Rogier Bos
email: r.d.bos@uu.nl

Action-property duality in embodied design

Keywords: Embodied design, Operational-structural duality, Action-property duality

For those working on embodied design it is a challenge to create tasks that enable students to develop abstract mathematical concepts. We approach this issue from the perspective of Sfard's notions of saming, encapsulation, and reification. We discuss a duality of properties and actions, and use this duality to review saming, encapsulation and reification from an action-and perceptionbased perspective. To illustrate the power of this theoretical contribution we discuss one new embodied task design and two from literature: MIT-P for proportion and a design for the gradient of a plane using the Augmented Reality Sandbox.

Introduction

In her 1991 paper on operational and structural conceptions in the formation of mathematical concepts Sfard writes that "we have good reasons to expect that in the process of concept formation, operational conceptions would precede the structural" (1991, p.10). Research into action-based embodied design seems to support and exploit this view that operations -in the form of goaloriented actions that develop in the context of motor problems -form a ground for developing mathematical concepts [START_REF] Abrahamson | The Future of Embodied Design for Mathematics Teaching and Learning[END_REF]. Operational-structural theory and Abrahamson's embodied design theories share a central role for the transitions from process to object, and the aim of this paper is to study in more depth how a further application of ideas of operational-structural theory can inform embodied design. In particular, we are interested to see how the terminology of object formation-saming, encapsulation, and reification-as it evolved in Sfard's later work (2008), apply to the context of embodied design. To this purpose we write about action-property duality, a duality we believe to be at the heart of students' discovery and development of new mathematics in embodied designs. Whereas in Sfard's later work emphasis lies on how the development of saming, encapsulation, and reification take place in communication-in the introduction of new discourse, through signifiers, like nouns-we would like to draw attention to how these developments take place and are observable in students' non-communicative actions in embodied learning environments, in particular their interaction with artifacts [START_REF] Shvarts | Embodied instrumentation in learning mathematics as the genesis of a body-artifact functional system[END_REF].

In the next section we present a theoretical perspective on operational-structural theory and embodied design, immediately adding our view on action-property duality. In this section these theoretical ideas are illustrated a new embodied design for studying quadrilaterals. The next section illustrates how the theory applies to embodied designs in two studies, a well-known example from design for the concept of proportion, the MIT-P [START_REF] Abrahamson | Making sense of movement in embodied design for mathematics learning[END_REF], and a more recent embodied design, using augmented reality, for the equation of a plane in a three-dimensional coordinate system and its relation to the gradient vector [START_REF] Bos | Embodiment through augmented reality: the case of the gradient[END_REF].

Theoretical contribution and background

Action-property duality

Let us postulate what we mean by a property in relation to actions:

A property is an invariant under constrained actions

The invariance means a property is perceived to be the same before and after (and possibly during) the constrained action. Trivial but crucial to us, a property is only invariant under actions that maintain the invariance. For example, an angle is only invariant under angle-preserving actions.

There is an interesting duality in this formulation: properties and invariant-preserving actions cannot exist without each other. A naïve perception of sameness underlies the ability to discern a property. Our main claim is that such a naïve sense of sameness co-develops with an ability to maintain the property. To be able to compare the sameness across instances of the property in different objects, transformations of one object to the other need to be performed or imagined. In short, action-property duality refers to the phenomenon that the sameness with respect to a property across objects can only be perceived by transforming those objects into each other while maintaining the property either physically or in the mind's eye.

To make sense of this, let us look at the example of the angles of polygons. Suppose a student is invited to manipulate a polygon by dragging the corners. A naïve perception of angle depends on a naïve perception of the sameness of angles, since different instances of, e.g., straight angles need to be recognized as the same. As a consequence, a naïve perception of the sameness of angles must codevelop with a naïve ability to maintain an angle while transforming one polygon into another.

Operational-structural development

A theory of operational-structural development within mathematics education developed in the 90s within the framework of traditional cognitive psychology. Sfard argues that operational understanding precedes structural understanding of mathematical concepts (1991). She describes three stages in object/concept development: interiorization, condensation, and reification. Later, Sfard elaborated her perspective on operational-structural issues within the commognitive framework ( 2008). In the first two columns of Table 1 we present this later view on object formation (cf. Sfard, 2008, p.170). We do not intend to embrace or study the whole theory of commognition here, but find its description of saming, encapsulation and reification most suited to apply in the embodied design context.

A role of the operational-structural perspective in embodied design

Over the last 20 years, ideas from the psychological theory of embodied cognition have gained currency in mathematics education research, see [START_REF] Abrahamson | Embodiment and embodied design[END_REF]) and references therein. Embodied designs allow students to develop mathematical concepts from naïve perceptions (perception-based designs) or actions (action-based designs) in embodied learning environments [START_REF] Abrahamson | The Future of Embodied Design for Mathematics Teaching and Learning[END_REF]. In perception-based design students are challenged to use their innate perceptive qualities to observe certain events with potential mathematical meaning. Similarly, in action-based designs students are challenged to solve a problem of motor control with potential mathematical meaning. These naïve perceptions and actions are then developed into more robust mathematical concepts with the guidance of a tutor, thus grounding the meaning of the concepts in embodied (perceptive and motoric) experiences [START_REF] Flood | Teaching with embodied learning technologies for mathematics: responsive teaching for embodied learning[END_REF].

