Matthieu Dussaule

Marc Peigné

Samuel Tapie

A LOCAL LIMIT THEOREM FOR CONVERGENT RANDOM WALKS ON RELATIVELY HYPERBOLIC GROUPS

We study random walks on relatively hyperbolic groups whose law is convergent, in the sense that the derivative of its Green function is finite at the spectral radius. When parabolic subgroups are virtually abelian, we prove that for such a random walk satisfies a local limit theorem of the form pn(e, e) ∼ CR -n n -d/2 , where pn(e, e) is the probability of returning to the origin at time n, R is the inverse of the spectral radius of the random walk and d is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate.

1. Introduction 1.1. General setting. Consider a finitely generated group Γ and a probability measure µ on Γ. The µ-random walk on Γ starting at γ ∈ Γ is defined as X γ n = γg 1 ...g n , where (g k ) are independent random variables of law µ in Γ. The law of X γ n is denoted by p n (γ, •). For γ = e, it is given by the convolution powers µ * n of the measure µ. The Local Limit problem consists in finding the asymptotic behaviour of p n (e, e) when n goes to infinity.

The action by isometries of a discrete group on a Gromov-hyperbolic space (X, d) is said to be geometrically finite if for any o ∈ X, the accumulation points of Γo on the Gromov boundary ∂X are either conical limit points or bounded parabolic limit points. We refer to Section 2.1 below for a definition of these notions. A finitely generated group Γ is relatively hyperbolic with respect to a collection of subgroups Ω = {H 1 , ..., H p } if it acts via a geometrically finite action on a proper geodesic Gromov hyperbolic space X, such that, up to conjugacy, Ω is exactly the set of stabilizers of parabolic limit points for this action. The elements of Ω are called (maximal) parabolic subgroups. We will often assume that parabolic subgroups are virtually abelian.

In this paper, we prove a local limit theorem for a special class of random walks on relatively hyperbolic groups. We will always assume in the sequel that µ is admissible, i.e. its support generates Γ as a semigroup, symmetric, i.e. for every g, µ(g) = µ(g -1 ), and aperiodic i.e. p n (e, e) > 0 for large enough n.

On the one hand, it is known that aperiodic random walks with exponential moments on virtually abelian groups of rank d satisfy the following local limit theorem,

The first author has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under the Grant Agreement No 759702.

see [START_REF] Woess | Random Walks on Infinite Graphs and Groups[END_REF]Theorem 13.12] and references therein : [START_REF] Ancona | Positive harmonic functions and hyperbolicity[END_REF] p n (e, e) ∼ CR -n n -d/2 , where C is a positive constant and R ≥ 1 is the inverse of the spectral radius of the random walk.

On the other hand, Gouëzel [START_REF] Gouëzel | Local limit theorem for symmetric random walks in Gromov-hyperbolic groups[END_REF] proved that for finitely supported, aperiodic and symmetric random walks on non-elementary hyperbolic groups, the local limit theorem has always the following form: [START_REF] Bingham | Regular variation[END_REF] p n (e, e) ∼ CR -n n -3/2 , where, again, C is a positive constant and R the inverse of the spectral radius of the random walk. Notice that R > 1 since non-elementary hyperbolic groups are non-amenable, see [START_REF] Kesten | Full Banach mean values on countable groups[END_REF].

On relatively hyperbolic groups, the first author proved in [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups II : the non spectrally degenerate case[END_REF] that the local limit theorem (2) still holds provided the random walk is spectrally non degenerate. This will be precisely defined in Definition 3.1 below, see also [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF]Definition 2.3]. Roughly speaking, the random walk is spectrally degenerate along a parabolic subgroup H if it gives enough weight to H. When it is spectrally non degenerate along all the parabolic subgroups in Ω, its is called spectrally non degenerate. The spirit of the result in [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups II : the non spectrally degenerate case[END_REF] is that a spectrally non degenerate random walk mainly sees the underlying hyperbolic structure of the group.

In contrast, for spectrally degenerate random walks, one would expect to see in the local limit theorem the appearance of a competition between the exponents d/2 and 3/2, related to the competition between parabolic subgroups and the underlying hyperbolic structure.

The simplest examples of relatively hyperbolic groups are free products. Candellero and Gilch [START_REF] Candellero | Phase transitions for random walk asymptotics on free products of groups[END_REF] gave a complete classification of local limit theorems that can occur for nearest neighbour random walks on free products of finitely many abelian groups; in this context, these free factors play the role of parabolic subgroups. They indeed proved that whenever the random walk gives enough weight to the free factors, the local limit theorem is given by [START_REF] Ancona | Positive harmonic functions and hyperbolicity[END_REF] as in the abelian case, whereas it is of the form (2) in the remaining cases, see in particular the many examples given in [6, Section 7].

Our paper is devoted to the general study of local limit theorems for the so called convergent random walks on a relatively hyperbolic group. In this case, the parabolic subgroups have the maximal possible influence on the random walk: we make a precise presentation in the next subsection. The main results of these paper are valid when parabolic subgroups are abelian. Nevertheless, let us emphasize that a large part of the following study remains valid for any convergent random walk, see Remark 1.1 below. 1.2. Main results. Let µ be a probability measure on a relatively hyperbolic group Γ. Denote by R µ the inverse of its spectral radius, that is the radius of convergence of the Green function G(x, y|r), defined as G(x, y|r) = n≥0 p n (x, y)r n . This radius of convergence is independent of x and y. Definition 1.1. Let Γ be a relatively hyperbolic group and let µ be a probability measure on Γ. We say that µ, or equivalently the random walk driven by µ, is convergent if d dr |r=Rµ G(e, e|r) < +∞.

Otherwise, µ is said to be divergent.

This terminology was introduced in [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF]. It comes from the strong analogy between randow walks on relatively hyperbolic groups on the one hand and geodesic flow on geometrically finite negatively curved manifolds on the other hand. We will describe this analogy in Section 1.3 below. Spectrally non degenerate random walks on relatively hyperbolic groups are always divergent as shown in [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF]Proposition 5.8]. All non-elementary cases presented above which show a local limit theorem of abelian type (1) come from a convergent random walk.

Hence, if µ is convergent, then it is necessarily spectrally degenerate along some parabolic subgroup. Moreover, whenever parabolic subgroups are virtually abelian, each of them has a well-defined rank. Definition 1.2. Let Γ be a relatively hyperbolic group with respect to virtually abelian subgroups and let µ be a convergent probability measure on Γ. The rank of spectral degeneracy of µ is the minimal rank of a parabolic subgroup along which µ is spectrally degenerate.

The central result of this paper is the following local limit theorem.

Theorem 1.3. Let Γ be a finitely generated relatively hyperbolic group with respect to virtually abelian subgroups. Let µ be a finitely supported, admissible, symmetric and convergent probability measure on Γ. Assume that the corresponding random walk is aperiodic. Let d be the rank of spectral degeneracy of µ. Then for every x, y ∈ Γ there exists C x,y > 0 such that p n (x, y) ∼ C x,y R -n µ n -d/2 . If the µ-random walk is not aperiodic, similar asymptotics hold for p 2n (x, y) if the distance between x and x ′ is even and for p 2n+1 (x, y) if this distance is odd.

Note that by [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF]Proposition 6.1], the rank of any virtually abelian parabolic subgroup along which µ is spectrally degenerate is at least 5. Therefore this local limit theorem cannot coincide with the one given by (2) when µ is spectrally non degenerate. We get the following corollary.

Corollary 1.4. Let Γ be a finitely generated relatively hyperbolic group with respect to virtually abelian subgroups. Let µ be a finitely supported, admissible, symmetric and convergent probability measure on Γ such that the corresponding random walk is aperiodic. Let d be the rank of spectral degeneracy of µ. Denote by q n (x, y) the probability that the first visit in positive time of y starting at x is at time n. Then for every x, y ∈ Γ there exists C ′

x,y > 0 such that

q n (x, y) ∼ C ′ x,y R -n µ n -d/2
. By the results of Candellero and Gilch [6, Section 7], convergent measures do exist. We do not attempt in this paper to systematically construct such measure on any relatively hyperbolic group with virtually abelian parabolic subgroups .

It follows from the proof that a local limit theorem analogous to our Theorem 1.3 can be shown (even without assuming virtually abelian parabolic subgroups) as soon as (Γ, µ) satisfy two conditions:

• The Martin boundary of (Γ, µ) is stable in the sense of Definition 2.6;

• The Martin boundary of the first return kernel to any dominant parabolic subgroup is reduced to a point at the spectral radius. We refer to Theorem 5.4 for a precise general statement. An important step in our study, which is of independent interest, is hence the following fact.

Theorem 1.5 (Theorem 4.1). Let Γ be a finitely generated relatively hyperbolic group with respect to virtually abelian subgroups. Let µ be a finitely supported, admissible and symmetric probability measure on Γ. Then the Martin boundary of (Γ, µ) is stable.

This completes the results of [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF]. Along Section 3, we prove precise results on the asymptotics of the Green function in a virtually abelian finitely generated group at the spectral radius which will show this stability.

Remark 1.1. Our proof relies on some important properties satisfied by the virtually abelian parabolic subgroups H 1 , . . . , H p , presented in Sections 3 and 4. Once these properties can be extended to the case where the H 1 , . . . , H p are virtually nilpotent parabolic subgroups, our main result Theorem 1.3 and its consequences may be extended to this more general setting, by using Theorem 5.4 and following the proofs of Section 5.

For all k ∈ N, we write

I (k) (r) = x1,...,x k ∈Γ G(e, x 1 |r)G(x 1 , x 2 |r)...G(x k-1 , x k |r)G(x k , e|r).
It follows from Lemma 5.1 that (Γ, µ) is convergent if and only if I (1) (R µ ) < +∞. For all parabolic subgroup H < Γ, we write

I (k) H (r) = x1,...,x k ∈H G(e, x 1 |r)G(x 1 , x 2 |r)...G(x k-1 , x k |r)G(x k , e|r).
The following terminology was introduced in [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF].

Definition 1.6. A symmetric admissible and finitely supported random walk µ on a relatively hyperbolic group Γ is said to be spectrally positive recurrent if:

(1) µ is divergent, i.e. I (1) (R µ ) = +∞;

(2) for all parabolic subgroup H < Γ, I

H (R µ ) < +∞. Any random walk which is spectrally non degenerate is spectrally positive recurrent, see [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF]Proposition 3.7]. The terminology positive recurrent is classical for the study of countable Markov shift, see for instance to [START_REF] Vere-Jones | Geometric ergodicity in denumerable Markov chains[END_REF], [START_REF] Gurevich | Thermodynamic formalism for countable symbolic Markov chains[END_REF] and [START_REF] Omri | Thermodynamic formalism for countable Markov shifts[END_REF]. The analogous of spectral non degeneracy is given for countable Markov shifts by the notion of strong positive recurrence, also called stable positive recurrence, see [START_REF] Gurevich | Thermodynamic formalism for countable symbolic Markov chains[END_REF] or [START_REF] Omri | Phase transitions for countable Markov shifts[END_REF]. In our setting, this terminology has been inspired by the close analogy with the study of the geodesic flow on negatively curved manifolds (see Section 1.3 below and [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF]Section 3.3] for more details).

We will discuss in Section 7 the relationships between the divergence of the random walk and its spectral positive recurrence. It happens that, when parabolic subgroups are virtually abelian, both are equivalent unless the random walk is spectrally degenerate along a parabolic subgroup of rank 5 or 6. This allows us to classify almost all possible behaviours for p n (e, e) on a relatively hyperbolic group whose parabolic subgroups are virtually abelian, as illustrated by the following corollary. Notation. For two functions f and g, we write f g if there exists a constant C such that f ≤ Cg. Also write f ≍ g if both f g and g f . If the implicit constants depend on a parameter, we will avoid this notation.

Corollary 1.7. Let Γ be a relatively hyperbolic group whose parabolic subgroups are virtually abelian. Let µ be a finitely supported admissible symmetric probability on Γ. Assume that µ is spectrally non degenerate along a parabolic subgroup of rank 5 or 6. Then one of the following possibilities occurs.

[11], Theorem 1.4: If µ is spectrally positive recurrent, then as n → +∞, p n (e, e) ≍ CR -n n -3/2 .

[12], Theorem 1.1: If µ is spectrally non degenerate, then as n → +∞,

p n (e, e) ∼ CR -n n -3/2 . Theorem 1.3: If µ is convergent, then as n → +∞, p n (e, e) ∼ CR -n n -d/2 ,
where d is the spectral degeneracy rank of µ.

We conjecture that for any spectrally positive recurrent walk (even spectrally degenerate), there should be a Local Limit theorem p n (e, e) ∼ CR -n n -3/2 . If Γ has parabolic subgroups which are virtually abelian of rank 5 or 6, it is possible to construct examples which are divergent but not spectrally positive recurrent, whose local limit theorem is not classified in this corollary. Examples of such groups with their corresponding local limit theorems are detailed in a forthcoming article [START_REF] Dussaule | Exotic local limit theorems at the phase transition in free products[END_REF]; see also the remark 7.1 at the end of the present paper.

