in a semi-distributed HH model based on (i) the rational method for rainfall-runoff transformation, (ii) a linear reservoir as a transfer function, and (iii) a spilled flow equation for CSO simulation. In each delineated sub-catchment, the percentage of impervious area was derived from land use maps (high-resolution Map Tiler Satellite imagery©).

Key results:

Approaches in which topography is unknown tend to include contributing areas on the catchment edges whose slope does not converge towards the sewer network inlets: therefore, they may overestimate the total catchment area. For instance, scenarios 1 and 3 lead to a total catchment area up to 30% higher than scenarios 2 and 4 (Figure 1), and consequently a runoff volume increased by a factor of ~2. On the hand, knowledge of the actual sewer network plan (scenarios 3 and 4) may lead to the identification of additional contributing urban areas outside the pre-identified catchment boundaries, as exemplified by the case of Vernaison with a south-western zone connected to the sewer system. Also, it was confirmed that mimicking the network following the main roads provides a good approximation of the main branches of the network. In Ecully, the fictitious network in scenarios 1 and 2 tends to be more detailed than the available information on the actual one (which only includes the downstream branches). However, when comparing results from topography-based inputs (scenarios 2 and 4), the simulated hydrographs show quite similar flow dynamics during rainfall events (Figure 1), and thus similar CSO characteristics (total volume and frequency).

Conclusion and perspectives:

This study explores different strategies to collect the required input data for HH models, depending on the available geographical information. Current results show a better estimation of total runoff volume when topography is available, even if the actual plans of the sewer network are unknown. Further research will include one more case study with different land characteristics, and will consider different data resolutions: DEM with a 5m x 5m grid vs. DEM with a 30m x 30m grid (assumed to be more easily accessible). Ongoing simulations are based on long-term rainfall series and will be compared to sewer flow monitoring data. The final objective will be to model the effects of SUDS dissemination on CSO.

(

  Sub-)Catchment delineation with different levels of geographical data: which effects on combined sewer overflow modelling? Violeta A. Montoya-Coronado, Pascal Molle, D Tedoldi, Gislain Lipeme Kouyi, L. Souillard, R. Gaubert To cite this version: Violeta A. Montoya-Coronado, Pascal Molle, D Tedoldi, Gislain Lipeme Kouyi, L. Souillard, et al.. (Sub-)Catchment delineation with different levels of geographical data: which effects on combined sewer overflow modelling?. 10th International Conference on Sewer Processes and Networks, Institute of Urban Water Management and Landscape Water Engineering; TU Graz Wissen Technik Leidenschaft; TU Graz University of Technology, Aug 2022, Graz University of Technology, France. hal-04330233

Figure 1 :

 1 Figure 1: Catchment and sub-catchment delimitation for scenarios 1, 2, 3, 4. Red delimitation represents delimitation made with minimal data, colour polygons correspond to sub-catchments, blue lines represent the fictitious or real network and red points correspond to catchment outlet. Ecully case study show the preliminary HH simulation for a 19 mm rainfall event.


Violeta A Montoya-Coronado

Pascal Molle

Damien Tedoldi

Hélène Castebrunet

Gislain Lipeme Kouyi

Decision support tool for stormwater disconnectionupport tool for stormwater disconnection

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

This project has received funding from the European Union's Horizon H2020 innovation action programme under grant agreement 101003527.

Decision support tool for stormwater disconnection

1) Delineation of (sub-)catchment areas according to available geographic information

A preliminary delimitation of contributing areas to CSO is made from aerial views (excluding natural or artificial frontiers).

Whenever necessary, we propose to mimic the real sewer network by means of a fictive gravity line following the main roads direction. Boundaries between sub-catchments are defined considering the natural path flow direction (D8 method) or by the equidistance between two pipes.

Ecully catchment delineation (All needed data available)

2) Hydrological & hydraulic model for CSO simulation

Aims to represent the physical processes of runoff and discharges in urban areas provided by a combined sewer system. Is a semidistributed model based on i) the rainfall-runoff model, ii) a linear reservoir as a transfer function, and iii) a spilled flow equation for CSO simulation. It Ipro vides indicators on CSO annual volume and frequency.

Model overview

3) Optimization of disconnection scenarios

Developement of an algorithm allowing to indicate : i) Where disconnection would be suitable for CSO mitigation ii) The volume of runoff and/or wastewater that should be disconnected from the sewer system to fulfill local compliances.

The disconnection concept focuses on the principle of correcting the consequences of previous centralized urban water management by reintegrating water cycle in the city. Solutions are based on infiltration, storage and evapotranspiration for stormwater and local reuse for wastewater.