Returning to operational-structural development, as presented in the second column of Table 1, we argue, firstly, that saming, encapsulation, and reification are not exclusively revealed through communicative acts, but additionally through non-communicative actions: the way artifacts are handled in an embodied learning environment. This point of view is elaborated in the third column in Table 1. Below we present support for the idea that students' actions evidence stages of saming, encapsulation, and reification, before those stages are communicated through speech or gesture.

Secondly, from an action-property duality perspective, we argue that saming, encapsulation, and reification can be interwoven in embodied design. Motoric fluency in action-based design indicates that the series of necessary actions to solve the motor problem is perceived as part of a single process. These transformations contribute to the discovery of a property (of a new object) and hence contribute to a process of saming and encapsulation. As a consequence, development towards saming/encapsulation and reification are made simultaneously; this rephrases the idea of action-property duality within operational-structural perspective. This way action-based design offers an opportunity for the simultaneous development of a new object and the associated constrained actions (transformations).

To illustrate this let us look again at quadrilaterals. We developed a task series in which a student is invited to move similar quadrilaterals on a multi-touch screen on top of each other by dragging the corners with their fingers (see Figure 2). The corners must be moved independently but simultaneously by four fingers. Moreover, the similarity of the quadrilaterals must be maintained while moving -this is supported by color feedback as in MIT-P. Recognizing similarity of types of quadrilaterals relies on recognizing the similarity of angles and proportions of side lengths. This, in turn, co-develops with the ability to mentally or physically transform one quadrilateral onto/into the other while maintaining those properties. Movements that maintain similarity are turning, dragging, and mirroring. While students try to "same" similar quadrilaterals, they inevitably stumble upon those transformations as naïve actions. Naturally adaptive motor control might lead students to develop distinguishable fluent transformations that could be developed into more rigorously mathematical concepts of rotation, translation, and reflection. This illustrates the main point of how new objects and the associated transformations potentially codevelop in a saming-task.

Examples of embodied designs: the role of transformations

In this section we present two examples of the role of the action-property duality and the operational-structural development in embodied design. We emphasize how fluent motor-action could be reified into mathematical transformations (seen as objects).

Example 1: embodied design for proportion A well-studied example of embodied design is the action-based task for proportion based on dragging two vertical bars [START_REF] Abrahamson | Making sense of movement in embodied design for mathematics learning[END_REF]. The student is encouraged to find positions where the heights of the bars are in a fixed proportion (e.g. 2: 3) by receiving green feedback when such a position is achieved, changing to red if the heights are not according to this proportion. Once a position has been achieved the student is invited to move the bars in a way that maintains the green feedback. In our interpretation from a procedural-structural perspective the student is hence invited to perform constrained transformations on the system of two vertical bars. The outcome is not only a naïve conception of proportion, but also a naïve conception of those transformations that leave a proportion invariant. [START_REF] Flood | Teaching with embodied learning technologies for mathematics: responsive teaching for embodied learning[END_REF] report on a student, Ben, who arrived at a solution where the bars move with constant, but different speeds. We infer that Ben not only explored the invariant proportion, but also the actions that maintain the invariant. The latter could be reified into the notion of geometric vertical multiplication, i.e. the transformation that leaves the proportion of height invariant (see Figure 3). In particular, Ben established how (what we call) vertical multiplication has properties different from vertical translation, a transformation that does not leave the proportion of heights invariant. This transformation alludes to the property of proportional variables that, increasing one variable with a factor, the other must increase with the same factor. In general, this again illustrates how the process of saming situations of two bars' heights, the defining property of proportion codevelops with an ability to perform fluent vertical transformation, which could be reified into a mathematical notion of vertical multiplication.

A transformation can also be associated with the eye movements from one bar to the other. For the task to make any sense the bars need to be considered 'the same' by the student: There must be reason to compare the heights. Some students tend to focus on an imaginary diagonal line between the tips of their two hand dragging the tops of the bars, a so-called attentional anchor [START_REF] Abrahamson | Making sense of movement in embodied design for mathematics learning[END_REF]. We argue that this diagonal line is a naïve conception that could (or even should) be reified into a more formal mathematical notion: the transformation of one bar into the other through enlargement with respect to a point (see Figure 4). The diagonal line is the essential ingredient for this transformation. This transformation alludes to the property of proportional variables 𝐻𝐻 1 ~𝐻𝐻2 that one is a multiple of the other: 𝐻𝐻 1 = 𝑐𝑐 𝐻𝐻 2 . So, yet another process of saming leads to a potential transformation to be reified.

Figure 2 .

 2 Figure 2. Action-based task: saming flexible quadrilateral by dragging four corners

Figure 3 .

 3 Figure 3. Vertical multiplication as a transformation that leaves the proportion of heights invariant

Table 1 : Operational-structural development from a commognitive and an embodied/action-based perspective Commognitive (Sfard, 2008) Embodied: Perception-Action Saming

 1

	Example: two different squares are manipulated

Creating a subsuming discourse on hitherto unrelated objects with the help of a single signifier. Example: "This is a square and Hitherto unrelated objects are perceived as similar and acted upon in similar ways.

