Pierre Andreoletti

The local time of a random walk on growing hypercubes

Keywords: random environment, reversible markov chain, Dirichlet method, recurrent regime, local time

The local time of a random walk on growing hypercubes

Introduction and results

Multidimensional RWRE have been studied a lot in the past thirty years in many different directions, so many directions that we can not be exhaustive here, we recommand [START_REF] Zeitouni | Lectures notes on random walks in random environment[END_REF] for a survey on some of those directions. One way to construct a RWRE is to start with a reversible markov chain, in our case it will be a reversible nearest neighborhood random walk. We also assume that the reversible measure is a function of a certain potential V , which is a function from Z d to R. Now if this potential also called environment is random we get a RWRE. Under additional assumptions on V , R. Durrett [START_REF] Durrett | Some multidimensional rwre with subclassical limiting behavior[END_REF] shows that unlike the simple random walk in Z d for d ≥ 3 , these RWRE can be recurrent (for almost all environment) and sub-diffusive. He also points out that the walk localize itself but due to the dimension d > 1 the point of localization is not easy to characterize contrary to the one-dimension case of Ya. G. Sinai [START_REF] Ya | The limit behaviour of a one-dimensional random walk in a random medium[END_REF]. Also he gets the joint distributions of the logarithm of the occupation times of the random walk (also called local time), before it reaches a certain level set of V . This is this last aspect we are interested in, in fact it is the link between the local time and the reversible measure, which is random for a RWRE, that we study in this paper. The random environment (V ) we built here is different from the ones of R. Durrett: we build a potential that creates d -1 dimensional valleys. Indeed the idea that a particle can be trapped on a 1 surface like a sphere or an hypercube, for example, by opposite random fields is pretty physical. We also want the particle to be able to escape from a surface to reach another one where stronger fields act. Finally, in addition to be random, we do not want the fields to be uniform on a given surface, but that the number of values taken on this surface is finite. The definition of V we will consider is the following: first V is a function of two elementary processes. The first one, denoted (S k , k ∈ Z), is a sum of i.i.d random variables. The second one, denoted (δ x , x ∈ Z d ), is a sequence of i.i.d random variable indexed by the vertices of Z d . This two processes are assumed to be independent. Then the potential V on a point x of Z d , is a sum of random variables (S) with random number of terms (x+ δ x ) where . is a norm on Z d . The way we construct this environment make it, at a given distance of the origin, strongly dependant from a site to the other. Naturally we add hypothesis on the random environment, that is to the increments of S, and the distribution of the δ ′ s. With this hypothesis detailed below we get an almost surely recurrent random walk for almost all environment. It appears also that the walk is trapped on a set of points, whose shape depends on the norm .. In fact, like in the one dimensional case, the random environment creates valleys, and as we restrict our analysis to the infinite norm on Z d the shape of the valleys are hypercubes.

An important fact is that these hypercubes, where the walk is trapped, are not level sets for the potential neither for the reversible measure denoted π, however they can be partitioned in level sets for π. A part of this work is devoted to the study of the local time of the walk on these level sets.

In the same way N. Gantert, Y. Peres and Z. Shi [GPS] (see also [START_REF] Andreoletti | Limit law of the local time for brox's diffusion[END_REF] for the one-dimension continuous case) give the limit behavior of the local time in the neighborhood of the coordinate of the depth of a specific valley we do it here but for a set of points. One of the main difference is the fact that we have to deal with the local time on a set of points and therefore several values of the reversible measure are involved.

The ingredient for the study of the local time of a random walk is mainly the computation of the moments of the excursions of the walk. Excursions are the amount of time the walk, starting from a point of the lattice (or a set of points), spends before it returns to this point (or this set of points). This moments can be written as an explicit function of probabilities for the walk to touch a point before another. It appears that for the (nearest neighborhood) 1-dimensionnal RWRE these probabilities can be expressed explicitly from V . In dimension larger than 1 it is not the case in general so other technics can be used, like Dirchlet methods for example. However this method leads in general to inequality instead of equality and therefore weaker results in term of precision and convergence. Our purpose is to show that we can be as precise as in the one dimensional case for non-trivial examples of multidimensional RWRE.

The paper is organized as follows, Sections 1.1 and 1.2 are devoted to the definitions of the basic processes together with a general description of the random environment. In Section 1.3 we state our main results, and in Section 1.4 we give some examples. In Section 2 the reader will find the proof of the results, it is devided into three sub-sections, the first one concerns the random environment, the second the quenched result and the last one the annealed result. Also we add an appendix making this paper quite self-containt.

Definitions and hypothesis

The random environment, is a random sum of random variables: first let us denote (S k , k ∈ Z) the following sum

S k := 1≤i≤k η i , if k = 1, 2, • • • k+1≤i≤0 -η i , if k = -1, -2, • • • S 0 := 0,
where (η i , i ∈ Z) is a i.i.d. sequence of random variables, we denote (Ω 1 , F 1 , P 1 ) the probability space associated to (S k , k ∈ Z). Notice that S . is the typical potential of a one-dimensional random walk in random environment. Also let (δ x , x ∈ Z d ) another sequence of i.i.d. random variables with integer values independent of the sequence (η i , i ∈ Z), and (Ω 2 , F 2 , P 2 ) is the associated probability space. Let d ∈ N * , the potential

V : Z d → R also called random environment is defined by, for all x ∈ Z d V (x) := S x+δx , if x + δ x ≥ 0, 0 instead.
where x is the infinite norm in Z d (x = max 1≤j≤d {|x j |}). Also we denote (Ω e , F e , P e ) the probability space induced by (V (x), x ∈ Z d ).

The random walk, let us fix V , define a reversible Markov chain like an electical network following for example [START_REF] Doyle | Random walks and electric networks[END_REF] or [START_REF] Lyons | Probability on trees and networks[END_REF]. First we assign to each edge (x,y) of Z d a conductance denoted π(x, y) defined as follows

π(x, y) := exp(-1/2V (x) -1/2V (y))1 1 {|x-y|=1} .
where 1 1 is the indicator function and |x -y| = 1 means that x and y differ only by one coordinate

(|x -y| = 1 ⇐⇒ x = y ± e i , 1 ≤ i ≤ d). Also define the capacitances π, for all x ∈ Z d π(x) := z∈Z π(x, z) = exp(-1/2V (x)) z,|x-z|=1 exp(-1/2V (z)).
Then, let (X n , n ∈ N) a markov chain with state space Z d and probability of transition p given by, for all x ∈ Z d and y ∈ Z d ,

p(x, y) := π(x, y) π(x) = exp(-1/2V (y)) z,|x-z|=1 exp(-1/2V (z)) 1 1 {|x-y|=1} .
By construction, (X n , n ∈ N) is reversible and the reversible measure is given by π(.), notice also that p(x, y) = p(y, x) so the walk is not symmetric. We denote (Ω 3 , F 3 , P V ) the probability measure associated to this random walk and we denote

P V z (•) := P V (•|X 0 = z), for all z ∈ Z d .
Finally the whole process, also denoted X . , is defined on a probability space denoted (Ω, F, P). In particular for all

A 1 ∈ F 1 , A 2 ∈ F 2 and A 3 ∈ F 3 , P(A 1 × A 2 × A 3 ) = A 3 dP V (w 1 ,w 2 ) (w 3 ) A 2 ,A 1 dP 2 (w 2 )dP 1 (w 1 ).
Hypothesis on the random environment, to study this walk we add hypothesis on V , we choose the following

E 1 [η 0 ] = 0, (1.1) Var[η 0 ] > 0, (1.2) |η 0 | is bounded P 1 .a.s, (1.3) δ 0 d is bounded P 2 .a.s, (1.4)
and 0 d is the origin of Z d . Under those hypothesis we can prove, see [START_REF] Andreoletti | Note on the localization of random walks in some basic surfaces[END_REF], that the walk is recurrent for almost all environnment, it is sub-diffusive, not localized in the neighborhood of a point of Z d but on an hypercube of Z d centred in 0 d , and at a random distance of the origin.

Local time on hypercubes, here we are interested in the asymptotic behavior of the local time L of X:

let x ∈ Z d and n ∈ N L(x, n) = n k=1 1 1 X k =x ,
it is the time spent by the walk at the point x within the intervall of time [[1, n]], moreover for any U ⊂ Z d define L(U, n) = x∈U L(x, n), the time spent by the walk in the sunbet U . More especially we are interested in the asymptotic behavior of the local time on the sets C k with k ∈ N * define as follows

B k := {y ∈ Z d , ȳ ≤ k}, C k := B k B k-1 .
Notice that for d = 1, C k is reduced to 2 points, so we treat that case separately in Section 1.4.3. As an application of our results, we will get the convergence in law of the supremum of the local time sup k∈Z L(C k , n). The results we present depend on some more definition on the random environment, we define it in the following section.

Values of π on C k and potential conditionned to stay positive

In this section we assume d ≥ 2.