The main objective is to identify the sub-catchment(s) and its associated degree of disconnection to reduce combined sewer overflows (CSO) volume, frequency and pollutants loads.

Water balance during a rainfall event (06/08/2007); Ecully catchment.

  Violeta A. Montoya-Coronado*, Pascal Molle**, Damien Tedoldi*, Hélène Castebrunet*, Gislain Lipeme Kouyi* * UNIV LYON, INSA Lyon, Waste Water Environment and Pollution laboratory, Villeurbanne, France **INRAE, Research Unit REVERSAAL, Villeurbanne, France


Ahmed Dermaj

Najat Hajjaji

Suzanne Joiret

Kamal Rahmouni

Abdellah Srhiri

Hisasi Takenouti

Vincent Vivier
email: vivivier@ccr.jussieu.fr

Electrochemical and spectroscopic evidences of corrosion inhibition of bronze by a triazole derivative

Keywords: Bronze corrosion, Raman spectroscopy, Electrochemical impedance spectroscopy, Phenyl-triazole-thione, Inhibitor adsorption

Introduction

Bronze, like brass, is a common structural material, and was one of the first alloys developed by ancient metalworkers. Bronze is a copper alloy containing tin, generally in the 1 to 10% range. The addition of tin increases the bronze hardness making it more resistant to wear when compared with pure copper metal. Nowadays, a large variety of bronze compositions is known, and they may contain various other elements, such as lead, zinc, manganese, aluminium, silicon or iron. The variations of bronze composition (both in proportion and in elemental composition) significantly affect its characteristics, such as providing a higher resistance to wear or improving its machining. For instance, the adjunction of small amount of lead in the bronze composition acts as a lubricant. Such bronzes are often used to make machine parts, like bearings, that must ensure a lot of sliding action against other parts.

However, like all other alloys, bronzes are subjected to corrosion, which results in the formation of an oxide or salt layer on its surface. The oxidized layers can result of either an intentional surface treatment, or a spontaneous origin due to the degradation of the object in various corrosive mediums, for instance due to a burying in the soil during many centuries. In this last case, corrosion products formed on bronze surface, once a sufficient thickness is reached, acquire the barrier property preventing a complete destruction of the object. In this way, the archaeological bronze objects are transmitted up to the present time, even in an aggressive media, such as seawater. The layer of corrosion products is generally named "patina", and when protection of the substrate material is efficient, it is frequently called "beautiful or noble patina" [1][2][3].

Nevertheless, after the excavation, the sudden change of the environment breaks off the equilibrium obtained over a long period, and sometimes induces an acceleration of the corrosion processes. As a disturbing media, one may mention the marine and/or urban atmosphere with a high level of pollution. This may lead to a complete decomposition of the archaeological object within a very short period. Consequently, it appears that the use of the protection methods for bronze substrate or the reinforcement of protective property of patina layer become necessary.

The electrochemical study of copper and various Cu-Sn alloys in aqueous solutions are reported by many workers [START_REF] Robbiola | Decuprification Phenomenon of Cu-Sn Alloys in Aqueous Solution at Nearly Neutral pH Condition in METAL 98[END_REF][START_REF] Sidot | [END_REF][6]. The corrosion inhibitors are easy to apply, reasonable in cost and often efficient to protect these materials. Moreover, the reversibility of the treatments is often required in the conservation of art object. The use of inhibitors is already currently applied as a protective treatment on ancient bronzes. Benzotriazole and its derivatives were shown to be efficient corrosion inhibitors for copper and copper base alloys through the formation of insoluble Cu-BTA complexes on the surface [7][8][9][10][11][12][13].

The main objective of this study was to understand the way a triazole inhibitor interacts with a bronze to protect it. Thus, the first part of this work is devoted to study of the corrosion inhibition of a bronze Cu-8%Sn (in wt%) by a triazole derivative, 3-phenyl-1,2,4-triazole-5-thione (PTS) in a NaCl 3% solution. The composition of this alloy is in fact well representative of those of outdoor bronze monuments obtained by casting process. The electrochemical behavior of this bronze was studied by voltammetry and electrochemical impedance spectroscopy (EIS). Its surface state was then characterized by scanning electron microscopy (SEM) coupled with X-ray energy dispersion spectroscopy (EDS) for elemental analysis, and by Raman micro-spectroscopy.

Experimental methods

The bronze used in this study, the electrochemical cell and the conditions at which the experiments were performed are briefly presented below.