Table 2 . Action-property duality for two properties that constitute the gradient vector qualitatively

 2

	Action	Invariant (property)	Property gradient	Changing
	Rotation round	Steepness, angle of	Length	Direction of steepest ascent,
	vertical axis	imagined triangle		direction of rolling marble
	Rotation round	Direction of steepest ascent,	Direction	Steepness, angle of imagined
	horizontal axis	direction of rolling marble		triangle, speed of marble

Acknowledgment

I would like to thank Anna Shvarts for many discussions and the organization of the Embodied Instrumentation Group meetings at the Freudenthal Institute. The ideas in this paper originate from the inspiring conversations on embodiment with all participants.

Example 2: embodied design for the gradient of a plane

In a recent study the author and collaborators investigated embodied design tasks for plane equations (𝑧𝑧 = 𝑎𝑎 𝑥𝑥 + 𝑏𝑏 𝑦𝑦 + 𝑐𝑐) and the relation to the gradient vector (𝑎𝑎, 𝑏𝑏) [START_REF] Bos | Embodiment through augmented reality: the case of the gradient[END_REF]. The embodied learning environment consisted of an Augmented Reality Sandbox (ARSB) -cf. https://web.cs.ucdavis.edu/~okreylos/ResDev/SARndbox/ -together with some plastic planes and bamboo sticks (see Figure 5). The ARSB consists of a box of sand of area roughly 0,6m 2 , a stereo camera and a projector. The stereo camera captures images of the height of the sand and objects in the box, and the projector projects height lines and color feedback onto the sand and objects accordingly.

The first few tasks of the teaching sequence aim for students to discover the relations between the parameters 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 in the equation 𝑧𝑧 = 𝑎𝑎 𝑥𝑥 + 𝑏𝑏 𝑦𝑦 + 𝑐𝑐 and the transformations rotation parallel to the 𝑦𝑦-axis, rotation parallel to the 𝑥𝑥-axis, and translations. In this paper we would like to highlight the next task in the sequence. This task aims to support development of the meaning of the gradient vector in a qualitative way. A vector is defined by its two properties: direction and length. In the case of the gradient vector these properties correspond to direction of the steepest ascent, and to the steepness, respectively. Applying the action-property perspective, each property has associated transformations that leave the property invariant. In the case of direction these are translations and rotations around an axis perpendicular to the direction; in case of steepness these are translations and rotations around a vertical axis. The task is divided into two sub-tasks accordingly: (1) move a circular plane in a way that the direction of the height lines stays the same, but the distance varies; meanwhile, roll a marble down the plane in several positions and try to explain the rolling direction;

(2) do the same, but keep the distance between the height lines the same, and explain the speed of the marble. Observing the marble adds a perception-based element to these action-based tasks. Below is an excerpt from a dialogue between tutor Rogier and student Tiago, the case student in the study presented in (Bos et al., in press). The fragment begins while Tiago is working on sub-task 2.

1 Tiago Like this you get [it] too, and then you can make all those movements again" [Tiago refers to horizontal and vertical translations] but then, in any case, it will remain just as steep. [Then Tiago begins to rotate the plane round the place where it touches the sand (see Figure 3 Tiago fluently performs a series of actions he calls rotation (line 4), "saming" the plane positions with equal steepness. In line 1 Tiago mentions that during his actions the plane "remain just as steep". This is the first time he associates this adjective "steep" to the plane as a property: one of the two which define the gradient vector. Moreover, Tiago's explanation in line 6 and remarks earlier in course of the experiment suggest he perceives an imagined triangle as depicted in Figure 4 as an attentional anchor, where the angle between the diagonal on plane and the horizontal line on the sand is a measure of steepness. In the light of Table 1 it is important to stress how the introduction of the noun "steep" (commognition perspective, column 2) is preceded and accompanied by the development of fluent action to solve the motor problem and a perception (attentional anchor) that facilitates this action (action-based perspective, column 3).

Table 2 summarizes the action-property duality for the two properties that constitute the gradient vector. The table highlights how each property is closely connected to an action that can be reified into a transformation as an object. We observed how the actions described in the first column are performed as single process with a clear goal, e.g. in line 6 during rotation "I keep it equally steep". The reification of the object "gradient vector" goes hand in hand with the reification of rotation round a vertical axis as a "keeping equally steep"-action and rotating round a horizontal axis as a "keeping same direction"-action.


Ë L Nathana

Christophe Enriquez
email: nenriquez@u-paris10.fr

Olivier Sabot
email: sabot@math.univ-lyon1.fr

Zindy

LIMIT LAWS FOR TRANSIENT RANDOM WALKS IN RANDOM ENVIRONMENT ON Z

Keywords: 2000 Mathematics Subject Classification. primary 60K37, 60F05; secondary 60F17, 60E07, 60E10 Random walk in random environment, stable laws, fluctuation theory of random walks, Beta distributions

We consider transient random walks in random environment on Z with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level n converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.

Introduction

One-dimensional random walks in random environment to the nearest neighbors have been introduced in the sixties in order to give a model of DNA replication. In 1975, Solomon gives, in a seminal work [START_REF] Solomon | Random walks in a random environment[END_REF], a criterion of transience-recurrence for these walks, and shows that three different regimes can be distinguished: the random walk may be recurrent, or transient with a positive asymptotic speed, but it may also be transient with zero asymptotic speed. This last regime, which does not exist among usual random walks, is probably the one which is the less well understood and its study is the purpose of the present paper.