1.3. Geodesic flow on negatively curved manifolds and random walks. For convenience of the reader, we present now some results on the ergodic properties of the geodesic flow on geometrically finite manifolds with negative curvature, which has influenced this work.

Let (M, g) = ( M , g)/Γ be a complete Riemannian manifold, where Γ = π 1 (M ) acts discretely by isometries on the universal cover ( M , g). Assume that M has pinched negative curvature, i.e. its sectional curvatures κ g satisfy the inequalities -b 2 ≤ κ g ≤ -a 2 < 0 for some constants b > a > 0. Also assume that the action of Γ on ( M , d g ) is geometrically finite (see above or Section 2.1 below). By definition Γ is hence relatively hyperbolic with respect to a finite family of parabolic subgroups H 1 , ..., H p , and the pinched curvature hypothesis implies that the H k are virtually nilpotent. The interested reader will find in [START_REF] Bowditch | Geometrical finiteness with variable negative curvature[END_REF] several other equivalent definitions of geometrical finiteness in the context of smooth negatively curved manifolds.

For each discrete group H acting by isometries on ( M , g), and for all s > 0, the Poincaré series of H at s is

P H (x, y|s) = γ∈H e -sdg (x,γy) ∈ (0, +∞].
There is a δ H ≥ 0 independent of x, y such that this series converges if s > δ H and diverges if s < δ H . This quantity is called the critical exponent of H. The action of H on ( M , g) is called convergent if P H (x, y|δ H ) < +∞, and divergent otherwise.

Let µ be a random walk on Γ. We define for x, y ∈ Γ the symmetrized r-Green distance by

(3) d r (x, y) = log G µ (x, y|r) G µ (e, e|r) + log G µ (y, x|r) G µ (e, e|r) .
This (signed) "distance" was introduced in [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF] and is an elaborated version of the classical Green distance defined by Blachère and Brofferio [START_REF] Blachère | Internal diffusion limited aggregation on discrete groups having exponential growth[END_REF]. By Lemma 2.7, for all x ∈ Γ,

P µ (x, y|r) = γ∈Γ e dr(x,γy) = 1 G(e, e|r) 2 z∈Γ G(x, z|r)G(z, x|r) ≍ d dr |r=Rµ G(x, x ′ |r).
As emphasized by our notation, the Poincaré series P Γ (x, y|s) of Γ is the analogous in this context of group action on a metric space to P µ (x, y|r) for the random walk on Γ, which is of same order as the r-derivative of the Green function. The Riemannian metric g has the role of the law µ. The (logarithm of the) critical exponent δ H of the Poincaré series has the same role as the radius of convergence of the Green function. The local limit theorem describes the asymptotic behavior as n → +∞ of the quantity p (n) (x, y) for any x, y ∈ Γ; in the geometrical setting, it is replaced by the orbital counting asymptotic, that is the asymptotic behavior as R → +∞ of the orbital function N Γ (x, y, R) defined by : for all x, y ∈ M , N Γ (x, y, R) := # {γ ∈ Γ ; d(x, γy) ≤ R} .

The following definition, which is the analogous in this context to our previous Definition 1.6, comes from the results of [START_REF] Dal | Séries de Poincaré des groupes géométriquement finis[END_REF] even though the terminology has been fixed in [START_REF] Pit | Finitude des mesures de Gibbs sur les variétés non compactes à courbure négative pincée[END_REF] (in the full general setting of negatively curved manifolds, not necessarily geometrically finite). Definition 1.8. Let (M, g) = ( M , g)/Γ be a geometrically finite Riemannian manifold with pinched negative curvature, where Γ = π 1 (M ). Let o ∈ M be fixed. The action of Γ on ( M , g) is said to be positive recurrent if [START_REF] Ancona | Positive harmonic functions and hyperbolicity[END_REF] the action of Γ on ( M , g) is divergent;

(2) for all parabolic subgroup H ⊂ Γ, h∈H d(o, h.o)e -sd(o,po) < +∞.

We refer to [START_REF] Pit | Finitude des mesures de Gibbs sur les variétés non compactes à courbure négative pincée[END_REF]Definition 1.3] for a definition of positive recurrence beyond geometrically finite manifolds. The action of Γ is said to be strongly positive recurrent -in the literature, one also says that γ has a critical gap -if for all parabolic subgroup H ⊂ Γ, we have δ H < δ Γ . This is the analogue of the notion of spectrally non degeneracy for random walks on relatively hyperbolic groups. Theorem A of [START_REF] Dal | Séries de Poincaré des groupes géométriquement finis[END_REF] shows that strongly positive recurrent actions are positive recurrent (only the divergence is non-trivial). This has later been shown for more general negatively curved manifolds in [START_REF] Schapira | Regularity of entropy, geodesic currents and entropy at infinity[END_REF]Theorem 1.7] and the analogous result for random walks is given by Proposition 3.7 of [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF]. Moreover, Theorem B of [START_REF] Dal | Séries de Poincaré des groupes géométriquement finis[END_REF] shows that the action is positive recurrent if and only if the geodesic flow admits an invariant probability measure of maximal entropy. This has been shown for general negatively curved manifolds in [START_REF] Pit | Finitude des mesures de Gibbs sur les variétés non compactes à courbure négative pincée[END_REF]Theorem 1.4]. Combined with Theorem 4.1.1 of [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF], it gives the following asymptotic counting. Theorem 1.9. Let (M, g) = ( M , g)/Γ be a negatively curved manifold.

• If the action of Γ is positive recurrent, then for all x, y ∈ M , there is

C xy > 0 such that, as R → +∞, N Γ (x, y, R) ∼ C xy e δΓR .
• If the action of Γ is not positive recurrent, then for all x, y ∈ M ,

N Γ (x, y) = o e δΓR .
Precising the asymptotics of N Γ (x, y, R) when the action of Γ is not positive recurrent is in general difficult. To the authors' knowledge, the only known examples are abelian coverings (cf [START_REF] Pollicott | Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature[END_REF]), which are not geometrically finite, and geometrically finite Schottky groups whose parabolic factors have counting functions satisfying some particular tail condition and for which asymptotics have been obtained in [START_REF] Dal'bo | Convergence et comptage en mesure infinie[END_REF], [START_REF] Vidotto | Ergodic properties of some negatively curved manifolds with infinite measure[END_REF] and [START_REF] Peigné | Comptage orbital pour certains groupes convergents[END_REF]. Recall that Schottky groups are free products of elementary groups whose limit sets are at a positive distance from each other, see for instance Section 2.4 of [START_REF] Peigné | Comptage orbital pour certains groupes convergents[END_REF] for a definition. In our analogy with random walks on relatively hyperbolic groups, we emphasize hence the following result from [START_REF] Peigné | Transition de phase de fonctions orbitales pour des groupes de schottky en courbure négative[END_REF], which gathers in some particular case results of [START_REF] Vidotto | Ergodic properties of some negatively curved manifolds with infinite measure[END_REF] and [START_REF] Peigné | Comptage orbital pour certains groupes convergents[END_REF]. It is a Riemannian analogous to the work of Candellero and Gilch [START_REF] Candellero | Phase transitions for random walk asymptotics on free products of groups[END_REF] already quoted.

Theorem 1.10. Let (M, g H ) = H 2 /Γ be a hyperbolic surface where Γ is a Schottky group with at least one parabolic free factor H = h . We fix a parameter b ∈ (1, 2). Then, there exists a family (g a,b ) a∈(0,+∞) of negatively curved Riemannian metrics on M obtained by perturbation of the hyperbolic metric g H in such a way:

-the metrics g a,b coincides with g H outside a small neighbourhood (controlled by the value a) of the cuspidal end associated with H;

-the distance d a,b induced by g a,b satisfies the following condition: for any fixed

point x ∈ H 2 , d a,b (x, p n x) = 2 ln |n| + b ln | ln |n|| + O(1)
.

Then, there exists a "critical value" a * > 0 such that :

• if a > a * then the action of Γ on (H 2 , g a ) is strongly positive recurrent. In particular for all x, y ∈ H 2 ,

N Γ (x, y, R) ∼ C xy e δΓR ;
• if a = a * , then the action of Γ on (H 2 , g a ) is divergent but non positive recurrent. Moreover, for all x, y ∈ H 2 ,

N Γ (x, y, R) ∼ C xy e δΓR R 2-b ;
• if a ∈ (0, a * ), then the action of Γ on (H 2 , g a ) is convergent and for all x, y ∈ H 2 ,

N Γ (x, y, R) ∼ C xy e δΓR R b .
It follows from Corollary 1.7 that the critical situation of a divergent but non positive recurrent action which appears above in the context of geometrically finite negatively curved surfaces when a = a * can occur in the context of random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups in the only case when the random walk is spectrally degenerate along some parabolic subgroups of rank 5 or 6.

We end this paragraph with a table that summarizes the different cases that arise in the study of local limit theorems of relatively hyperbolic groups. We also indicate the corresponding results obtained in the framework of geometrically finite non-compact surfaces endowed with the metric g a defined in Theorem 1.10.

Local Limit Theorem

Counting problem Γ spectrally non degenerate critical gap property δ Γ > δ H (µ is spectrally positive recurrent) (Γ is positive recurrent)

p n (x, y) ∼ C x,y R -n µ n -3/2 N Γ (x, y, R) ∼ C xy e δΓR
see [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups II : the non spectrally degenerate case[END_REF] see [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF], [

p n (x, y) ≍ R -n µ n -3/2 N Γ (x, y, R) ∼ C xy e δΓR
see [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF] see [30] [25] Conjecture:

p n (x, y) ∼ C x,y R -n µ n -3/2 µ spectrally degenerate Γ exotic i.e δ Γ = δ H + divergent + + divergent + not spectrally positive recurrent not positive recurrent degeneracy rank 5 or 6 N Γ (x, y, R) ∼ C xy e δ Γ R R 2-b
possible exotic Local limit theorem see [START_REF] Dussaule | Exotic local limit theorems at the phase transition in free products[END_REF] see [START_REF] Vidotto | Ergodic properties of some negatively curved manifolds with infinite measure[END_REF], [START_REF] Peigné | Transition de phase de fonctions orbitales pour des groupes de schottky en courbure négative[END_REF] µ convergent Γ convergent (thus µ spectrally degenerate) (thus Γ exotic)

p n (x, y) ∼ C x,y R -n µ n -dµ/2 N Γ (x, y, R) ∼ C xy e δ Γ R R b see Theorem 1.3 see [26]
On the left column, Γ is a relatively hyperbolic group with respect to virtually abelian parabolic subgroups H 1 , . . . , H p (up to conjugacy). We consider an aperiodic µ-random walk on Γ where µ is a finitely supported, admissible and symmetric probability measure on Γ with rank d µ . On the right column, Γ is a geometrically finite Fuchsian group with parabolic subgroups H 1 , . . . , H p (up to conjugacy). We assume that H 2 /Γ is endowed with the metric g a,b and set δ H = max(δ H1 , . . . , δ Hp ).

1.4. Organization of the paper. In Section 2, we present our setting: relatively hyperbolic groups, transition kernels with their Green function and their Martin boundary, and Ancona inequalities which roughly state that the random walk tracks relative geodesics with high probability in a relatively hyperbolic group.

In Section 3, we introduce the first return kernel p H (., .|r) to a parabolic subgroup H of rank d. Assuming that the random walk is spectrally degenerate along H we give asymptotics for the j-th derivative of the Green function associated to p H where j = ⌈d/2⌉ -1 (see Proposition 3.13).

In Section 4, we assume that parabolic subgroups are virtually abelian and show that the Martin boundary is stable in the sense of Definition 2.6 below (see Theorem 4.1). This had already been shown in [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF] when the random walk is spectrally non degenerate; it still holds for convergent (hence spectrally degenerate) random walks.

In Section 5, we assume that the Martin boundary of the full random walk is stable and the Martin boundary of the walk restricted to parabolic subgroups is reduced to a point. Under these conditions, we prove that asymptotics for the j-ith derivative of the Green function of the full random walk are given by the analogous asymptotics for the transition kernels of the first return to the parabolic subgroups along which the walk is spectrally degenerate (see Theorem 5.4).

In Section 6, we gather the ingredients of the three previous section which give asymptotics for the j-th derivative of the full Green function, where j = ⌈d/2⌉ -1 and d is the rank of spectral degeneracy of the walk. Theorem 1.3 follows, applying a Tauberian type theorem shown in [START_REF] Gouëzel | Random walks on co-compact Fuchsian groups[END_REF].

Eventually, in Section 7, we show that, if the parabolic subgroups are virtually abelian and the random walk is divergent, then the random walk is automatically spectrally positive recurrent.

Random walks on relatively hyperbolic groups

2.1. Relatively hyperbolic groups and relative automaticity.

2.1.1. Limit set. Consider a discrete group Γ acting by isometries on a Gromovhyperbolic space X. Let o ∈ X be fixed. Define the limit set ΛΓ as the adherence of Γo in the Gromov boundary ∂X of X. This set does not depend on o.

