Nathalie Krell
email: .nathalie.krell@upmc.fr

Multifractal spectra and precise rates of decay in homogeneous fragmentations

Keywords: Interval fragmentation, Lévy process, Multifractal spectrum. A.M.S. Classification. 60G09, 60J25, 60J80

Multifractal spectra and precise rates of decay in homogeneous fragmentation.

Fragmentation appears in a wide range of phenomena in science and technology, such as degradation of polymers, colloids, droplets, rocks,... See the proceedings [START_REF]Proceedings of the workshop : Fragmentation phenomena[END_REF] for some applications in physics, for example [START_REF] Krapivsky | Traveling waves, front selection, and exact nontrivial exponents in random fragmentation problem[END_REF] for computer science, [START_REF] Bertoin | Fragmentation energy[END_REF] for mineral crushing, and works quoted in [START_REF] Berestycki | Multifractal spectra of fragmentation processes[END_REF] for some further references. This work is a contribution to the study of 1 the rates of decay of fragments. More precisely, our aim is to investigate the set of locations which have an exact exponential decay (see [START_REF] Basdevant | Fragmentation of ordered partitions and intervals[END_REF] below for a precise definition).

Roughly a homogeneous fragmentation of intervals F (t) can be seen as a family of nested open sets in (0, 1) such that each interval component is spill independently of the others, independently of the way that spill before, and with the same law as that of the initial fragmentation (up to spatial rescaling). We will suppose that no loss of mass occurs during the process.

Let x ∈ (0, 1) and I x (t) be the interval component of the fragmentation F (t) which contains x, and |I x (t)| its length. Bertoin showed in [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF] that if V is a uniform random variable on [0, 1] which is independent of the fragmentation, then ξ(t) :=log |I V (t)| is a subordinator entirely determined by the fragmentation characteristics. By the SLLN for a subordinator, there exists v typ such that ξ (t) t → v typ a.s., which means that |I V (t)| ≈ e -vtypt . Berestycki [START_REF] Berestycki | Multifractal spectra of fragmentation processes[END_REF] computed the Hausdorff dimension of the set

G v := x ∈ (0, 1) : lim t→∞ 1 t log |I x (t)| = -v
for all v > 0. In this article we shall rather consider for some 0 < a < b the set

G (v,
Our goal is to compute the Hausdorff dimension of the set G (v,a,b) . Our approach relies on some results on Lévy processes constrained to stay in a given interval.

Firstly we will recall background on fragmentations and Lévy processes. Secondly we will consider an additive martingale M which is naturally associated to the problem and obtain a criterion for uniform integrability. This is used in Section 4 to derive some limit theorems which may be of independent interest (see Engländer and Kyprianou [START_REF] Engländer | Local extinction versus local exponential growth for spatial branching processes[END_REF] for a related approach in the setting of spatial branching processes). Finally we will compute the Hausdorff dimension of G (v,a,b) in Section 5.

2 Preliminaries.

Definition of fragmentation.

We will recall some facts about homogeneous interval fragmentations, which are mostly lifted from [START_REF] Berestycki | Multifractal spectra of fragmentation processes[END_REF] , [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF] and [START_REF] Bertoin | The asymptotic behavior of fragmentation processes[END_REF]. More precisely, we will consider fragmentations defined on the space U of open subsets of (0, 1). We shall use the fact that every element U of U has an interval decomposition, i.e. there exists a collection of disjoint open intervals (J i ) i∈I , where the set of indices I can be finite or countable, such that U = ∪ i∈I J i . Each interval component is viewed as a fragment.

A homogeneous interval fragmentation is a Markov process with values in the space U which enjoys two keys properties. First the branching property: different fragments have independent evolutions. Second, the homogeneity property: up to an obvious spacial rescaling, the law of the fragment process does not depend on the initial length of the interval.

Specifically, if P stands for the law of the interval fragmentation F started from F (0) = (0, 1), then for s, t ≥ 0 conditionally on the open set F (t) = ∪ i∈I J i (t), the interval fragmentation F (t+ s) has the same law as F 1 (s) ∪F 2 (s) ∪... where for each i, F i (s) is a subset of J i (t) and has the same distribution as the image of F (s) by the homothetic map (0, 1) → J i (t).

Poissonian construction of the fragmentation.

Recall that U denotes the space of open subsets of (0, 1), and set 1 = (0, 1). For U ∈ U, |U| ↓ := (u 1 , u 2 , ...) will be the decreasing sequence of the interval component lengths of U. For U = (a 1 , b 1 ) ∈ U, we define the affine transformation g U : (0, 1) → U given by g U (x) = a 1 + x(b 1a 1 ).

In this article we will only consider proper fragmentations (which means that the Lebesgue measure of F (t) is equal to 1). In this case, Basdevant [START_REF] Basdevant | Fragmentation of ordered partitions and intervals[END_REF] has shown that the law of the interval fragmentation F is completely characterized by the so-called dislocation measure ν (corresponding to the jump-component of the process) which is a measure on U which fulfills the conditions ν(1) = 0,

U (1 -u 1 )ν(dU) < ∞, (2)
and

∞ i=1 u i = 1 f or ν -almost every U ∈ U.
This last assumption is imposed by the hypothesis of length-conservation and means that when a sudden dislocation occurs, the total length of the intervals is unchanged. Specialists will notice that the erosion rates of the fragmentation c r and c l are here equal to 0 for the same reason. We now recall the interpretation of sudden dislocations of the fragmentation process in terms of atoms of a Poisson point process (see [START_REF] Basdevant | Fragmentation of ordered partitions and intervals[END_REF], [START_REF] Berestycki | Ranked fragmentations[END_REF]). Let ν be a dislocation measure fulfilling the preceding conditions. Let K = ((∆(t), k(t)), t ≥ 0) be a Poisson point process with values in U × N, and with intensity measure ν ⊗ ♯, where ♯ is the counting measure on N. As in [START_REF] Berestycki | Ranked fragmentations[END_REF], we can construct a unique U-valued process F = (F (t), t ≥ 0) started from (0, 1), with paths that jump only for times t ≥ 0 at which a point (∆(t), k(t)) occurs, and then F (t) is obtained by replacing the k(t)-interval J k(t) (t-) by g J k(t) (t-) (∆(t)). This point of view will be used in Section 3. Some information about the dislocation measure ν and therefore about the distribution of the homogeneous fragmentation F is contained in the function:

κ(q) := U 1 - ∞ j=1 u q+1 j ν(dU) ∀q > p (3)
with p the smallest real number for which κ remains finite :

p := inf p ∈ R : U ∞ j=2 u p+1 j ν(dU) < ∞ . We have that -1 ≤ p ≤ 0 (because U (1 -u 1 )ν(dU) < ∞ and ∞ i=1 u i = 1 for ν-almost every U ∈ U).
This point of view is the same as in [START_REF] Berestycki | Multifractal spectra of fragmentation processes[END_REF] and [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF], which deal with ranked fragmentation instead of interval fragmentation. In the latter the space U is replaced by the space of mass partitions

S ↓ := x = (x 1 , x 2 , ...) | x 1 ≥ x 2 ≥ ... ≥ 0 , ∞ i=1 x i ≤ 1 .
For the precise link between these two fragmentations see [START_REF] Basdevant | Fragmentation of ordered partitions and intervals[END_REF].

An important subordinator.

Let x ∈ (0, 1) and I x (t) be the interval component of the random open set F (t) which contains x, and |I x (t)| its length. Let V be a uniform random variable on [0, 1] which is independent of the fragmentation.

Bertoin showed in [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF] that

ξ(t) := -log |I V (t)|, t ≥ 0, (4)
is a subordinator, with Laplace exponent κ(q) defined in (3) (i.e. E(e -λξ(t) ) = e -tκ(λ) for all λ > p). In order to interpret this as a Lévy-Khintchine formula, we introduce the measure

L(dx) := e -x ∞ j=1 ν(-log u j ∈ dx), x ∈ (0, ∞).
It is easy to check that min(1, x)L(dx) < ∞, thus L is the Lévy measure of a subordinator, and we can check that κ(q) = (0,∞) (1e qx ) L(dx).

In this article we shall consider the Lévy process Y t = vtξ(t). In order to apply certain results to this process, we will need to assume that its one-dimensional distributions are absolutely continuous. Let L ac be the absolutely continuous part of the measure L. Tucker has shown in [START_REF] Tucker | Absolute continuity of infinitely divisible distributions[END_REF] that

R + 1 1 + x 2 L ac (dx) = ∞, (5)
ensures the absolute continuity of one-dimensional distribution of the Lévy process evaluated at any t > 0. As min(1, x)L(dx) < ∞, the condition ( 5) is equivalent to :

L ac ([0, ǫ)) = ∞ for any ǫ > 0. ( 6
)
Let ν 1 be the image of the measure ν by the map U → u 1 (recall that u 1 is the length of the longest interval component of the open set U) and ν ac 1 be the absolutely continuous part of the measure ν 1 . Throughout this work we will make the following assumption, which is easily seen to imply (6) (in fact we can even show that the two are equivalent):

ν ac 1 ([0, ǫ)) = ∞ for any ǫ > 0. (7)
In the next subsection, we will give some results about Lévy processes that will be needed in the sequel, and apply for Y t = vtξ(t).

An estimate for completely asymmetric Lévy processes.

For the next sections, we will need some technical notions about completely asymmetric Lévy processes. Therefore we recall some facts mostly lifted from [START_REF] Bertoin | Lévy processes[END_REF] and [START_REF] Bertoin | Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval[END_REF]. Let Y = (Y t ) t≥0 be a Lévy process with no positive jumps and (E t ) t≥0 the natural filtration associated to (Y t ) t≥0 . The case where Y is the negative of a subordinator is degenerate for our purpose and therefore will be implicitly excluded in the rest of the article. The law of the Lévy process started at x ∈ R will be denoted by P x (so bold symbols P and E refer to the Lévy process while P and E refer to the fragmentation), its Laplace transform is given by E 0 (e λYt ) = e tψ(λ) , λ, t ≥ 0, where ψ : R + → R is called the Laplace exponent. Let φ : R + → R + be the right inverse of ψ (which exists because ψ : R

+ → R is convex with lim t→∞ ψ(λ) = ∞), i.e. ψ(φ(λ)) = λ ∀λ ≥ 0.
Let us recall some important features on the two-sided exit problem (which is completely solved in [START_REF] Bertoin | Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval[END_REF]). For β > 0 we denote the first exit time from (0, β) by

T β = inf{t : Y t / ∈ (0, β)}. (8)
Let W : R + → R + be the scale function, that is the unique continuous function with Laplace transform:

∞ 0 e -λx W (x)dx = 1 ψ(λ)
, λ > φ(0).