Let us first remind the main existing results concerning the other regimes. In his paper, Solomon computes the asymptotic speed of transient regimes. In 1982, Sinai states, in [START_REF] Sinai | The limiting behavior of a one-dimensional random walk in a random medium[END_REF], a limit theorem in the recurrent case. It turns out that the motion in this case is unusually slow since the position of the walk at time n has to be normalized by (log n) 2 in order to present a non trivial limit. In 1986, the limiting law is characterized independently by Kesten [START_REF] Kesten | The limit distribution of Sinai's random walk in random environment[END_REF] and Golosov [START_REF] Golosov | Limit distributions for random walks in random environments[END_REF]. Let us notice here that, beyond the interest of his result, Sinai introduces a very powerful and intuitive tool in the study of one-dimensional random walks in random environment. This tool is the potential, which is a function on Z canonically associated to the random environment. It turns out to be an usual random walk when the transition probabilities at each site are independent and identically distributed (i.i.d.).

Let us now focus on the works about the transient walk with zero asymptotic speed. The main result was obtained by Kesten, Kozlov and Spitzer in [START_REF] Kesten | A limit law for random walk in a random environment[END_REF] who proved that, when normalized by a suitable power of n, the hitting time of the level n converges towards a positive stable law whose index corresponds to the power of n lying in the normalization. Recently, Mayer-Wolf, Roitershtein and Zeitouni [START_REF] Mayer-Wolf | Limit theorems for one-dimensional transient random walks in Markov environments[END_REF] generalized this result to the case where the environment is defined by an irreducible Markov chain.

Our purpose is to characterize the positive stable law in the case of i.i.d. transition probabilities. Let us mention here that the stable limiting law has been characterized in the case of diffusions in random potential when the potential is either a Brownian motion with drift [START_REF] Kawazu | A diffusion process in a Brownian environment with drift[END_REF], [START_REF] Hu | Rates of convergence of diffusions with drifted Brownian potentials[END_REF] or a Lévy process [START_REF] Singh | Rates of convergence of a transient diffusion in a spectrally negative Lévy potential[END_REF], but we remind here that despite the similarities of both models one cannot transport results from the continuous model to the discrete one.

The proof chooses a radically different approach than previous ones dealing with the transient case. The proofs in [START_REF] Kesten | A limit law for random walk in a random environment[END_REF] and [START_REF] Mayer-Wolf | Limit theorems for one-dimensional transient random walks in Markov environments[END_REF] were mainly based on the representation of the trajectory of the walk in terms of branching processes in random environment (with immigration). This encoding was also used by Alili [START_REF] Alili | Persistent random walks in stationary environment[END_REF] in its study of transient persistent random walks in random environment having zero asymptotic speed. In contrast with these works, our approach relies heavily on Sinai's interpretation of a particle living in a random potential. However, in the recurrent case, the potential one has to deal with is a recurrent random walk and Sinai introduces a notion of valley which does not make sense anymore in our setting where the potential is a (let's say negatively) drifted random walk. Therefore, we introduce a different notion of valley which is closely related to the excursions of this random walk above its past minimum. It turns out that a result of Iglehart [START_REF] Iglehart | Extreme values in the GI/G/1 queue[END_REF] provides the asymptotic for the distribution of the tail of the height of these excursions. Now, as soon as one can prove that the hitting time of the level n can be reduced to the time spent by the random walk to cross the high excursions of the potential above its past minimum, between 0 and n, which are well separated in space, an i.i.d. property comes out, and the problem is reduced to the study of the tail of the time spent by the walker to cross a single excursion.

It turns out that the distribution of this tail can be expressed in terms of the expectation of the functional of some meander associated with the random walk defining the potential. Now, this functional is itself related to the constant that appears in Kesten's renewal theorem [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF]. These last two facts are contained in [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF]. Now, in the case where the transition probabilities follow some Beta distribution a result of Chamayou and Letac [START_REF] Chamayou | Explicit stationary distributions for compositions of random functions and products of random matrices[END_REF] gives an explicit formula for this constant which yields finally an explicit formula for the parameter of the positive stable law which is obtained at the limit.

The same technics also allow to derive the convergence of the normalized process to the inverse of a standard stable subordinator. This result can be compared with the scaling limits obtained for the trap model of Bouchaud, see [START_REF] Ben Arous | Dynamics of trap models, Ecole d' Éte de Physique des Houches[END_REF] for a review. Soon after finishing this article, we learnt of an independent work, by Peterson and Zeitouni [START_REF] Peterson | Quenched limits for transient, zero speed one-dimensional random walk in random environment[END_REF], which, by the study of the fluctuations of the potential, showed that a quenched stable limit law is not possible in the zero asymptotic speed regime.

The paper is organized as follows: the results are stated in Section 2, and the rest of the paper is devoted to the proofs.