A point ξ ∈ ΛΓ is called conical if there is a sequence (γ n ) n in Γ and distinct points ξ 1 , ξ 2 in ΛΓ such that, for all ξ = ζ in ΛΓ, the sequences (γ n ξ) n and (γ n ζ) n converge to ξ 1 and ξ 2 respectively. A point ξ ∈ ΛΓ is called parabolic if its stabilizer Γ ξ in Γ is infinite and the elements of Γ ξ fix only ξ in ΛΓ. A parabolic limit point ξ in ΛΓ is said bounded if Γ ξ acts cocompactly on ΛΓ \ {ξ}. The action of Γ on X is said to be geometrically finite if ΛΓ only contains conical limit points and bounded parabolic limit points.

Relatively hyperbolic groups.

There are in the literature several equivalent definitions of relatively hyperbolic groups. Let us present the two characterizations which we will use in this paper. We refer to [START_REF] Bowditch | Relatively hyperbolic group[END_REF], [START_REF] Farb | Relatively hyperbolic groups[END_REF] and [START_REF] Osin | Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems[END_REF] for more details.

Let Γ be a finitely generated groups and S be a fixed generating set. Let Ω 0 be a finite collection of subgroups, none of them being conjugate. Let Ω be the closure of Ω 0 under conjugacy.

The relative graph Γ = Γ(S, Ω 0 ) is the Cayley graph of Γ with respect to S and the union of all H ∈ Ω 0 [START_REF] Osin | Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems[END_REF]. It is quasi-isometric to the coned-off graph introduced by Farb in [START_REF] Farb | Relatively hyperbolic groups[END_REF]. The distance d in Γ is called the relative distance. We also denote by Ŝn the sphere of radius n centered at e in Γ. Eventually, we will call relative geodesic a geodesic in Γ.

Theorem 2.1 ([5]

). Using the previous notations, the following conditions are equivalent.

(1) The group Γ has a geometrically finite action on a Gromov hyperbolic space X such that the parabolic limit points are exactly the fixed points of elements in Ω.

(2) The relative graph Γ(S, Ω 0 ) is Gromov hyperbolic for the relative distance d, and for all L > 0 and all x ∈ Γ, there exists finitely many closed loop of length L > 0 which contains x.

When these conditions are satisfied, the group Γ is said to be relatively hyperbolic with respect to Ω 0 .

Assume now that Γ is relatively hyperbolic with respect to Ω, and let X be a Gromov hyperbolic space on which Γ has a geometrically finite action whose parabolic subgroups are the element of Ω. The limit set ΛΓ ⊂ ∂X is called the Bowditch boundary of Γ. It is unique up to equivariant homeomorphism.

Archetypal examples of relatively hyperbolic groups with respect to virtually abelian subgroups are given by finite co-volume Kleinian groups. In this case, the group acts via a geometrically finite action on the hyperbolic space H n and the Bowditch boundary is the full sphere at infinity S n-1 .

2.1.3. Automatic structure. The notion of relative automaticity was introduced by the first author in [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF]. Definition 2.2. A relative automatic structure -or shortly an automaton -for Γ with respect to the collection of subgroups Ω 0 and with respect to some finite generating set S is a directed graph G = (V, E, v * ) with a distinguished vertex v * called the starting vertex, where the set of vertices V is finite and with a labelling map φ : E → S ∪ H∈Ω0 H such that the following holds. If ω = e 1 , . . . , e n is a path of adjacent edges in G, define φ(e 1 , ..., e n ) = φ(e 1 )...φ(e n ) ∈ Γ. Then,

• no edge ends at v * , except the trivial edge starting and ending at v * , • every vertex v ∈ V can be reached from v * in G, • for every path ω = e 1 , ..., e n , the path e, φ(e 1 ), φ(e 1 e 2 ), ..., φ(γ) in Γ is a relative geodesic from e to φ(γ), i.e. the image of e, φ(e 1 ), φ(e 1 e 2 ), ..., φ(γ) in Γ is a geodesic for the metric d, • the extended map φ is a bijection between paths in G starting at v * and elements of Γ. This statement is proved by first constructing an automaton that encodes relative geodesics, then showing that there exist finitely many relative cone-types, see [11, Definition 4.7, Proposition 4.9] for more details. To obtain a bijection between paths in the automaton and elements of Γ, one fixes an order on the union of S and all the H ∈ Ω 0 , which allows to choose the smallest possible relative geodesics for the associated lexicographical order.

Relative automaticity is a key point in [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups II : the non spectrally degenerate case[END_REF] to prove a local limit theorem in the spectrally non degenerate case. It is again of crucial use in this paper, see Section 5. When the total mass is 1 (which we will not require), we call it a probability transition kernel. It then defines a Markov chain on E, i.e. a random process (X n ) n≥0 such that P (X n+1 = b|X n = a) = p(a, b). In general, we will say that p defines a chain on E.

If µ is a probability measure on a finitely generated group Γ, then the kernel p µ (g, h) = µ(g -1 h) is a probability transition kernel and the corresponding Markov chain is the µ-random walk. Definition 2.5. Let p : E × E → R + be a transition kernel on E.

• The Green function associated to p is defined by

G p (x, y) = n≥0 p (n) (x, y) ∈ [0, +∞],
where p (n) is the nth convolution power of p, i.e.

p (n) (x, y) = x1,...,xn-1∈E p(x, x ′ 1 )p(x 1 , v 2 ) • • • p(x n-1 , y).
• The chain defined by p is finitely supported if for every x ∈ E, the set of y ∈ E such that p(x, y) > 0 is finite. • The chain is admissible (or irreducible) if for every x, y ∈ E, there exists n such that p (n) (x, y) > 0. • The chain is aperiodic (or strongly irreducible) if for every x, y ∈ E, there exists n 0 such that ∀n ≥ n 0 , p (n) (x, y) > 0. • The chain is transient if the Green function is everywhere finite.

Consider a transition kernel p defining an irreducible transient chain. For y ∈ E, define the Martin kernel based at y as

K p (x, y) = G p (x, y) G p (o, y) .
The Martin compactification of E with respect to p and o is a compact space containing E as an open and dense space, whose topology is described as follows.

A sequence (y n ) n in E converges to a point ξ in the Martin compactification if and only if the sequence (K(•, y n )) n converges pointwise to a function which we write K(•, ξ). Up to isomorphism, it does not depend on the base point o and we denote it by E p . We also define the p-Martin boundary (or Martin boundary, when there is no ambiguity) as ∂ p E = E p \ E. We refer for instance to [START_REF] Sawyer | Martin boundaries and random walks[END_REF] for a complete construction of the Martin compactification.

The Martin boundary contains a lot of information. It was first introduced to study non-negative harmonic functions. We will use it here to prove our local limit theorem.

Let us now define the notion of stability for the Martin boundary, following Picardello and Woess [START_REF] Picardello | Examples of stable Martin boundaries of Markov chains[END_REF]. Assuming that p is irreducible, the radius of convergence of the Green function G p (x, y) is independent of x and y. Denote it by R p and for all 0 ≤ r ≤ R µ let us set G p (x, y|r) = G rp (x, y), i.e.

G p (x, y|r) = n≥0 r n p (n) (x, y).
Also set K(x, y|r) = K rp (x, y). The Martin compactification, respectively the boundary associated with K(•, •|r), is called the r-Martin compactification, respectively the r-Martin boundary, and is denoted by E rp , respectively by ∂ rp E.

Definition 2.6. The Martin boundary of E with respect to p is stable if the following conditions hold.

(1) For every x, y ∈ E, we have G p (x, y|R p ) < +∞ where R p is the radius of convergence of the Green function. (2) For every 0 < r 1 , r 2 < R p , the sequence (K(•, y n |r 1 )) n converges pointwise if and only if (K(•, y n |r 2 )) n converges pointwise, i.e. the r 1 and r 2 -Martin compactifications are homeomorphic. For simplicity we then write ∂ p Γ for the r-Martin boundary whenever 0 < r < R p .

(3) The identity on Γ extends to a continuous and equivariant surjective map

φ p from E ∪∂ p E to E ∪∂ Rp p E. We then write K(x, ξ|R p ) = K(x, φ p (ξ)|R p ) for ξ ∈ ∂ p E. (4) The map (x, ξ, r) ∈ E ×∂ p E ×(0, R p ] → K(x,
ξ|r) is continuous with respect to (x, ξ, r). We say that the Martin boundary is strongly stable if it is stable and the second condition holds for every 0 < r 1 , r 2 ≤ R p ; in this case, the map φ p induces a homeomorphism from the r-Martin boundary to the R p -Martin boundary.

If p is the transition kernel of an admissible random walk on a finitely generated group which is non-amenable, it has been shown by Guivarc'h in [20, p. 20, remark b] that the condition (1) is always satisfied. Note that non-elementary relatively hyperbolic groups are always non-amenable.

The Martin boundary of any finitely supported symmetric admissible random walk on a hyperbolic group is strongly stable. More generally, the Martin boundary of a finitely supported symmetric and admissible random walk on a relatively hyperbolic group is studied in [START_REF] Gekhtman | Martin boundary covers Floyd boundary[END_REF], [START_REF] Dussaule | The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups[END_REF] and [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF]. In particular, whenever the parabolic subgroups are virtually abelian, the homeomorphism type of the r-Martin boundary is described in [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF]. It is proved there that the Martin boundary is strongly stable if and only if the random walk is spectrally non non degenerate. We will prove in Section 4 that stability (but not strong stability) still holds in the spectrally degenerate case.

Let us eventually mention a central computation which we use many times.

Lemma 2.7. Let p : E × E → R + be a transition kernel and for all r ∈ [0, R p ], write again G p (x, y|r) = n≥0 r n p (n) (x, y). Then

d dr (rG p (x, y|r)) = z∈E G(x, z|r)G(z, y|r).
We refer to [11, Lemma 3.1] for a proof, which is a standard manipulation of power series. The generalization to higher derivatives is given by Lemma 5.1.

Standing assumptions. From now, and until the end of this paper, we fix a finitely generated group Γ relatively hyperbolic with respect to a finite collection of parabolic subgroups Ω 0 = {H 1 , ..., H N }. We fix a finitely supported symmetric probability measure µ on Γ whose associated random walk is admissible and irreducible. Eventually, we assume that the support S of µ is a generating set, which is fixed from now on; the distance on Γ is the word distance induced by S. This implies in particular that for all x, y ∈ Γ, if x and y are on the same geodesic in Γ, then there exists n > 0 such that p (n) (x, y) > 0.

In the sequel, we denote by Γ the relative graph Γ(S, Ω 0 ), by R µ the inverse of the spectral radius of µ and by G(x, y|r) the Green function, where 0 ≤ r ≤ R µ and x, y ∈ Γ. As already mentioned, since Γ is not amenable and µ is admissible, it follows from [START_REF] Guivarc | Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire[END_REF] that G(x, y|R µ ) < +∞ for all x, y ∈ Γ.

r n µ(h -1 g 1 )µ(g -1 1 g 2 )...µ(g -1 n-2 g n-1 )µ(g -1 n-1 h ′ );
this quantity is called the relative Green function of paths staying in A except maybe at their beginning and end. Writing A c = Γ\A, the relative Green function p A,r (., .) := G(., .; A c |r) is called the first return kernel to A. For all y ∈ Γ and η > 0, we write

B η (y) = {z ∈ Γ | d(y, z) ≤ η}.
We will use repeatedly the following results.

Proposition 2.8. [13, Corollary 3.7] For every ǫ > 0 and every R ≥ 0, there exists η such that the following holds. For every x, y, z such that y is within R of a point on a relative geodesic from x to z and for every r ≤ R µ ,

G(x, z; B η (y) c |r) ≤ ǫG(x, z|r).
This proposition can be interpreted as follows : with high probability, a random path from x to z has to pass through a neighborhood of y, whenever y is on a relative geodesic from x to z. As a consequence, we have the following. Proposition 2.9. (Weak relative Ancona inequalities) For every R ≥ 0, there exists C such that the following holds. For every x, y, z such that y is within R of a point on a relative geodesic from x to z and for every r ≤ R µ , 1 C G(x, y|r)G(y, z|r) ≤ G(x, z|r) ≤ CG(x, y|r)G(y, z|r).

This is proved by decomposing a trajectory from x to z according to its potential first visit to B η (y), where η is chosen such that G(x, z; B η (y) c |r) ≤ 1/2G(x, z|r) from Proposition 2.8, see precisely [17, Theorem 5.1, Theorem 5.2].

These inequalities were first proved by Ancona [START_REF] Ancona | Positive harmonic functions and hyperbolicity[END_REF] in the context of hyperbolic groups for r = 1. The uniform inequalities up to the spectral radius were proved by Gouezel and Lalley [START_REF] Gouëzel | Random walks on co-compact Fuchsian groups[END_REF] for co-compact Fuchsian groups and then by Gouezel [START_REF] Gouëzel | Local limit theorem for symmetric random walks in Gromov-hyperbolic groups[END_REF] in general. For relatively hyperbolic groups, Gekhtman, Gerasimov, Potyagailo and Yang [START_REF] Gekhtman | Martin boundary covers Floyd boundary[END_REF] proved them for r = 1. The uniform inequalities up to the spectral radius were then proved by the first author and Gekhtman [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF]. They play a key role in the identification of the Martin boundary, see [START_REF] Gekhtman | Martin boundary covers Floyd boundary[END_REF] for several other applications.