Materials and electrode preparation

The alloy used was an industrial bronze, the composition of which was determined by EDS analysis (Fig. 1). This alloy was an as-cast leaded bronze exhibiting a dendritic structure, which was characterized by a low tin (8 wt%) content. However, Fig. 1 clearly evidences that the studied alloy was corresponding to a diphase (Fig. 1b andc) also containing non-miscible lead mostly located in interdendritic spaces (Fig. 1d). The elemental composition of these two phases is reported in Table 1. The amount of tin roughly varies from 5 wt% in the phase I to reach 21 wt% in the phase II. It should also be mentioned that the amount of white globules in Fig. 1d is corresponding to a very low Pb content bronze. From these analyses, the cartography of phase repartition could be drawn (Fig. 1e). However, some impurities (Zn and Ni) were also detected by EDX analysis, but their amount remains low (Table 1).

For the electrochemical studies, the sample was manufactured to form a right cylinder, the cross-section of which was a disk of 1 cm in diameter (surface area S = 0.78 cm 2 ). The lateral part of the bronze rod was firstly protected by a cataphoretic paint (PPG, W781-I292 + W975-G292), heat treated at 150 • C for 30 min, and then molded into an epoxy resin (Buhler, Epoxycure TM ). The overall diameter of the electrode, bronze and epoxy resin was 18 mm in diameter. The bronze cylinder was assembled along its axis of rotation allowing its use as a common rotating disk electrode (RDE). Before each experiment, the electrode was abraded with emery papers (from grade 600 to grade 1200), thoroughly rinsed with distilled water, and dried with argon or nitrogen flow. All electrochemical experiments were performed between twice and five times. No significant discrepancy of the results was observed.

The corrosion inhibitor used in this work was a triazole derivative, the 3-phenyl-1,2,4-triazole-5-thione (PTS), synthesized, purified and characterized according to a protocol describes elsewhere [14]. Briefly, this compound was prepared in two steps; a condensation of the thiosemicarbazide with the benzoyl chloride in pyridinic medium to form the 1benzoylthiosemicarbazide as an intermediate. This reaction was followed by an intra-molecular cyclization in methanol in presence of the sodium methalate. The resulting product was the PTS.

Electrochemical device

The electrochemical cell was a conventional three-electrodecell. The reference electrode and the counter electrode were a saturated calomel electrode (SCE) and a large surface platinum grid, respectively. Both the electrochemical cell and the electrometer were placed in a Faraday cage, and all experiments were performed at room temperature. All electrochemical measurements were carried out with a Gamry potentiostat (model PC4/300). The EIS experiments were performed using a 10 mV rms ac-signal at the open circuit corrosion potential (E oc ) from 50 kHz to 10 mHz with 10 points per decade. The results presented in this study were obtained in 3% NaCl solution prepared from doubly distilled water.

Techniques

Scanning electron microscopy (SEM) studies were performed with a Leica Stereoscan 440 coupled with EDS elemental semi-quantitative analyses (Princeton Gamma-Tech) at 20 keV and 5.3 nA.

The Raman spectroscopy was carried out with a Labram-Jobin-Yvon spectrometer. The samples were irradiated with a helium neon laser at λ = 632.8 nm. The laser power was varied between 0.1 and 1 mW to avoid any thermal effect on sample during the analyses. A confocal microscope was used and the investigated area was limited to 1.2 m 2 using an 80 ULWD objective lens.

Results and discussion

Voltammetry experiments

The cathodic and anodic polarization curves of the bronze obtained with a RDE at 1000 rpm in 3% NaCl solution with various PTS concentrations are presented in Figs. 2 and3, respectively. The range of PTS concentrations investigated in this work was limited since the solubility of the synthesized compound was quite low (between 1 and 1.2 mM). All of these curves were obtained after 1 h immersion of the electrode in the electrolytic solution at open circuit potential (E oc ). Then cathodic and anodic polarization curves were recorded from independent experiments. The initial potential was stated at a slightly more positive potential from E oc for the cathodic scans, and conversely, it was set at a slightly more negative value from E oc for the anodic scans.

In absence of PTS, the cathodic current (Fig. 2) follows the Tafel law for E > -0.5 V/SCE, and then the corrosion the corrosion current density can be determined readily (I corr ≈ 7 Acm -2 ). For potential more negative than -0.7 V/SCE, the current plateau observed is ascribed to the limiting current of the dissolved oxygen reduction [15]. The adding of 0.1 or 0.5 mM of PTS into the corrosive medium is accompanied by both a shift of E corr toward more positive potential, and by a decrease of I corr . It is important to point out that for potential more positive than -0.5 V/SCE, the Tafel slope is independent of the inhibitor concentration. In contrast, the cathodic curves in the most negative potential domain (Fig. 2) are different in presence and in absence of PTS, namely, the limiting current corresponding to the oxygen reduction is always observed but its value is three to five times lower than in absence of PTS. This may be explained, for instance, by the formation of surface film which hinders the diffusion of dissolved oxygen towards the electrode surface in addition to the hydrodynamic diffusion layer.