Notations and main results

Let ω := (ω i , i ∈ Z) be a family of i.i.d. random variables taking values in (0, 1) defined on Ω, which stands for the random environment. Denote by P the distribution of ω and by E the corresponding expectation. Conditioning on ω (i.e. choosing an environment), we define the random walk in random environment (X n , n ≥ 0) as a nearest-neighbor random walk on Z with transition probabilities given by ω:

(X n , n ≥ 0) is the Markov chain satisfying X 0 = 0 and for n ≥ 0,

P ω (X n+1 = x + 1 | X n = x) = ω x = 1 -P ω (X n+1 = x -1 | X n = x) .
We denote by P ω the law of (X n , n ≥ 0) and E ω the corresponding expectation. We denote by P the joint law of (ω, (X n ) n≥0 ). We refer to Zeitouni [START_REF] Zeitouni | Random Walks in Random Environment, XXXI summer school in probability[END_REF] for an overview of results on random walks in random environment.

In the study of one-dimensional random walks in random environment, an important role is played by the sequence of variables

ρ i := 1 -ω i ω i , i ∈ Z.
We now introduce the hitting time τ (x) of level x for the random walk (X n , n ≥ 0),

(2.1) τ (x) := inf{n ≥ 1 : X n = x}, x ∈ Z.
For α ∈ (0, 1), let S ca α be a completely asymmetric (actually positive) stable random variable of index α with Laplace transform, for λ > 0,

E[e -λS ca α ] = e -λ α .
Moreover, let us introduce the constant C K describing the tail of Kesten's renewal series, see [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF], defined by R := k≥0 ρ 0 ...ρ k :

P {R > x} ∼ C K x κ , x → ∞. (2.2)
Then the main result of the paper can be stated as follows. The symbols " law -→" denotes the convergence in distribution.

Theorem 1. Let ω := (ω i , i ∈ Z) be a family of independent and identically distributed random variables such that (a) there exists 0 < κ < 1 for which E [ρ κ 0 ] = 1 and E ρ κ 0 log + ρ 0 < ∞, (b) the distribution of log ρ 0 is non-lattice.

Then, we have, when n goes to infinity,

τ (n) n 1/κ law -→ 2 πκ 2 sin(πκ) C 2 K E[ρ κ 0 log ρ 0 ] 1 κ S ca κ , X n n κ law -→ sin(πκ) 2 κ πκ 2 C 2 K E[ρ κ 0 log ρ 0 ] 1 S ca κ κ .
Remark 1. Note that several probabilistic representations are available to compute C K numerically, which are equally efficient. The first one was obtained by Goldie [START_REF] Goldie | Implicit renewal theory and tails of solutions of random equations[END_REF], a second was conjectured by Siegmund [START_REF] Siegmund | Note on a stochastic recursion[END_REF], and we obtained a third one in [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF], which plays a central role in the proof of the theorem.

Remark 2. We think that the method used in this paper could also treat the case κ = 1 (see Section 9 for conjecture and comments).

This theorem takes a remarkably explicit form in the case of Dirichlet environment, i.e. when the law of the environment satisfies ω 1

( dx) = 1 B(α,β) x α-1 (1-x) β-1 1 [0,1] (x) dx, with α, β > 0 and B(α, β) := 1 0 x α-1 (1 -x) β-1 dx
, things can be made much more explicit. The assumption of Theorem 1 corresponds to the case where 0 < α -β < 1 and an easy computation leads to κ = α -β.

Thanks to a very nice result of Chamayou and Letac [START_REF] Chamayou | Explicit stationary distributions for compositions of random functions and products of random matrices[END_REF] giving the explicit value of C K in this case, we obtain the following corollary: Corollary 1. In the case where ω 1 has a distribution Beta(α, β), with 0 < α -β < 1, Theorem 1 applies with κ = α -β. Then, we have, when n goes to infinity,

τ (n) n 1/κ law -→ 2 π sin(π(α -β)) ψ(α) -ψ(β) B(α, β) 2 1 α-β S ca κ , X n n κ law -→ sin(π(α -β)) 2 α-β π B(α, β) 2 ψ(α) -ψ(β) 1 S ca κ κ , where ψ denotes the classical Digamma function, ψ(z) := (log Γ) ′ (z) = Γ ′ (z) Γ(z) .
Remark 3. Our technics also allow to derive the convergence of the normalized process. More precisely, under the assumption (a)-(b) of Theorem 1, the law of the process n -κ X ⌊nt⌋ , t ≥ 0 , defined on the space of càdlàg functions equipped with the uniform topology, converges to the law of sin(πκ)

2 κ πκ 2 C 2 K E [ρ κ 0 log ρ 0 ] Z t , t ≥ 0 ,
where Z is the inverse of the κ-stable subordinator Y satisfying E[e -λYt ] = e -tλ κ , for all λ > 0. This result can be compared with the scaling limits obtained for the trap model of Bouchaud, see [START_REF] Ben Arous | Dynamics of trap models, Ecole d' Éte de Physique des Houches[END_REF] for a review.

In the following, the constant C stands for a positive constant large enough, whose value can change from line to line.

Two notions of valleys

Sinai introduced in [START_REF] Sinai | The limiting behavior of a one-dimensional random walk in a random medium[END_REF] the notion of valley in a context where the random walk defining the potential was recurrent. We have to do a similar job in our framework where the random walk defining the potential is negatively drifted.