Let us mention that there exist strong relative Ancona inequalities (cf [12, Definition 2.14]), that are a key ingredients in [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups II : the non spectrally degenerate case[END_REF] to prove a local limit theorem in the spectrally non degenerate case. However, we do not need them in the present paper.

Asymptotics of the first return to parabolic Green functions

Throughout this section, we fix a parabolic subgroup H ∈ Ω 0 . For η ≥ 0, the η-neighbourhood of H is noted N η (H) .

We introduce below the first return transition kernel to N η (H). The main goal of this section is showing asymptotics for the derivatives of the Green function associated to this first return kernel.

3.1. First return transition kernel and spectral degeneracy. For r ≤ R µ , let p H,η,r (h, h ′ ) = G(h, h ′ ; N η (H) c |r) be the first return kernel to N η (H), i.e. p H,η,r (h, h ′ ) = n≥1 g1,...,gn-1

/ ∈Nη (H) r n µ(h -1 g 1 )µ(g -1 1 g 2 )...µ(g -1 n-2 g n-1 )µ(g -1 n-1 h ′ ).
For simplicity, when η = 0, we write p H,r = p H,0,r .

The n th -convolution power of p H,η,r is noted p

(n)
H,η,r and the associated Green function, evaluated at t, is

G H,η,r (h, h ′ |t) := n≥1 p (n) H,η,r (h, h ′ )t n .
The radius of convergence R H,η (r) of this power series is the inverse of the spectral radius of the associated chain.

For simplicity, we write

R H,η = R H,η (R µ ) and R H = R H,0 (R µ ).
Recall the following definition from [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF].

Definition 3.1. The measure µ, or equivalently the random walk, is said to be spectrally degenerate along

H if R H = 1.
Since H is fixed for the remainder of the section, we drop the index H in the notations. We now enumerate a list of properties satisfied by p η,r and G η,r .

• Since the µ-random walk on Γ is invariant under the action of Γ, the kernel p η,r is H-invariant: in other words, p η,r (hx, hy) = p η,r (x, y) for any h ∈ H and any x, y ∈ N η (H). • By definition, the first return transition kernel satisfies p η,r (x, y) > 0 if and only if there is a first return path in N η (H) from x to y, i.e. if there exists n ≥ 0 and a path x, g 1 , ..., g n , y in Γ with positive probability such that g i / ∈ N η (H) for i = 1, ..., n. • The chain with kernel p η,r is admissible, i.e. for every x, y ∈ Γ, there exists n such that p (n) η,r (x, y) > 0, see [14, Lemma 5.9] for a complete proof. The following lemma shows that when x and y are in N η (H), the relative Green function equals the full Green fucntion. The proof is straightforward (see [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF]Lemma 4.4]). It will be frequently used. Lemma 3.2. Under the previous notations, for all x, y ∈ N η (H), any r ≤ R µ and any η ≥ 0, G η,r (x, y|1) = G(x, y|r), For x, y ∈ Γ, we write d H (x, y) the distance between the projections π H (x) and π H (y) of x and y respectively onto H. Since projections on parabolic subgroups are well-defined up to a uniformly bounded error term, see [START_REF] Sisto | Projections and relative hyperbolicity[END_REF]Lemma 1.15] ,d H (x,y) is also defined up to a uniformly bounded error term. Letting M ≥ 0, we say that p η,r has exponential moments up to M if for any x ∈ N η (H), we have y∈Nη(H) p η,r (x, y)e MdH(x,y) < +∞.

The following lemma is the main reason for introducing p H,η,r with η > 0.

Lemma 3.3. [13, Lemma 4.6] For every M ≥ 0, then exists η M such that for every η ≥ η M and for every r ≤ R µ , the kernel p η,r has exponential moments up to M .

Note that η M does not depend on r; hence, choosing the neighborhood of H large enough, the kernels p η,r have exponential moments up to M , uniformly in r.

3.2.

Vertical displacement transition matrix. Until the end of Section 3, we assume that H is virtually abelian of rank d and that µ is spectrally degenerate along H. According to [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF]Lemma 4.16], it implies that R η = 1 for any η ≥ 0, i.e. µ is spectrally degenerate along N η (H).

Our first goal is to obtain asymptotics of the ⌈d/2⌉ -1 th derivative of the Green function G η,r at 1; see Proposition 3.13 below.

We fix α ∈ (0, 1) and consider the transition kernel pη,r defined by pη,r (x, y) = αδ x,y + (1α)p η,r (x, y).

Let Gη,r be the corresponding Green function. Then, by [39, Lemma 9.2],

Gη,r (e, e|t) = 1 1 -αt G η,r e, e (1 -α)t 1 -αt .
Hence, up to a constant that only depends on α and j, the jth derivative of Gη,r and G η,r coincide at 1. Therefore, up to replacing p η,r by pη,r we can assume that, p η,r (x, x) > 0 for every x so that the transition kernel p η,r is aperiodic. We keep this assumption for all this section.

By definition, there exists a subgroup of H of finite index which is isomorphic to Z d . Any section H/Z d → H allows us to identify H with Z d × F for some finite set F . As in [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF] and [START_REF] Dussaule | The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups[END_REF], the group Γ can be H-equivariantly identified with H × N. Indeed, the parabolic subgroup H acts by left multiplication on Γ and the quotient is countable. We order elements in the quotient according to their distance to H.

It follows that (1) N η (H) can be Z d -equivariantly identified with Z d × {1, ..., N η }, (2) if η ≤ η ′ , then N η ≤ N η ′ . In other words, the set Z d × {1, ..., N η }, identified with N η (H), is a subset of Z d × {1, ..., N η ′ }, identified with N η ′ (H).
Each element of N η (H) can be written as (x, j), where x ∈ Z d , j ∈ {1, ..., N η }. We also write p j,j ′ ;r (x, x ′ ) = p η,r ((x, j), (x ′ , j ′ )) for simplicity.

Definition 3.4. Letting u ∈ R d .. The vertical displacement transition matrix F r (u) is defined by: for all j, j ′ ∈ {1, ..., N η }, its (j, j ′ ) entry equals

F j,j ′ ;r (u) := x∈Z d p j,j ′ ;r (0, x)e u•x .
This transition matrix was introduced in [START_REF] Dussaule | The Martin boundary of a free product of abelian groups[END_REF] for r = 1. Many properties are derived there from the fact that it is strongly irreducible, i.e. there exists n such that every entry of F r (u) n is positive. Since by [10, Lemma 3.2],

F j,j ′ ;r (u) n = x∈Z d p (n) j,j ′ ;r (0, x)e u•x , where p (n) j,j ′ ;r (0, x) = p (n)
η,r ((0, j), (x, j ′ )), strong irreducibility is deduced from the fact that p η,r is aperiodic. Denote by F r ⊂ R d the interior of the set of u ∈ R d where F r (u) has finite entries. By the Perron-Frobenius Theorem [35, Theorem 1.1], the matrix F r (u) has a positive dominant eigenvalue λ r (u) on F r . Also, by [10, Proposition 3.5], the function λ r is continuous and strictly convex on F r and reaches its minimum at some value u r . Moreover, by Lemma 3.3, uniformly in r, the transition kernel p η,r has arbitrary large exponential moments, up to taking η large enough.

Denote by B(0, M ) the closed ball of radius M and center 0 in R d . It then follows from [14, (5), Proposition 4.6] that for large enough η, there exists a constant M such that, for every u ∈ B(0, M ), the matrix F r (u) has finite entries and the minimum of the function λ r is reached at some u ∈ B(0, M ). In other words, u r ∈ B(0, M ) ⊂ F r ; note that M is independent of r.

We now fix such η large enough so that the size of the matrices F r (u) is a fixed number, say K. We endow M K (R d ) with a matrix norm. For fixed r, the function F r is continuous in u. We then endow the space of continuous functions from B(0, M ) to M K (R d ) with the norm • ∞ . We also choose an arbitrary norm on R and endow the space of continuous functions from B(0, M ) to R with the norm • ∞ . According to [13, Lemma 5.4, Lemma 5.5], the functions r → F r and r → λ r are continuous for these norms.

(r) = R η (r) -1 satisfies (5) ρ η (r) = inf λ r (u) = λ r (u r ).
Actually, [39, Theorem 8.23] only deals with finitely supported transition kernels on Z d × {1, ..., N }, but the statement remains valid for the transition kernel p η,r since this condition of finite support can be dropped, see [39, (8.24)]. In what follows, we need to take the derivative of the fonction r → ρ η (r), hence we first prove that this function is differentiable.

Lemma 3.5. For every v ∈ N η (H), the function r → p η,r (e, v) is continuously differentiable on [0, R µ ].

Proof. Recall that p η,r (e, v) = G(e, v; N η (H) c |r), so it can be expressed as a power series in r with positive coefficients a n . These coefficients are at most equal to µ * n (v). Since the random walk on Γ is convergent, it follows that

n≥0 na n r n-1 ≤ n≥0 nµ * n (x)R n µ < +∞.
By monotonous convergence, the fonction r → p η,r (e, v) is continuously differentiable.

For simplicity, we write p r = p η,r and denote by p ′ r the derivative of p r at r; the kernels p r and p ′ r are both Z d -invariant transition kernels on Z d × {1, ..., N η }. By Lemma 2.7, it holds :

d dr rp r (x, y) = p r (x, y) + rp ′ r (x, y) = z / ∈Nη (H) G(x, z; N η (H) c |r)G(z, y; N η (H) c |r). (6)
The following statement can be thought of an enhanced version of relative Ancona inequalities. This is the first time we use the fact that parabolic subgroups are virtually abelian. Proof. For simplicity, we have assumed that the generating set S of Γ equals the support of µ. Let us fix x, y in Γ, a geodesic from y to H and ỹ satisfying the previous hypotheses.

If G(x, y; N η (H) c |r) = 0, i.e. if there is no trajectory of the random walk from x to y staying outside N η (H), then there is nothing to prove.

Otherwise there exists such a trajectory from x to y and we denote by x the first point on this trajectory outside N η (H).

Denote by H(x) the set of h ∈ H such that there is a trajectory outside N η (H) from x to hx. Since the random walk is symmetric, the set H(x) is a subgroup of H. Note that there exists finitely many connected components c 1 , ..., c m of trajectories of the random walk lying in the the set of points z / ∈ N η (H) that project on H at e. For any index j such that there exists trajectories from x to some hc j , h ∈ H, we choose such a particular trajectory and denote by z j its endpoint and by h j the corresponding point in H. Consider now a trajectory γ starting at x and staying outside N η (H); let z be its endpoint. This point z lies in some connected component hc j , hence we can find a trajectory from z to hh -1 j z j staying outside N η (H). We now concatenate the translated trajectory from hh -1 j z j to hh -1 j x. In other words, we can add a trajectory of fixed length to γ to reach some h ′ x. In particular, the endpoint of γ projects on H within a bounded distance of H(x).

The following notion was introduced in [13, Definition 3.11].

Definition 3.7. Let k, c > 0, A ⊂ Γ and y ∈ Γ. The set A is (k, c)-starlike around y if for all z ∈ A, there exists a path of length at most kd(y, z) + c staying in A.

Denote by ỹ′ the point on the chosen geodesic from y to H which is at distance η + 1 from H. We now prove the following. Lemma 3.8. There exist positive constants k, c only depending on η such that the connected component of x in Γ\N η (H) is (k, c)-starlike around ỹ′ .

Proof. We have to prove that, for every z / ∈ N η (H) that can be reached by a trajectory starting at x, there exists a path of length at most kd(z, ỹ′ ) + c which stays outside N η (H) and joins ỹ′ .

Let z be such a point and z be a point on a geodesic from z to H at distance η + 1 from H. Denote by z 0 and y 0 , the respective projection of z and y on H. By the above, we can find a trajectory of fixed length starting at some point hx and ending at z. Similarly, we can find a trajectory of fixed length starting at ỹ′ and ending at some h ′ x.

Since H is virtually abelian, the subgroup H(x) is quasi-isometrically embedded in Γ. Therefore, we can find a path (h j ) j from h to h ′ staying inside H(x) with length at most k 0 d(h, h ′ ) + c 0 for some fixed constants k 0 , c 0 . By concatenating successive trajectories from h j x to h j+1 x, we can thus find a trajectory from hx to h ′ x of length at most k 1 d(z 0 , y 0 ) + c 1 staying outside N η (H), where k 1 , c 1 > 0 only depend on η. Hence, there exist c 2 , k 2 > 0 and a trajectory γ from z to ỹ′ of length at most k 2 (d(z, z) + d(z 0 , y 0 )) + c 2 and staying outside N η (H). The distance formula [START_REF] Sisto | Projections and relative hyperbolicity[END_REF]Theorem 3.1] shows that there exist positive constants c 3 and k 3 such that d(z, ỹ′ ) ≥ k -1 3 (d(z, z 0 ) + d(ỹ ′ , y 0 ) + d(z 0 , y 0 ))c 3 . Hence, the length of γ is at most kd(z, ỹ′ ) + c, where k and c only depend on η.