1.2.1 Possible values of V on C k , k ∈ N
Let E 0 the space state of δ 0 d and |E 0 | its cardinal, from the definitions we give above it is clear that V can take, at most, |E 0 | values on a C k . Figure 1 is a graphical representation of the values of V on C k if d=2 and if we assume, as an example, that δ can only take 3 values: -1, 0 and 1. A large segment outside the square means that at this point of the square δ . = 1, a large segment inside the main square means that at this point of the square δ . = -1, no segment means that at this point of the square δ . = 0. Unfortunatly, we will see that it is not the number of different values that takes the potential that is of importance for our results but the one of the reversible measure:

1.2.2 Values of the reversible measure on C k , k ∈ N Let k ∈ Z, define V π k,d := {disctinct π(x), x ∈ C k }.
n d := |V π k,d |. Let, 1 ≤ j ≤ n d , we denote (π j k , 1 ≤ j ≤ n d ) the elements of V π k,d and
C j k := {z ∈ C k , π(z) = π j k }, k S k+1 S k-1 S k-1 (0, 0) S k S k+1 S k Figure 1: d = 2, different values of V on C k a subset of C k which is a level set for π. Of course C k = n d j=1 C j k
, moreover we can give a simple expression for the π j k , let x ∈ Z d we recall that

π(x) = exp(-1/2V (x)) d i=-d,i =0 exp(-1/2V (x + sign(i)e |i| ))
where

(e i , 1 ≤ i ≤ d) is the canonical base of Z d . Let us define Ck the sub-set of C k without the edge i.e. Ck = {z ∈ Z d , z = k, #{z i , 1 ≤ i ≤ d : z i = k} = 1}
, also denote " • " the standard Euclidean scalar product. If we assume that x ∈ Ck with for example x • e 1 = k, then x ± e 1 = k ± 1, whereas x ± sign(i)e |i| = k for the other i = 1. Therefore following this example we get for all x ∈ Ck with

x • e 1 = k, π(x) = exp(-1/2S k+δx ) ×   exp(-1/2S k+1+δ x+e 1 ) + exp(-1/2S k-1+δ x-e 1 ) + d i=-d,i =0,1 exp(-1/2S k+δ x+sign(i)e |i| )   .
When x ∈ C k Ck , we can get similar expressions. The conclusion of this is that, for all k and j, there exists a sequence of integers (a j 0 , a j 1 , • • • , a j 2d ) such that

π j k = exp(-1/2S k+a j 0 ) 2d l=1 exp(-1/2S k+a j l
).

(1.5)

Note that a j 0 takes its values in E 0 and the other a j i take their values on one of the following three sets, E 0 , E + := {l + 1, l ∈ E 0 } and E -:= {l -1, l ∈ E 0 }. We also define E := {l -1, l, l + 1, l ∈ E 0 }.

To resume we can say that the d -1 hypercubes of Z d C k can be partitioned on level sets for π. Moreover, due to the δ's, the C j k have a random geometry, and the values of π on these level sets are given by 1.5.

Limit law of S conditioned to stay positive

As we can see in the above expression of π, the reversible measure on the hypercubes involves the knowledge of random sums of random variables. Now looking at the expression of the π j k , we see that for fixed δ . , the reversible measure only depends on a sum of i.i.d. random variables (S . ) which pretty looks like the one dimensional case (see for example [GPS]). Let us recall basic random variable used in the one-dimensional case:

M n := inf{k > 0 : S k -min 0≤i≤k S i ≥ log n + (log n) 1/2 } m n := inf{k > 0 : S k = min 0≤i≤Mn S i }
These two random variables correspond to a the coordinate of the top (M n ) and the bottom (m n ) of a one dimensionnal localization valley. In what follows we also need to introduce ( Sk , k) a sequence of random variables with the same distrbution as S and conditioned to stay non-negative for x > 0 and strictly positive for x < 0. [Gol84] and [START_REF] Bertoin | Spliting at the infimum and excursions in hal-lines for random walks and Lévy processes[END_REF] has shown independently that under 1.1 S is well defined and also that +∞ x=-∞ exp(-Sx ) < +∞, a. s. We are ready to define πj k for all 1 ≤ j ≤ n d and k ∈ Z

πj k := exp(-1/2 Sk+a j 0 ) 2d l=1 exp(-1/2 Sk+a j l
).

Results -Local time on hypercubes

In this section we also assume that d ≥ 2 and we start with the annealed results

Annealed results

Our first result deals with the limit distribution of the supremum of the local time taken over all the hypercubes (C l , l ∈ N):

Theorem 1.1. Assume hypothesis 1.1-1.4 are satisfied, there exists a sequence of non-negative real

(p j , 1 ≤ j ≤ n d ) with n d j=1 p j = 1 and a sequence of vectors (a j 0 , a j 1 , • • • , a j 2d , 1 ≤ j ≤ n d ) belonging to E ⊗n d such that sup l∈N L(C l , n) n L -→ sup i∈Z Π(i),
where for all i ∈ Z

Π(i) = n d j=1 p j πj i +∞ l=-∞ n d j=1 p j πj l , (1.6)
and L -→ is the convergence in law under P.

This result shows that the behavior of the local time depends on the different values taken by the invariant (reversible) measure on the hyper cubes C k . Of course the π's depends itself of the values taken by the δ and S. If we enter in the details, we see that like in the one dimensionnal case the reversible measure and the porcess S are the important ingredients, and that the δ's act like a noise.

The new things is first that as we are looking on a set of points, several values of the reversible measure appear (the numerator of π j i ). The second new things is that the values of the reversible measure do not have the same importance, indeed the distribution of the δ's can give more importance to some of them and neglect other. This information is contained in the p j , in particular, for this general case, we allow them to be nul. In the next section, we take some examples for the distribution of the δ's and show that all the sequences (the (p j , j) and the (a j 0 , a j 1 , • • • , a j 2d , j)) that appears here become explicit. The above result is an easy consequence of the following result, in that one we look at the local time as a process on specific (random) hypercubes.

Theorem 1.2. Assume hypothesis 1.1-1.4 are satisfied, there exists a sequence of non-negative real

(p j , 1 ≤ j ≤ n d ) with n d j=1 p j = 1 and a sequence of vectors (a j 0 , a j 1 , • • • , a j 2d , 1 ≤ j ≤ n d ) belonging to E ⊗n d such that L(C mn+i , n) n , i ∈ Z L -→ Π(i), i ∈ Z
where the Π, is given by 1.6.

Quenched result

The importance of the quenched results, a part from the fact that they are the key results to get the annealed ones, is that it contains the information that leads from the local time to the reversible measure.

Proposition 1.3. Assume hypothesis 1.1-1.4 are satisfied. Let δ > 0, there exists a sequence of vectors

(a j 0 , a j 1 , • • • , a j 2d , 1 ≤ j ≤ n d )
belonging to E ⊗n d , also for all n there exists a subset A n ∈ F e with lim n→+∞ P e (A n ) = 1 such that

lim n→+∞ inf V ∈An P V 0   l∈{-k,••• ,k}    L(C mn+l , n) n - n d j=1 R j mn+l ≤ δ n d j=1 R j mn+l      = 1,
where

R j p = |C j p |π j p x∈B Mn π(x)
.

This proposition tells us that looking at the local time on the hypercubes (C mn+l , l) is like looking at the weighted distinct values that can take the reversible measure in this set. The weight are given by the amount of points taken by the reversible measure on the hypercube (|C j mn+l |) and normalized by the "total mass" ( x∈B Mn π(x)). Notice that R j p can be re-written in the following way

R j p = |C j p |π j p Mn i=0 n d j=1 |C j i |π j i .
In fact we can be more precise, and replace the denominator of R j p by

(log n) 2-ǫ i=-(log n) 2-ǫ n d
j=1 |C j mn+i |π j mn+i , and then understand the real p j of the annealed result. Indeed, thanks to the fact that the δ . 's are i.i.d., |C j mn+i |/|C mn | → p j , P e .a.s. so the p j are just the proportion of a given reversible measure on an hypercube. By changing the A n for an A ′ n with the same property as A n we can replace R j p by the following:

Rj p = p j π j p (log n) 2-ǫ i=-(log n) 2-ǫ n d j=1 p j π j i .
(1.7)

Notice also that we are looking to the walk on moving hypercubes: they are randomly growing because their distance from the origin is m n . Finally, the constants (p j , 1 ≤ j ≤ n d ) and vectors (a j 0 , a j 1 , • • • , a j 2d , 1 ≤ j ≤ n d ) are preserved from an hypercube to an other, they purely depend on the random environment, especially the δ . 's.

To the question: what can we learn about one weighted reversible measure from the local time ? The above proposition is not very useful, because it mixes different values of the reversible measure. However the above proposition is a corollary of the following statement

lim n→+∞ inf V ∈An P V 0   l∈{-k,••• ,k} n d j=1 L(C j mn+l , n) n -R j mn+l ≤ δR j mn+l   = 1,
also we can compare the strength of the different weighted π's from the local time by the following:

lim n→+∞ inf V ∈An P V 0   n d j=1 i =j L(C j mn , n) L(C i mn , n) - p j π j mn p i π i mn ≤ δ   = 1.

Examples

In this section we give a short analysis of three cases. The first one is a trivial case: we assume that the δ 0 d ≡ 0, Theorem 1.1 get of course simpler, and can be seen as a one-dimensional case. The second example is the simplest we can build with random and non trivial δ's, we assume that they are Bernoulli trials with parameter p, in this case we easily get explicit formula for the p i 's and the π j i 's. Our third case is the one-dimensional case, we just point out that it is not exactly Sinai's walk ( [START_REF] Ya | The limit behaviour of a one-dimensional random walk in a random medium[END_REF], [GPS]), even if it pretty looks like it.

1.4.1 The trivial case, δ ≡ 0, d ≥ 2
In this case the reversible measure on C k can only takes d different values and by definition C k are level sets for V , however C k is not a level set for the reversible measure π. Indeed there is some edges exception: for example when d=2 π takes two different values on

C k : ∀x ∈ Ck , π(x) = (2 exp(-S k ) + exp(-S k+1 ) + exp(-S k-1 )), but ∀x ∈ C k \ Ck , π(x) = (2 exp(-S k ) + 2 exp(-S k+1 )). In general it is easy to check that ∀x ∈ Ck , π(x) = π(k, 0 d-1
). We can represent the level sets in the 3-dimensionnal case as in figure 2: surfaces delimited by the black lines represent the level sets for V (on the left) and the level sets of π (on the right).