Let us define precisely the potential, denoted by V = (V (x), x ∈ Z). We recall first the following notation

ρ i = 1 -ω i ω i , i ∈ Z.
Then, the potential is a function of the environment ω and is defined as follows:

V (x) :=    x i=1 log ρ i if x ≥ 1, 0 if x = 0, -0 i=x+1 log ρ i if x ≤ -1.
Furthermore, we consider the weak descending ladder epochs for the potential defined by e 0 := 0 and

e i := inf{k > e i-1 : V (k) ≤ V (e i-1 )}, i ≥ 1,
which play a crucial role in our proof. Observe that (e i -e i-1 ) i≥1 is a family of i.i.d. random variables. Moreover, classical results of fluctuation theory (see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF], p. 396), tell us that, under assumptions (a)-(b) of Theorem 1,

E[e 1 ] < ∞. (3.1)
Now, observe that the ((e i , e i+1 ]) i≥0 stand for the set of excursions of the potential above its past minimum. Let us introduce H i , the height of the excursion (e i , e i+1 ] defined by

H i := max e i ≤k≤e i+1 (V (k) -V (e i )) ,
for i ≥ 0. Note that the (H i ) i≥0 's are i.i.d. random variables.

The principle of the proof is to notice that the random walk in random environment spends most of its time climbing the high excursions. In order to quantify what "high excursions" are, we need a key result of Iglehart [START_REF] Iglehart | Extreme values in the GI/G/1 queue[END_REF] which provides the asymptotic for the distribution of the tail of H i , namely

(3.2) ∀i ≥ 0, P {H i > h} ∼ C I e -κh , h → ∞,
where

C I = (1 -E[e κV (e 1 ) ]) 2 κE[ρ κ 0 log ρ 0 ]E[e 1 ] . (3.3)
Iglehart's result is actually deduced from a former well-known result of Cramer, whose proof was later simplified by Feller [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF], concerning the tail of the maximum S := sup{V (k); k ≥ 0} which claims that (3.4)

P {S > h} ∼ C F e -κh , h → ∞.
Since S is stochastically bigger than H 0 , C I must be smaller than C F , and a rather straight argument of Iglehart shows that the ratio between both constants is equal to 1 -E[e κV (e 1 ) ].

Our strategy will be to compute the Laplace transform of the hitting time τ (e n ) (where τ (x) is defined by (2.1)) which at the end will be related to τ (n) by the strong law of large numbers via

E[e 1 ].
Moreover, it appears that the times needed to cross an excursion of height h is roughly of order e h . Combined with Iglehart's result, it implies that the time to cross an excursion is heavy tailed for κ < 1. As we know, from classical phenomena arising in the sum of heavy tailed i.i.d. random variables, the particle will spend most of the time at the foot of the very few high excursions, namely those whose height has order log n κ . (Note that, by Iglehart's result, with an overwhelming probability, there are no excursions of height larger than (1+ε) log n κ , among the n-first excursions.) This explains why the deep valleys we define later are constructed from excursions higher than the critical height h n = (1-ε) log n κ . These valleys consist actually in some portion of potential including these excursions. The high excursions are quite seldom and the valleys are likely to be disjoint. In order to deal with almost sure disjoint valleys, we also introduce * -valleys which coincide with deep valleys with high probability.

3.1. The deep valleys. Let us define the maximal variations of the potential before site x by

V ↑ (x) := max 0≤i≤j≤x (V (j) -V (i)), x ∈ N, V ↓ (x) := min 0≤i≤j≤x (V (j) -V (i)), x ∈ N.
By extension, we introduce

V ↑ (x, y) := max x≤i≤j≤y (V (j) -V (i)), x < y, V ↓ (x, y) := min x≤i≤j≤y (V (j) -V (i)), x < y.
In order to define deep valleys, we extract from the first n excursions of the potential above its minimum, these whose heights are greater than a critical height h n , defined by

(3.5) h n := (1 -ε) κ log n,
for some 0 < ε < 1/3, see Figure 1. Let (σ(i)) i≥1 be the successive indexes of excursions, whose heights are greater than h n . More precisely, We consider now some random variables depending only on n and on the environment, which define the deep valleys.

σ(1) := inf{i ≥ 0 : H i ≥ h n , }, σ(j) := inf{i > σ(j -1) : H i ≥ h n }, j ≥ 2, K n := max{j ≥ 0 : σ(j) ≤ n}.
Definition 1. For 1 ≤ j ≤ K n + 1, let us introduce b j := e σ(j) , a j := sup{k ≤ b j : V (k) -V (b j ) ≥ D n }, T ↑ j := inf{k ≥ b j : V (k) -V (b j ) ≥ h n }, d j := e σ(j)+1 , c j := inf{k ≥ b j : V (k) = max b j ≤x≤d j V (x)}, d j := inf{k ≥ d j : V (k) -V (d j ) ≤ -D n }.
where D n := (1 + 1 κ ) log n. We call (a j , b j , c j , d j ) a deep valley and denote by H (j) the height of the j-th deep valley.

Note that all the random variables introduced in this section depend on n, see Figure 2.