Close to the open circuit corrosion potential, that is where a linear behavior is expected, the Stern-Geary relationship [16] was considered to be valid:

I = I corr {exp[b a (E -E corr )] -exp[b c (E -E corr )]} (1)
where b a and b c are the anodic and cathodic activation coefficients, respectively. A non-linear regression calculation was applied to evaluate these kinetic parameters, and the results of calculation are reported in Table 2. In all cases, the correlation factor R 2 was greater than 99% indicating a good fitting result. The confident interval for I corr is in the order of one percent whereas E corr was determined with an error margin lower than 1 mV. As stated above, the cathodic Tafel constant b c remains essentially constant with PTS concentration whereas the anodic Tafel constant b a becomes very huge in presence of the inhibitor. This phenomenon can be explained by desorption of the inhibitor when the potential shifted towards more anodic direction as previously reported by Epelboin et al. [17].

The inhibitor efficiency (IE) in percent was evaluated from the following relationship:

IE (%) = 100 × I 0 corr -I corr I 0 corr ( 2
)
where I 0 corr and I corr denote the corrosion currents densities in absence and in presence of inhibitor, respectively. The inhibiting efficiency increases with the PTS concentration for a given temperature and immersion time, and it reaches 96% for the addition of 1 mM PTS.

In absence of inhibitor, the anodic curve (Fig. 3) exhibits two distinct domains: the first zone is located in the vicinity of the corrosion potential and is characterized by a steep Tafel slope whereas the second zone located towards more positive potentials is characterized by a quite constant current density. This limiting current density is likely governed by the solubility of CuCl salt layer [18].

The addition of corrosion inhibitor into the solution is accompanied by a significant decrease of the anodic current density which is more marked near E corr . For instance, at 0 V/SCE, the anodic current density is divided by a factor 20 for a 0.5 mM PTS and by a factor 300 when the PTS concentration reaches 1 mM. These results are fully compatible with the presence of a compact surface film composed of the corrosion products formed during the initial stage of the experiment, that is, when the sample is left at its rest potential for 1 h.

E corr and the kinetic parameters determined by a non-linear regression calculation from the anodic polarization scans close Table 2 Values determined from the non-linear regression calculation for the Stern-Geary relationship for the experimental results presented in Fig. 2 [PTS] (mM) E corr (V vs. SCE) to the open circuit potential are similar to those determined for the cathodic scans, though I corr is slightly higher. The inhibiting efficiency at different PTS concentrations is also similar to those presented in Table 2.

I corr (Acm -2 ) b c (V -1 ) b a (V -

Electrochemical impedance spectroscopy (EIS)

We will firstly determine the effect of the inhibitor concentration for a given immersion time, here 1 h, and then the influence of immersion period will be estimated.

Influence of PTS concentration

Fig. 4 shows EIS spectra collected after 1 h of immersion of the bronze in the electrolytic solution at the rest potential with and without PTS.

In absence of inhibitor, the Nyquist diagram presents a flattened semi-circle with a deformation in the high frequency domain as shown on the insert of Fig. 4. Assuming that this impedance data can be described by two R and C ladder circuits (Fig. 5), the impedance spectrum could be fitted suitably. As a matter of fact, the cross symbol on Fig. 4, obtained by a non-linear regression calculation is close to experimental data. The addition of the PTS at various concentrations also displays a capacitive behavior of the alloy in the NaCl medium (Fig. 4). The polarization resistance R p determined from the extrapolation of the impedance spectra towards the low frequency limit, increases with the inhibitor concentration. R p is about 2 k cm 2 in 3% NaCl solution without PTS, whereas it reaches 100, 600 and 900 k cm 2 for PTS concentrations of 0.1, 0.5 and 1 mM, respectively. However, these diagrams cannot be explained with only two R and C circuits displayed in Fig. 5, and thus, a third R-C circuit was added. The use of such an electrical equivalent circuit is in agreement with the previous results presented and discussed in the references [19,20]. Those R-C couples are ascribed to the following contributions: The high frequency contribution (C f -R f ) is ascribed to the dielectric character of the corrosion products (C f ) due to the formation of thin surface film that is reinforced by the presence of the inhibitor and by the ionic conduction through the pores of the film (R f ). The medium frequency contribution is attributed to the double layer capacitance (C d ) at the electrolyte/bronze interface at the bottom of the pores coupled with the charge transfer resistance (R t ), whereas the low frequency elements are related to the oxidation-reduction contribution of the corrosion products (R F -C F ). There is no contribution of the latter couple in the absence of inhibitor. Moreover, the values determined for this couple from the numerical simulations of the ac diagrams are not discussed in the text since it appears that they are scattered probably due to the fact that they are likely ascribed to corrosion products.