For q ∈ R, let W (q) : R + → R + be the continuous function such that for every x ∈ R +

W (q) (x) := ∞ k=0 q k W * k+1 (x),
where W * n = W * ... * W denotes the nth convolution power of the function W (for more details about this see [START_REF] Bertoin | Lévy processes[END_REF] or [START_REF] Bertoin | Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval[END_REF]). So that

∞ 0 e -λx W (q) (x)dx = 1 ψ(λ) -q , λ > φ(q).
The next statement is about the asymptotic behavior of the Lévy process killed when it exits (0, β) (point 1 and 2), which is taken from [START_REF] Bertoin | Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval[END_REF], and about the Lévy process conditioned to remain in (0, β) (point 3, 4 and 5), which is taken from Theorem 3.1 ( ii) and Proposition 5.1 (i) and (ii) in [START_REF] Lambert | Completely asymmetric Lévy processes confined in a finite interval[END_REF] :

Theorem 1 Let us define the transition probabilities

P t (x, A) := P x (Y t ∈ A, t < T β ) for x ∈ (0, β) and A ∈ B((0, β)),
and the critical value

ρ β := inf{q ≥ 0 ; W (-q) (β) = 0}, (9)
Suppose that the one-dimensional distributions of the Lévy process are absolutely continuous. Then the following holds:

1. ρ β ∈ (0, ∞) and the function W (-ρ β ) is strictly positive on (0, β)

2. Let Π(dx) := W (-ρ β ) (βx)dx. For every x ∈ (0, β):

lim t→∞ e ρ β t P t (x, .) = cW (-ρ β ) (x)Π(.)
in the sense of weak convergence, where

c := β 0 W (-ρ β ) (y)W (-ρ β ) (β -y)dy -1
.

The process

D t := e ρ β t 1 {t<T β } W (-ρ β ) (Y t ) W (-ρ β ) (x) ( 10
)
is a (P x , (E t ))-martingale.

The mapping

(x, q) → W (q) (x) is of class C 1 on (0, ∞) × (-∞, ∞).
5. The mapping β → ρ β = inf{q > 0 : W (-q) (β) = 0} is strictly decreasing and of class C 1 on (0, ∞).

Remark 1 The definition of ρ β is of course complicated, however in the simple case when Y is a standard Brownian motion, we have:

ρ β = π 2 /β 2 and W (-ρ β ) (x) = β π sin π β x .
In the case where Y is a standard stable process, the mapping of β → ρ β is depicted in [START_REF] Bertoin | On the first exit time of a completely asymmetric stable process from a finite interval[END_REF].

We also point at the more explicit lower bound (see Lemma 5 in [START_REF] Bertoin | Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval[END_REF]):

ρ a ≥ 1/W (a),
Another lower bound will be given in Remark 4 below.

Remark 2

The formula for the constant c in part 2. of Theorem 1 stems from the relation

e ρ β t W (-ρ β ) (y) W (-ρ β ) (x) P t (x, dy) ∼ t→∞ cW (-ρ β ) (β -y)W (-ρ β ) (y)dy.
Integrating over (0, β) and using the fact that D t is a martingale yields the given expression.

We also refer to the recent article of T. Chan and A. Kyprianou [START_REF] Chan | Smoothness of scale function for spectrally negative Lévy processes[END_REF] for further properties of W (-ρ β ) . Now we have recalled the background that is needed to solve our problem.

3 An additive martingale.

Now we turn our attention to the main purpose of this article and consider a homogeneous interval fragmentation (F (t), t ≥ 0) and some real numbers v > 0 and 0 < a < b. We are interested in the asymptotic set: In order to do that, we will have to consider first the non asymptotic set:

G (v,a,b) = x ∈ (0,
Λ (v,a,b) = x ∈ (0, 1) : ae -vt < |I x (t)| < be -vt ∀t ≥ 0 , for 0 < a < 1 < b.
In this section and in the next we will assume that 0 < a < 1 < b.

We introduce some notation, that we will need in the rest of the article: define the set of the "good" intervals at time t as G(t) := {I x (t) : x ∈ (0, 1) and ae -vs < |I x (s)| < be -vs ∀ s ≤ t}. [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF] Let (F t ) t≥0 be the natural filtration of the interval fragmentation (F (t), t ≥ 0). Let (G t ) t≥0 be the enlarged filtration defined by G t = F t ∨ σ(I V (t)) where V is a uniform variable independent of the fragmentation). We can remark that for all t we have G t F t ∨ σ{V }, and G ∞ = F ∞ ∨ σ{V }.

We recall that ξ(t) =log |I V (t)| is a subordinator. More precisely we are interested in the Lévy process with no positive jump Y t := vtξ(t) + log(1/a), and use the results of preceding subsection for this Lévy process. We remark that its Laplace exponent ψ(λ) is equal to vλκ(λ), with κ defined in Subsection 2.3. Since we have supposed [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF], the one-dimensional distributions of the Lévy process Y t are absolutely continuous and we can apply Theorem 1.

For this Lévy process Y let

T := T log(b/a) and ρ := ρ log(b/a) ,
where T β is defined in ( 8) and ρ β is defined in [START_REF] Bertoin | Asymptotic laws for nonconservative self-similar fragmentations[END_REF]. We stress that ρ depends on v, a, b and κ.

To simplify the notation, let also

h(t) := W (-ρ) (t -log a)1 {t∈(log a,log b)}
for all t ∈ R, and h(-∞) = 0. By rewriting [START_REF] Bertoin | Fragmentation energy[END_REF] with the new notation we get a (G t )-martingale

D t = e ρt 1 {t<T } h(vt + log |I V (t)|) h(0) , t ≥ 0.
If I is an interval component of F (t), we define the "killed" interval I † by I † = I if I is good (i.e. I ∈ G(t) with G(t) defined in ( 11)), else by I † = ∅. Projecting the martingale D t on the sub-filtration (F t ) t≥0 , we obtain an additive martingale

M t := e ρt h(0) 1 0 h(vt + log |I † x (t)|) dx , t ≥ 0.
We notice that if y ∈ I x (t), then I y (t) = I x (t). Now we will consider the interval decomposition (J i (t), J 2 (t), ...) of the open F (t) (see subsection 2.1). We can rewrite M t as:

M t = e ρt h(0) i∈N h vt + log |J † i (t)| |J † i (t)|. ( 12
)
We will use this expression in the rest of the article. Finally, let the absorption time of M t at 0 be

ζ := inf{t : M t = 0} = inf{t : G(t) = ∅}, with the convention inf ∅ = ∞.
Our first result is:

Theorem 2 In the previous notation, with the assumptions [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF] and if v > ρ holds, then:

1. The martingale M t is bounded in L 2 (P).

2. Conditionally on ζ = ∞, we have: lim t→∞ M t > 0.

Remark 3 We stress that as ρ depends on v, a, b and κ, the condition v > ρ involves implicitly the parameters a and b. In particular it forces b > 2a, otherwise there would never be more than one "good" interval (as a fragment of size x will split into at least two different fragments and the smallest one will have a size at most equal to x/2), and as a consequence we would have M ∞ = 0 a.s., in contradiction with the uniform integrability of M . .

The proof of Theorem 2.1. is given in the appendix.

In order to prove Theorem 2.2 we will first introduce some notation, then prove two lemmas, and after we will conclude.

Let I be an interval of (0, 1). The law of the homogeneous interval fragmentation started at I will be denoted by P I . We remark that P I (M ∞ = 0|ζ = ∞) only depends on the length of I. Therefore we define g(x)

:= P I (M ∞ = 0|ζ = ∞),
where I is an interval such that |I| = x. Let N be the integer part of (2ba)/a. As we assume v > ρ, we have necessarily b > 2a (see Remark 3), thus N ≥ 2. Let η := (ba)N -1 . We remark that η < a and ba = Nη. Denote the first time when there are at least two good intervals by

T F := inf{t : ♯G(t) ≥ 2},
with the convention inf ∅ = ∞. We notice that T F is an (F t ) stopping time as ♯G(t) is F t -adapted.

Lemma 1 In the previous notation, supposing that (7) and v > ρ hold, we get: for every open interval

I P I (T F = ∞|ζ = ∞) = 0.
Proof We notice that, as the martingale M t is not identically 0 and is uniformly integrable, we have

P I (T F = ∞|ζ = ∞) < 1 (because M ∞ = 0 when T F = ∞).
Let I be an open interval such that |I| ∈ (a, b), t 0 := log(2b/a)/v and ǫ := a 2 /(2b 2 ). Thus

|I|(1ǫ) > a/2 ≥ be -vt 0 and |I|ǫ < bǫ ≤ ae -vt 0 therefore, if the dislocation of I produces at time t 0 an interval of length at least |I|(1ǫ) then this interval is too large to be good and the remaining ones are too small to be good either. As a consequence we have

P I (M t 0 = 0) ≥ P log |I| (e -ξ(t 0 ) > e -log |I| (1 -ǫ)) = P(ξ(t 0 ) < -log(1 -ǫ)),
by the homogeneous property of the fragmentation. Moreover since ξ(t) is a subordinator, we get p := P(ξ(t 0 ) <log(1ǫ)) > 0, therefore

P I (M t 0 = 0) ≥ p > 0. ( 13
)
Additionally for every open interval I such that |I| ∈ (a, b):

P I (♯G(t) = 1 ∀t ≤ t 0 ) ≤ 1 -P I (M t 0 = 0) ≤ 1 -p.
Using the strong Markov property of the fragmentation and ( 13) we find by induction that for all k ∈ N:

P I (♯G(t) = 1 ∀t ≤ kt 0 ) ≤ (1 -p) k .
Therefore lim

t→∞ P I (♯G(s) = 1 ∀s ≤ t) = 0
and as a consequence

P I (T F = ∞|ζ = ∞) = 0.
Lemma 2 In the previous notation, supposing that (7) and v > ρ hold, we get:

sup a<x<b g(x) = max 1≤k≤N g(a + kη),
where N = ⌊(2ba)/a⌋ and η = (ba)/N.

Proof We will prove this lemma by induction. The hypothesis of induction is for n ≤ N:

(H) n : sup x∈(a,a+nη) g(x) = max 1≤k≤n
g(a + ηk). * The case n = 1: let I be an open interval such that |I| ∈ (a, a + η). We work under P I conditionally on "non-extinction" (which means conditionally on the event ζ = ∞). Let

T 1 := inf{t ≥ 0| ∃J(t) ∈ G(t) : e vt |J(t)| / ∈ (a, a + η)},
with G(t) defined in [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF]. The random time T 1 is an (F t ) stopping times. As the quantity vt -(-log |J(t)|) creeps upwards with probability equals to 1 and as J(t) ∈ G(t) implies that e vt |J(t)| > a, we get

T 1 = inf{t ≥ 0| ∃J(t) ∈ G(t) : e vt |J(t)| = a + η}.
Moreover by the choice of η we have a + η < 2a, which implies that there is at most one good interval whose length is always in (a, a + η). Recall from Lemma 1 that

P I (T F < ∞|ζ = ∞) = 1, thus P I (T 1 < ∞|ζ = ∞) = 1.
Using the strong Markov property at the stopping times T 1 , we get

g(x) ≤ g(a + η) , x ∈ (a, a + η),
thus (H) 1 holds. * The case n + 1 (with n + 1 ≤ N): we suppose that the hypothesis of induction holds for all k ≤ n.

Let I be an open interval such that |I| ∈ (a + nη, a + (n + 1)η). We work under P I conditionally on "non-extinction". Let

T n := inf{t ≥ 0| ∃J(t) ∈ G(t) : e vt |J(t)| / ∈ (a + nη, a + (n + 1)η)},
with G(t) defined in [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF]. The random time T n is an (F t ) stopping times. As the quantity e vt |J(t)| grows only continuously, we get

T n = inf{t ≥ 0| ∃J(t) ∈ G(t) : e vt |J(t)| = a + (n + 1)η or e vt |J(t)| ∈ (a, a + nη]}.
Moreover by the choice of η we have a + η < 2a, which implies that there is at most one good interval which length is always in (a + nη, a + (n + 1)η). Additionally by Lemma 1, we get P I (T F < ∞|ζ = ∞) = 1, thus

P I (T n < ∞|ζ = ∞) = 1.
Using the strong Markov property at the stopping times T n , we get

g(|I|) ≤ max g(a + (n + 1)η), sup y∈(a,a+nη] g(y) .
As this holds for every open interval I such that |I| ∈ (a+nη, a+(n+1)η), by the hypothesis of induction, we have established (H) n+1 .