Remark 4. It may happen that two different deep valleys are not disjoint, even if this event is highly improbable as it will be shown in Lemma 4 and Lemma 5 in Subsection 4.1. 3.2. The * -valleys. Let us introduce now a subsequence of the deep valleys defined above. It will turn out that both sequences coincide with probability tending to 1 as n goes to infinity. This will be specified in Lemma 6. Let us first introduce

b j c j a j d j d j T ↑ j h n D n D n
γ * 1 := inf{k ≥ 0 : V (k) ≤ -D n }, T * 1 := inf{k ≥ γ * 1 : V ↑ (γ * 1 , k) ≥ h n }, b * 1 := sup{k ≤ T * 1 : V (k) = min 0≤x≤T * 1 V (x)}, a * 1 := sup{k ≤ b * 1 : V (k) -V (b * 1 ) ≥ D n }, d * 1 := inf{k ≥ T * 1 : V (k) ≤ V (b * 1 )}, c * 1 := inf{k ≥ b * 1 : V (k) = max b * 1 ≤x≤d * 1 V (x)}, d * 1 := inf{k ≥ d * 1 : V (k) -V (d * 1 ) ≤ -D n }.

Let us define the following sextuplets of points by iteration

(γ * j , a * j , b * j , T * j , c * j , d * j , d * j ) := (γ * 1 , a * 1 , b * 1 , T * 1 , c * 1 , d * 1 , d * 1 ) • θ d * j-1 , j ≥ 2,
where θ i denotes the i-shift operator.

Definition 2. We call a * -valley any quadruplet (a * j , b * j , c * j , d * j ) for j ≥ 1. Moreover, we shall denote by K * n the number of such * -valleys before e n , i.e. K * n := sup{j ≥ 0 :

T * j ≤ e n }.
It will be made of independent and identically distributed portions of potential (up to some translation).

Reduction to a single valley

This section is devoted to the proof of Proposition 1 which tells that the study of τ (e n ) can be reduced to the analysis of the time spent by the random walk to cross the first deep valley. To ease notations, we introduce λ n := λ n 1/κ .

Proposition 1. For all n large enough, we have

E e -λn τ (en) ∈ E E b 1 ω,|a 1 e -λnτ (d 1 ) Kn + o(1) , E E b 1 ω,|a 1 e -λnτ (d 1 ) K n + o(1) .
where

K n := ⌊nq n (1 -n -ε/4 )⌋, K n := ⌈nq n (1 + n -ε/4 )⌉, q n := P {H 0 ≥ h n }
and where E x ω,|y denotes the quenched law of the random walk in the environment ω, starting at x and reflected at site y.

4.1.

Introducing "good" environments. Let us define the four following events, that concern exclusively the potential V. The purpose of this subsection is to show that they are realized with an asymptotically overwhelming probability when n goes to infinity. These results will then make it possible to restrict the study of τ (e n ) to these events.

A 1 (n) := {e n < C ′ n} , A 2 (n) := ⌊nq n (1 -n -ε/4 )⌋ ≤ K n ≤ ⌈nq n (1 + n -ε/4 )⌉ , A 3 (n) := ∩ Kn j=0 σ(j + 1) -σ(j) ≥ n 1-3ε , A 4 (n) := ∩ Kn+1
j=1 {d j -a j ≤ C ′′ log n} , where σ(0) := 0 (for convenience of notation) and C ′ , C ′′ stand for positive constants which will be specified below.

In words, A 1 (n) allows us to bound the total length of the first n excursions. The event A 2 (n) gives a control on the number of deep valleys. The event A 3 (n) ensures that the deep valleys are well separated, while A 4 (n) bounds finely the length of each of them. Before proving that the A i 's are typical events, let us first give a preliminary result concerning large deviations that we will use throughout the paper.

Lemma 1.

Under assumption (a), large deviations occur for the potential seen as a sum of i.i.d. random variables. Indeed for all x ≥ m := E [log ρ 0 ] (recall that (a) implies m < 0) and all j ≥ 1, we have

P {V (j) ≥ jx} ≤ exp{-jI(x)}, (4.1) with I(x) := sup t≥0 {tx -Λ(t)} and Λ(t) := log E[ρ t 0 ]
. Moreover, the rate function I is lower semicontinuous, satisfies I(0) > 0 and

inf x>0 I(x) x ≥ κ. (4.2)
Proof. Let us first prove (4.1) which is the upper bound in Cramer's theorem in R, see [START_REF] Dembo | Large deviations techniques and applications[END_REF]. Observe first that for all x and every t ≥ 0, an application of Markov's inequality yields

P {V (j) ≥ jx} = E[1 {V (j)-jx≥0} ] ≤ E[e t(V (j)-jx) ] = e -jtx E[e t log ρ 0 ] j = e -j{tx-Λ(t)} . (4.3)
Then, we get (4.1) by taking the infimum over t ≥ 0 in (4.3).

To prove that I(0) > 0, observe first that I(0) =inf t≥0 Λ(t). Now since the function g(t) := E[ρ t 0 ] satisfies g(0) = g(κ) = 1 (by assumption (a)) and g ′ (0) < 0 (indeed

g ′ (0) = E [log ρ 0 ] < 0), we get that inf 0≤t≤κ g(t) < 1, which implies -inf t≥0 Λ(t) > 0.
The proof of (4.2) is straightforward. Indeed, recalling that

I(x) = sup t≥0 {tx-Λ(t)} for x > 0, we have I(x) ≥ κx -Λ(κ) = κx, since Λ(κ) = 0.
Note that the claim of (4.2) appears on page 236 in [START_REF] Zeitouni | Random Walks in Random Environment, XXXI summer school in probability[END_REF] and that [START_REF] Zeitouni | Random Walks in Random Environment, XXXI summer school in probability[END_REF] claims an equality under certain assumptions. Now, let us introduce the following hitting times (for the potential)

T h := min{x ≥ 0 : V (x) ≥ h}, h > 0, T A := min{x ≥ 0 : V (x) ∈ A}, A ⊂ R.
and prove that the A i (n)'s occur with an overwhelming probability when n tends to infinity.