Note that ỹ′ is within a uniform bounded distance, depending only on η, of a point on a relative geodesic from x to y. From [13, Proposition 3.12], it follows that [START_REF] Dal | Séries de Poincaré des groupes géométriquement finis[END_REF] G(x, y;

N η (H) c |r) ≤ CG(x, ỹ′ ; N η (H) c |r)G(ỹ ′ , y; N η (H) c |r).
Finally, the existence of one-step paths from ỹ′ to ỹ and x to x yields

G(x, y; N η (H) c |r) G(x, y; N η (H) c |r), G(x, ỹ′ ; N η (H) c |r) G(x, ỹ; N η (H) c |r) and G(ỹ ′ , y; N η (H) c |r) G(ỹ, y; N η (H) c |r
). Therefore, one can replace x by x and ỹ′ by ỹ in [START_REF] Dal | Séries de Poincaré des groupes géométriquement finis[END_REF]. The proof of Proposition 3.6 is complete. Proposition 3.9. Let M ≥ 0. There exists η M such that for every η ≥ η M and for every r ≤ R µ , the transition kernel p ′ r has exponential moments up to M . Proof. In view of [START_REF] Candellero | Phase transitions for random walk asymptotics on free products of groups[END_REF], it is enough to prove that for large enough η,

y / ∈Nη (H) G(e, y; N η (H) c |r)G(y, v; N η (H) c |r) ≤ Ce -2M x0 ,
where x 0 is the projection of x on Z d k . For every y / ∈ N η (H), denote by y 0 its projection on Z d and let ỹ be the point at distance η from H on a geodesic from y to H. Also, let P y0 be the set of points in Γ whose project onto Z d at y 0 . Proposition 3.6 shows that y / ∈Nη (H) G(e, y; N η (H) c |r)G(y, v; N η (H) c |r)

y0∈Z d y∈Py 0 G(e, ỹ; N η (H) c |r)G(ỹ, v; N η (H) c |r) G(ỹ, y; N η (H) c |r)G(y, ỹ; N η (H) c |r).
Since the µ-random walk is convergent, equality (6) yields This concludes the proof, since y0∈Z d e -2M y0 < +∞.

y∈Py 0 G(ỹ, y; N η (H) c |r)G(y, ỹ; N η (H) c |r) G ′ (e, e|r) < +∞. Consequently, y / ∈Nη (H) G(e, y; N η (H) c |r)G(y, v; N η (H) c |r) y0∈Z d G(e, ỹ; N η (H) c |r)G(ỹ, v; N η (H) c |r).
Proposition 3.9 now allows us to describe the regularity of the map (r, u) → F r (u) on (0, R µ ) × B(0, M ) where M is the constant which appears at the end of the previous subsection and B(0, M ) is the open ball with center 0 and radius M . We fix η such that both p r and p ′ r have exponential moments up to M .

Lemma 3.10. The function (r, u)

→ F r (u) is continuously differentiable on the open set (0, R µ ) × B(0, M ).
Proof. It suffices to prove that for every j, j ′ , F j,j ′ ;r (u) is continuously differentiable. By definition,

F j,j ′ ;r (u) = x∈Z d p j,j ′ ;r (0, x)e u•x .
For all x ∈ Z d , the function f : (r, u) → p j,j ′ ;r (0, x)e u•x is continuously differentiable and its derivative is given by

∇ r,u f (r, u) = p ′ j,j ′ ;r (0, x)e u•x v r + p j,j ′ ;r (0, x)e u•x v u (x),
where v r = (1, 0, ...0) and v u (x) = (0, x). Then,

∇ r,u f ∞ ≤ sup r p ′ j,j ′ ;r (0, x)e M ′ x + x sup r p ′ j,j ′ ;r (0, x)e M ′ x .
Lemma 3.3 and Proposition 3.9 show that this quantity is summable. Hence, by dominated convergence, the function (r, u) → F j,j ′ ;r (u) is continuously differentiable and its derivative equals

∇ r,u F j,j ′ ;r (r, u) = x∈Z d p ′ j,j ′ ;r (0, x)e u•x v r + p j,j ′ ;r (0, x)e u•x v u (x),
which concludes the proof.

We can now prove that the function r → ρ η (r) is differentiable on (0, R µ ) and compute the value of its derivate.

Proposition 3.11. The function r → ρ η (r) is continuously differentiable and its derivative is given by ρ ′ η (r) = λ ′ r (u r ). Proof. The function F → λ is on the set where F has a unique dominant eigenvalue. Thus, (r, u) → λ r (u) is continuously differentiable on (0, R µ )× B(0, M ). Moreover, it follows from [START_REF] Dussaule | The Martin boundary of a free product of abelian groups[END_REF]Proposition 3.5] that for all r, the Hessian of the map u → λ r (u) is positive definite. Therefore, the implicit function theorem shows that the function r → u r is continuously differentiable on (0, R µ ), and so is the function r → λ r (u r ). Moreover,

ρ ′ η (r) = ∇ u λ r (u r ) • u ′ r + λ ′ r (u r )
. Since λ r is stricly convex and reaches its minimum at u r , we have ∇ u λ r (u r ) = 0, hence the desired formula.

We can now extend by continuity the function ρ ′ η (r) on [0, R µ ], so that r → ρ η (r) is differentiable on the closed set [0, R µ ] and its derivative is given by λ ′ r (u r ). Lemma 3.12. For any r ≤ R µ , we have ρ ′ η (r) = 0. Proof. We just need to show that λ ′ r (u) = 0 for any u ∈ B(0, M ). For a strongly irreducible matrix F , denote by C and by ν right and left eigenvectors associated to the dominant eigenvalue λ. By the Perron-Frobenius Theorem [35, Theorem 1.1], they both have positive coefficients. Moreover, one can normalize them such that we have ν • C = 1 and such that F → C and F → ν are analytic functions, see [START_REF] Dussaule | The Martin boundary of a free product of abelian groups[END_REF]Lemma 3.3]. In particular, denoting by C r (u) and ν r (u) right and left eigenvectors of F r (u) associated with the eigenvalue λ r (u), we get that the maps (r, u) → C r (u) and (r, u) → ν r (u) are continuously differentiable and satisfy ν r (u)

• C r (u) = 1. Therefore, λ r (u) = ν r (u) • F r (u) • C r (u) and so λ ′ r (u) = λ r (u) ν ′ r (u) • C r (u) + ν r (u) • C ′ r (u) + ν r (u) • F ′ r (u) • C r (u).
Differentiating in r the expression ν r (u)

• C r (u) = 1, we get ν ′ r (u) • C r (u) + ν r (u) • C ′ r (u) = 0 and so λ ′ r (u) = ν r (u) • F ′ r (u) • C r (u). Since p ′
r (e, v) is non-negative for every v ∈ Γ, the matrix F ′ r has non-negative entries. Also, it cannot be equal to the null matrix since p ′ r (e, v) is positive for at least some v. Moreover, C r (u) and ν r (u) both have positive entries. Hence, λ ′ r (u) is positive.

Asymptotics of G (j)

r . We set j = ⌈d/2⌉ -1. Applying the previous results, we show the following statement; this is a crucial step in the proof of our main theorem. Proposition 3.13. Assume that µ is spectrally degenerate along H. If η is large enough, then the following holds. As r ր R µ ,

• if d is even then

G (j) η,r (e, e|1) ∼ C Log 1 R µ -r .
• if d is odd then

G (j) η,r (e, e|1) ∼ C R µ -r .
Proof. By [10, Proposition 3.14] applied to the kernel R η (r)p η,r , there exists a constant C r such that

C -1 r R η (r) n p (n)
η,r n d/2 -1 converges to 0 as n → +∞. Moreover, the convergence is uniform on r and the function r → C r is continuous; consequently, the quantity C r remains bounded away from 0 and infinity. Fix ǫ > 0. Assume first that d is even, so that j = d/2 -1. Then, for large enough n, say n ≥ n 0 , independently of r, we have

C -1 r n j p (n) η,r -n -1 ρ η (r) n ≤ ǫn -1 ρ η (r) n . Consequently, n≥n0 n j p (n) η,r -C r n -1 ρ η (r) n ≤ C r ǫ n≥0 n -1 ρ η (r) n .
Hence,

n≥0 n j p (n) η,r -C r n≥0 n -1 ρ η (r) n ≤ n≤n0-1 n j p (n) η,r + C r n≤n0-1 n -1 ρ η (r) n + ǫC r n≥0 n -1 ρ η (r) n . Note that C r n≥0 n -1 ρ η (r) n = C r Log 1 1 -ρ η (r)
.

Since ρ η (r) converges to 1 as r ր R µ , this last quantity tends to infinity as r converges to R µ . In particular, this proves that

n≥0 n j p (n) η,r ∼ rրRµ C r Log 1 1 -ρ η (r) ,
and so ( 8)

G (j) η,r (e, e|1) ∼ rրRµ C ′ r Log 1 1 -ρ η (r)
.

By Proposition 3.11, there exists α ∈ R such that ρ η (r) = 1+α(r-R µ )+o (r -R µ ); Lemma 3.12 yields α = 0. Hence 1ρ η (r) ∼ α(R µr). Combined with [START_REF] Dal'bo | Convergence et comptage en mesure infinie[END_REF], this concludes the proof. The case where d is odd is treated in the same way.

Stability of the Martin boundary

This section is dedicated to the proof of the stability of the Martin boundary (see Definition 2.6) in the case where the µ-random walk is convergent; this had not been dealt with before. We adapt here the arguments [START_REF] Dussaule | The Martin boundary of a free product of abelian groups[END_REF], [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF] and [START_REF] Dussaule | The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups[END_REF] in this context. The central result of this section is the following one. Theorem 4.1. Let Γ be a relatively hyperbolic group with respect to virtually abelian subgroups. Let µ be an admissible, symmetric and finitely supported probability measure on Γ. Then, the Martin boundary of (Γ, µ) is stable.

Proof. Recall that the Martin boundary is stable if it satisfies four conditions given by Definition 2.6. Let us first explain why the three first conditions are already known to be satisfied.

• As already mentioned, for all x, y ∈ Γ, since µ is admissible and Γ is non amenable it follows from Guivarc'h [START_REF] Guivarc | Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire[END_REF] that G(x, y|R µ ) < +∞, i.e. conditon (1) of stability is satisfied. • From [13, Theorem 1.2], conditions (2) and (3) of stability are satisfied for any admissible random walk on a relatively hyperbolic group with virtually abelian parabolic subgroups: the homeomorphism type of ∂ rµ Γ does not depend on r ∈ (0, R µ ) (we denote it by ∂ µ Γ) and there exists an equivariant surjective and continuous map φ µ :

∂ µ Γ → ∂ Rµµ Γ. • Again from [13, Theorem 1.2] the r-Martin boundary of the µ-random
walk is identified with the r-geometric boundaries of Γ, defined as follows.

When r < R µ , the r-geometric boundary is constructed from the Bowditch boundary of Γ where each parabolic limit point ξ is replaced with the visual boundary of the corresponding parabolic subgroup. Equivalently, it is the Gromov boundary ∂ Γ to which has been attached at each parabolic fixed point ξ P fixed by P the visual boundary of P. At r = R µ , the r-geometric boundary is given by the same construction, with the following change: the parabolic limit points are replaced with the visual boundary of the corresponding parabolic subgroup only when the random walk is spectrally non degenerate along the underlying parabolic subgroup.

We are hence left with showing that the map

(x, y, r) ∈ Γ × Γ ∪ ∂ µ Γ × (0, R µ ] → K(x, y|r) is continuous, where for ξ ∈ ∂ µ (Γ), we write K(x, ξ|R µ ) = K(x, φ µ (ξ)|R µ ).
Notice that this property is proved in [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF]Theorem 1.3] in the case where the random walk is spectrally non degenerate. By using the geometric interpretation of the r-Martin boundaries mentioned before, we need to check that, for any x ∈ Γ, any sequence (y n ) n in Γ ∪ ∂ µ Γ which converges to a point ξ in the geometric boundary of a parabolic subgroup H along which the random walk is spectrally degenerate and any (r n ) n which converges to R µ , the sequence (K(x, y n |r n )) n converges to K(x, ξ|R µ ).

As in [14, Section 5], we can assume without loss of generality that (y n ) n stays in N η (H) for some η > 0. This neighborhood can be identified with Z d × {1, ..., N } as in Section 3.2 above. Theorem 4.1 hence appears as a direct consequence of Proposition 4.2 below which yields the convergence of (K(x,

y n |r n )) n . Proposition 4.2. Let p be a Z d -invariant transient kernel on E = Z d × {1, ..., N }.
Assume that p is irreducible and aperiodic and has exponential moments. Then, the Martin boundary is stable and the function

(x, y, r) ∈ E × E ∪ ∂ p E × (0, R p ] → K(x, y|r) is continuous.
The proof of this theorem will rely on two lemmas. For any v ∈ R d , we write v ∈ Z d the vector with integer entries which is closest (for the Euclidean distance) to v, chosing the first in lexicographical order in case of ambiguity.