Proof[Proof of Theorem 2.2.] With Lemma 2, we get that there exists a integer k 0 in [1, N] such that g(a + ηk 0 ) = sup x∈(a,b) g(x) (if two or more values of k, are possible, we choose the smallest one). Let x 0 be a + ηk 0 .

Additionally, with Lemma 1, we get P (0,x 0 ) (T F < ∞|ζ = ∞) = 1. Using the strong property of Markov for the stopping times T F , and with n ≥ 2 the random number of good intervals of the fragmentation at time T F and with α 1 , ..., α n the length of those intervals, we get:

g(x 0 ) ≤ E(g(α 1 )...g(α n )) ≤ E(g(x 0 ) n ) ≤ g(x 0 ) 2 .
As g(x 0 ) < 1 by the uniformly integrability of M t , we get that g(x 0 ) = 0 and finally that g ≡ 0.

4 Limit theorems.

In this section, we establish two corollaries of Theorem 2, which will be useful in the sequel.

Bertoin and Rouault (Corollary 2 in [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF]) proved that lim

t→∞ 1 t log ♯{I x (t) : ae -vt < |I x (t)| < be -tv } = C(v), (14)
where 1Here we deal with the more stringent requirement: ∀s ≤ t, |I x (s)| ∈ (ae -sv , be -sv ), and the next proposition gives the rates that we find in that case.

C(v) := (Υ v + 1)v -κ(Υ v ) and Υ v is the reciprocal of v by κ ′ i.e, κ ′ (Υ v ) = v for v ∈ (v min , v max ).
Proposition 1 In the notation of the previous sections, with the assumptions [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF] and if v > ρ we get that conditionally on ζ = ∞ (i.e. M is not absorbed at 0, or in a equivalent way Λ (v,a,b) = ∅):

lim t→∞ 1 t log ♯G(t) = v -ρ a.s. ( 15
)
Before proving this corollary we make the following remark Remark 4 It is interesting to compare the estimate found by Bertoin and Rouault and the present one (of course we have not considered the same set, nevertheless the two estimates are related). For this we show that for all v ∈ (v min , v max ) and a and b such that ρ ≥ v min we have

C(v) ≥ v -ρ.
In this direction we use results from [START_REF] Berestycki | Multifractal spectra of fragmentation processes[END_REF] Section 1. Let Ψ(p) := pκ ′ (p)κ(p) for all p > 0 with κ ′ the derivative of κ (this function is well defined because of the definition of p in Section 2 and because p ≤ 0). For every p > 0, Ψ

′ (p) = pκ ′′ (p) ≤ 0 since κ is concave. As a consequence Ψ is decreasing. With the definition of Υ v , we get that the function v ∈ (v min , v max ) → Υ v ∈ R is decreasing, additionally Υ v min > 0, therefore the function v ∈ (v min , v max ) → g(Υ v ) ∈ R is increasing. Moreover Ψ(Υ v ) = C(v) -v, hence for all v ∈ (v min , v max ): C(v) -v ≥ C(v min ) -v min = -v min .
Additionally as ρ ≥ v min , we finally obtain:

∀v ∈ (v min , v max ) C(v) ≥ v -ρ.
As a consequence, we have checked that the rate of growth of ♯G(t) (defined in [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF]) is lower that of ♯{I x (t) : |I x (t)| ∈ (ae -tv , be -vt )}, which was of course expected.

Proof In this proof we work conditionally on ζ = ∞ ( i.e M is not absorbed at 0). Applying Theorem 2, we get M ∞ > 0. In order to show that (15) holds, we will first look at the lower bound of the inequality, and then at the upper bound.

• With the definition of M t in [START_REF]Proceedings of the workshop : Fragmentation phenomena[END_REF], of G(t) and of J † i (t) at the beginning of Section 3 and by the conditioning, there exists t ′ > 0 such that for all t ≥ t ′ :

M ∞ 2 ≤ e ρt h(0) i∈N h(vt + log(|J † i (t)|)) |J † i (t)| ≤ e ρt h(0) i∈N C 4 be -vt 1 {J i (t)∈G(t)} ,
with C 4 as maximum of h(.) on [log a, log b]. Hence for all t ≥ t ′ :

♯G(t) ≥ e (v-ρ)t h(0) 2C 4 b M ∞ ,
and as a consequence, conditionally on ζ = ∞,

lim inf t→∞ 1 t log ♯G(t) ≥ v -ρ. (16)
• Secondly we will show the converse inequality. Let 0 < a ′ < a < 1 < b < b ′ , and ρ ′ := ρ log(b ′ /a ′ ) . Denote the set of "good" intervals associated to a ′ and b ′ by:

G ′ (t) := {I x (t) : x ∈ (0, 1) and |I x (s)| ∈ (a ′ e -vs , b ′ e -vs ) ∀ s ≤ t}.
Let M ′ t be the martingale defined at the beginning of Section 3 (and denoted there by M) associated to a ′ , b ′ instead of a, b. Plainly, if M t is not absorbed at 0, then a fortiori M ′ t is not absorbed at 0 either. Additionally, since log(b ′ /a ′ ) > log(b/a), and ρ . is strictly decreasing (see Theorem 1.5), we get v > ρ > ρ ′ and we may apply Theorem 2 for a

′ , b ′ instead of a, b. We get lim t→∞ M ′ t = M ′ ∞ > 0.
With the definition (12) of M t and with an analogue of the function

h(t), namely t ∈ R ϕ(t) := W (-ρ ′ ) (t + log(1/a ′ ))1 {t∈(log a ′ ,log b ′ )} ,
we get:

M ′ ∞ = lim t→∞ e ρ ′ t ϕ(0) i∈N ϕ(vt + log |J i (t)|)|J i (t)| 1 {J i (t)∈G ′ (t)} .
Therefore there exists t ′ > 0 such that for every t ≥ t Combining this with

′ 2M ′ ∞ ≥ e ρ ′ t ϕ(0) i∈N ϕ(vt + log |J i (t)|)|J i (t)| 1 {J i (t)∈G ′ (t)} ≥ e ρ ′ t ϕ(0) i∈N ϕ(vt + log |J i (t)|) a ′ e -vt 1 {J i (t)∈G(t)} . Since (ae -vt , be -vt ) (a ′ e -vt , b ′ e -vt
C 5 := 2M ′ ∞ ϕ(0)/ a ′ inf x∈[log a,log b] ϕ(x) < ∞,
we get for all t ≥ t ′ :

C 5 ≥ e (ρ ′ -v)t i∈N 1 {J i (t)∈G(t)}
and thus

C 5 e (v-ρ ′ )t ≥ ♯G(t).
Hence for all a ′ , b

′ such that 0 < a ′ < a < 1 < b < b ′ : lim sup t→∞ 1 t log ♯G(t) ≤ v -ρ ′ .
For a ′ → a and b ′ → b we get by the continuity of ρ . :

lim sup t→∞ 1 t log ♯G(t) ≤ v -ρ.
Now we will give an other corollary, using the same method as that of Bertoin and Gnedin in [START_REF] Bertoin | Asymptotic laws for nonconservative self-similar fragmentations[END_REF]. We encode the configuration J † (t) = {|J † i (t)|} of the lengths of good intervals into the random measure

σ t := e ρt h(0) i∈N h vt + log |J † i (t)| |J † i (t)|δ log(1/a)+vt+log |J † i (t)|
which has total mass M t .

The associated mean measure σ * t is defined by the formula

∞ 0 f (x)σ * t (dx) = E ∞ 0 f (x)σ t (dx)
which is required to hold for all compactly supported continuous functions f . Since M t is a martingale, σ * t is a probability measure. More precisely the next proposition establishes the convergence of the mean measure σ * t , and then of σ t itself.

Proposition 2 In the notation of the previous sections, with the assumptions [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF], and v > ρ we get:

1. The measures σ * t converge weakly, as t → ∞, to the probability measure

̺(dy) := ch(y + log a)h(log(b) -y)dy
where c > 0 is the constant that appears in Theorem 1.5.

For any bounded continuous

f L 2 -lim t→∞ ∞ 0 f (x)σ t (dx) = M ∞ ∞ 0 f (x)̺(dx). (17)
Proof 1. Firstly we prove the convergence of the mean measures σ * t → ̺. Let f be a bounded continuous function. By definition we get: Therefore the measure σ * t converge weakly to the probability measure ̺.

∞ 0 f (y)σ * t (dy) = E 1 0 f (log(1/a) + vt + log |I † x (t)|) e ρt h(0) h vt + log |I † x (t)| 1 {I † x (t)∈G(t)} dx = E log(1/a) f (Y t )e ρt h(Y t + log a) h(0) 1 {t<T } ,
2. Now we show that the scaled empirical measures induced by J(t) converge in the L 2 -sense to the random measure M ∞ ̺.

Let f 1 and f 2 be two continuous functions bounded from above by 1, and

S t = i,j f 1 (log(1/a) + vt + log |J † i (t)|) e ρt h(0) h vt + log |J † i (t)| |J † i (t)| ×f 2 (log(1/a) + vt + log |J † j (t)|) e ρt h(0) h vt + log |J † j (t)| |J † j (t)|.
We need to show that

E (S t ) → ∞ 0 f 1 (x)̺(dx) ∞ 0 f 2 (x)̺(dx) E M 2 ∞ ( 18
)
for f 1 and f 2 positive and bounded from above by 1. Indeed, suppose ( 18) is shown. Denote

A t = j f 1 (log(1/a) + vt + log |J † i (t)|) e ρt h(0) h vt + log |J † j (t)| |J † j (t)|. Take f 2 = 1 to conclude from (18) that lim t→∞ E(A t M t ) = ∞ 0 f 1 (x)̺(dx)E M 2 ∞ .
Similarly, by setting

f 1 = f 2 we get lim t→∞ E A 2 t = ∞ 0 f 1 (x)̺(dx) 2 E M 2 ∞ .
Recalling that E(M 2 t ) → E(M 2 ∞ ) and combining the above we get the desired

lim t→∞ E A t -M t ∞ 0 f 1 (x)̺(dx) 2 = 0.
To prove [START_REF] Krapivsky | Traveling waves, front selection, and exact nontrivial exponents in random fragmentation problem[END_REF] let us replace t by t + s and condition on J † = (|J † i (s)|) i∈N . We have two cases: write i ∼ s j for the case where at time t + s two coexisting intervals J † i (t + s) and J † j (t + s) stem from the same interval at time s, and i ≁ s j for the case these intervals are not included into the same interval component at time s. Therefore, with the notation

S (1) t+s := E i∼sj S t+s | J † (s)
and S

(2)

t+s := E i≁sj S t+s | J † (s)
we get:

S (1) t+s + S (2)
t+s = E(S t+s |J † (s)).