The values determined from a non-linear regression calculation for C f -R f and R t -C d couples are summarized in Table 3.In this calculation, in stead of a pure capacitance, the distribution of the time constant for R-C couple was used.

Z = R 1 + (j × ω × τ) n , where τ = R × C (3)
Here, n is a fractional number comprised between 0 and 1. The use of this parameter, similar to the constant phase element (CPE), allowed the depressed feature of Nyquist plot to be reproduced readily. However, in contrast to CPE, the distribution of time constant τ would be involved in the capacitance as well as in the resistance. It is shown that the capacitance C f decreases with the PTS concentration. In absence of inhibitor, C f is about 10 Fcm -2 , which corresponds to a contribution of a very thin layer. Assuming that the permittivity ε of the patina is similar to that of the cuprite Cu 2 O, that is, ε =12 [START_REF] Tamamushi | Chemical Handbook[END_REF], the thickness of film, d, can Table 3 Parameters determined by non-linear regression of the results presented in Fig. 3 [PTS] (mM) be evaluated by the following relationship:

C f (Fcm -2 ) R f (k cm 2 ) C d (Fcm -2 ) R t (
C f = ε × ε 0 × A d ( 4
)
where A is the electrode surface area and ε 0 is the permittivity of the vacuum. In this calculation, the surface roughness was neglected, and a thickness of 1.2 nm was determined. Such a value corresponds to about three monolayers of cuprite. In presence of 1 mM PTS, C f decreased by a factor of 24, compared with the solution without inhibitor. In the same time, R f increased by 1000. That is, the ionic conductivity of surface film significantly decreases in the presence of the inhibitor, much more than that expected for the mere change of the film thickness.

Assuming that the ionic conduction through film is ensured by pores crossing the both sides of film, and taking account of the increase of film thickness, the total surface of pores should then be divided by 40. As a result, the capacitance C d corresponds to the contribution of the surface of the bronze in contact with the electrolytic solution at the pore bottom would be divided by the same factor. Table 3 indicates C d decreased by the factor of 30. Thus, these calculations corroborate the model of porous layer used in this study. The value of n for the surface film is close to 1, which indicates relatively uniform surface film.

The charge transfer resistance increases by a ratio of 400 in the presence of 1 mM PTS. This indicates that the inhibiting effect of PTS is not only due to the decrease of the surface area in contact with electrolyte, but also to the slowing down of the charge transfer process at the interface. The inhibiting effect of the PTS can also be assessed, assuming that I corr is reciprocally proportional to R t :

IE (%) = 100 × R t -R 0 t R t (5)
IE(%) is then larger than 99% when 1 mM of PTS is added in the 3% NaCl solution. However, one should keep in mind that Eq. ( 5) gives a way to evaluate the inhibitor efficiency taking the R t value without inhibitor as reference. Thus, even if the difference between IE = 98.8% (obtained for 0.1 mM PTS) and IE = 99.8% (obtained for 1 mM PTS) seems to be small, it corresponds in reality to a significant improvement of the corrosion protection by the PTS.

Influence of the immersion time

Fig. 6 presents the effect of the immersion time on the impedance spectra at the corrosion potential. The inhibitor concentration was set at 1 mM. The shape of these curves is very similar to that obtained when varying the inhibitor concentration. The polarization resistance (determined at the low frequency limit of the impedance spectrum) changes from about 1 M cm 2 after 1 h to approximately 5.5 M cm 2 after 20 h immersion. This increase illustrates the protective effect of the PTS, which is reinforced with the immersion time.

The Fig. 7a shows the surface state of bronze used before the experiment, that is, before the immersion of the electrode in the electrolytic solution. As discussed in the experimental section, its surface is inhomogeneous. The white spots correspond to Pb-rich zones as it was already observed on an archaeological bronze coin of the Post Roman Empire discovered in Morocco [START_REF] Serghini-Idrissi | [END_REF]. After the immersion of the bronze electrode in a 3% NaCl solution (Fig. 7b), corrosion products form a cracked film on the electrode surface. However, in presence of PTS (Fig. 7c), the whole surface of the bronze appears to be poorly affected by corrosion and it should be mentioned that even the lead globules appear to be not dissolved. The surface analysis by EDS (not presented here) shows the presence of oxygen and chloride in addition to the three major elements (Cu, Sn and Pb) when bronze was immersed in NaCl 3% with or without inhibitor. The corrosion products would be thus mainly in the oxide form. Moreover, in the presence of inhibitor, sulphur characterizing the presence of PTS at the film surface was evidenced.