Remark 1.4. A few words on the size of the sets C k and Ck , by size we mean the number of points in this set, we have

|C k | = k d -(k -2) d , also we easily check that |C k Ck | = k d -(k -2) d -2d(k -2) d-1 therefore | Ck | = 2d(k -2) d-1 . Finally for large k we have |C k | ≈ | Ck | ≈ k d-1 , and |C k Ck | ≈ k d-2 . (0, 0) Ck k k C k k Figure 2: d = 3
, on the left level sets for V , on the right for π .

Thanks to the preceding Remark there is only one value of the reversible measure associated with a strictly positive probability p 1 , which is of course equal to one. For example Theorem 1.1 is given by Corollary 1.5. Assume hypothesis 1.1-1.3 are satisfied and δ 0 d = 0, then

sup k∈N L(C k , n) n L -→ sup l∈Z Π(l),
where for all l ∈ Z Π(l) = exp(-1/2 Sl ) (2d -2) exp(-1/2 Sl ) + exp(-1/2 Sl-1 ) + exp(-1/2 Sl+1 ) +∞ l=-∞ exp(-1/2 Sl ) (2d -2) exp(-1/2 Sl ) + exp(-1/2 Sl-1 ) + exp(-1/2 Sl+1 )

.

When we look at the expression of Π(l), we understand that this case can be seen as a one dimension case: see Section 1.4.3.

1.4.2 The Bernoulli case δ ∼ B(p), d ≥ 2
This is the simplest case we can define with non trivial random δ's. In this paragraph we show that we can get a quite simple expression for Π. Theorem 1.1 becomes:

Corollary 1.6. Assume hypothesis 1.1-1.3 are satisfied and that δ 0 d is a Bernoulli with parameter p, then

sup k∈N L(C k , n) n L -→ sup i∈Z Π(i),
where for all i ∈ Z

Π(i) = Γ i +∞ l=-∞ Γ l , Γ i = 1 i 0 =0 0 i 1 =-1 2 i 2 =1 2d-2 k=0 p B (i 0 , i 1 , i 2 , k)π B (i, i 0 , i 1 , i 2 , k), p B (i 0 , i 1 , i 2 , k) = p i 0 +i 1 +i 2 (1 -p) 3-i 0 -i 1 -i 2 2d -2 k p k (1 -p) 2d-2-k , π B (i, i 0 , i 1 , i 2 , k) = e -Si+i 0 e -Si+i 1 + e -Si+i 2 + ke -Si+1 + (2d -2 -k)e -Si .
Like for the trivial case, we are less interested in the value of n d (the number of distinct reversible measures we can obtained on a surface C k for some k large enough), than in ñd which is the number of distinct reversible measure associated with a non-nul p i . Observing the expression for Γ i we easily get that ñd = 8(2d -2). Notice that the p i 's are replaced by the p B (i 0 , i 1 , i 2 , k), and of course we can check that 1

i 0 =0 0 i 1 =-1 2 i 2 =1 2d-2 k=0 p B (i 0 , i 1 , i 2 , k) = 1.
If we assume for the moment that the general case is proved, we can get the above corollary, just by showing the Lemma 1.7. Assume δ 0 d is a Bernoulli with parameter p, then the only values, associated with nonnul p i 's, that can take the reversible measure π on a C k are given by the π B , moreover the associated number p . are given by the p B .

Proof. The proof is pretty easy, however it may be used for more general case so we give some details. First thanks to Remark 1.4, we only have to consider the number of distinct π on Ck . Let us define a vector v such that its coordinate

(v 1 , v 2 , • • • , v 2d+1 ) ∈ {0, 1} × {1, 2} × {-1, 0} × {0, 1} 2d-2 . Now assume that x ∈ Ck with x • e 1 = k, we recall that π(x) = e (-1/2S x+δx )   e (-1/2S x+1+δ x+e 1
) + e

v = (0, 2, -1, 1, 0 2d-2 )) means that δ x = 0, δ x+e 1 = 1, δ x-e 1 = 0, • • • .
To get the result we can use, for example, a tree representation (see also Figure 3): the root is given by the vector v 0 = (0 or 1, -1, 0 2d-2 ), the first coordinate of the vector v having no importance. For the first generation we fix or increment the second and third coordinate (in Figure 3 we underlined the fixed coordinates). Once all the second and third coordinate are fixed, each of these vectors give 2 d distinct vectors with fixed coordinate. Each of this last vectors leads to 2d -2 distinct reversible measures and we are done. ñd is given by 2 (0 or 1) times 4 times the number of this vectors. We can read the expression of π B and the probability p B from the tree.

About dimension one (d=1)

With the definition of the RWRE we give we do not get exactly Sinai's random walk, even if we assume δ 0 d ≡ 0. First because the probability of transition are not the same and also because in our case the random environment is almost symmetric with respect to the origin. Almost because if δ 0 d ≡ 0 the random variables δ 0 d differs at each site. In [START_REF] Andreoletti | Note on the localization of random walks in some basic surfaces[END_REF], it is shown that we do not have necessarily a localization in one point but in one or two points each with a positive probability. Here, by definition, we have C k = {-k, k} so unlike the d-dimensional with d > 1, the size of C k does not grow with k. To compare, with the result in [GPS] we assume that the walk is reflected at the origin so C k = {k} first we recall Theorem 1.8. ( [GPS]) Assume that L S is the local time of Sinai's walk (reflected at 0), then

2 d children v 0 = (0 or 1, 1, -1, 0 2d-2 ) (0 or 1, 2, 0, 0 2d-2 ) (0 or 1, 1, -1, 0 2d-2 ) (0 or 1, 1, 0, 0 2d-2 ) (0 or 1, 1, 0, 0 2d-2 ) (0 or 1, 2, -1, 0 2d-2 ) 2 d children 2 d children 2 d children
sup l∈N L S (l, n) n L -→ sup i∈Z e -Si-1 + e -Si +∞ k=-∞ e -Sk-1 + e -Sk .
In our case, we can say something like Corollary 1.9. Assume d = 1, hypothesis 1.1-1.4 are satisfied then

sup l∈N L(l, n) n L -→ sup i∈Z Π(i),
where

Π(i) = e -1/2 Si+δ i e -1/2 Si-1+δ i-1 + e -1/2 Si+1+δ i+1 +∞ k=-∞ e -1/2 Sk+δ k e -1/2 Sk-1+δ k-1 + e -1/2 Sk+1+δ k+1 .
To compare with the result of [GPS], just take δ 0 d = 0 and get

Π(i) = e -1/2 Si e -1/2 Si-1 + e -1/2 Si+1 +∞ k=-∞ e -1/2 Sk e -1/2 Sk-1 + e -1/2 Sk+1
.

Notice that the one dimensional case do not behave like the multi-dimensional one, in particular we see that the δ appears explicitly when d = 1. This just comes from the fact that the C k are reduced to one point. We will not give any details of the proof of this result, indeed we can get it just by mimic the proof in [GPS].

Ending remarks

In [GPS] the result in law (equivalent of our Theorem) leads quite easily to the almost sure behavior of the lim sup of the supremum of the local time. Here, except in some simple cases including the preceeding examples, some more work are needed that the reason why we treat this aspect in a seperate paper (see [START_REF] Andreoletti | A concentration result of a random walk in dimension d[END_REF]).

The assumption 1.4, about the δ's, allows us to keep the structure of valleys, things are different if the support of the δ's depends on x ∈ Z d . For example for δ's not i.d. and with a support depending on x it is easy to construct an environment without valleys. However for a small support comparing to a typical fluctuation of the walk we believe that we can get a quite similar behavior. Also we choose the infinite norm for the definition of the random environment which naturally leads to our hypercubes, however for other norms one can get similar results.

The rest of the paper is devoted to the proof of the results.

Proof of the results

In all this section we assume that d ≥ 2. We start with some notions and results for the random potential V ,

Some properties of the random potential

We need some basic definitions and elementary properties on the random environment, we start with definition well known for the one dimensionnal case: Let n ∈ N, we define

∆ n = max 0≤k≤l≤mn max x∈C k max y∈C l (V (y) -V (x)) ∨ max mn≤k≤l≤Mn max x∈C k max y∈C l (V (x) -V (y))
In words ∆ n is the maximum between the highest height of potential the walk can encountered, when going from 0 d to C mn and the one when going from C Mn to C mn . We continue with the Remark 2.1. thanks to hypothesis 1.3 and 1.4 ( the δ ′ s and the increments of S are bounded) we have, for all m > 0 S mconst ≤ S m+δ. ≤ S m + const.

We are now ready to state a lemma for the random environment V :

Lemma 2.2. Assume 1.1-1.4 are satisfied, let n ∈ N, and A n ∈ F e be such that ∀ω ∈ A n , S(ω) and V (ω) satisfy the following 3 properties, 1. For all x ∈ C mn and all y ∈ C Mn V (y)

-V (x) -log n -(log n) 1/2 ≤ const, 2. Let ǫ > 0, (log n) 2-ǫ ≤ m n ≤ (log n) 2+ǫ , (log n) 2-ǫ ≤ M n ≤ (log n) 2+ǫ , M n /m n ≤ (log n) ǫ , 3. ∆ n ≤ log n(1 -ǫ n ), with ǫ n = (log log n) 2 log n , then lim n→+∞ P e (A n ) = 1.
The proof is very basic, thanks to the above Remark we move from V to S easily, we are left to prove a one-dimensionnal result that can be found in the literature (see for example [START_REF] Andreoletti | On the concentration of Sinai's walk[END_REF] page 1389). Notice that Lemma 2.2 takes only into account the basic properties of the environment we need, some more advance statements are needed especially to obtain the annealed result. We present and show them Section 2.3. As usual, we write V ∈ A n to say that we are looking at an event ω ∈ F e such that V (ω) satisfies the three conditions.