For the studies of S

(1)

t+s we use the homogeneous property of the fragmentation and the notation I 0 = (0, log(b/a)), and get

|S (1) t+s | ≤ i |J † i (s)| 2 e 2ρs E j |J † j (t)|e ρt 2 sup x∈I 0 h(x + log a) h(0) 2 sup x∈I 0 |f 1 (x)| sup x∈I 0 |f 2 (x)| ≤ be (ρ-v)s C 6 , with
C 6 := i |J † i (s)|e ρt E j |J † j (t)|e ρt 2 sup x∈I 0 h(x + log(a)) 2 sup x∈I 0 |f 1 (x)| sup x∈I 0 |f 2 (x)|/h(0) 2
which is finite because

E j |J † j (t)|e ρt = E 1 {t<T } e ρt < ∞.
Thus S

t+s → 0 as s → ∞ uniformly in t. Now we look at S Therefore by dominated convergence

E S (2) t+s ∼ s→∞ ∞ 0 f 1 (x)̺(dx) ∞ 0 f 2 (x)̺(dx) E k =l e ρs h(0) |J † k (s)| h vs + log(|J † k (s)|/a) e ρs h(0) h vs + log(|J † l (s)|/a) |J † l (s)| .
Moreover with

C 7 := b sup x∈I 0 |h(x + log a)| 2 /h(0) 2 , we get E    k e 2ρs h vs + log |J † k (s)|/a 2 h(0) 2 |J † k (s)| 2    ≤ C 7 E k e ρs |J † k (s)| e (ρ-v)s
which goes to 0 when s → ∞, as a consequence

E   k =l e ρs h vs + log |J † k (s)|/a h(0) |J † k (s)|e ρs h vs + log |J † l (s)|/a h(0) |J † l (s)|   ∼ s→∞ E M 2 s .
5 The Hausdorff dimension.

In this section we use the notation and definitions of the previous sections. We recall that ρ = ρ log(b/a) , where ρ . is define in [START_REF] Bertoin | Asymptotic laws for nonconservative self-similar fragmentations[END_REF]. Let dim be the Hausdorff dimension. The aim of this section would be to proof the main theorem:

Theorem 3 : Multifractal spectrum. Assume ( 7):

• if ρ > v holds, then: G (v,a,b) = ∅ a.s. • if ρ < v holds, then: dim(G (v,a,b) ) = 1 -ρ/v a.s. ( 19
)
Remark 5 1. Berestycki in [START_REF] Berestycki | Multifractal spectra of fragmentation processes[END_REF] has computed the Hausdorff dimension of the set

G v = x ∈ (0, 1) | lim t→∞ 1 t log |I x (t)| = -v . He found that for v ∈ (v min , v max ), dim(G v ) = C(v)/v (with C(v)
defined at the beginning of section 4). In Remark 4 we have shown that for all v ∈ (max(v min , ρ), v max ) we have C(v) ≥ vρ and we can notice that the inequality is strict for ρ > v min . As a consequence the set G (v,a,b) has a Hausdorff dimension smaller than that of G v , and also smaller than that one could have infer from equality [START_REF] Engländer | Strong law of large numbers for branching diffusions[END_REF].

2. In the case v > v typ , we have Y t /t → t→∞ vv typ > 0 a.s. and

P log(1/a) (inf{t : Y t ≤ 0} = ∞) > 0.
Thus W (-q) (∞) = 0 for all q ≥ 0 and then lim β→∞ ρ β = 0. Moreover using the fact that, lim β→0 ρ β = ∞ and ρ . is decreasing, we get that for all v > v typ , there exist a and b such that ρ log(b/a) < v and thus the fact that the set of good intervals is not empty.

The proof of this theorem use the non-asymptotic set Λ (v,a,b) . In particular the key of the proof is the next proposition: Proposition 3 Assume (7) and 0 < a < b < 1:

• if ρ > v holds, then:

Λ (v,a,b) = ∅ a.s. • if ρ < v holds, then: P(Λ (v,a,b) = ∅) > 0, and conditionally on Λ (v,a,b) = ∅, dim(Λ (v,a,b) ) = 1 -ρ/v. ( 20
)
Proof 1. Let v > 0 and a and b such that v < ρ. We define

N(t) := ♯G(t),
with G(t) defined in [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF]. We remark that

N(t) = 1 0 1 |I x (t)| 1 {Ix(t)∈G(t)} (x)dx.
and in particular

E(N(t)) = E 1 0 1 |I x (t)| 1 {Ix(t)∈G(t)} (x)dx .
Additionally by ( 4), we get

E(N(t)) = e vt E e ξ(t)-vt 1 {vs-ξ(s)-log a∈(0,log(b/a)) ∀ s≤t} .
With the notation Y t = vtξ(t) and P t defined in Theorem 1 we rewrite the previous equality as: Finally as the function y → e -y h(log(b)y) is continuous, the integral above is a finite constant. Thus if ρ > v then lim t→∞ E(N(t)) = 0, from which one concludes that lim t→∞ N(t) = 0 , i.e. Λ (v,a,b) = ∅ a.s.

E(N(t)) = e vt E log(1/a) e -Yt-log a 1 {t<T } = 1 a e (v-
2. Now we deal with the case where a and b are such that v > ρ. We work conditionally on Λ (v,a,b) = ∅ (or, equivalently, on the event ζ = ∞, which has a positive probability by Theorem 2).

• Firstly, in order to prove the lower bound of the Hausdorff dimension of Λ (v,a,b) , we will use the same method as Berestycki in [START_REF] Berestycki | Multifractal spectra of fragmentation processes[END_REF] . We will divide this proof into three steps. Each step will begin with a star (⋆). In the first step we will construct a subset

∩ n∈N G δ (n) of Λ (v,a,b
) , which will be defined latter on (see [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]). In the second we shall obtain a lower bound of the Hausdorff dimension of this subset. In order to do that we will construct an increasing process indexed by t ∈ (0, 1), which only increases on ∩ n∈N G δ (n), and which is Hölder continuous. In the last step we will conclude. ⋆ As in [START_REF] Berestycki | Multifractal spectra of fragmentation processes[END_REF] for δ > 0 we define for all n ∈ N, H δ (n) as a multi-type branching process with each particle corresponding to a segment of G(δn) and

G δ (n) := ∪ I∈H δ (n) I,
with G(t) defined in [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF]

(i.e. G δ (n) = G(δn)).
We notice that the family (G δ (n)) n∈N is nested and that

∩ n∈N G δ (n) = Λ (v,a,b) .
Let ǫ > 0, and fix ǫ ′ > 0 and η > 0 such that η < min(ǫ, vρ). By Proposition 1, for this ǫ ′ > 0 and η > 0, we may find t 0 > max((1+| log(1-ǫ ′ )|)/(ǫ-η), log(2)/(v-ρ-η)) such that for all t > t 0 :

P(|t -1 log(♯G(t)) -(v -ρ)| > η|ζ = ∞) < ǫ ′ .
For each t > 0, we consider a variable ∼ χ(t) whose law is given by

P( ∼ χ(t) = 0) = ǫ ′′ , and
P( ∼ χ(t) = ⌊e [(v-ρ)-η]t ⌋) = 1 -ǫ ′′ ,
where ⌊.⌋ is the integer part and ǫ

′′ := P(|t -1 log(♯G(t)) -(v -ρ)| > η|ζ = ∞) < ǫ ′ .
Moreover by using that for all x ≥ 2: log(x) -1 ≤ log(⌊x⌋), we notice that

|t -1 log(E( ∼ χ(t))) -(v -ρ)| ≤ η + t -1 (| log(1 -ǫ ′ )| + 1).
Plainly ∼ χ(t) is stochastically dominated by ♯G(t). Exactly as in [START_REF] Berestycki | Multifractal spectra of fragmentation processes[END_REF] we can construct a true Galton-Watson tree H by thinning H δ where δ > t 0 . More precisely the offspring distribution of H is given by the law of ∼ χ(δ). Let m := E( ∼ χ(δ)) be the expectation of the number of children of a particle. Therefore, we get

(a) |δ -1 log m -(v -ρ)| < ǫ. ( 21
) (b) The family (G(n) := ∪ I∈H(n) I) n∈N (22)
is nested. The G(n) is the union of the interval of the n generation of H.

(c) ∩ n∈N G(n) ⊆ Λ (v,a,b) .
This last point makes sense because we work conditionally on ζ = ∞.

⋆ We fix ǫ > 0. We choose δ > t 0 as shown above and consider the tree H. We define Z(n) as the number of nodes of H at height n. By the theory of Galton-Watson processes, as we are working conditionally on the event Λ (v,a,b) = ∅, we have that almost surely

m -n Z(n) → W > 0.
Let σ be a node of our tree (thus it is also a subinterval of (0, 1)). Fix an interval I ⊂ (0, 1) and introduce

H I (n) := {σ ∈ H(n), σ ∩ I = ∅}, Z I (n) := ♯ H I (n). Define x → L x := lim n m -n Z (0,x) (n),
x ∈ (0, 1).

We will now state a lemma that we will use to conclude: Proof[Proof of Lemma 3.] (a) Exactly as in [START_REF] Berestycki | Multifractal spectra of fragmentation processes[END_REF] , we show the first point by verifying Kolmogorov's criterium (see [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] Theorem 2.1 p.26). Let W (σ) be the "renormalized weight" of the tree rooted at σ, i.e.,

Lemma 3 For each ǫ > 0, (a)
W (σ) := lim n→∞ m -n ♯{σ ′ ∈ H(|σ| + n), σ ′ ⊂ σ},
where |σ| is the generation of σ.

By the definition of L we have for all x > y ∈ (0, 1):

|L x -L y | = lim n→∞ m -n Z (x,y) (n), x ∈ (0, 1).
For any J open subinterval of (0, 1), let

η(J) := sup{n ∈ N : e -vδn ≥ |J|} = ⌊-log(|J|)/vδ⌋.
For all x, y such that x < y by the definition of L . , we get:

|L x -L y | = lim n m -η((x,y)) m -n+η((x,y)) σ∈H (x,y) (η((x,y))) ♯{σ ′ ∈ H(|σ| + n -η((x, y))), σ ′ ⊂ σ} ≤ m -η((x,y)) σ∈H (x,y) (η((x,y))) W (σ),
and by the definition of η(.):

|L x -L y | ≤ e log m( 1 vδ log(y-x)+1) σ∈H (x,y) (η((x,y))) W (σ) ≤ m|x -y| 1-ǫ-ρ/v σ∈H (x,y) (η((x,y)))
W (σ), by using [START_REF] Lyons | Conceptual proofs of L log(L) citeria for mean behaviour of branching processes[END_REF]. Moreover by the definition of good intervals, we have that for each n the sizes of intervals in H(n) have a lower bound given by ae -vδn , so a|J|e -vδ is a lower bound for the sizes of the intervals of H(η(J)), and thus Z J (η(J)) ≤ e vδ /a. Therefore for all γ > 1 and all J ⊂ (0, 1) we have:

E   ( σ∈H J (η(J)) W (σ)) γ   ≤ E((W 1 + ... + W ⌊e vδ /a⌋+1 ) γ ) ≤ E((W 1 + ... + W η(J)+2 ) γ ) < ∞,
where the W i are i.i.d. with the same law as W . The finiteness comes from the existence of finite moments of all orders for W (see for example Theorem 3.4 p. 479 of Harris [START_REF] Harris | Branching processes[END_REF]).

(b) The second point is clear by the choice of L.