Raman spectroscopy

The three Raman spectra collected on the bronze electrode when immersed in NaCl solution with 1 mM L -1 of PTS, with real intensity in counts s -1 coming from the nearest part of the metal (a) up to the interface of corrosion products with solution (c) are depicted in Fig. 8. These in-depth profiles were obtained thanks to the confocal microscope.

Raman spectrum of PTS in solid state is shown in Fig. 9a, the same spectrum is re-plotted in Fig. 9b in enlarged scale by truncating the three main bands marked by "*" and ascribable to the phenyl group. The spectrum of methyl-triazole thione (MTS) is presented in Fig. 9c.

Reference spectra

Due to the scarceness of reference data for the attribution of the Raman bands to the vibration modes of PTS, Raman spectra of reference products were recorded. As this work is not devoted to an assignment of PTS vibration modes, only the bands useful for the interpretation of the compounds found on bronze electrode were analyzed. For simplicity, various bands reported in the literature for similar compounds are summarized in Table 4: the triazole [23], the triazolethione [24], the 3-mercaptotriazole [25] and the triazolethione anion for the triazole moiety [26].

The two products used as references are clearly in thione form since the band characteristic of thiol function [25] at 2555 cm -1 is not observed in the Raman spectra (not shown for methyl). According to literature data [24][25][26] the band at Triazolethione anion [19] Mercaptotriazole [18] Triazole [16] 1600 422 cm -1 (phenyl)-427 cm -1 (methyl) is to be attributed to C S vibration.

The four main bands at 1002, 810, 795 and 618 cm on PTS spectrum are due to phenyl vibration as they are absent in methyl spectrum. Moreover, their value are in accordance with those reported for substituted benzene and their modes, respectively, trigonal ring breathing ν 12 , umbrella ν 11 and ν 10 (both C H out of plane) and in plane ring bending ν 6b , are deduced from references [27,28].

In order to compare with the methyl spectrum these bands are reduced in Fig. 9b. The common features of the two spectra have to be attributed to triazole unit: a red shift of triazole bands is observed when changing from phenyl to methyl substitute more pronounced comparing them with non-substituted triazole. The six main ring vibrations are observed and the same frequency shift (100 cm -1 ) between single (ν 2 ) and double (ν 1 ) bonded C N stretching vibrations is measured as for all the ref- erences compounds from Table 4. Values for N H bending are given without any confidence as they are clearly dispersed in the reference data.

Corrosion layer

The two bands at 470 and 620 cm -1 are characteristic of the passive "oxide" layer on copper in neutral solution [29,30] showing that the inhibitive property of the molecule is not complete as already determined from electrochemical results. These bands appear in the three spectra.

However, Raman bands characteristic of organic compounds are clearly detected on the three spectra. It is important to recall that the inhibitor solution (at the concentration used in this study) does not give any evidence of Raman signal coming from organics. Therefore, the Raman spectra observed here is indeed related to the species adsorbed on the bronze surface, only.

The huge intensity of spectrum (Fig. 8a) can be attributed to a surface enhanced Raman scattering (SERS) effect coming from the roughness of the copper part of the metal still suffering a corrosion process. The fact that this spectrum is characteristic of the part of the corrosion product nearest to the surface is also in favor of a SERS mechanism as the enhancement of Raman spectra is confined to a layer thickness of a few nanometers. In this case, the vibration modes having dipole changes perpendicular to the surface are expected to show the greatest enhancement [31].

The three main bands observed in the spectrum correspond to phenyl vibration with a red shift due to interaction with the surface through -electrons from the ring. The bands corresponding to C H out of plane vibrations are the greatest in this spectrum suggesting that phenyl ring is almost flat on the copper surface leading to the greatest enhancement for vibrations perpendicular (out of plane) to it.