Proof of Proposition 1.2 (quenched results)

We start with some basic statements, one saying that the walk is trapped in a specific valley, the other one that the walk reaches a level set C j mn on a negligible amount of time comparing to n. In all this section we assume that V ∈ A n .

Lemma 2.3. Let 1 ≤ j ≤ n d , we have

lim n→∞ P V 0 T C j mn ≤ n 1-ǫn/2 = 1, (2.1)
for all z ∈ C mn

P V z (T BMn > n) ≥ 1 -(log n) 2(d-1) n -(log n) -1/2 . (2.2)
Proof.

The proof is based on the estimation of certain probability involving hitting times, we recall what we need at the end of the Appendix. Proof of 2.2 we use the classical way in that situation: we show that the walk prefers to return n times to C mn instead of reaching BMn the complementary of B Mn in Z d (notice also that

T BMn = T C Mn+1 ).
Let us define the following stopping times, let p ≥ 1, A ⊂ Z d ,

T A,p ≡ inf{k > T A,p-1 , X k ∈ A}, +∞, if such k does not exist. T A ≡ T A,1 ; T A,0 = 0.
For simplicity, we also denote T + A the first return to A starting from A. Let y ∈ C mn , the strong Markov property yields:

P V y (T BMn > T Cm n ,n ) ≥ min z∈Cm n P V z T BMn > T + Cm n n ≥ 1 -max z∈Cm n P V z T BMn > T + z n .
3.7 together with item 1. of Lemma 2.2 yields

P V z T + z < T CMn ≤ const|B Mn | exp(-log n-(log n) 1/2
). So we get 2.2 thanks to the facts |B mn | = (2m n + 1) d , T Cm n ,n ≥ n, and item 2 of Lemma 2.2. Proof of 2.1 we do that in two steps, first we show that

lim n→∞ P V 0 (T Cm n ≤ n 1-ǫn/2 ) = 1, (2.3) We have T Cm n = L(B mn , T Cm n ), Markov inequality implies P V (L(B mn , T Cm n ) ≥ n 1-ǫn ) ≤ n -1+ǫn x∈Bm n E V (L(x, T Cm n )) moreover E V (L(x, T Cm n )) = P V (T x < T Cm n )/P V x (T + x > T Cm n ) ≤ 1/P V x (T + x >
T Cm n ). Thanks to 3.9, for all x ∈ B mn ,

P V x (T + x > T Cm n ) ≥ const exp(-∆ n )|B mn | -1
, finally thanks to item 2 and 3 of Lemma 2.2 we get

P V 0 (L(B mn , T Cm n ) ≥ n 1-ǫn ) ≤ const (log n) 2d(2+ǫ) exp((log log n) 2 /2) ,
with ǫ > 0 and we 2.3 comes. To get 2.1 from 2.3 we only have to show that for all ǫ > 0 for any j : 1 ≤ j ≤ n d and z ∈ C mn

lim n→+∞ P V z (T C j mn ≤ n 1-ǫ ) = 1. (2.4)
There is nothing to do if z ∈ C j mn , if not we get it with a similar argument used for 2.2.

Our first key result is the following d-dimensional equivalent of Theorems 3.8 and 3.14 in [And06] or equation 2.10 in [GPS],

Proposition 2.4. Let ǫ > 0. For all 1 ≤ j ≤ n d , lim n→+∞ y 0 ∈C j mn P V y 0 L(C j mn , n) n -R j mn ≥ ǫR j mn = 0, (2.5)
where

R j p = |C j p |π j p x∈B Mn π(x)
, moreover

lim n→+∞ P V 0   L(C mn , n) n - n d j=1 R j mn ≥ ǫ n d j=1 R j mn   = 0, (2.6) and if k ∈ N * lim n→+∞ P V 0   l∈{-k,••• ,k}    L(C mn+l , n) n - n d j=1 R j mn+l ≥ ǫ n d j=1 R j mn+l      = 0. (2.7) Proof. Proof of 2.5 Let us denote A n = L(C j mn ,n) n -R j mn ≥ ǫR j mn and D n = {T BMn > n}. It is easy to check that A n ∩ D n ⊂ A + n ∪ A - n , where A ± n = {±L(B Mn , T C j mn ,nR j mn (1∓ǫ)
) ≥ ±n} (assuming for simplicity that nR j mn (1 ± ǫ) are integers). Note that thanks to 2.2 lim n→+∞ y 0 ∈C j mn

P V y 0 (A n ) ≤ lim n→+∞   y 0 ∈C j mn P V y 0 (A + n ) + y 0 ∈C j mn P V y 0 (A - n )   .
Let us give an estimate of P V y 0 (A + n ). Denoting n ǫ = nR j mn (1 -ǫ) and applying Markov inequality:

P V y 0 (A + n ) ≡ P V y 0 L(B Mn , T C j mn ,nǫ ) -n ǫ (R j mn ) -1 ≥ ǫn , ≤ E V y 0 L(B Mn , T C j mn ,nǫ ) -n ǫ (R j mn ) -1 2 (ǫn) -2 .
(2.8)

Note that, as C j mn is not a singleton, L(B Mn , T C j mn ,j ) can not be written as a sum of i.i.d random variables and E V y 0 L(B Mn , T C j mn ,nǫ ) -n ǫ (R j mn ) -1 2 is not the variance of L(B Mn , T C j mn ,nǫ ). For any w ∈ C j mn let us denote

E w = E V w (L(B Mn , T + C j mn ) -(R j mn ) -1
) , from Lemma 3.4, we have:

y 0 ∈C j mn E V y 0 L(B Mn , T C j mn ,nǫ ) -n ǫ (R j mn ) -1 2 = n ǫ y 0 ∈C j mn E V y 0 L(B Mn , T + C j mn ) -(R j mn ) -1 2 + 2 n ǫ -1 i=1 (n ǫ -i) v i ∈C j mn v∈C j mn E v i E v P V v (X T + C j mn ,i = v i ).
For the first sum we apply Cauchy-Schwarz inequality and get

E V y 0 L(B Mn , T + C j mn ) -(R j mn ) -1 2 ≤ |B Mn | z∈B Mn E V y 0   L(z, T + C j mn ) - π(z) π j mn |C j mn | 2   ,
then with the help of Lemma 3.2 of the Appendix, and Remark 2.1,

y 0 ∈C j mn E V y 0 L(B Mn , T + C j mn ) -(R j mn ) -1 2 ≤ |B Mn | z∈B Mn π(z) π j mn 2 P V z (T z > T + C j mn ) ≤ const|B Mn | z∈B Mn 1 P V z (T z > T + C j mn ) .
Finally thanks to Lemma 3.6 equations 3.9 and 3.10 together with Lemma 2.2 statement 3

y 0 ∈C j mn E V y 0 L(B Mn , T + C j mn ) -(R j mn ) -1 2 ≤ const|B Mn | 3 n 1-ǫn , (2.9)
and we recall that ǫ n = ((log log n) 2 )/ log n.

For the second sum, Lemma 3.5 shows that for n large enough and i ≥ |C j mn | 1+ǫ 1 with ǫ 1 > 0

P V v X T + C j mn ,i = v i - 1 |C j mn | ≤ exp(-|C j mn | ǫ 1 ), so asympoticaly when n increases P V v X T + C j mn ,i
= v i does not depend on v, moreover thanks to Lemma 3.2 v∈C j mn E v = 0, therefore we get for n large enouh

nǫ-1 i=1 (n ǫ -i) v i ∈C j mn v∈C j mn E v i E v P V v (X T + C j mn ,i = v i ) = |C j mn | 1+ǫ 1 i=1 (n ǫ -i) v i ∈C j mn v∈C j mn E v i E v P V v (X T + C j mn ,i = v i ).
Finally for n large enough

nǫ-1 i=1 (n ǫ -i) v i ∈C j mn v∈C j mn E v i E v P V v (X T + C j mn ,i = v i ) ≤ n ǫ |C j mn | 1+ǫ 1   y 0 ∈C j mn E V y 0 L(B Mn , T + C j mn )   2 ,
and by using Lemma 3.1 and Remark 2.1 we obtain

nǫ-1 i=1 (n ǫ -i) v i ∈C j mn v∈C j mn E v i E v P V v (X T + C j mn ,i = v i ) ≤ constn ǫ |C j mn | 1+ǫ 1 |B Mn | 2 .
(2.10)

To finish we collect 2.8, 2.9 and 2.10, we get

y 0 ∈C j mn P V y 0 (A + n ) ≤ constR j mn |B Mn | 3 ǫ 2 n ǫn + constR j mn |C j mn | 1+ǫ 1 |B Mn | 2 ǫ 2 n .
By definition R j mn ≤ 1, so we get lim n→+∞ y 0 ∈C j mn P V y 0 (A + n ) = 0 by applying Remark 1.4 and statement 2 of Lemma 2.2. We get the same estimate with the same method for y 0 ∈C j mn P V y 0 (A - n ). Proof of 2.6 First notice that n d j=1 R j mn ≤ 1 and n d is bounded, with ∪ n d j=1 C j mn = C mn therefore to get 2.6 from 2.5, we only have to check that:

lim n→+∞ P V 0 L(C j mn , n) n -R j mn ≥ ǫR j mn = 0.
Only the very begining of the computations differs from what we did above because the walk starts from 0, by using 2.1 we easily get that

P V 0 L(C j mn , n) n -R j mn ≥ ǫR j mn ≤ z∈C j mn P V z L(C j mn , n) n -R j mn ≥ ǫR j mn + P V z L(C j mn , n ′ ) n -R j mn ≥ ǫR j mn + o(1).
with n ′ = n(1 -1/n ǫn ), and lim n→+∞ o(1) = 0. Then the computations remain the same as above.