⋆ To prove that dim ∩ n∈N G(n) ≥ 1 -ρ/v -ǫ, it is enough to show that i diam(U i ) 1-ρ/v-ǫ > 0 (23)
for any cover

{U i } of ∩ n∈N G(n), where diam(U i ) is the diameter of U i .
Clearly, it is enough to assume that the {U i } are intervals, and by expanding them slightly and using the compactness of the closure of ∩ n∈N G(n), we only need to check [START_REF] Tucker | Absolute continuity of infinitely divisible distributions[END_REF] if

{U i } is a finite collection of open subintervals of [0, 1]. Let ∪ N i=0 (l i , r i ) be a cover of ∩ n∈N G(n) (where the (l i , r i ) are disjoints open intervals). Therefore N i=1 | ∼ L r i - ∼ L l i | = W.
Thus for all such covers with max i (r il i ) small enough

W ≤ k N i=0 (r i -l i ) 1-ρ/v-ǫ and hence dim(Λ (v,a,b) ) ≥ dim( ∩ n∈N G(n)) ≥ 1 -ρ/v -ǫ.
To get the lower bound of the Hausdorff dimension of Λ (v,a,b) , we let ǫ tend to 0.

• Secondly, the upper bound for ( 20) is an easy corollary of the fact that the Hausdorff dimension is smaller than the box-counting dimension (see [START_REF] Falconer | Fractal geometry[END_REF] p.36-43), using the cover

∪ n≥N ∪ i∈θ v,a,b (n) J i (n), with θ v,a,b (t) = {i ∈ N | J i (t) ∈ G(t)} (with G(t) defined in Section 3).
Then we have the next corollary, which deals with the general case for a and b:

Corollary 1 For t ′ ≥ 0 set Λ (v,a,b) (t ′ ) := x ∈ (0, 1) : ae -vt < |I x (t)| < be -vt ∀t ≥ t ′ .
Assume [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF], 0 < a < b and ρ < v, then

P(Λ (v,a,b) (t ′ ) = ∅) → t ′ →∞ 1,
and

P dim(Λ (v,a,b) (t ′ )) = 1 -ρ/v Λ (v,a,b) (t ′ ) = ∅ = 1. Proof 1.
The first part of the proof is a consequence of the homogeneity of the fragmentation and of Proposition 3.

2. Fix ρ ′ > ρ. As lim β→0 ρ β = ∞, and, by Theorem 1.5, the application β → ρ β is continuous and strictly decreasing, therefore there exists β 0 ∈ (1, b/a) such that ρ ′ = ρ log(β 0 ) . Let ǫ := (β 0 -1)/(1 + β 0 ), a ′ := 1ǫ, b ′ := 1 + ǫ, x 0 := (β 0 + 1)(a + b/β 0 )/4 (notice that x 0 ∈ (a, b)) and

p 0 := P(dim(Λ (v,a ′ ,b ′ ) ) ≥ 1 -ρ log(b ′ /a ′ ) /v).
By Proposition 3, we get that p 0 > 0. We notice that by the choice of a ′ and of b ′ , we have

ρ log(b ′ /a ′ ) = ρ log(β 0 ) = ρ ′ .
Let I be an interval of (0, 1). The law of the homogeneous interval fragmentation started at I will be denoted by P I . We remark that P I (dim(Λ (v,a,b) ) ≥ 1ρ ′ /v) only depends on the length of I. Thus we define

g a,b (x) := P I (dim(Λ (v,a,b) ) ≥ 1 -ρ ′ /v),
where I is an interval such that |I| = x.

Let x ∈ (x 0 a ′ , x 0 b ′ ). We remark that by the choice of x 0 and as 1 < β 0 < b/a we have that (x 0 a ′ , x 0 b ′ ) ⊂ (a, b) and thus

g a,b (x) ≥ g x 0 a ′ ,x 0 b ′ (x).
Moreover by the scaling property of the fragmentation we get that

g x 0 a ′ ,x 0 b ′ (x) = P(dim(Λ (v,a ′ /x,b ′ /x) ) ≥ 1 -ρ log((b ′ /x)/(a ′ /x)) /v) = p 0 Therefore inf x∈(x 0 a ′ ,x 0 b ′ ) g a,b (x) ≥ p 0 . ( 24
) Let B(t) = {i : x 0 a ′ < e vt |J i (t)| < x 0 b ′ } , n t = ♯B(t),
where (J 1 , J 2 , ...) is the interval decomposition of F (t).

Fix t ′ ≥ 0. By applying the Markov property at time t ′ we get that

P(dim(Λ (v,a,b) (t ′ )) < 1 -ρ ′ /v)) ≤ E   i∈B(t ′ ) P J i (t ′ ) (dim(Λ (v,x 0 a ′ ,x 0 b ′ ) ) < 1 -ρ ′ /v)   ≤ E((1 -p 0 ) n t ′ ),
by using (24). Therefore as p 0 > 0, n t ′ → t ′ →∞ ∞ (see [START_REF] Engländer | Strong law of large numbers for branching diffusions[END_REF]) and with the first part of the proof we can conclude.

Now we are able to proof our main result:

Proof[Proof of Theorem 3.] Observe that for all n ∈ N, we have

Λ (v,a,b) (n) ⊂ G (v,a,b) ⊂ ∩ ǫ>0 ∪ m∈N Λ (v,a-ǫ,b+ǫ) (m). ( 25
)
We can notice that the second inclusion is actually an equality.

• First we consider the case where ρ > v. As the application β → ρ β is continuous and strictly decreasing (see Theorem 1.5), there exists ǫ 0 > 0 such that v < ρ log((b+ǫ 0 )/(a-ǫ 0 )) < ρ. Moreover by ( 25)

G (v,a,b) ⊂ ∪ m∈N Λ (v,a-ǫ 0 ,b+ǫ 0 ) (m),
therefore thanks to Proposition 3 and the homogeneous property of the fragmentation, we get the first part of the proof. compatible (i.e. for every n, Π(n, t) is a partition of [n] such that the restriction of Π(n+1, t) to [n] coincide with Π(n, t)), as a consequence, there exists a unique partition Π(t), such that for all n ≥ 0 we have Π(t) |[n] = Π(t, n). With the terminology of [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF], it is shown in [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF] that this process Π is a (partition valued) homogeneous fragmentation.

One says that a block B ⊂ N has an asymptotic frequency, if the limit

|B| := lim n→∞ n -1 card(B ∩ [n])
exists. When every block of some partition π ∈ P has an asymptotic frequency, we write |π| = (|π 1 |, ...) and then |π| ↓ = (|π 1 | ↓ , ...) ∈ S ↓ for the decreasing rearrangement of the sequence |π|. In the case where some block of the partition π does not have an asymptotic frequency, we decide that |π| = |π| ↓ = ∂, where ∂ stands for some extra point added to S ↓ . We stress that the process of ranked asymptotic frequencies |Π| ↓ is a ranked fragmentation.

Moreover, let ν be the dislocation measure associated to this ranked fragmentation (see Subsection 2.2). According to Theorem 2 in [START_REF] Bertoin | Homogeneous fragmentation processes[END_REF], there exists a unique measure µ on P which is exchangeable (i.e. invariant by the action of finite permutations on P), and such that ν is the image of µ by the map that associate the decreasing rearrangement |π| ↓ of the sequence of the asymptotic frequencies of the blocks of π, to π. Thanks to exchangeability, we get that for all measurable function f :

[0, 1] → R + such that f (0) = 0. P f (|π 1 |)µ(dπ) = S * ∞ i=1 s i f (s i )ν(ds).
We denote the sigma-field generated by the restriction to [0, t] × P × N by G 0 (t). So (G 0 (t)) t≥0 is a filtration, and the nested partitions (Π(t), t ≥ 0) are (G 0 (t)) t≥0 -adapted. We define also the sigma-field (F 0 (t)) t≥0 generated by the decreasing rearrangement |Π(r)| ↓ of the sequence of the asymptotic frequencies of the blocks of Π(r) for r ≤ t. Of course (F 0 (t)) t≥0 is a sub-filtration of (G 0 (t)) t≥0 .

Let G 1 (t) the sigma-field generated by the restriction of the discrete point measure w to the fiber [0, t] × P × {1}. So (G 1 (t), t ≥ 0) is a sub-filtration of (G 0 (t), t ≥ 0), and the first block of Π is (G 1 (t), t ≥ 0)-measurable. Let D 1 ⊆ R + be the random set of times r ≥ 0 for which the discrete point measure has an atom on the fiber {r} × P × {1}, and for every r ∈ D 1 , denote the second component of this atom by π(r).

We define the probability measure P as the h-transform of P based on the martingale D t (defined in Theorem 1 (3)):

dP x | Et = D t dP x | Et . (26)
To simplify the notation, as in the section 3 we define for all t ∈ R h(t) = W (-ρ) (t + log(1/a))1 {t∈(log(a),log(b))} . This function is well defined thanks to Theorem 1.

Let P i (t) the block of Π(t) which contains i at time t. Similarly as in Section 3, for a homogeneous fragmentation, we define the killed partition

Π † j (t) = Π j (t)1 {∃i∈N * | Π j (t)=P i (t); ∀s≤t |P i (s)|∈(ae -vs ,be -vs )} .
When we project the martingale D t of (10) on the sub-filtration (G 0 (t)) t≥0 , we obtain an additive martingale e ρt h(0)

∞ i=1 h(vt + log(|Π † i (t)|)) |Π † i (t)| .
As |Π| is a ranked fragmentation with dislocation measure ν, this martingale is the same as this of Section 3. From now on, we denote this martingale by M t too.

Observe that the projection (26) on the sub-filtration G 0 (t) give the identity:

dP x | G 0 (t) = M t dP x | G 0 (t) .
Like in lemma 8 (ii) [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF], with the probability measure P we get:

Lemma 4 Under P , the restriction of w to R + × P × {2, 3, ...} has the same distribution as under P and is independent of the restriction to the fiber R + × P × {1}.

It follows immediately from Theorem 1 that

Remark 6 For x ∈ [0, log(b/a)], let F x (t) := E x e ρt 1 {T >t}} for t ∈ [0, ∞), then F x (t) converges when t → ∞ to a finite limit, and

F x (.) : [0, ∞) → [0, ∞) is càdlàg. In particular we have sup x∈[0,log(b/a)] sup t≥0 |F x (t)| < ∞.

Remark 7

We have for all t ≥ 0:

|M t -M t-| ≤ e (ρ-v)t b 2 ah(0) sup x∈[log a,log b] h(x) a.s. If v > ρ, there exists 0 < C ′ < ∞ such that sup t≥0 |M t -M t-| < C ′ a.s. Let c t := e ρt h(0) h vt + log(|Π † 1 (t)|) |Π † 1 (t)| and d t := e ρt h(0) ∞ i=2 h vt + log |Π † i (t)| |Π † i (t)|.
Now we have the background that we need to study

M t = c t + d t ,
and we will show that M is bounded in L 2 (P). In order to do that, as

E(M 2 t ) = E (M t ), it is enough to prove that lim t→∞ E (M t ) < ∞ .
6.2 The proof of Theorem 2.1.