The second spectrum still shows the main band of phenyl part of the inhibitor but fairly less intense. Now, the bands corresponding to the triazole moiety are clearly displayed in this spectrum with two new features: a band at 1520 cm -1 and another one on the side of copper passive layer spectrum at 701 cm -1 . This last value is close to the band measured at 707 cm -1 in mercaptotriazole and attributed to C SH unit. Formation of a thiol complex between copper and PTS (or adsorption of the molecule by sulphur through a thiol bonding) is already indicated this band. The new band at 1520 cm -1 , in the range of C N vibration values, is also in favor of this mechanism together with the frequency decrease of the band at 1485 cm -1 appearing now at 1440 cm -1 in the spectrum. The band at 1520 cm -1 can be attributed to the new C N bond formed in the thiol form of the molecule and frequency shift to the weakening of C NH though those bands are known as not pure vibration modes [32]. The Raman spectrum observed in the intermediate part of the corrosion layer shows that a bond exists between copper and the inhibitor in the thiol form.

The third Raman spectrum (the less intense one), observed at the external part of corrosion products is much more difficult to interpret due to the lack of definition of bands in the range 1200-1600 cm -1 .However,itisknown [33] that triazole form with copper ions (cuprous and cupric forms) polymeric complexes where two copper ions are bounded with the same triazole molecule (with two different nitrogen in the case of triazole and one sulphur and one nitrogen in the case of triazolethione). Raman spectrum in this case is expected to give a large band extending from 1100 to 1600 cm -1 like in the case of polyaniline [34] but in this polymer, resonance allows the separation of the various contributions.

Conclusions

Thanks to the combination of electrochemical impedance and Raman spectroscopies, the behavior of bronze electrode in NaCl solution in presence of 3-phenyl-1,2,4-triazole-5-thione inhibitor has been elucidated. In presence of PTS, it was shown that both anodic and cathodic processes are inhibited. Electrochemical impedance measurements also evidenced the formation of a compact thin-film. The good inhibitive properties of the PTS molecule were ascribed to its adsorption on the metal surface in combination with the release of ions in solution increasing the patina compactness and decreasing the ingress of aggressive anions. Electrochemical measurements gave a quantification of these phenomena through the inhibition efficiency coefficient and the pore dimensions.

Fig. 1 .

 1 Fig. 1. SEM image of the bronze (a), quantitative EDX analyses element by element Cu (b), Sn (c) and Pb (d). Image (e) shows the different phases of the bronze and was obtained from EDX results (phase analysis).

Fig. 2 .

 2 Fig. 2. Cathodic polarization curves of the bronze in a 3% NaCl solution with various PTS concentrations (1000 rpm, ν =1mVs -1 ).

Fig. 3 .

 3 Fig. 3. Anodic polarization curves of the bronze in a 3% NaCl solution with various PTS concentrations (1000 rpm, ν =1mVs -1 ).

Fig. 4 .

 4 Fig. 4. EIS spectra of bronze in a 3% NaCl solution for various PTS concentrations obtained after 1 h immersion at rest potential (1000 rpm). The crosses (+) on each diagram correspond to the simulated spectrum.

Fig. 5 .

 5 Fig. 5. Electrical equivalent circuit used to simulate EIS data for the bronze corrosion in 3% NaCl solution without inhibitor.

Fig. 6 .

 6 Fig. 6. EIS spectra of bronze in a 3% NaCl solution after various immersion time in 3% NaCl + 1 mM PTS solution at rest potential (1000 rpm). The crosses (+) on each diagram correspond to the simulated spectrum.

Fig. 7 .

 7 Fig. 7. Back-scattered electron image of the bronze before corrosion experiment (a), and after 10 h immersion (1000 rpm) in 3% NaCl solution (b) and in 3% NaCl solution+1mMdePTS(c).

Fig. 8 .

 8 Fig. 8. Raman spectra collected on the bronze electrode after 1 h of immersion in NaCl + 1 mM L -1 PTS (1000 rpm), coming from the nearest part of the metal (a and b) up to the interface of corrosion products with solution (c) (real intensity in counts s -1 ).

Fig. 9 .

 9 Fig. 9. (a) Raman spectrum of PTS in solid state, (b) same spectrum re-plotted in in enlarged scale by truncating the three main bands marked by "*" and (c) spectrum of methyl-triazole thione (MTS).

Table

  Composition of the bronze determined by EDX analysis (only element with a significant percentage were reported)

		Sn	Pb	Ni	Zn	Cu
	Phase I (wt%)	5.12	0.59	0.39	0.79	93.11
	Phase II (wt%)	21.29	1.69	0.38	0.29	76.35

Table 4

 4 Raman vibration modes of various groups for the attribution of PTS vibration modes Methyl-triazole thione Phenyl-triazolethione PhenylComplex Phenyl ads Triazolethione[17]

Acknowledgements

The authors gratefully acknowledge Stephan Borensztajn (CNRS-UPR 15) for his efficient help for SEM and EDS observations and interpretations.