Proof of 2.7, we use the same method as above and we get the following inequality, for all 1 ≤ j ≤ n d and l such that |l| ≤ k:

lim n→+∞ y 0 ∈C 1 mn P V y 0 L(C j mn+l , T C 1 mn ,nR 1 mn ) n -R j mn+l ≥ ǫR j mn+l = 0.
This leads easily to the result by considering 2.5 and the above discussion about 2.6.

Proof of Theorem 1.2 (annealed results)

In this part we, prove a result in law for the random environment, we introduce the potential conditioned to remain positive S, the constants p i , this leads to our second theorem.

The main result of this section is the following Proposition 2.5. For any k ∈ N,

(L(n, C mn+l ), -k ≤ l ≤ k)) L -→ ( Π(l), -k ≤ l ≤ k), recall that Π is defined in 1.6.
Considering what we did in the previous paragraph, we only need to prove the following Lemma 2.6. Let k ∈ N * , there exists a sequence (p j , 1 ≤ j ≤ n d ) of non negative terms satisfying

n d j=1 p j = 1 such that   n d j=1 R j mn+i , -k ≤ i ≤ k   l -→ n d j=1 p j πj i +∞ l=-∞ n d j=1 p j πj l , -k ≤ i ≤ k ,
recall that π.

. is defined Section 1.2.3, and l -→ is the convergence in law under P e .

Proof.

First we prove that for all k ∈ N * , ǫ > 0, there exists γ > 0 such that

lim n→+∞ P e   -k≤l≤k    n d j=1 R j mn+l - n d j=1 F j (m n + l, m n ) (log n) 2-ǫ i=-(log n) 2-ǫ n d j=1 F j (m n + i, m n ) > const (log n) γ      = 0, (2.11)
where

F j (r, m n ) := |C j r | |C mn | πj r , πj r = π j r exp(S mn ).
Recalling the definition of R

| |i|≥(log n) 2-ǫ z∈C mn+i ∩B Mn π(z) exp(S mn ) ≤ const(log n) ǫ 0 (d-1)
|i|≥(log n) 2-ǫ ,-mn≤i≤Mn-mn exp(-(S mn+i -S mn )), with ǫ 0 > 0 to be chosen. Moreover we know (see for example the Appendix of [START_REF] Andreoletti | Almost sure estimates for the concentration neighborhood of Sinais walk[END_REF]) that for all r

lim n→∞ P 1   |i|≥r,-mn≤i≤Mn-mn e -(S mn +i -Sm n ) > ǫ ′ 0   ≤ const ǫ ′ 0 √ r , (2.12)
with ǫ ′ 0 > 0. Assembling what we did above leads to: choosing ǫ ′ 0 = (log n) -1/4 there exists a γ > 0 such that

lim n→+∞ P e   1 |C mn | x∈B Mn π(x) - (log n) 2-ǫ i=-(log n) 2-ǫ z∈C mn+i π(z) exp(S mn ) ≥ (log n) -γ   = 0.
Notice that the above normalized double sum is P e .a.s. larger than a strictly positive constant and it can be re-written in the following way,

1 |C mn | (log n) 2-ǫ i=-(log n) 2-ǫ z∈C mn +i π(z) = 1 |C mn | (log n) 2-ǫ i=-(log n) 2-ǫ n d j=1 |C j mn+i |π j mn+i
moreover we can check that the numerator of R j mn+l : |C j mn+l |/|C mn |π j mn+l exp(S mn ) is bounded P e .a.s. so we get 2.11.

Lemma 2.7. There exists a sequence of non-negative numbers (p j , 1 ≤ j ≤ n d ), with n d j=1 p j = 1 such that

lim n→+∞ P e   (log n) 2-ǫ r=-(log n) 2-ǫ n d j=1 |C j mn+r |/|C mn | -p j ≤ g(n)   = 1,
where g is a positive decreasing function with lim n→+∞ g(n) = 0.

Proof.

First, by independence of the δ ′ s , it is easy to show that

lim n→+∞ P e   (log n) 2-ǫ r=-(log n) 2-ǫ {||C mn+r |/|C mn | -1| ≤ g 1 (n)}   = 1,
where g 1 = (log n) -γ 0 , with some γ 0 > 0. We recall that n d is bounded and that n d j=1 |C j mn+r |/|C mn+r | = 1, moreover by Kolmogorov's zero-one law, for all j and r, lim n→∞ |C j mn+r |/|C mn+r | exists P 2 .a.s and we call this limit p j . This finish the proof. The above Lemma yields (2.14) So 2.11 and 2.13 leads to, for all ǫ ′′ 0 > 0

lim n→+∞ P e   1 |C mn | (log n) 2-ǫ i=-(log n) 2-ǫ n d j=1 |C j mn+i |π j mn+i - ˜ (log n) 2-ǫ ≥ g(n) ˜ (log n) 2-ǫ   = 0, ( 2
lim n→+∞ P e   -k≤l≤k    n d j=1 R j mn+l -Fn (l)/ ˜ (log n) 2-ǫ ≥ ǫ ′′ 0      = 0.
We are now moving to the convergence in law, we need to prove the following, let α ≥ 0

lim n→+∞ P 1 Fn (l)/ ˜ (log n) 2-ǫ ≤ α = P 1 F (l)/ ¯ ∞ ≤ α .
(2.15)

First step Let us denote A n,n 1 ,ǫ 1 = Fn (l)/ ˜ n 1 -Fn (l)/ ˜ (log n) 2-ǫ ≤ ǫ 1 , with ǫ 1 > 0. We show that lim n 1 →+∞ lim n→+∞ P 1 (A n,n 1 ,ǫ 1 ) = 1.
(2.16)

Thanks to hypothesis 1.3, we only have to check that lim n 1 →+∞ lim n→+∞ P 1 1/ ˜ n 1 -1/ ˜ (log n) 2-ǫ > ǫ 1 = 0, keeping the same ǫ 1 even if it changes a little bit. Assume n 1 < (log n) 2-ǫ , a little of computations yields

P 1 1/ ˜ n 1 -1/ ˜ (log n) 2-ǫ > ǫ 1 ≤ P 1 ˜ n 1 ,(log n) 2-ǫ > ǫ 1 ( ˜ n 1
) 2 , then thanks to hypothesis 1.3 again, ˜ n 1 ≥ const > 0, statement 2 of Lemma 2.2 and 2.12 we get

P 1 ˜ n 1 ,(log n) 2-ǫ > ǫ 1 ( ˜ n 1 ) 2 ≤ 1 constǫ 1 1 √ n 1 .
To finish we take the limit for n, and finally for n 1 so we get 2.16. Second step By using 2.16, we easily get that lim

n 1 →+∞ lim n→+∞ P 1 Fn (l) ˜ n 1 ≤ α -ǫ 1 ≤ lim n→+∞ P 1 Fn (l) ˜ (log n) 2-ǫ ≤ α ≤ lim n 1 →+∞ lim n→+∞ P 1 Fn (l) ˜ n 1 ≤ α + ǫ 1 ,
moreover from [Gol84] Lemma 4 we know that the finite distribution of (S mn+i -S mn , i ∈ Z) converge to those of ( Sl , l) defined Section 1.2.3, that is to say

Fn (l) ˜ n 1 , -k ≤ l ≤ k → F (l) ¯ n 1 , -k ≤ l ≤ k , in P 1 law
where F (resp. ¯ ) is given by 2.14, replacing Fn by F (resp. ˜ by ¯ ). The above inequality becomes lim

n 1 →+∞ P 1 F (l) ¯ n 1 ≤ α -ǫ 1 ≤ lim n→+∞ P 1 Fn (l) ˜ (log n) 2-ǫ ≤ α ≤ lim n 1 →+∞ P 1 F (l) ¯ n 1 ≤ α + ǫ 1 . (2.17)
We are almost done: let

γ 1 > 0, denote B n 1 := (1 -γ 1 ) ¯ +∞ ≤ ¯ n 1 ≤ ¯ +∞ ,
for all µ ≥ 0, we have

P 1 F (l)/( ¯ +∞ (1 -γ 1 )) ≤ µ, B n 1 ≤ P 1 F (l)/ ¯ n 1 ≤ µ, B n 1 ≤ P 1 F (l)/ ¯ +∞ ≤ µ thanks to Bertoin [Ber93], lim n 1 →+∞ P (B n 1 ) = 1, so P 1 F (l)/( ¯ +∞ (1 -γ 1 )) ≤ µ ≤ lim n 1 →+∞ P 1 F (l)/ ¯ n 1 ≤ µ ≤ P 1 F (l)/ ¯ +∞ ≤ µ
letting γ 1 goes to zero, inserting the result in 2.17 and letting ǫ 1 goes to zero we get 2.15 Proposition 2.5 is a consequence of 2.7 and Lemma 2.6.

Proof of Theorem 1.1

The proof can be deduced from Proposition 2.5 and the same method exposed in [GPS] pages 6 and 7.