• First we show that lim t→∞ E (c t ) = 0. With the subordinator ξ(t) :=log(|Π 1 (t)|), whose Laplace exponent is κ (exactly the same as this defined in Subsection 2.3), with the Lévy Process Y t = vtξ(t) + log(1/a), and T := T log(b/a) defined in (8) associated to this Lévy Process, under P log(1/a) we get:

c t = e (ρ-v)t h(0) W (-ρ) (Y t ) e Yt 1 {t<T } .
As a consequence,

E log(1/a) W (-ρ) (Y t ) h(0) e Yt 1 {t<T } = E log(1/a) W (-ρ) (Y t ) 2 h(0) 2 e Yt e ρt 1 {t<T } ≤ sup x∈[log a,log b] (h(x)) 2 b ah(0) 2 F log(1/a) (t)
which is bounded by a constant independent of t by Remark 6, and as ρ < v, we have lim t→∞ e (ρ-v)t = 0. Therefore:

lim t→∞ E (c t ) = 0 . (27)
• Now we consider d t . As shown in [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF] with B(r, j) = {i ≥ 2 : Π i (t) ⊆ π j (r)∩Π 1 (r-)}, we get, for every r ∈ [0, t] and j ≥ 2, conditionally on r ∈ D 1 , Π 1 (r-) and π j (r), the partition (Π i (t) : i ∈ B(r, j)) can be written in the form Π(j) (tr) |π j (r)∩Π 1 (r-) . Here ( Π(j) ) j∈N is a family of i.i.d. homogeneous fragmentations distributed as Π under P and independent of the sigma-field G 1 (t). As a consequence: which is finite by [START_REF] Berestycki | Ranked fragmentations[END_REF]. Moreover as ρ < v, the term e (ρ-v)r is integrable on [0, ∞), so that we have lim t→∞ N t < ∞.

As both X t := A t -N t and M t are martingales, by Theorem 4.50 of [START_REF] Jacod | Limit theorems for stochastic processes[END_REF], we get that XM -[X, M] is a local martingale.

A sequence (τ n = (T (m, n)) m∈N ) n∈N of adapted subdivisions is called a Riemann sequence if sup m∈N [T (m + 1, n) ∧ t -T (m, n) ∧ t] → 0 for all t ∈ R + . By Theorem 4.47 of [START_REF] Jacod | Limit theorems for stochastic processes[END_REF], for any Riemann sequence {τ n = (T (m, n)) m∈N } n∈N of adapted subdivisions, the processes S τn (X, M) defined by S τn (X, M) t := m∈N (X T (m+1,n)∧t -X T (m,n)∧t )(M T (m+1,n)∧t -M T (m,n)∧t ) converge to the process [X, M], in measure, uniformly on every compact interval.

We will now bound S τn (X, M) t uniformly in t. As Therefore by the previous study of A t and N t we get that there exist C 10 < ∞ independent of t such that:

  a,b) := x ∈ (0, 1) : a ≤ lim inf t→∞ e vt |I x (t)| ≤ lim sup t→∞ e vt |I x (t)| ≤ b .

  ), we get by Theorem 1.1, that for all x ∈ [log a, log b]: ϕ(x) > 0. Because [log a, log b] is compact and ϕ(.) is a continuous function, inf x∈[log a,log b] ϕ(x) > 0.

f

  with the definition of Y t . Thus by the definition of P t in Theorem 1, we get (y)h(y + log a)h(log(b)y)dy.

( 2 )ef 2

 22 t+s . We introduce the notation y k = |J † k (s)|. Write i ց k if the length |J † i (t + s)| stems from y k . By independence, the intervals which are included in the interval with length y k and those which are included in the interval with length y l evolve independently, thus gathering the lengths |J † i (t + s)| by the ancestors at time s yields S On the other hand, by self-similarity and convergence of the mean measures E iցk e ρs h (vs + log (y k /a)) h(0) y k f 1 (vt + log(|J † i (t)|) + vs + log(y k /a)) e ρt h vt + log |J † i (t)| + vs + log(y k /a) h (vs + log (y k /a)) |J † i (t)| J † (s) ρs h (vs + log (y l /a)) h(0) y l f 2 (vt + log(|J † j (t)|) + vs + log(y l /a)) e ρt h vt + log |J † j (t)| + vs + log(y l /a) h (vs + log (y l /a)) |J † j (t)| J † (s) (x)̺(dx) .

  ρ)t log(b/a) 0 e -y+ρt P t (log(1/a), dy).

  There exists a version ∼ L of (L x ) x∈[0,1] which is Hölder continuous of order α for any α < 1ρ/vǫ for every ǫ > 0. (b) The process ∼ L only grows on the set ∩ n∈N G(n).

1 Π 1 ∞ 1 ∞

 111 (j) (tr) |π j (r)∩Π 1 (r-) .Moreover |π j (r)||Π 1 (r-)| is G 1 (t)measurable, and we have that for alli ∈ N | Πi (j) (tr) |π j (r)∩Π 1 (r-) | = | Πi (j) (tr)||π j (r)||Π 1 (r-)|so that we get:E (d t |G 1 (t)) j=2 |π j (r)||Π † 1 (r-)|1l {a≤|π j (r)||Π † 1 (r-)|e vr ≤b} ∞ i=1 E | Πi (j) (tr)1l {a≤| Πi (j) (t ′ -r)|e v(t ′ -r) |π j (r)||Π † 1 (r-)|e vr ≤b ∀t ′ ∈[r,t]} |G 1 (t) j=2 |π j (r)||Π † 1 (r-)|1l {a≤|π j (r)||Π † 1 (r-)|e vr ≤b} ∞ i=1 E | Πi (j) (tr)1l {a/b≤| Πi

(

  j) (t ′ -r)|e v(t ′ -r) ≤b/a ∀t ′ ∈[r,t]} |G 1 (t) ,with C 8 the maximum of h(t) on the compact [log(a), log(b)]. As Π is independent of the sigma-field G 1 (t), Π has the same distribution under P as under P . Π(j) is also distributed as Π under P. Thus,∞ i=1 E | Πi (j) (tr)1l {a/b≤| Πi (j) (t ′ -r)|e v(t ′ -r) ≤b/a ∀t ′ ∈[r,t]} | G 1 (t) = ∞ i=1 E | Πi (j) (tr)|1l {a/b≤| Πi (j) (t ′ -r)|e v(t ′ -r) ≤b/a ∀t ′ ∈[r,t]}. Now we have by size-biased sampling:∞ i=1 E e ρ(t-r) |Π i (tr)|1l {a/b≤|Π i (t ′ -r)|e v(t ′ -r) ≤b/a ∀t ′ ∈[r,t]} = E e ρ(t-r) 1 {t-r<inf{s: |Π 1 (s)| / ∈ ( a b e -vs , b a e -vs )}} = E log(1/a) e ρ(t-r) 1 {T 2 log(b/a) >t-r}} , as ρ . is decreasing ρ 2 log(b/a) ≤ ρ, thus ∞ i=1 e ρ(t-r) E | Πi (j) (tr)1l {a/b≤| Πi (t ′ -r)|e v(t ′ -r) ≤b/a ∀t ′ ∈[r,t]} | G 1 (t) ≤ E log(1/a) e ρ 2 log(b/a) (t-r) 1 {T 2 log(b/a) >t-r}} . Therefore with F ′ x (t) := E x e ρ 2 log(b/a) t 1 {T 2 log(b/a) >t}} and since |Π † 1 (r-)| = ae Y r--vr 1l {r-<T } under P log(1/a) , we get:E (d t |G 1 (t)) ≤ a e (ρ-v)r C 8 h(0) r∈[0,t]∩D 1 ∞ j=2 |π j (r)|e Y r- sup x∈[0,log(b/a)] sup t≥0 |F ′ x (t)|.Moreover we have by definition e Y r-1l {r-<T } ≤ b/a. We letC 9 := sup x∈[0,log(b/a)] sup t≥0 |F ′ x (t)|bC 8 /h(0) according to Remark 6 we have C 9 < ∞. Thus E (d t |G 1 (t)) ≤ C 9 r∈[0,t]∩D 1 ∞ j=2 e (ρ-v)r |π j (r)|.Under P, the G 0 (t-) predictable compensator ofA t := r∈[0,t]∩D 1 ∞ j=2e (ρ-v)r |π j (rs i .As ∞ j=1 s j = 1 νa.s., we achieve:

S

  τn (X, M) t ≤ sup l∈N |M T (l+1,n)∧t -M T (l,n)∧t | m∈N |X T (m+1,n)∧t -X T (m,n)∧t | (28) we will first focus on m∈N |X T (m+1,n)∧t -X T (m,n)∧t |: m∈N |X (m+1)/n∧t -X T (m,n)∧t | ≤ m∈N   r∈[T (m,n)∧t,T (m+1,n)∧t]∩D 1 ∞j=2e (ρ-v)r |π j (r)| +

  lim n→∞ E m∈N |X T (m+1,n)∧t -X T (m,n)∧t | ≤ C 10 for all t.Moreover limn→∞ sup l∈N |M T (l+1,n)∧t -M T (l,n)∧t | ≤ sup r≤t |M r -M r-| is a.s.bounded by C ′ (see remark 7) independently of t. Consequently by (28) lim t→∞ E([X, M] t ) < ∞.

  Thus as XM -[X, M] is a local martingale, we get that lim t→∞ E (d t ) < ∞ .Finally according to (27), we get lim

t→∞ E (M t ) = lim t→∞ E (E (d t + c t |G 1 (t))) < ∞.

Where v min is the maximum of the function p → κ(p -1)/p on (p + 1, ∞) and v max := κ ′ (p + ) (see[START_REF] Berestycki | Multifractal spectra of fragmentation processes[END_REF] ).

Acknowledgments: I wish to thank J. Bertoin for his help, and especially for his patience.

I am very grateful to A. Kyprianou, who kindly pointed out an original mistake in the proof of Theorem 2.1. and for his valuable comments and advices.

I also wish to thank an anonymous referee of an earlier draft for various interesting suggestions.

• Second we consider the case where ρ < v. Thanks the second inclusion and the corollary 1, we get that: for all ǫ ∈ (0, a),

Then by the continuity of ρ . (see Theorem 1.5), we get the uper bound of the Hausdorff dimension of G (v,a,b) .

The lower bound of the Hausdorff dimension is a consequence of the first inclusion of (25), as dim(Λ (v,a,b) (n)) = 1ρ/v with a probability which goes to 1 when n goes to infinity.

Appendix 6.1 A partition fragmentation.

In this appendix we give a proof of Theorem 2.1. (Section 3).

For this, we use the method of Bertoin and Rouault in [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF] for fragmentation, which goes back to Lyons and al. [START_REF] Lyons | Conceptual proofs of L log(L) citeria for mean behaviour of branching processes[END_REF] for Galton-Watson processes, and tools taken from the article of Engländer, Harris and Kyprianou [START_REF] Engländer | Strong law of large numbers for branching diffusions[END_REF].

We first introduce the notations that we need and we define what a partition fragmentation Π is. Let P the space of partition of N, and for every integer k, the block {1, ..., k} is denoted by [k]. As in [START_REF] Bertoin | Discretization methods for homogeneous fragmentations[END_REF], we call discrete point measure on the space Ω := R + × P × N, any measure :

where D is a subset of R + × P × N such that

and for all t ∈ R w({t} × P × N) ∈ {0, 1}.