The next lemma is technical and is inspired by [START_REF] Dussaule | The Martin boundary of a free product of abelian groups[END_REF]Lemma 3.28]. It will be used in a particular case, where p n will be the convolution power of the transition kernel of first return to N η (H), β v will be the derivative of the eigenvalue λ v , α v will be some suited factor and Σ v will be the Hessian matrix associated with p n . The fact that the quantity a n defined in the lemma uniformly converges to 0 will then be a consequence of [START_REF] Dussaule | The Martin boundary of a free product of abelian groups[END_REF]Proposition 3.14].

Lemma 4.3. Let p n (x) be a sequence of real numbers, depending on x ∈ Z d . Let K ⊂ R N be a compact set and v → α v and v → β v two continuous functions on K, with α v ∈ R and β v ∈ R d . Let Σ v be a positive definite quadratic form on R d , that depends continuously on v ∈ K. Define a n (x, v, γ) = x -nβ v √ n γ (2πn)
d 2 p n (x)e v•x -α v e -1 2n Σv (x-nβv) .
Denote by g(x) the sum over n of the p n (x). If (a n ) n converges to 0 uniformly in x ∈ Z d , v ∈ K and γ ∈ [0, 2d], then, for x ∈ Z d and for v ∈ K such that β v = 0, it holds as t tends to infinity,

(2πt) d-1 2 Σ v (β v ) g( tβ v -x)e v•( tβ(v) -x) = α v + o(1)
where the term o(1) is bounded by an asymptotically vanishing sequence which does not depend on v.

Proof. It is proved in [START_REF] Dussaule | The Martin boundary of a free product of abelian groups[END_REF]Lemma 3.28] that with the same assumptions, assuming moreover that

β v = 0 for every v ∈ K, (2πt)
d-1

2 g( tβ v -x)e v•( tβ(v) -x) -→ t→∞ α v Σ v (β v ) ,
the convergence being uniform in v. The proof is the same as in [START_REF] Ney | The Martin boundary for random walk[END_REF]Theorem 2.2]. ; the key ingredient is that Σ v (β v ) is uniformly bounded from below by some constant β, so that (9) e -y 2 Σv (βv) ≤ e -y 2 β .

The function on the right-hand side is integrable on [0, +∞), and this estimate in turn allows one to prove uniform convergence in v, using the dominated convergence theorem.

We will use this lemma in a setting where we cannot assume anymore that β v > 0. We need therefore to prove that

Σ v (β v )(2πt) d-1 2 g( tβ v -x)e tβ(v) -x
converges uniformly in v. We replace (9) by

Σ v (β v )e -y 2 Σv (βv) ≤ C 1 1 + y 2 .
The right-hand side is again an integrable function and so we can conclude exactly like in the proof of [START_REF] Ney | The Martin boundary for random walk[END_REF]Theorem 2.2]. This result yields the necessary asymptotics of the Green function for every r, u such that ∇λ r (u) = 0. When ∇λ r (u) = 0, we need the following lemma. Lemma 4.4. With the same notations as in Proposition 4.3, assume that for every γ ∈ [0, d -1], a n converges to 0, uniformly in x ∈ Z d and v ∈ K. Then, for x ∈ Z d and for v ∈ K such that β v = 0, as y tends to infinity, we have

g(y -x)e v•(y-x) ∼ α v C d 1 Σ(y) d-2 2
,

where C d only depends on the rank d.

Proof. Define

g(y) = n≥1 1 (2πn) d/2 α v e -1
2n Σv (y) .

Setting

t n = n Σv (y-x) , we have ∆ n := t n -t n-1 = 1
Σv (y-x) which tends to 0 as y tends to infinity. Hence,

1 α v (2π) d/2 Σ v (y -x) d-2 2 g(y -x) = n≥1 t -d/2 n e -1 2tn ∆ n is a Riemannian sum of +∞ 0 t -d/2 e -1 2t dt = C ′ d . Consequently, we just need to show that g(y -x)e v•(y-x) g(y -x) -→ y→∞ 1.
Equivalently, we prove that

g(y -x)e v•(y-x) -g(y -x) = o y -(d-2) .
We set

α n = sup y∈Z d sup γ∈[0,d-1] y -x √ n γ (2nπ) d/2 p n (y -x)e v•(y-x) -α v e -1 2n Σv (y-x) .
By assumption, (α n ) n converges to 0 as n tends to infinity. Let ǫ > 0. Then for n ≥ n 0 , α n ≤ ǫ. We have

y -x d-2 g(y -x)e v•(y-x) -g(y -x) ≤ 1 y -x (2π) d/2 n0-1 n=1 α n n 1/2 + 1 y -x (2π) d/2 y-x 2 n=n0 α n n 1/2 + y -x d-2 (2π) d/2 n> y-x 2 α n n d/2 .
The first term in the right-hand side converges to 0 as y tends to infinity. The second term is bounded by

ǫ y -x (2π) d/2 y-x 2 0 t -1/2 dt ǫ.
The last term is bounded by

ǫ y -x d-2 (2π) d/2 +∞ y-x 2 t -d/2 dt ǫ.
This concludes the proof.

All the ingredients are gathered to achieve the demonstration of Proposition 4.2. Lemma 4.3 yields the convergence of the Martin kernels, as r tends to R µ , r < R µ , and y tends to a point ξ in the Martin boundary. Lemma 4.4 gives in turn the convergence of the Martin kernels for r = R µ and y converging to ξ. Since these two limits coincide, this yield s the continuity of the map (x, y, r) → K(x, y|r).

Proof of Proposition 4.2. For all u ∈ R d and r ∈ [0, R µ ], let F r (u) be the vertical displacement transition matrix defined in Section 3.2 and let λ(r, u) be its dominant eigenvalue. Let K be the set of pairs (r, u) such that r ∈ [0, R µ ] and u satisfies that λ(u) = 1. Since r → λ r is a continuous function from [0, R µ ] to the set of continuous functions of u, the set K is compact.

Recall that for fix k, j ∈ {1, ..., N }, we write p k,j (x, y) = p((x, k), (y, j)), for every x, y ∈ Z d . According to [10, Proposition 3.14, Proposition 3.16], for every k, j in {1, ..., N }, the kernel p k,j satisfies the assumptions of Lemma 4.3 and Lemma 4.4 if we define

α (r,u) := 1 det(Σ u ) C r (u) k ν r (u) j ,
β u,r = ∇λ r (u) and Σ (r,u) to be the inverse of the quadratic form associated to the Hessian of the eigenvalue λ(u) and where C r (u) and ν r (u) are right and left eigenvectors associated to λ r (u).

Consider ξ in the boundary ∂H of H and let ((y n , j n )) n a sequence in E which converges to ξ, i.e. (y n ) n tends to infinity and (y n / y n ) n converges to ξ. Also write e = (0, k 0 ) for the basepoint of E.

Assuming that r < R µ , [START_REF] Dussaule | The Martin boundary of a free product of abelian groups[END_REF]Lemma 3.24] shows that the map

u ∈ {u, λ r (u) = 1} → ∇λ r (u) ∇λ r (u)
is a homeomorphism between {u, λ r (u) = 1} and S d-1 . Thus, there exists u r,n such that y n y n = ∇λ r (u r,n ) ∇λ r (u r,n ) .

As explained in the previous section, [14, (5), Proposition 4.6] shows that the set {u, λ r (u) = 1} is contained in a fixed ball B(0, M ). Therefore, u r,n is bounded and so is ∇λ r (u r,n ) . Setting t n = y n / ∇λ r (u r,n ) , we see that (t n ) n tends to infinity and that y n = t n ∇λ r (u r,n ).

Consider now a sequence (r n ) n converging to R µ , r n < R µ . By Lemma 4.3,

(2πt n ) d-1 2 G((x, k), (y n , j n )|r n )e ur n ,n •(y-x) = 1 Σ rn,ur,n (∇λ r (u r,n )) C rn (u rn,n ) k ν rn (u rn,n ) jn det(Σ rn,ur n ,n ) + o(1)
and

(2πt n ) d-1 2 G(e, (y n , j n )|r n )e ur n ,n •y = 1 Σ rn,ur,n (∇λ r (u r,n )) C rn (u rn,n ) k0 ν rn (u rn,n ) jn det(Σ rn,ur n ,n ) + o(1)
Consequently, + o [START_REF] Ancona | Positive harmonic functions and hyperbolicity[END_REF] .

K((x, k), (y n , j n )|r n ) = e
By [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF]Lemma 5.6], the sequence (u rn,n ) n converges to u Rµ as (y n / y n ) n tends to ξ and (r n ) n tends to R µ . Note that the limit does not depend on ξ. Indeed, u Rµ is a point such that λ Rµ (u Rµ ) = 1 and by ( 5), the minimum of λ Rµ is 1. Since λ Rµ is strictly convex, the point u Rµ is unique. The o(1) term above is uniform in r,

hence (K((x, k), (y n , j n )|r n )) n converges to e u Rµ,ξ •x CR µ (u Rµ ,ξ ) k CR µ (u Rµ ,ξ ) k 0
. Assume now that r = R µ is fixed and (y n ) n converges to ξ. We apply Lemma 4.4 to the same parameters α v , β v , Σ v to deduce that

K((x, k), (y n , j n )|R µ ) ∼ e u Rµ,ξ •x C Rµ (u Rµ,ξ ) k C Rµ (u Rµ,ξ ) k0 .
Thus, as (y n , j n ) → ξ and r n → R µ with r n < R µ , the limit of the two sequences (K((x, k), (y n , j n )|r n )) n and (K((x, k), (y n , j n )|R µ )) n coincide.

Asymptotics of the full Green function

The purpose in this section is to show that, for a convergent random walk on a relatively hyperbolic group whose Martin boundary is stable, the asymptotics of the (derivatives of) the full Green function are given by the asymptotics of the (derivatives of the) Green functions associated to the first return kernels to dominant parabolic subgroups. The precise statement is given in Theorem 5.4 below. This is the last crucial step before the proof of the local limit theorem. Note that throughout this section, we do not need to assume that parabolic subgroups are virtually abelian.

If x ∈ Γ, we define I (k)

x (r) by

I (k) x (r) = x1,...,x k ∈Γ G(e, x 1 |r)G(x 1 , x 2 |r)...G(x k-1 , x k |r)G(x k , x|r).
For x = e, we write I (k) (r) = I (k) e (r). These quantities are related to the derivatives of the Green function by the following result. We inductively define

F 1,x (r) = d dr (rG r (e, x))
and

F k,x (r) = d dr (r 2 F k-1,x (r)), k ≥ 2.
The following Lemma generalizes Lemma 2.7 and is valid for any kernel. Moreover, if k is the smallest integer such that I (k) (R µ ) = +∞ -or equivalently such that G (k) (e, e|R µ ) = +∞ -then, as r ր R µ ,

I (k) x (r) ∼ CG (k) (e, x|r).
For any parabolic subgroup H of Γ and any η ≥ 0, we also set: for every x ∈ N η (H) and r ∈ [0, R µ ],

I (k) H,η,x (r) = x1,...,x k ∈Nη(H) G(e, x 1 |r)G(x 1 , x 2 |r)...G(x k-1 , x k |r)G(x k , x|r).
Again, if x = e, we write

I (k) H,η (r) = I (k) H,η,e (r). Since G(x, x ′ |r) = G H,η,r (x, x ′ |1) for any x, x ′ ∈ N η (H), the quantities I (k)
H,η,x (r) are related to the derivatives of G H,η,r at 1 by the same formulae as in Lemma 5.1.

We now fix a finite set {H 1 , ..., H N } of representatives of conjugacy classes of the parabolic subgroups. For η ≥ 0, we set ( 10)

J (k) η (r) = N p=1 I (k)
Hp,η (r)

and J (k) (r) = J (k) 0 (r), k ≥ 1. Proposition 5.3. Consider a finitely generated relatively hyperbolic group Γ and a finitely supported symmetric and admissible probability measure µ on Γ. Assume that the µ-random walk is convergent, i.e. I (1) (R µ ) is finite. Let k be the smallest integer such that J (k) (R µ ) is infinite. Then, the quantity I (j) (R µ ) is finite for every j < k and for every η ≥ 0, I (k) (r) ≍ J (k) η (r) where the implicit constant only depends on η.

Proof. Clearly, we have J (j) (r) I (j) (r) for every j. Also, by [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF]Lemma 5.7], the sum I (j) is bounded by some quantity that only depends on all the I (l) (r), l < j and on all the J (l) (r), l ≤ j. Thus, by induction, I (j) (R µ ) is finite for every j < k and I (k) (r) J (k) (r). Finally, J (j) (r) ≍ J Theorem 5.4. Consider a finitely generated relatively hyperbolic group Γ and a finitely supported symmetric and admissible probability measure µ on Γ. Assume that the random walk is convergent, i.e. I (1) (R µ ) is finite. For any parabolic subgroup H of Γ such that the random walk is spectrally degenerated along H and any r ≤ R µ , let p H,r be the first return kernel to H associated with rµ.

Assume that the following holds.