3 Appendix

Basic formula for reversible random walks

We recall and shortly prove basic results on the moments of the local time of nearest neighborhood reversible random walks. Assume that A and A ′ ⊂ Z d are two sets where the reversible measure π is constant, also we recall that the different hitting times T are defined at the begining of Section 1.2, we have

Lemma 3.1. Let x ∈ Z d , y ∈ A, v ∈ A ′ then z∈A E V z (L(x, T + A )) = π(x) π(y) , (3.1) z∈A E V z (L(A ′ , T + A )) = |A ′ | π(v) π(y) , (3.2)
where |A ′ | is the size (cardinal) of A ′ .

For completness, a few words of the proof Proof.

For 3.1 It is easy to get that

E V z (L(x, T + A )) = P V z (T x < T + A ) P V x (T + x > T + A )
, moreover the chain is reversible therefore

P V x (T + x > T + A ) =
1 P V z (T + z > T + A )
.

(3.3)

Proof.

With the same idea as the proof above we easily get

y∈A E V y L(z, T + A ) 2 = π(z) π(x) 1 P V z (T + z > T + A ) + π(z) π(x) ,
and using Lemma 3.1 for the other term we finish the proof.

For the study of the excursion of the walk we also need the following elementary results: Proof.

If A is a singleton then it is trivial because the sequence (T + A,l -T + A,l-1 , l) is i.d. When A is not a singleton we can get the result recursively, indeed, we easily have:

E V z (L(x, T + A,l )) = u∈A E V z (L(x, T + A )1 1 X T A =u ) + u∈A E V u (L(x, T + A,l-1 ))P z (X T A = u) ≡ E V z (L(x, T + A )) + u∈A E V u (L(x, T + A,l-1 ))P z (X T A = u)
and thanks to the reversibility and the definition of A, P z (X T A = u) = P u (X T A = z), therefore (l -i)

v i ∈A v∈A E v i L(x, T + A ) - 1 |A| e x P v (X T + A,i = v i )E v L(x, T + A ) - 1 |A| e x .
Proof.

The Lemma can be proven recursively, adding and substrating L(x, T + A,l-1 ), we get that recursively on l we also get that

v∈A E V v L(x, T + A,l-1 ) - l -1 |A| e x E V v L(x, T + A ) - 1 |A| e x = l-1 i=1 v i ∈A v∈A E V v i L(x, T + A ) - 1 |A| e x P V v (X T + A,i = v i )E V v L(x, T + A ) - 1 |A| e x .
Putting together what we did above gives the Lemma.

Lemma 3.5. For all l ∈ N * and u, u 0 ∈ A, |P V u 0 X T A,l = u -1/|A|| ≤ (1 -1/|A|) l .

Proof.

We have a reversible Markov chain with finite space state and symmetric distribution, proof is basic, see [START_REF] Lyons | Probability on trees and networks[END_REF] for example.

where the last inequality comes from Cauchy-Schwarz inequality, to finish notice that by definition Φ(g k ) ≤ Φ(h k ) and g k (z) = 1. Finally to prove 3.9, by following what we did above we get

Φ(g k ) ≥ 1 m ′ inf 1≤l≤m ′ π(z ′ l-1 , z ′ l )
because g k (z) = 1 and g k (w) = 0 for all w ∈ Bk , the proof is complete because Φ(g k ) = 2π(z)P V z (T + z > T Bk ). Proof of 3.10 is the same as above.



  2S x+δ x+sign(i)e i )  So the vector v represents one configuration for π(x), example (

Figure 3 :

 3 Figure 3: Tree representation for the Bernoulli case.

  x < T + A ), so we get the lemma by definition of A. 3.2 follows immediatly from 3.1. Lemma 3.2. Let x ∈ A and z /

Lemma 3. 3 .

 3 Let x ∈ Z d , y ∈ A and v ∈ A ′ , and l ∈ N * then z∈A

  the Lemma using 3.1.Let us denote ex = u∈A E V u (L(x, T + A )).Lemma 3.4. Let x ∈ Z d , y ∈ A and l ∈ N * then z∈A

+|A| e x 2 + 2

 22 L(x, T + A,l ) -L(x, T + A,l-1 ) -1 L(x, T + A,l-1 ) -l -1 |A| e x L(x, T + A,l ) -L(x, T + A,l-1 ) -1 |A| e x let us denote F (l) = z∈A E V z L(x, T + A,l ) -l |A| e x 2and let F (0) = 0, using the above decomposition the reversibility of the Markov chain, the definition of A and the strong Markov propertyF (l) = F (l -

  Thanks to Hypothesis 1.4, we know that |V π k,d | is bounded and constant for k large enough, for simplicity we denote this constant

  |C mn | x∈B Mn π(x) exp(S mn ) we split x∈B Mn π(x) exp(-S mn ) into two parts (log n) 2-ǫ z∈C mn+i ∩B Mn π(z) exp(S mn ).For the second sum, with the help of Remark 2.1, we have P e .a.s.

	(log n) 2-ǫ
	π(x) =	π(z) exp(S mn ) +
	j mn+l , |C j mn+l |π j mn+l x∈B Mn π(x) i=-(log n) 2-ǫ z∈C mn+i R j mn+l = z∈C mn+i ∩B Mn π(z) exp(S mn ) ≤ const|C Mn | exp(-(S mn+i -S mn )), ≡ |C j mn+l |/|C mn |π j mn+l then thanks to Remark 1.4 and statement 2. of Lemma 2.2, with a P e probability converging to one 1 1/x∈B Mn |C mn

|i|≥

The Dirchlet method

The Dirchlet method (see for example [START_REF] Thomas | Interacting Particle Systems[END_REF]) allows us to get a part of the estimate we need in this paper, we recall that for a reversible Markov chain, we have the following elementary result : Let z ∈ Z d , and

Let (z q , 0 < q < m), a self avoiding path from z 0 = z to z m = y then

Proof.

The proof can be found, for example, between pages 98 to 101 of [START_REF] Durrett | Some multidimensional rwre with subclassical limiting behavior[END_REF], for completness we recall the main steps. The first equality and inequality is a direct consequence of what we recalled above, notice that

For the third inequality, we have

π(z q-1 , z q )(g k (z q ) -g k (z q-1 )) 2 ≥ inf 1≤q≤m (π(z q-1 , z q )) m q=1 (g k (z q ) -g k (z q-1 )) 2 ≥ 1 m inf 1≤q≤m (π(z q-1 , z q )) (g k (z) -g k (y)) 2


Intersectionality: Reception in France and Findings from a study of the transition from childhood to adolescence 1

Intersectionality: its reception in France

In France, the notion of intersectionality appeared in [2004][2005], with the translation of an article by Kimberlé Crenshaw (2004) in a well-known French publication on gender studies, the Cahiers du genre. This article was defined by a number of French commentators on intersectionality as Act I of the movement, with Crenshaw in the role of the inventor of the term. The article brought about a great change in approaches to women's studies: Crenshaw attacked these studies for their ethnocentrism; feminism was nothing more than 'white' feminism which, in so far as it related to the United States, excluded consideration of the plurality of "racial" groups of American women and the situation of those who were both Black and female. In France, the article was well received and introduced the idea that gender relations should be articulated with national origin, colour and social class, that is, those other social relations referred to in the triptych "gender, class, race".

In my view, as an anthropologist, the term "race" is problematical because it creates 1 Translation from French to English by Philippa Norriss. 2 Research Program CORAGE, Experiencing Bodily Changes and Crossing Age: the case of 9 to 13 year-old Children (France -Italy), funded by the French National Agency of Research (ANR), directed by

Introduction

The notion of intersectionality is now used in many fields in sociological research in France, a sort of methodological "must" frequently invoked in research on gender studies. My paper is in two parts: the first focuses on the way the notion of intersectionality was received in France in the early 2000s and how it was used; the second is a presentation of a study on the transition from childhood to adolescence 2 that a team of researchers including myself carried out and during which we noted instances of intersectionality at play. their contribution to discussions on intersectionality; the work of contemporary French theorists, on the other hand, tends to be in line with English perspectives on the subject. Among the many publications since 2004, special mention should be made of those by Elsa Dorlin, a philosopher and specialist in the history of ideas. She has edited and annotated articles published in Black feminism (2007), thus making these available to a wide public; she has also published essays on the joint historical construction of gender and population/racerelated hierarchies through the history of colonialism and slavery from the 17 th to the 19 th century [START_REF] Dorlin | La matrice de la race. Généalogie sexuelle et coloniale de la nation française[END_REF].

3 For example the conférences Race, classe, genre comme catégories de la différence et de l'inégalité. Comment profiter des approches de l'intersection: Colloque junior du Ciera (Histoire), September 2009; À l'intersection des situations de domination: Perspectives transnationales et transdisciplinaires, Eleni Varikas (GTM/CRESPPA-CNRS, PARIS 8) and Patricia Paperman (LABTOP, PARIS 8) (Etudes politiques, Sociologie), 7-8th June 2011. 4 Cahiers du Cedref, (Ré)articulation des rapports sociaux de sexe, classe et "race". Repères historiques et contemporains. 14 |2006 ; Genre et perspectives postcoloniales 17 |2010 ; Hamel Christelle and Siméant Johanna, "Genre et classes populaires ", Genèses, 2006/3 no. 64, p. 2-4. The synthesis in French edited by Sirma [START_REF] Bilge | De l'analogie à l'articulation: théoriser la différenciation sociale et l'inégalité complexe[END_REF], a Professor of sociology at the University of Montreal, made a significant contribution to thinking on theory and methodology. She identifies three major approaches to intersectionality in English and French speaking research circles. The first approach is analytical: one relation or term takes precedence over the other; this other term is presented as an extension, or development of the first one and so is soluble in it. For instance, "racism is an extension of sexism". The second approach to intersectionality is from the arithmetic perspective. Here, intersection is presented either as an addition or as a multiplication of distinct forms of domination, the number of which can be infinite (Bilge, ibidem). For example, addition is involved when discrimination by gender is compounded by discrimination by national origin (female immigrants from North Africa); multiplication when interaction between axes of domination occurs (this is frequently expressed as "reciprocal attachment"). Third, according to Bilge, there is a more holistic approach to the interweaving of social relations and power which demonstrates how categories can be constructed together. Elsa [START_REF] Dorlin | La matrice de la race. Généalogie sexuelle et coloniale de la nation française[END_REF], whom we mentioned earlier, adopts this approach. She shows how, historically, the construction of the concepts of race and gender emerged in parallel, in other words, when a hierarchy of gender was developed alongside a hierarchy of populations: in the 19 th century, the body of a Black was thought of as pathogenic -like the body of a woman -and so Blacks were considered to be naturally inferior to white Europeans.