This work was in part financially supported by the CNRS-CNRST under contract no. 15753.


M-C Bernard

A Hugot-Legoff

S Joiret

N N Dinh

N N Toan

Polyaniline layer for Iron Protection in sulfate medium

Keywords: Polyaniline, elecrvsition, oxalic acid, phosphoric acid, iron corr osion 1. Introduction

Polyaniline has been potentiostatically elecrueposited on iron in acidic medium to protect the electrode from corrosion. Two acids have been tested: oxalic and phosphoric acids to look for the influence of the passive layer on the rpstem behavior. Influence of additives in the deposition have been examined together with temperature. Careful control of temperatr}e deposition is necessary due to the high activation energy of the process. Metanilic and camphorsulfonic acids act as plasticisers for the polymer layer but also have beneficial effects on electrodeposition by lowering the passive potential and active cr~rents.

iron oxalate layer). We compared a layer elecrtdeposited by a Electroactive polymers have been considered for non-noble metal protection against corrosion [l,2,3,4] for two decades.

Recent works report polyaniline coating on iron by electrodeposition of the polymer layer in oxalic media by cyclic voltamerw [5] or galvanostatic deposition [6]. However no evidence of beneficial effect on iron protection is given in these two papers except the diminution of corro sion crr ents which may be attributed to the presence of a varnish type layer of P ANI which blocks the iron dissolution and not to an electroactive interaction between the polymer and the metal.

Experimental

Iron rods ($=5mm) from Goodfellow have bee n annealed for one week at 1000°C under vacuum, then a cataphoretic lacquer is applied and cured at I 80°C for 2 hours to insure a good protection of the electrode edges. The electrochemical experiments are performed rth an Ecochemie Autolab system.

The deposition solution is temperature controlled with a Neslab thermostatic bath. After deposition the polyaniline layer is insulated on its edges by a varnish application as we have noticed that the layer is thinner near the cataphoretic layer. This may be arxbuted to a repulsive effect between the two polrers or due to the potential distribution effect. Corrosion solution is O. lM K2S04acidifi ed to pH 4.5 or 2 by sulfuric acid.

J. Results

I Deposition technique

Cyclic voltametry is forbidden to elecrropolymerize polyaniline in the case of iron (iron electrode reactivation takes place before the polymer reduction, therefore a deposit of this rye is a multilayered system alternating oxidized polymer and

Influence of anions in solution

As it is kno\\n that sulfonated anions have a positive effect on redox behavior of polyaniline [8] we have tried to introduce two of them in deposition solution by addition of the 0.05M corresponding acid. One is metanilic (3-aminobenzenesulfonic acid), the other one is camphorsulfonic (HCSA). These anions have two beneficial effects in the case of PANI formation on iron.

4.Conclusion

We have defmed conditions for PANI electropolymerization on iron and shown that the deposition method has a great infl uence on the electroactivity of the polymer. The nature of the passive layer is also infl uent on the protection offered by the polymer. Studies of interactions between the passive layer and polyaniline are now under investigation.

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: (a) potential versus time for a galvanostatic deposit; (b) charge versus time for a potentiostatic deposit at 300m V/SSE; ( c) cyclic voltammogram at l Om V/sec in pH 4. 5 solution, solid line: potentiostatic, dotted line: galvanostatic We can see in fig (le) that the galvanostatic deposit is non electroactive, this is due to the high potential of deposition, inside the oxygen evolution range. To get an electroactive polymer deposition potential must be controlled. With this technique the obtained layer is thin but further increase of thickness is possible if the electrode is cycled in the solution of deposition. The layer already present on the electrode stabilizes iron and no further reactivation of the passive layer is seen.3.2 Influence of temperatureAs we can see in figure1b the charge versus time presents a sharp increase corresponding to iron passivation (riro followed

Figure 4 :

 4 Figure 4: cyclic voltametry on iron electrode at lmV/sec, rotation speed 125 rpm in: -2M phosphoric acid+ 0.5M aniline ; •••••• 2M phosphoric acid + 0. SM aniline + 0. IM metanilic acid ; o.o.o.o2M phosphoric acid+ 0.5M aniline+ O. lM HCSA We can see in figure 4 that the presence of sulfonated ions in the deposition solution lowered the potential of passivation of iron in phosphoric acid. Since this potential must be as low as possible to avoid fonnation of overoxidized monomers during PANI electropolymerization, these anions have benefi cial effects. A comparative study by scanning electron microscopy reveals that the polymer is more compact when these anions are present.