• The Martin boundary is stable and the function

(x, y, r) ∈ Γ × Γ ∪ ∂ µ Γ × (0, R µ ] → K(x, y|r) is continuous. • The 1-Martin boundary of (H, p H,Rµ ) is reduced to a point.
Let k be the smallest integer such that J (k) (R µ ) is infinite. Then, for every η ≥ 0, there exists a constant C η such that as r ր R µ ,

I (k) η (r) ∼ C η J (k) (r).
The next two subsections are dedicated to the proof of this theorem. 5.1. Asymptotics of the second derivative. We start with showing Theorem 5.4 when k = 2, i.e. J (1) (R µ ) is finite and J (2) (R µ ) is infinite. We first consider the case η = 0.

Claim. Under the assumptions of Theorem 5.4, if k = 2, then there exists a positive constant C such that I (2) (r) ∼ CJ (2) (r).

5.1.1.

Step 1. I (2) (r) from a transfer operator. The purpose of this paragraph is to prove Proposition 5.6, which shows that I (2) (r) can be written as the image of a suitable transfer operator and a remainding term which is bounded as r → R µ . Many computations are analogous to [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups II : the non spectrally degenerate case[END_REF]Section 4]; nevertheless we apply this operator to functions which may not be continuous and we cannot use thermodynamical formalism as in [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups II : the non spectrally degenerate case[END_REF] to control the convergence of our estimates. We present detailed computations.

We We fix a finite generating set S. Using the automaton G encoding relative geodesics given by Theorem 2.3, for any x ∈ Γ, we choose a relative geodesic [e, x] from e to x. Also, we will write Ω 0 = {H 1 , ..., H N } and H 0 = S so that each increment of a relative geodesic is in one of the H j .

Let α : Z → Γ be a relative geodesic such that α(0) = e and Γ α be the set of elements x ∈ Γ such that x 1 and α 1 lie in the same H j . We define Ψ r (α) by

Ψ r (α) = y∈Γα G(α -, y|r)G(y, α + |r) G(α -, α + |r) ,
where α -and α + are the left and right extremities of α. We prove the following. Let T be the left shift on relative geodesics, so that

T k α, k ∈ Z, is the relative geodesic α(k) -1 α. Proposition 5.5. Let x ∈ Ŝn . Then Φ r (x) = n-1 k=0 Ψ r T k [e, x] + O(n).
Proof. Write [e, x] = (e, x 1 , x 2 , ..., x n ). Fix k ≤ n -1. Let Γ k be the set of elements y such that the projection of y on [e, x] in the relative graph Γ is at x k . If there are several projections, we choose the one which is the closest to e. Let j k be such that x - Proof. We decompose x ∈ Ŝn as x = x 2 hx 1 , where x 1 ∈ Ŝn-k-1 , h ∈ H j for some j and x 2 ∈ Ŝk . If y is fixed, we write X j y (resp. X y j ) for the set of elements z of relative length j that can precede (resp. follow) y in the automaton G. We also write X y , respectively X y for the set of all elements z that precede, respectively follow y. We thus need to study

:= k≥0 n≥k+1 x1∈ Ŝn-k-1 h∈X 1 x 1 x2∈X k h H(e, x 1 |r) H(e, x 2 hx 1 |r) H(e, x 1 |r) h ′ ∈Γ h x ′ ∈X h ′ G(x -1 2 , h ′ x ′ |r)G(h ′ x ′ , hx 1 |r) G(x -1 2 , hx 1 |r)
.

We reorganize the sum over h, x 2 , h ′ , x ′ as

h∈X 1 x 1 h ′ ∈Γ h G(e, h ′ |r)G(h ′ , h|r)G(h, e|r) x2∈X k h x ′ ∈X h ′ H(e, x 2 |r)H(e, x ′ |r) χ r (h, h ′ , x 1 , x 2 , x ′ ),
where the function χ r is defined by

χ r (h, h ′ , x 1 , x 2 , x ′ ) = G(hx 1 , x -1 2 |r) G(e, x -1 2 |r)G(h, e|r)G(x 1 , e|r) G(x -1 2 , h ′ x ′ |r) G(x -1 2 , e|r)G(e, h ′ |r)G(e, x ′ |r) G(h ′ x ′ , hx 1 |r) G(x ′ , e|r)G(h ′ , h|r)G(e, x 1 |r) (13) so that = k≥0 n≥0 x1∈ Ŝn H(e, x 1 |r) h∈X 1 x 1 h ′ ∈Γ h G(e, h ′ |r)G(h ′ , h|r)G(h, e|r) x2∈X k h x ′ ∈X h ′ H(e, x 2 |r)H(e, x ′ |r) χ r (h, h ′ , x 1 , x 2 , x ′ )
Proposition 5.8. The functions χ r are bounded uniformly in r ∈ [0, R µ ]. Moreover, as r tends to R µ , the family (χ r ) r uniformly converges to χ Rµ .

Proof. Let us note that every quotient in the definition of χ r is uniformly bounded. Hence, according to the weak relative Ancona inequalities (Proposition 2.9), we just need to prove that each of them uniformly converges as r tends to R µ . We start with the first term. and fix ǫ > 0. By Proposition 2.8, there exists η, independent of r, such that

G(hx 1 , x -1 2 ; B η (h) c |r) ≤ ǫG(hx 1 , x -1 2 |r). v∈B η ′ (e) G(hu, v; B η ′ (v) c |r)G(v, v -1 2 |r),
where η ′ only depends on ǫ and η. As above, we check that both

G(hu,v;B η ′ (e) c |r) G(h,e|r) and G(v,v -1 2 |r) G(e,v -1 2 |r) uniformly converge, as r → R µ .
This shows uniform convergence of the first quotient in the definition of χ r . We deal similarly with the two other ones to conclude.

Let ǫ > 0. Since x H(e, v|R µ ) is finite, if |R µ -r| is small enough, we have

k≥0 n≥0 x1∈ Ŝn H(e, x 1 |r) h∈X 1 x 1 h ′ ∈Γ h G(e, h ′ )G(h ′ , h)G(h, e) x2∈X k h H(e, x 2 |r) x ′ ∈X h ′ H(e, x ′ |r) χ r (h, h ′ , x 1 , x 2 , x ′ ) -χ Rµ (h, h ′ , x 1 , x 2 , x ′ ) ≤ ǫJ (2) (r).
We can thus replace χ r by χ Rµ in the expression of . Proposition 5.10. Consider a parabolic subgroup H along which the random walk is spectrally degenerate. As h, h ′ ∈ H tend to infinity and d(h, h ′ ) tends to infinity, the function

k≥0 x2∈X k h x ′ ∈X h ′ H(e, x 2 |r)H(e, x ′ |r)χ Rµ (h, h ′ , x 1 , x 2 , x ′ )
converges to a function χH (x 1 ). Moreover, the convergence is uniform in x 1 and r ∈ [0, R µ ] and the function χH is bounded.

We first prove the following lemma. The same proof does not apply to elements x ′ that can follow h ′ . However, decomposing elements of Γ as h ′ x ′ and choosing the inverse lexicographical order rather than the original lexicographical order, we get similarly the following result. Lemma 5.12. If D is large enough and if d(e, h ′ ) > D, the set of x ′ that can follow h ′ lying in a fixed parabolic subgroup H is independent of h ′ .

We can now prove Proposition 5.10.

Proof. By Lemmas 5.11 and 5.12, it is enough to prove that χ Rµ (h, h ′ , x 1 , x 2 , x ′ ) converges to a function, as h, h ′ tend to infinity and d(h, h ′ ) tends to infinity, uniformly in x 1 , x 2 , x ′ . Uniformity is proved using Proposition 2.8, as in the proof of Proposition 5.8. Hence, we just need to prove that for fixed u and v, the ratio

We also write H h for the parabolic subgroup containing h. When h lies in several parabolic subgroups, we arbitrarily choose one of them and, if possible, we choose one along which the random walk is spectrally degenerate. Recall that the intersection of two parabolic subgroups is finite, see [ On the one hand, the sub-sum in ( 14) over the h and h ′ such that, either d(e, h) < D ǫ or d(e, h ′ ) < D ǫ or d(h, h ′ ) < D ǫ , is uniformly bounded. Hence, it can be bounded by ǫJ (2) (r) if r is close enough to R µ . On the other hand, the sub-sum over the h and h ′ such that d(e, h), d(e, h ′ ), d(h, h ′ ) ≥ D ǫ but the random walk is spectrally non degenerate along H h is also uniformly bounded and can thus be bounded as well by ǫJ (2) (r) if r is close enough to R µ .

To conclude the proof, we only need to prove that n≥0 x1∈ Ŝn H(e, x 1 |r)

h∈X 1 x 1 h ′ ∈Γ h
G(e, h ′ |r)G(h ′ , h|r)G(h, e|r) χH h (x 1 ) ∼ CJ (2) (r).

The double sum over x 1 and h that can precede x 1 is exactly the sum over every element of relative length 1+ d(e, x 1 ), so we can replace this double sum by a double sum over h and x 1 that can follow h. We thus need to prove that By Lemma 5.12, if d(e, h) is large enough, then the set of x 1 that can follow h is independent of h. This concludes the proof, since for fixed D, the sub-sum over the h such that d(e, h) ≤ D is uniformly bounded.

This concludes the proof of the Claim 5.1. To complete the proof of Theorem 5.4 in the case k = 2, we need to show the following. Proposition 5.14. Under the assumptions of Theorem 5.4, if k = 2, then for every η ≥ 0, there exists a positive constant C η such that J (2) η (r) ∼ C η J (2) (r). Denote by E η the set of x ∈ Γ such that x is in N η (H) for some H. Recall that for a relative geodesic α such that α(0) = e, the set Γ α contains all the elements x ∈ Γ such that x 1 and α 1 lie in the same H j . Setting 5.2. Higher derivatives. We now consider the general case, i.e. J (j) (R µ ) is finite for every j < k and J (k) is infinite. We introduce the function Φ As above, it holds

I (k) (r) = x∈Γ
H(e, v|r)Φ (k) r (x).

We have the following.

Proposition 5.15. There exists D k such that, for any x ∈ Ŝn ,

Φ (k) r (x) = n-1 j=0 Ψ (k) r T j [e, v] + O(n D k ).
Proof. Like in the proof of Proposition 5.5, we consider the set Γ l of elements y such that the projection of y on [e, x] is at x l . If all the y j do not lie in the same Γ l , then and conclude like in the case k = 2. This proves Theorem 5.4.

Proof of the local limit theorem

In this section, we prove Theorem 1.3. Recall that for fixed x in Γ,

I (k) x (r) =
I (k) x (r) ∼ C η,x J (k) η (r).
more precisely, from the analogous of Proposition 5.2 for H) that I

H (R µ ) < +∞. We gather these observations in the following proposition. Proposition 7.1. Let Γ be a a finitely generated relatively hyperbolic group Γ with respect to virtually abelian subgroups. Let µ be a finitely supported, admissible, symmetric probability measure on Γ. Assume that for every parabolic subgroup H of rank 5 or 6, the random walk is spectrally non degenerate along H. Then the random walk is spectrally positive recurrent if and only if it is divergent.

In a work in preparation [START_REF] Dussaule | Exotic local limit theorems at the phase transition in free products[END_REF], we show that there exist examples of spectrally degenerate random walks on relatively hyperbolic groups with parabolic subgroups being virtually abelian of rank d = 5 or 6, which are divergent but not spectrally positive recurrent. They show exotic local limit theorems which are neither of the form (1) nor [START_REF] Bingham | Regular variation[END_REF]. We construct such examples on free products of abelian groups.

Remark 7.1. Note that these exotic examples contradict [START_REF] Candellero | Phase transitions for random walk asymptotics on free products of groups[END_REF]Lemma 4.5]. Unfortunately the proof of this Lemma shows a subtle gap. A previous version of our article contained a similar gap, which leaded to the (wrong) conclusion that any divergent random walk is spectrally positive recurrent as soon as parabolic subgroups are virtually abelian.

Theorem 2 . 3 .

 23 [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF] Theorem 4.2] If Γ is relatively hyperbolic with respect to Ω, then for any finite generating set S and for any choice of a full family Ω 0 of representatives of conjugacy classes of elements of Ω, Γ is relatively automatic with respect to S and Ω 0 .

2. 2 .

 2 Transition kernels and Martin boundaries. Let us make a general presentation of the notion of Martin boundaries. In what follows, E is a countable space endowed with the discrete topology and o is a fixed base point in E Definition 2.4. A transition kernel p on E is a positive map p : E × E → R + with finite total mass, i.e. such that ∀x ∈ E, y∈E p(x, y) < +∞.

2. 3 .

 3 Relative Ancona inequalities. For any set A ⊂ Γ, we set (4) G(x, y; A|r) := n≥1 g1,...,gn-1∈A

3. 3 .

 3 Differentiability of the parabolic spectral radius. By [39, Theorem 8.23], the spectral radius ρ η

Proposition 3 . 6 .

 36 Let η ≥ 0 be fixed. There exists C = C η such that the following holds. Let x ∈ N η (H) and let y / ∈ N η (H). Consider a geodesic in Γ from y to H and denote by ỹ the point in Γ at distance η from H on this geodesic. Then, G(x, y; N η (H) c |r) ≤ CG(x, ỹ; N η (H) c |r)G(ỹ, y; N η (H) c |r).