Starting from an arbitrary discrete point measure ω on R + × P × N, we will construct a nested partition Π = (Π(t), t ≥ 0) (which means that for all t ≥ t ′ Π(t) is a finer partition of N than Π(t ′ )). We fix n ∈ N, the assumption that the point measure ω is discrete enables us to construct a step path (Π(t, n), t ≥ 0) with values in the space of partitions of [n], which only jumps at times t at which the fiber {t} × P × N carries an atom of ω, say (t, π, k), such that π |[n] = ([n], ∅, ∅, ...) and k ≤ n. In that case, Π(t, n) is the partition obtained by replacing the kth block of Π(t-, n), denoted Π k (t-, n), by the restriction π |Π k (t-,n) of π to this block, and leaving the other blocks unchanged. Of course for all t ≥ 0, (Π(t, n), n ≥ 0) is


Florent Benaych-Georges
email: florent.benaych@gmail.com

ON A SURPRISING RELATION BETWEEN RECTANGULAR AND SQUARE FREE CONVOLUTIONS

Keywords: . MSC 2000 subject classifications. 46L54, 15A52 Key words. free probability, random matrices, free convolution

published or not. The documents may come

Introduction

Free convolutions are operations on probability measures on the real line which allow to compute the spectral or singular empirical measures of large random matrices which are expressed as sums or products of independent random matrices, the spectral measures of which are known. More specifically, the operations ⊞, ⊠, called respectively free additive and multiplicative convolutions are defined in the following way [START_REF] Voiculescu | Free random variables CRM Monograghs Series[END_REF]. Let, for each n, M n , N n be n by n independent random hermitian matrices, one of them having a distribution which is invariant under the action of the unitary group by conjugation, which empirical spectral measures 1 converge, as n tends to infinity, to non random probability measures denoted respectively by τ 1 , τ 2 . Then τ 1 ⊞ τ 2 is the limit of the empirical spectral law of M n + N n and, in the case where the matrices are positive, τ 1 ⊠ τ 2 is the limit of the empirical spectral law of M n N n . In the same way, for any λ ∈ [0, 1], the rectangular free convolution ⊞ λ is defined, in [START_REF] Benaych-Georges | Rectangular random matrices, related convolution[END_REF], in the following way. Let M n,p , N n,p be n by p independent random matrices, one of them having a distribution which is invariant by multiplication by any unitary matrix on any side, which symmetrized 2 empirical singular measures 3 tend, as n, p tend to infinity in such a way that n/p tends to λ, to non random probability measures ν 1 , ν 2 . Then the symmetrized empirical singular law of M n,p + N n,p tends to ν 1 ⊞ λ ν 2 . These operations can be explicitly computed using either a combinatorial or an analytic machinery (see [START_REF] Voiculescu | Free random variables CRM Monograghs Series[END_REF] and [START_REF] Nica | Roland Lectures on the combinatorics of free probability[END_REF] for ⊞, ⊠ and [START_REF] Benaych-Georges | Rectangular random matrices, related convolution[END_REF] for ⊞ λ ). In the cases λ = 0 or λ = 1, i.e. where the rectangular random matrices we consider are either "almost flat" or "almost square", the rectangular free convolution with ratio λ can be expressed with the additive free convolution: ⊞ 1 = ⊞ and for all symmetric laws ν 1 , ν 2 , ν 1 ⊞ 0 ν 2 is the symmetric law which push-forward by the map t → t 2 is the free convolution of the push forwards of ν 1 and ν 2 by the same map. However, though one can find many analogies between the definitions of ⊞ and ⊞ λ and still more analogies have been proved [START_REF] Benaych-Georges | Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation Probability Theory and Related Fields[END_REF], no general relation between ⊞ λ and ⊞ had been proved until a paper of Debbah and Ryan [START_REF] Debbah | Multiplicative free Convolution and Information-Plus-Noise Type Matrices[END_REF] (which submitted version, more focused on applications than on this result, is [START_REF] Debbah | Free Deconvolution for Signal Processing Applications Second round review[END_REF]). It is to notice that this result is not due to researchers from the communities of Operator Algebras or Probability Theory, but to researchers from Information Theory, working on communication networks. In [START_REF] Debbah | Multiplicative free Convolution and Information-Plus-Noise Type Matrices[END_REF], Debbah and Ryan proved a result about random matrices which can be interpreted as an expression, for certain probability measures ν 1 , ν 2 , of their rectangular convolution ν 1 ⊞ λ ν 2 in terms of ⊞ and of another convolution, called the free multiplicative deconvolution and denoted by . In this note, we present this result with a new approach and we give a new and shorter proof, where the hypothesis are more general. This generalization of the hypothesis answers a question asked by Debbah and Ryan in the last section of their paper [START_REF] Debbah | Multiplicative free Convolution and Information-Plus-Noise Type Matrices[END_REF]. The question of a more general relation between square and rectangular free convolutions is considered in a last "perspectives" section.

The result of Debbah and Ryan

Let us define the operation on certain pairs of probability measures on [0, +∞) in the following way. For µ, µ 2 probability measures on [0, +∞), if there is a probability measure on [0, +∞) such that µ = µ 1 ⊠ µ 2 , then µ 1 is called the free multiplicative deconvolution of µ by µ 2 and is denoted by µ 1 = µ µ 2 . Let us define, for λ ∈ (0, 1], µ λ be the law of λX for X random variable distributed according to the Marchenko-Pastur law with parameter 1/λ, i.e. the law with support [(1 -√ λ) 2 , (1 + √ λ) 2 ] and density

x → 4λ -(x -1 -λ) 2 2πλx .
Theorem 1 of [START_REF] Debbah | Multiplicative free Convolution and Information-Plus-Noise Type Matrices[END_REF] states the following result. λ ∈ (0, 1] is fixed and (p n ) is a sequence of positive integers such that n/p n tends to λ as n tends to infinity. δ 1 denotes the Dirac mass at 1.

Theorem 1 (Debbah and Ryan). Let, for each n, A n , G n be independent n by p n random matrices such that the empirical spectral law of A n A * n converges almost surely weakly, as n tends to infinity, to a compactly supported probability measure µ A and such that the entries of G n are independent N (0, 1 pn ) random variables. Then the empirical spectral law of (A n + G n )(A n + G n ) * converges almost surely to a compactly supported probability measure ρ which, in the case where µ A µ λ exists, satisfies the relation

(1) ρ = [(µ A µ λ ) ⊞ δ 1 ] ⊠ µ λ .
Remark 2. Note that in the case where µ A µ λ doesn't exist, the relation (1) stays true in the formal sense. More specifically, for µ A probability measure such that µ A µ λ exists, the moments of ρ = [(µ A µ λ ) ⊞ δ 1 ] ⊠ µ λ have a polynomial expression in the moments of µ A (this can easily be seen by the theory of free cumulants [START_REF] Nica | Roland Lectures on the combinatorics of free probability[END_REF]). It happens that this relation between the moments of the limit spectral law ρ of (A n + G n )(A n + G n ) * and the ones of µ A stays true even when µ A µ λ doesn't exist. It follows from the original proof of Theorem 1 and it will also follow from our proof (see Remark 3).

Note that by the very definition of the rectangular free convolution ⊞ λ with ratio λ recalled in the introduction and since the limit empirical spectral law of GG * is µ λ (it is a well known fact, see, e.g. Theorem 4.1.9 of [START_REF] Hiai | The semicircle law, free random variables, and entropy[END_REF]), this result can be stated as follows: for all compactly supported probability measure µ on [0, +∞) such that µ µ λ exists,

(2) ( √ µ ⊞ λ √ µ λ ) 2 = [(µ µ λ ) ⊞ δ 1 ] ⊠ µ λ ,
where for any probability measure ρ on [0, +∞), √ ρ denotes the symmetrization of the pushforward by the square root functions of ρ and for any symmetric probability measure ν on the real line, ν 2 denotes the push-forward of ν by the function t → t 2 . This formula allows to express the operator ⊞ λ √ µ λ on the set of symmetric compactly supported probability measures on the real line in terms of ⊞ and ⊠: for all symmetric probability measure on the real line ν,

(3) ν ⊞ λ √ µ λ = [(ν 2 µ λ ) ⊞ δ 1 ] ⊠ µ λ .
2. A proof of the generalized theorem of Debbah and Ryan λ ∈ (0, 1] is still fixed. In this section, we shall give a new shorter proof of the theorem of Debbah and Ryan, under weaker hypothesis: we shall prove (3) without supposing the support of ν to be compactly supported. The proof is based on the machinery of the rectangular free convolution and of the rectangular R-transform. Some analytic transforms. Let us first recall a few facts about the analytic approach to ⊠ and ⊞ λ . Let us define, for ρ probability measure on [0, ∞),

M ρ (z) := t∈R zt 1 -zt dρ(t), S ρ (s) = 1 + z z M -1 ρ (z),
where, as it shall be in the rest of the text, the exponent -1 stands for the inversion of analytic functions on C\[0, +∞) with respect to the composition operation •, in a neighborhood of zero. By [START_REF] Voiculescu | Free random variables CRM Monograghs Series[END_REF], for all pair µ 1 , µ 2 of probability measures on [0, +∞),

µ 1 ⊠ µ 2 is characterized by the fact that S µ 1 ⊠µ 2 = S µ 1 S µ 2 .
In the same way, the rectangular free convolution with ratio λ can be computed with an analytic transform of probability measures. Let ν be a symmetric probability measure on the real line. Let us define H ν (z) = z(λM ν 2 (z) + 1)(M ν 2 (z) + 1). Then the rectangular R-transform with ratio λ of ν is defined to be

C ν (z) = U z H -1 ν (z) -1 , where U (z) = -λ-1+[(λ+1) 2 +4λz] 1/2 2λ
. By theorem 3.12 of [START_REF] Benaych-Georges | Rectangular random matrices, related convolution[END_REF], for all pair ν 1 , ν 2 of symmetric probability measures, ν 1 ⊞ λ ν 2 is characterized by the fact that

C ν 1 ⊞ λ ν 2 = C ν 1 + C ν 2 .
2.2. Some preliminary computations. Note that by [START_REF] Nica | Roland Lectures on the combinatorics of free probability[END_REF], the S-and R-transforms of a probability measure µ on [0, +∞) are linked by the relation S µ (s) = 1 z R -1 µ (z), thus since the free cumulants of the Marchenko-Pastur law with parameter 1/λ are all equal to 1/λ (see [START_REF] Nica | Roland Lectures on the combinatorics of free probability[END_REF]), we have S µ λ (z) = 1 1+λz . Moreover, since by [START_REF] Nica | Roland Lectures on the combinatorics of free probability[END_REF] again, S µ (s) = 1+z z M

-1 µ (z), for any law σ on [0, +∞), (4) M -1

σ⊠µ λ = z z + 1 S σ 1 + λz = M -1 σ 1 + λz and M -1 σ µ λ = (1 + λz)M -1 σ .
At last, since ⊞δ 1 = * δ 1 , which implies that M σ⊞δ 1 (z) = [(z + 1)M σ (z) + z] • z z+1 , for any symmetric law ν, we have

(5) M -1 ((ν 2 ⋉µ λ )⊞δ 1 )⊠µ λ = 1 1 + λz × z 1 + z • (z + 1) (1 + λz)M -1 ν 2 -1 + z -1 .
2.3. Proof of the result. So let us consider a symmetric probability measure ν such that ν 2 µ λ exists and let us prove (3). As proved in the proof of Theorem 3.8 of [START_REF] Benaych-Georges | Rectangular random matrices, related convolution[END_REF], for any symmetric probability measure τ , H τ characterizes τ , thus it suffices to prove that

H ν⊞ λ √ µ λ = H m for m = [(ν 2 µ λ ) ⊞ δ 1 ] ⊠ µ λ .
By Theorem 4.3 and the paragraph preceding in [START_REF] Benaych-Georges | Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation Probability Theory and Related Fields[END_REF],

C √ µ λ (z) = z. Thus Lemma 4.1 of [BBA07]
applies here, and it states that in a neighborhood of zero in C\[0, +∞),

H ν⊞ λ √ µ λ = H ν • H ν T (H ν + M ν 2 ) -1
, where T (z) = (λz + 1)(z + 1). So it suffices to prove that in such a neighborhood of zero,

H m = H ν • H ν T (H ν + M ν 2 ) -1 , i.e. H m • H ν T (H ν + M ν 2 ) = H ν .
Using the fact that for any symmetric law τ , H τ (z) = zT (M τ 2 (z)), it amounts to prove that

H ν T (H ν + M ν 2 ) × T • M m 2 • H ν T (H ν + M ν 2 ) = H ν (z), i.e. T • M m 2 • H ν T (H ν + M ν 2 ) = T (H ν (z) + M ν 2 (z)),
which is implied, simplifying by T and using again H τ (z) = zT (M τ 2 (z)), by

M m 2 • zT (M ν 2 (z)) T [zT (M ν 2 (z)) + M ν 2 (z)] = zT [M ν 2 (z)] + M ν 2 (z).
It is implied, composing by M -1 ν 2 on the right and by M -1 m 2 on the left, by

M -1 ν 2 × T = (T × M -1 m 2 ) • (M -1 ν 2 (z)T (z) + z).