The large number of publications and conferences in France on intersectionality since 2004 shows that researchers have been keen to find a frame of analysis that would both articulate and bring together otherwise disconnected research work conducted during the 1990s on the subject of gender relations, social class, racialization and ethnicization. The notion of intersectionality came at just the right time for thinking forms of inequality in their complexity. But a review of works, conferences and seminars over the last five years shows that research groups today invest the term with different meanings and for different purposes.

First, some research groups emphasize the forms of oppression women are subjected to, particularly in the context of globalization, and how women distance themselves from these through engaging in social movements; others expose the oppressions that come from linking sexuality and race, such as the discriminations experienced by black homosexual males. These groups are in the tradition of critical feminism. On this point, it is interesting to note that S. [START_REF] Bilge | Intersectionality Undone; Saving Intersectionality from Feminist Intersectionality Studies[END_REF] should recently have published an article criticizing the depoliticization of the concept of intersectionality. Other research groups are highly critical of the notion: they contend that, in wanting to capture the plurality of power relations populations find themselves involved in, intersectionality completely effaces or dilutes those fundamental power relations of social class and gender domination 5 . Other publications based on empirical field studies on migrant, transnational populations in particular, have provided insights into those forms of categorization and discrimination certain populations are subjected to, and how individuals identify with categories of gender, and ethnic group according to the situation they find themselves in. They have also shown how negatively-viewed populations (women, migrants, those in employment low down in the social hierarchy) also have strategies and are not simply the "victims" of power systems: as, for example, in the case of female childcareers from the Philippines in Europe [START_REF] Fresnoza-Flot | Mères migrantes sans frontières[END_REF] or of transnational African trades-women [START_REF] Catarino | Femmes, genre, migration et mobilités[END_REF]. A number of sociologists have emphasized that social relations are not limited to the triptych "gender, race, class", and that to these should be added hierarchies by age and generation, which are also naturalized social orders [START_REF] Achin | Age, intersectionnalité, rapports de pouvoir. Table ronde avec Christelle Hamel[END_REF][START_REF] Legrand | Genre et parcours de vie : vers une nouvelle police des corps et des âges ? Nancy: PUN[END_REF][START_REF] Diasio | La préadolescence existe-t-elle?[END_REF].

The main thrust of the conference we organized in June 2015 was to define methodology for observing and reporting back on the interconnexions between power relations that take all these elements into account 6 . Our research on preadolescence was conducted with the same end in view.

A study on the transition from childhood to adolescence

In a four year study that a team of researchers and I carried out from 2010 to 2013 on discourses and practices on the subject of preadolescence, the intersections of gender, racialization and age were observed. Today, preadolescence is considered to be a period of transition between childhood and adolescence; we set out to examine attitudes towards this period and how it is handled on three different levels of involvement: one team 7 focused on the discourse and practice of educational professionals and P.E. teachers; the second 8 analyzed a corpus of medical literature and the discourse of practicing professionals on puberty, more particularly on the issue of early onset of puberty; the third team 9 concentrated on the children themselves and their parents.

Our research was based on a qualitative methodological approach which included: semistructured interviews with the children (N=69), the parents, educational professionals (N=40), school doctors and nurses, family doctors and pediatricians (N=52); ethnographic observations within the families concerned and in the areas where the children were active (in the neighborhood, the garden and the street), and an analysis of medical and epidemiological literature on puberty (N=110 articles).

The question of intersectionality appeared at several levels during our study:

-in discourses on and representations of children -in the ways children construct their identity -in the interactions between the children themselves -in the way the researchers themselves categorize children.

In this article, I shall focus on the first two points. We examine these issues from the double perspective of materialist gender theories (Guillaumin, 1992;Mathieu, 1991;Tabet, 1979) and English interactionist theories. These latter take their inspiration from the approaches to borderwork developed by Frederick Barth (1969). They pointed to the similarities that exist in the way borders of age, gender and ethnic grouping are produced. Here, I shall draw more particularly on the work of Barrie Thorne (1987;1993;2004). In her main book, Gender play: Girls and boys in school, she shows how divisions by gender, age, country or origin and ethnic grouping are continuous, active processes that can be observed in given situations: in the practices of children (who plays with whom in the yard or in the neighborhood), and in speech ("Dolls? They're for girls"; "Boys are stronger"), words that label groups of individuals by gender or age or colour. My first example is taken from an analysis of a corpus of medical literature on the notion of precocious puberty among Black girls. The second focuses on the discourse of community workers on the children they supervise; the third and last is on the ways children identify with an age or gender, or cultural or ethnic differences. 9 Nicoletta Diasio, Marie-Pierre Julien, Simona Tersigni, Louis Mathiot and myself.

Representations of young Black girls in medical discourse 10

My study focused on a medical controversy surrounding the early onset of puberty -more particularly, early breast development-American researchers had observed, among Afro-American girls. (Hermann-Giddens et al., 1997). The Hermann-Giddens findings were well received by a number of French doctors and transposed to France.

An American study published in 1997 showed that there had been a significant drop in the age of onset of puberty among girls since the 1960s (using the Marshall and Tanner scale (1969) breast development was put at 11,5 years and menarchy at 13,5). The study was given wide coverage in the English and French speaking worlds and in medical journals, but the subject was highly controversial. For example, later French and Danish articles on the issue (Gaudineau et al., 2010) pointed out that the American studies did not apply in the European context. Even so, other French specialists made indiscriminate use of the findings in the articles they put out on internet, thereby encouraging the emergence of stereotypes that came from linking gender, age and racialization.

The article entitled "La puberté avant l'âge. Nouveaux aspects" (Puberty before its time. New aspects) (Pienkowski & Granjean, 2008) and put out on internet by the Toulouse Children's Hospital is a case in point. We read: "If the age at which Caucasian girls have their first period seems to have stabilized in the mid-20 th century, different studies carried out since the 1970s show that the first signs are appearing earlier and earlier". (ibidem, 6). The article made no mention of discussion of the issue in America, or to the dietary and social differences between the United States and France; it simply took as given the findings of the Hermann-Giddens study (1997); it spoke of a "racial difference" with no critical regard for what the notion might imply. The reference to "race" is repeated several times: "There is hardly any difference between the races for the age at which periods first occur" (ibidem, 13) and "In Europe, for Caucasians, discounting exceptional cases, the lower age limit of puberty is put at 8 for girls and 9 for boys" (ibidem).

In another article, the notion of ethnic group is evoked in relation to those of African origin: "What is new", says the article, "is the physiological variability and the many factors influencing the age of onset of puberty. A drop in the age of onset over the last 100 years in the industrialized countries (…); ethnic variations (precocious puberty among girls of African 10 Cf. [START_REF] Cozzi | Risky, early, controversial. Puberty in medical discourses[END_REF]. origin (…)" (Cartauld, Edouard, Pienkowski & Tauber 2008, 1311). But we know, first, that the African ethnic group does not exist -there are many ethnic groups in Africa -and second, that in the United States (the study on puberty originated there), division by colour of the skin, a legacy of the slavery era, is based on social hierarchies that come from naturalizing intellectual and physical skills and associating these with skin colour [START_REF] Pap | Questions de couleur. Histoire, idéologie et pratiques du colorisme[END_REF][START_REF] Bosa | Plus Blanc que blanc. Une étude critique des travaux sur la whiteness in Fassin, D. Les nouvelles frontières de la société française[END_REF] 11 . Today, at census time, individuals define themselves in terms of ethnic group (Latinos, Indians) or racial group (black, white, half-caste). It is thus impossible to transfer these racialized or ethicized American social groupings to European populations. The notion of Afro-Americans is not relevant to France. Even so some practicing professionals have transposed it, replacing it by the term "black girls", or the more politically correct expression "girls of African origin," in spite of the fact that the Afro-American populations studied by American epidemiologists today have little in common with those African groups who, in many cases, have recently emigrated to France and Europe generally.

This confusion was apparent in what an endocrinologist I interviewed said: "What I mean is, the onset of puberty will always occur earlier among black girls than among young whites. So it also depends on the racial context." But when I asked him if girls living in Africa also experienced early onset of puberty he said "No", and seemed confused by the distortions that arose when relating the notion of race to place of residence, social conditions of living and diet.

In intersectional terms this analysis shows that certain medical representations are based on stereotypes linking race, gender and age: the Black girl is tall, with developed breasts, and sexually mature at an early age. This figure is a legacy of the eroticized, sexually attractive black woman, and is as much in evidence in French colonial literature [START_REF] Dorlin | La matrice de la race. Généalogie sexuelle et coloniale de la nation française[END_REF][START_REF] Boëtsch | Corps mauresques[END_REF]Tarraud, 2003) as in American pro-slavery literature.