By Lemma 3 . 3 ,

 33 for η large enough, it holds G(e, ỹ; N η (H) c |r) e -4M y0 and Gỹ, v; N η (H) c |r) e -2M y0-x0 e 2M y0 -2M x0 ; therefore, y / ∈Nη (H) G(e, y; N η (H) c |r)G(y, v; N η (H) c |r)

Lemma 5 . 1 . [ 11 , Lemma 3 . 2 ]

 511132 For every x ∈ Γ and r ∈ [0, R µ ],F k,x (r) = k!r k-1 I (k)x (r). As a direct consequence, it holds. Proposition 5.2. For every k ≥ 1, x ∈ Γ and r ≤ R µ , I (k) x (r) ≍ G(e, x|r) + G ′ (e, x|r) + ... + G (k) (e, x|r).

η

  (r), where the implicit constant only depends on η.The purpose of this section is to prove the following theorem. Its assumptions are satisfied as soon as the parabolic subgroups are virtually abelian, according to Theorem 4.1 and [13, Proposition 4.3].

  write H(x, y|r) = G(x, y|r)G(y, x|r). By definition I (1) (r) = x∈Γ H(e, x|r). We introduce the function Φ r defined by Φ r (x) = y∈Γ G(e, y|r)G(y, x|r) G(e, x|r) . By definition, I (2) (r) = x∈Γ H(e, x|r)Φ r (x).

  h h ′ ∈Γ h G(e, h ′ |r)G(h ′ , h|r)G(h, e|r) n≥0 x1∈X h n H(e, x 1 |r) χH h (x 1 ) ∼ CJ (2) (r).

  -, y|r)G(y, α + |r) G(α -, α + |r) 1 α -1 -α+∈Eη 1 y∈Eηthe proof of Proposition 5.14 is exactly the same as the one of Claim 5.1, replacing Ψ r by Ψ η r .

  any x ∈ Γ, Φ (k) r (x) = y1,...,y k-1 G(e, y 1 |r)G(y 1 , y 2 |r)...G(y k-1 , v|r) G(e, v|r) .and, for any relative geodesic α such that α(0) = e,Ψ (k) r (α) = y1,...,y k-1 ∈Γα G(α -, y 1 |r)G(y 1 , y 2 |r)...G(y k-1 , α + |r) G(α -, α + |r) .

y1,...,y k- 1 G

 1 (e, y 1 |r)G(y 1 , y 2 |r)...G(y k-1 , x|r) G(e, x|r)is bounded by a quantity only involving the J (j) (r), j < k, which are uniformly bounded. Hence we can restrict the sum in the definition of Φ (k) r to the y j lying in the same Γ l and the remainder of the proof of Proposition 5.5 can be reproduced to conclude.As in Proposition 5.6, we getI (k) (r) =n≥0 n-1 k=0 x∈ Ŝn H(e, x|r)Ψ (k) r T k [e, x] + O(1)

Proposition 6 . 1 .

 61 y1,...,y k G(e, y 1 |r)...G(y k , x|r). Under the assumptions of Theorem 5.4, for every η ≥ 0 and every x ∈ Γ, there exists C η,x such that

  ur,n•x Cr n (ur n ,n ) k νr n (ur n ,n )j n det(Σr n ,ur n ,n ) Cr n (ur n ,n ) k 0 νr n (ur n ,n)jn det(Σr n ,ur n ,n )

	+ o(1)

  1 k x k+1 ∈ H j k . P (R µ ) < 0. Therefore, the sum n≥0 n x∈ Ŝn H(e, x|R µ ) is finite. This concludes the proof, since H(e, x|r) ≤ H(e, x|R µ ).

	5.1.2. Step 2. J (2) (r) from the transfer operator. By Proposition 5.6, the Claim 5.1
	is a direct consequence of the following statement.
	Proposition 5.7. Under the previous notations,
		n-1
	n≥0	k=0 x∈ Ŝn	H(e, x|r)Ψ r T k [e, x] ∼ CJ (2) (r).

  Lemma 5.11. If D is large enough and if d(e, h) > D, the set of x 2 that can precede h lying in a fixed parabolic subgroup H is independent of h.Proof. Let h 1 and h 2 be in the same parabolic subgroup H, such that both have length bigger than D. Assume that x can precede h 1 . By[START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF] Lemma 4.11], if D is large enough, the concatenation of x and h 2 is a relative geodesic. Now, consider elements x and h in H such that xh = xh 2 . Hence xh h -1 2 h 1 = xh 1 so that x ≥ x in the lexicographical order. If x > x, then the concatenation of x and h is bigger than the concatenation of x and h 2 . Otherwise, x = x and so h = h 2 . Therefore x can precede h 2 .

  Both hu and h -1 tend to infinity. Since we assume that the Martin boundary of the first return kernel to H is reduced to a point, both ratiosG(e, h ′ |r)G(h ′ , h|r)G(h, e|r) k≥0 x2∈X k h x ′ ∈X h ′H(e, x 2 |r)H(e, x ′ |r)χ Rµ (h, h ′ , x 1 , x 2 , x ′ ).

	G(hu,v|Rµ) G(h,e|Rµ) converges as h tends to infinity. We write
	G(hu, v|R µ ) G(h, e|R µ )	=	G(hu, v|R µ ) G(hu, e|R µ )	G(hu, e|R µ ) G(h, e|R µ )	=	G(hu, v|R µ ) G(hu, e|R G(hu,v|Rµ) G(hu,e|Rµ) and
	To simplify notations, we set
	Ĩ(2) (r) =	n≥0 x1∈ Ŝn	H(e, x 1 |r)	h∈X 1 x 1	h ′ ∈Γ h

µ ) G(u, h -1 |R µ ) G(e, h -1 |R µ ) . G(u,h -1 |Rµ) G(e,h -1 |Rµ)

converge, as h tends to infinity.

  [START_REF] Druţu | Tree graded spaces and asymptotic cones of groups[END_REF] Lemme 4.7]; hence H h is uniquely defined if d(e, h) is large enough. Finally, if the random walk is spectrally non degenerate along H, we set χH (x 1 ) = 1.Proposition 5.13. Let ǫ > 0. If |r -R µ | is small enough, then G(e, h ′ |r)G(h ′ , h|r)G(h, e|r) χH h (x 1 ) ǫJ (2) (r).Proof. There exists D ǫ such that if the random walk is spectrally degenerate alongH h , then if d(e, h), d(e, h ′ ), d(h, h ′ ) ≥ D ǫ . k≥0 x2∈X k h x ′ ∈X h ′ H(e, x2|r)H(e, x ′ |r)χ Rµ (h, h ′ , x 1 , x 2 , x ′ ) -χH h (x 1 ) ≤ ǫ.

	Ĩ(2) (r) -	n≥0 x1∈ Ŝn	H(e, x 1 |r)	h∈X 1 x 1	h ′ ∈Γ h

|r)G(h,e|r) as r → R µ , we use Proposition 2.8 and replace G(hu, v -1 2 |r) by

Consider some y ∈ Γ k , so that x -1 k y projects on T k [e, x] = x -1 k [e, x] at e. Consider the sub-relative geodesic [e, x -1

k y] and write [e, x -1 k y] = (e, z 1 , ..., z m ) and assume that z 1 / ∈ H j k . By [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF]Lemma 4.16], any relative geodesic from x -1 k to z m = x -1 k y passes through a point z within a bounded distance of e. Also, by [START_REF] Sisto | Projections and relative hyperbolicity[END_REF]Lemma 1.15], there exists L ≥ 0 such that if the projection in the Cayley graph of Γ of z m on H j k is at distance at least L from e, then the relative geodesic [e, x -1 k y] contains an edge in H j k . If this edge is not z 1 , this contradicts the fact that [e, x -1

k y] is a relative geodesic. Hence, the projection of z m on H j k is within a bounded distance of e. By [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF]Lemma 4.16], we also know that any relative geodesic from z m to x -1 k x passes within a bounded distance of

) is large enough, then such a geodesic has an edge in H j k . By [START_REF] Sisto | Projections and relative hyperbolicity[END_REF]Lemma 1.13], the entrance point in H j k is within a bounded distance of the projection of z m on H j k . In any case, a relative geodesic from z m to x -1 k x passes within a bounded distance of e. Consequently, weak relative Ancona inequalities (Proposition 2.9) yield [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF] G Combining [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups I : rough estimates[END_REF] and ( 12) yields

H(e, y|r) n, which is the desired bound.

We deduce the following.

Proposition 5.6. We have

Proof. By Proposition 5.5,

H(e, x|r).

Following [START_REF] Dussaule | Local limit theorems in relatively hyperbolic groups II : the non spectrally degenerate case[END_REF], the sum x∈ Ŝn H(e, x|r) can be written as the the value at the empty sequence of the nth iterate of a suited transfer operator L r defined on the path-space of the automaton G encoding relative geodesics and applied to a function f , see [ (r) , where P (r) is the maximal pressure of L r . Since the random walk is convergent, we necessarily have Hence, for every r ≤ R µ ,

ǫ and so we just need to prove that u∈Bη (e)

G(h,e|r)G(x1,e|r) converges as r tends to R µ , uniformly in x 1 , h, x 2 . We study separately the two ratios which appear in this sum. Similarly, to study the behavior of the ratio

Sketch of proof. We use the same arguments as in Theorem 5.4. Let us briefly outline the case k = 2.

We have

G(e, y|r)G(y, z|r)G(z, e|r) G(z, x|r) G(z, e|r) .

We introduce the function Φ r,x defined by i.e. Φ r,x (z) = Φ r (z) G(z,x|r) G(z,e|r) . We then have

x (r) = z∈Γ H(e, z|r)Φ r,x (z).

We can then reproduce the same proof, replacing the function χ r defined in ( 13) by the function χ r,x defined by

Hence, χ r,x = χ r G(z2hz1,x|r) G(z2hz1,e|r) and is bounded by a constant that only depends on x. Since we assume that the Martin boundary is stable and the function

is continuous, the family (χ r,x ) r uniformly converges to χ Rµ,x , as r tends to R µ .

We now prove Theorem 1.3. By Theorem 4.1 and [13, Proposition 4.3], the assumptions of Theorem 5.4 are satisfied. Combining Proposition 3.13 and Theorem 5.4, we get the following. Corollary 6.2. For every x, there exists

In both cases, we deduce that p (n) (e, x) ∼ C x R -n µ n -d/2 ; let us explain this last step. The odd case is proved exactly like [START_REF] Gouëzel | Random walks on co-compact Fuchsian groups[END_REF]Theorem 9.1]. The method is based on a Tauberian theorem of Karamata and also applies to the even case.

Let us give a complete proof of the even case for sake of completeness. A function f is called slowly varying if for every λ > 0, the ratio f (λx)/f (x) converges to 1 as x tends to infinity. Combining Corollary 6.2 and [2, Corollary 1.7.3] (with to the slowly varying function log) one gets

Moreover, by [START_REF] Gouëzel | Random walks on co-compact Fuchsian groups[END_REF]Corollary 9.4],

(15)

, where c > 0 and the sequence (q n (x)) n is non-increasing. We first deduce that

The same proof as in [19, Lemma 9.5] yields n j R n µ q n (e) ∼ C 1 n -1 and so, using again [START_REF] Dussaule | Exotic local limit theorems at the phase transition in free products[END_REF],

. Since G (j) (e, e|r) + G (j) (e, x|r) ∼ (C e + C x ) log 1/(R µr) as r → R µ , we can again apply [START_REF] Bingham | Regular variation[END_REF]Corollary 1.7.3] to deduce that

The fact that (q n (x)) n is non-increasing readily implies

x n -1 as above; finally, by [START_REF] Dussaule | Exotic local limit theorems at the phase transition in free products[END_REF], µ * n (x) ∼ C (2) x R -n µ n -d/2 . Since C 1 is positive and the random walk is admissible, the quantity C

Divergence and spectral positive recurrence

In this section, we describe the relationships between divergence and spectral positive recurrence of random walks on relatively hyperbolic groups. As before, assume that Γ is a finitely generated relatively hyperbolic group with respect to virtually abelian subgroups. Fix a finite set Ω 0 of representatives of conjugacy classes of parabolic subgroups.

Recall that a random walk on Γ is called spectrally positive recurrent if it is divergent and has finite Green moments, i.e. using the notations defined in [START_REF] Dussaule | The Martin boundary of a free product of abelian groups[END_REF], if J (2) (R µ ) is finite. By definition, this occurs when I

H (R µ ) < +∞ for all parabolic subgroup H ∈ Ω 0 .

If the random walk driven by µ is spectrally non degenerate along the parabolic subgroup H, then we automatically have I H (R µ ) < +∞. Since the random walk cannot be spectrally degenerate along virtually abelian parabolic subgroups of rank at most 4 (see [START_REF] Dussaule | Stability phenomena for Martin boundaries of relatively hyperbolic groups[END_REF]Proposition 6.1]), other behaviors only occur when parabolic subgroups have rank at least 5.

Note that the Local limit theorem (1) implies that when the rank of H is at least 7, then G H (e, e|r), G ′ H (e, e|r) and G ′′ H (e, e|r) are finite at the inverse of the spectral radius of G H , which is at least 1. It follow then from Proposition 5.2 (or