Using the expression of M

-1 m 2 given by (5), it amounts to prove that

M -1 ν 2 (z)T (z) = (z + 1) × z 1 + z • (z + 1) (1 + λz)M -1 ν 2 -1 + z -1 • (M -1 ν 2 (z)T (z) + z), i.e. that M -1 ν 2 (z)T (z) M -1 ν 2 (z)T (z) + z + 1 = z 1 + z • (z + 1) (1 + λz)M -1 ν 2 -1 + z -1 • (M -1 ν 2 (z)T (z) + z). Now, composing by (z + 1) (1 + λz)M -1 ν 2 -1 + z • z 1-z on the left, it gives (z + 1) (1 + λz)M -1 ν 2 -1 + z • [(1 + λz)M -1 ν 2 (z)] = M -1 ν 2 (z)T (z) + z, i.e. [M -1 ν 2 (z)(λz + 1) + 1]z + [M -1 ν 2 (z)(λz + 1)] = M -1
ν 2 (z)(λz + 1)(z + 1) + z, which is easily verified.

Remarks on this result.

Remark 3. Note that we did not use the fact that ν 2 µ λ exists to prove that H ν⊞ λ √ µ λ = H m . It means that if ν 2 µ λ doesn't exist, there is no more probability measure µ on [0, +∞) such that

M -1 µ = (1+λz)M -1 ν 2
as in (4), but the polynomial expression of the moments of ν ⊞ λ √ µ λ (i.e. of the limit symmetrized singular law of the matrix A n + G n of Theorem 1) in the moments of ν following from

H ν⊞ λ √ µ λ = H m for m = [(ν 2 µ λ ) ⊞ δ 1 ] ⊠ µ λ stays true (see Remark 2).
Remark 4 (Case λ = 0). A continuous way to define µ λ for any λ ∈ [0, 1] is to define it to be the probability measure with free cumulants k n (µ λ ) = λ 1-n for all n ≥ 1 (see [START_REF] Nica | Roland Lectures on the combinatorics of free probability[END_REF]). This definition gives µ 0 = δ 1 . Note that by definition of the rectangular free convolution with null ratio ⊞ 0 (which is recalled in the introduction), the relation (3) stays true for λ = 0.

Remark 5. Note that the original proof of Debbah and Ryan in [START_REF] Debbah | Multiplicative free Convolution and Information-Plus-Noise Type Matrices[END_REF] is based on the combinatorics approach to freeness, via the free cumulants of Nica and Speicher [START_REF] Nica | Roland Lectures on the combinatorics of free probability[END_REF], whereas our proof is based on the analytical machinery for the computation of the rectangular R-transform, namely the rectangular R-transform. It happens sometimes that combinatorial proofs can be translated on the analytical plan by considering the generating functions of the combinatorial objects in question. Notice however that it is not what we did here. Indeed, the rectangular Rtransform machinery is actually related to other cumulants than the ones of Nica and Speicher. These are the so-called rectangular cumulants, defined in [START_REF] Benaych-Georges | Rectangular random matrices, related convolution[END_REF].

2.5. Remarks about the free deconvolution by µ λ . The following corollary is part of the answer given in the present paper to the question asked in the last section of the paper of Debbah and Ryan [START_REF] Debbah | Multiplicative free Convolution and Information-Plus-Noise Type Matrices[END_REF]. Let us endow the set of probability measures on the real line with the weak topology [B68].

Corollary 6. The functional ν → [(ν 2 µ λ ) ⊞ δ 1 ] ⊠ µ λ ,
defined on the set of probability measures ν on [0, +∞) such that ν µ λ exists, extends continuously to the whole set of probability measures on [0, +∞).

Proof. We just proved, in section 2.3, that the formula

ν ⊞ λ √ µ λ = [(ν 2 µ λ ) ⊞ δ 1 ] ⊠ µ λ
is true for any probability measure ν on [0, +∞). Since the operation ⊞ λ is continuous on the set of symmetric probability measures on the real line (Theorem 3.12 of [START_REF] Benaych-Georges | Rectangular random matrices, related convolution[END_REF]) and the bijective corespondance between symmetric laws on the real line and laws on [0, +∞), which maps any symmetric law to its push-forward by the map t → t 2 , is continuous with continuous inverse, the corollary is obvious.

The functional µ λ , which domain is contained in the set of probability measures on [0, +∞), plays surprisingly a key role here. It seems natural to try to study its domain. The first step is to notice that this domain is the whole set of probability measures on [0, +∞) if and only if δ 1 is in this domain, and that in this case, the functional µ λ is simply equal to ⊠(δ 1 µ λ ). However, the following proposition states that despite the previous corollary, the domain of the functional µ λ is not the whole set of probability measures on [0, +∞).

Proposition 7. The Dirac mass δ 1 at 1 is not in the domain of the functional µ λ .

Proof.

Suppose that there is a probability measure τ on [0, +∞) such that δ 1 = τ ⊠ µ λ . Such a law τ has to satisfy S τ (z) = 1 + λz. It implies that for z small enough,

M τ (z) = z-1+[(1-z) 2 +4λz] 1/2 2λ
. Such a function doesn't admit any analytic continuation to C\[0, +∞), thus no such probability measure τ exists.

Relations between square and rectangular matrices/convolutions

The theorem of Debbah and Ryan gives an expression of the empirical singular measure of the sum of two rectangular random matrices in terms of operations related to hermitian square random matrices. Two other results relate empirical singular measures of (non hermitian) square or rectangular random matrices to the operations devoted to hermitian random matrices.

The first one can be resumed by ⊞ 1 = ⊞. Concretely, it states that denoting by ESM(X) the symmetrization of empirical singular measure of any rectangular matrix X, for any pair M, N of large n by p random matrices, one of them being invariant in law by the left and right actions of the unitary groups, for n/p ≃ 1, Note that the matrices M, N are not hermitian, which makes (6) pretty surprising (since ⊞ was defined with hermitian random matrices). It means that for ε, ε 1 , ε 2 independent random variables with law δ -1 +δ 1 2 independent of M and N , we have

(7) Spectrum(ε|M + N |)≃ Spectrum(ε 1 |M | + ε 2 |N |)
The second one can be resumed by for any pair ν, τ of symmetric probability measures on the real line, (ν ⊞ 0 τ ) 2 = ν 2 ⊞ τ 2 Concretely, it states that for any pair M, N of n by p unitarily invariant random matrices, for 1 << n << p,

(8) Spectrum[(M + N )(M + N ) * ]≃ Spectrum(M M * + N N * ).
The advantage of the result of Debbah and Ryan on those ones is that it works for any value of the ratio λ, but its disadvantage is that it only works when one of the laws convoluted is µ λ , i.e. one of the matrices considered is a Gaussian one. In fact this sharp restriction can be understood by the fact that among rectangular random matrices which are invariant in law under multiplication by unitary matrices, the Gaussian ones are the only ones which can be extended to square matrices which are also invariant in law under multiplication by unitary matrices.

It could be interesting to understand better how relations like (7), (8) or like the one of the Debbah and Ryan's theorem work and can be generalized. Unfortunately, until now, even though nice proofs (see [START_REF] Benaych-Georges | Rectangular random matrices, related convolution[END_REF] for (7) and (8) or Theorem 4.3.11 of [START_REF] Hiai | The semicircle law, free random variables, and entropy[END_REF] and Proposition 3.5 of [START_REF] Haagerup | Brown's spectral distribution measure for R-diagonal elements in finite von Neumann algebras[END_REF] for the n = p case of (7)) relying in free probability have been given for these results relating rectangular convolutions and "square non hermitian convolutions" with the "square hermitian convolution" (i.e. ⊞), no "concrete" explanation has been given, and no generalization (to any λ, to any pair of probability measures) neither. Such a generalization could be the given of a functional f λ on the set of symmetric probability measures such that for all ν, τ symmetric probability measures, ν ⊞ λ τ is the only symmetric probability measure satisfying

f λ (ν ⊞ λ τ ) = f λ (ν) ⊞ f λ (τ ).
Note that in the case λ = 1, the functional f λ (ν) = ν works, and in the case λ = 0, the functional which maps a measure to its push-forward by the square function works.

Remark 8. Let (A, ϕ) be a * -non commutative probability space and p 1 , p 2 be two self-adjoint projectors of A such that p 1 + p 2 = 1 such that λ = ϕ(p 1 )/ϕ(p 2 ). As explained in Proposition-Definition 2.1 of [START_REF] Benaych-Georges | Rectangular random matrices, related convolution[END_REF], ⊞ λ can be defined by the fact that for any pair a, b ∈ p 1 Ap 2 free with amalgamation over Vect(p 1 , p 2 ), the symmetrized distribution of |a + b| in (p 1 Ap 1 , 1 ϕ(p 1 ) ϕ |p 1 Ap 1 ) is the rectangular free convolution with ratio λ of the symmetrized distributions of |a| and |b| in the same space.

Moreover, it is easy to see that for all a ∈ p 1 Ap 2 , the symmetrized distribution τ of |a| in (p 1 Ap 1 , 1 ϕ(p 1 ) ϕ |p 1 Ap 1 ) is linked to the distribution ν of a + a * in (A, ϕ) by the relation ν = where f λ is the function which maps a probability measure τ on the real line to 2λ 1+λ τ + 1-λ 1+λ δ 0 . But looking at fourth moment, it appears that (9) isn't true.

  + N ) ≃ ESM(M ) ⊞ ESM(N ).

2λ 1+λ τ + 1-λ 1+λ δ 0 .

 0 When λ = 1, the equation ⊞ = ⊞ λ can be summurized in the following way: for a, b ∈ p 1 Ap 2 free with amalgamation over Vect(p 1 , p 2 ), the distribution of (a + b) + (a + b) * in (A, ϕ) is the free convolution of the distributions of a + a * and b + b * . If this had stayed true for other values of λ, it would have meant that for all ν, τ compactly supported symmetric probability measures on the real line, we have (9) f λ (ν ⊞ λ τ ) = f λ (ν) ⊞ f λ (τ ),

Acknowledgments:

The author would like to thank Raj Rao for bringing the paper [DR07] to his attention and Mérouane Debbah for his encouragements and many useful discussions.