Intersectional stereotypes on the children are observed too in the discourse of other professionals: community workers.

Boys and girls from the housing estates

11 Persons with the darkest skin found themselves at the bottom of the social hierarchy and were relegated to the hardest work up until the middle of the 20 th century only Anglo-Saxons were considered as Whites [START_REF] Pap | Questions de couleur. Histoire, idéologie et pratiques du colorisme[END_REF][START_REF] Bosa | Plus Blanc que blanc. Une étude critique des travaux sur la whiteness in Fassin, D. Les nouvelles frontières de la société française[END_REF]. Categorization by colour of the skin continues to stratify American society but American racial groupings are presented as social entities not as biological or genetic types.

This example is based on interviews, observations and publications made by my colleague

Ingrid Voléry (2014Voléry ( , 2016)). Two expressions are used by community workers for distinguishing children who are difficult to work with from those who are not. These figures are used when the social workers speak of children living in areas with a large immigrant population. The terms in French are the "caïd de cite" and the "fille-de-quartier" which in substance mean, respectively, "the gang leader", and "the girl from the housing estate".

The community worker speaks of the boys from these areas as if they had a form of sexualization different from other boys. Indeed, in our study, generally, the adults -and this includes doctors -say that girls grow up more quickly than boys (which echoes what we said earlier on puberty). However, when talking of the boys from the estates, the community workers say that sexualized behavior begins very early, sometimes at the age of 8. They frequently talk of boys who have come to be sexually active through access to pornographic material. When talking of 12 year-olds, Maxime, a 26 year-old community worker said: "The problem you can come up against is unrestrained sexuality and then you've got no choice, which means you talk with them to try and get them to understand things like self-respect and so on." (Voléry, 2014, 85). These young boys are also characterized by the difficulty they have in controlling their bodies for example, controlling their emotions or channeling their energy: "Actually, it's a bit like young people in the disadvantaged areas. You get the impression that they can't contain their energy, but as soon as you give them a tip or two, they find they're able to direct it!" (ibidem, 86). These boys are considered to have been badly brought up, to be incapable of respecting the public and the private spheres alike; their parents are considered to have failed to educate them in exercising self-control.

My colleague Ingrid Voléry and I think that the intersection of age, gender, ethnicization stereotypes comes into play here: the young boys of immigrant descent are thought of as being uncivilized 12 ; they are thought to move on from childhood earlier than other boys.

The way the community workers consider the girls is totally different. In contrast to those popular images constructed around the figure of the young North African female immigrant (alluring, wild, loud-mouthed), the community workers in our survey spoke of the girls they knew as discreet and reserved. First, because they reveal their bodies with restraint -that is, in conforming to the norms of gender and age the professionals live by. The girls from the estates are said to be serious and well within the age norms of growing up to womanhood.

They stand in contrast to the "Lolitas" who are often represented as girls from the lower social classes (cf. infra).

The two figures used when referring to young people from these housing estates are stereotypes that do not take into account the diversity of girls and boys of immigrant families living there. The community workers have constructed norms of what girls and boys should be at a given age. These reflect the middle-class social norms of the native population: thus, young boys will be virile but not violent; young girls, reasonable, measured, tempered.

Intersectionality in identification and labelling

My last example is on the ways children identify with an age, gender, cultural or ethnic group in a given time and place, and on how they are labelled by other children using the same criteria.

First, the case of Simon 13 . Simon (10 years old) lives in a socially mixed suburb of Strasbourg. When the interviewer first went to see him, he demonstrated his culinary skills by preparing and serving a chicken-based Mauritian dish. This visit was the occasion for him to explain that his grandparents on his father's side are, like his great-uncle, all Mauritian, but that his father and mother are French. His mother (divorced) says that Simon likes going to see these relations: "They are nice and friendly. They have also brought him close to the Mauritian culture and I think he needs that."

In everyday life, such as we were able to observe and also from what his mother said, Simon has many friends and extra-curricular activities. At primary school and at secondary school, he was a good pupil (in primary school he even skipped a class). His mother describes him as being in advance for his age, both physically and intellectually, and in terms of independence.

His friends are the boys in his class, French, and -because of his skipping a class -often a year older than him. He does not like girls because, as he says, "they're silly." His mother says that at primary school he was the object of racist talk, linked to the colour of his skin, and so she chose to send him to a private secondary school where he has not experienced the same problems. This school is a selective one; the children, like Simon, are mainly good pupils and come from middle or upper-middle class families.

In terms of intersectionality and borderwork, the elements used for identifying and labeling Simon are a function of the social situation (Barth, 1969). At school and in his extra-curricular activities, Simon identifies himself with boys and insists on maintaining gender borders by downgrading girls of his age as when he says "they're silly." In the private sphere, he demonstrates his identification with the Mauritian culture through his interest in cooking and his attachment to his grandparents. Being intellectually and physically advanced for his age, and given he shows independence, he manifests his desire to identify with children a little older than him. The educational establishment endorses this when it invited him to skip a class in primary school. But, at the primary school, in a socially and ethnically mixed district, some children erected colour borders when, through their racist comments, they made him conscious of the colour of his skin. This case study points to the need to make situation by situation observations of the production of intersectional identification or labelling by gender, age and colour of the skin.

From other cases in our study we found that foreign children who had recently settled in France were also labelled; skin colour was not the issue here. For example, Louis (13 years old) is from Chechnya and had been living in France for two years when he was interviewed.

He said he had grown a lot since arriving in France; now he was 1,75 m tall. He told us that the boys in his class made fun of him because he was -in their words -"too" tall and developing facial hair. Another case, analyzed by Simona Tersigni (2014), was a Philippine girl who was 11 and in a class of ten year-olds. She was considered to be under-height, a dwarf, even though she was not the smallest in the class.

These features (being over-height, under-height, over-developed or using too much make-up) frequently appeared in the interviews with children who were quick to label other children who did not quite correspond to the average norm for a particular age group. But what is striking here is that these marks of differentiation and stigmatization are used in relation to foreign children: as if there were a need to find in those children's bodies differences leading to their exclusion.

In the children's discourse, and even more in that of the parents, a negative figure was frequently mentioned: that of the bimbo or the "Lolita" type. One characteristic of these young girls (in the discourse of the other children and their parents) is that they are taller on average than the others. Generally speaking, too, pubertal physical changes are apparent in these girls, particularly in terms of breast development. They wear heeled shoes that some call "stilettos" and then add: "Well, that's maybe an exaggeration." They are said to use too much make-up and pay too much attention to the kind of clothes they wear. They have often redone a year at school and so are older that most of the other girls in the class; they are looked up to by the younger girls, which stands in contrast to the institutional sanction. The other children, particularly the smaller and younger girls, and the same goes for adults, too, see in these taller girls who adorn themselves with the marks of adult femininity (high-heeled shoes, make-up) persons who transgress the codes of childhood and femininity during the age of childhood. This stereotype, which intersects gender and age, stands in contrast to the discreet girl mentioned earlier.

Boys, on the other hand, are not stigmatized for their over-masculinity. When boys talk of other boys, censure by social class comes into play: very low-slung "baggy" pants are considered by boys from the central areas to be yobbish (the term has ethnic undertones and opposes the middle-class town center to the suburbs/housing estates where a large proportion of descendants of immigrants live); the nicely dressed boy is "chicos", that means smart, classy, bourgeois. But, neither the children nor their parents mention sexualization or overmasculinity when talking about boys: an illustration of the fact that, on such gender-related issues, girls are negatively viewed, more readily criticized, stigmatized than boys of the same age.

These cases studies show that intersectional elements (gender, class, colour of the skin, ethnicity) are not mobilized in all social situations. Conversely, a particular factor may not be present in some situations but serve to stigmatize in others. The sociologist has to analyze each social situation and how instances of intersectionality are at stake.

In conclusion, the notion of intersectionality opens up the field for further research and reflexion. In my view, it should be factored into sociological and anthropological studies on different counts: first, because, as our study shows, intersectional elements in discourse on individuals are present in instances of labelling; second, because through observing the ways people construct their identity as a process in a social situation we can see how intersection is at play in the interactions between the actors themselves (who is friends with whom, and who exclude whom); finally, because intersectionality provides a frame for analyzing the way researchers themselves categorize people. The study of preadolescence we carried out shows that we need now to identify those social situations in which the qualifiers gender, age and colour, or social and cultural difference are mobilized: in other words, determine those elements which individuals use, and in what situations, to construct the "us" group and the "them" group -those we do not associate with, those we stigmatize, exclude, downgrade.

Detecting intersectional relationships at play can be usefully applied in schools. Future teachers would do well to espouse the notion. This would enable them to override prejudices and promote equality precisely through being able to pick up on those expressions of discrimination that intersect the elements of gender, age, colour, origin.

  University of Franche-Comté, Laboratory of Sociology and Anthropology, University of Strasbourg, Centre conflit, corps, créativité, and University of Lorraine, Laboratory of Social Sciences Lorraine, 2 nd of June 2015. 7 Ingrid Voléry, Caroline Hérasse, Benoît Dejaiffe and Simona Tersigni. 8 Donatella Cozzi, Nicoletta Diasio and myself.

5 For example: Comment le genre trouble la classe, Agone 43, June 2010. 6 Conference Age, genre, corps: approches empiriques et entrecroisements des rapports sociaux [Age, gender, body: field work and intersectionality],

It reminds a former French minister in 1999 talked of boys from the suburbs as "sauvageons" id. est "savages".