Lemma 2. The probability P {A 1 (n)} converges to 1 when n goes to infinity.

Proof. It is a direct consequence of the law of large numbers as soon as

C ′ is taken bigger than E[e 1 ].
Lemma 3. The probability P {A 2 (n)} converges to 1 when n goes to infinity.

In words, Lemma 3 means that K n "behaves" like C I n ε , when n tends to infinity. In particular, (3.2), which yields q n ∼ C I n 1-ε , and Lemma 3 imply

P {K n + 1 ≥ 2C I n ε } → 0, n → ∞. (4.4) Proof. At first, observe that P K n nq n ≥ 1 + n -ε/4 = P {K n -nq n ≥ n 1-ε/4 q n } ≤ Var(K n ) n 2(1-ε/4) q 2 n ,
the inequality being a consequence of Markov inequality and the fact that K n follows a binomial distribution of parameter (n,

q n ). Moreover, Var(K n ) = nq n (1 -q n ) ≤ nq n implies P { K n nq n ≥ 1 + n -ε/4 } ≤ 1 n 1-ε/2 q n . Now, Iglehart's result (see (3.2)) implies q n ∼ C I n 1-ε , n → ∞.
Therefore we get that P { Kn nqn ≤ 1 + n -ε/4 } converges to 1 when n goes to infinity. Using similar arguments, we get the convergence to 1 of P { Kn nqn ≥ 1 -n -ε/4 }.

Lemma 4. The probability P {A 3 (n)} converges to 1 when n goes to infinity.

Proof. We make first the trivial observation that

P {A 3 (n)} ≥ P {σ(j + 1) -σ(j) ≥ n 1-3ε , 0 ≤ j ≤ ⌊2C I n ε ⌋ ; K n ≤ 2C I n ε } ≥ P {σ(j + 1) -σ(j) ≥ n 1-3ε , 0 ≤ j ≤ ⌊2C I n ε ⌋} -P {K n ≥ 2C I n ε },
the second inequality being a consequence of P {A ; B} ≥ P {A} -P {B c }, for any couple of events A and B. Therefore, recalling (4.4) and using the fact that (σ(j + 1) -σ(j)) 0≤j≤⌊2C I n ε ⌋ are i.i.d. random variables, it remains to prove that

P {σ(1) ≥ n 1-3ε } ⌊2C I n ε ⌋ → 1, n → ∞.
Since σ( 1) is a geometrical random variable with parameter q n , P {σ(1) ≥ n 1-3ε } is equal to (1 -q n ) ⌈n 1-3ε ⌉ , which implies

P {σ(1) ≥ n 1-3ε } ⌊2C I n ε ⌋ = (1 -q n ) ⌊2C I n ε ⌋ ⌈n 1-3ε ⌉ ≥ exp -Cn 1-2ε q n .
Then, the conclusion follows from (3.2), which implies that

q n ∼ C I /n 1-ε , n → ∞.
Lemma 5. For C ′′ large enough, the probability P {A 4 (n)} converges to 1 when n goes to infinity.

Proof. Looking at the proof of Lemma 4, we have to prove that P {d j -

a j ≥ C ′′ log n} is equal to a o(n -ε ), n → ∞. Moreover, observing that d j -a j = (d j -d j ) + (d j - T ↑ j ) + (T ↑ j -b j ) + (b j -a j )
, the proof of Lemma 5 boils down to showing that, for C ′′ large enough,

P {d j -d j ≥ C ′′ 4 log n} = o(n -ε ), n → ∞, (4.5)
P {d j -T ↑ j ≥ C ′′ 4 log n} = o(n -ε ), n → ∞, (4.6) P {T ↑ j -b j ≥ C ′′ 4 log n} = o(n -ε ), n → ∞, (4.7) P {b j -a j ≥ C ′′ 4 log n} = o(n -ε ), n → ∞. (4.8)
To prove (4.5), we apply the strong Markov property at time d j such that we get P {d j -d j ≥ C ′′ 4 log n} ≤ P {T (-∞,-Dn] ≥ C ′′ 4 log n}. Therefore, we have

P {d j -d j ≥ C ′′ 4 log n} ≤ P { inf 0≤x≤ C ′′ 4 log n V (x) > -D n } ≤ P {V ( C ′′ 4 log n) > -D n }.
Recalling that

D n := (1+ 1 κ ) log n, we can use Lemma 1, which implies P {V ( C ′′ 4 log n) > -D n } ≤ e -C ′′ 4 log n I(-4 C ′′ (1+ 1 κ ))
. Then, this inequality implies (4.5) by choosing

C ′′ large enough such that C ′′ 4 I(-4 C ′′ (1 + 1 κ )) > ε, which is possible since I(0) > 0. To prove (4.6), observe first that (3.2) implies P {H (j) > (1+ε ′ ) κ log n} ∼ n -(ε ′ +ε) = o(n -ε ), n → ∞. Therefore, we obtain that P {d j -T ↑ j ≥ C ′′ 4 log n} is less or equal than P {T (-∞,-1+ε ′ κ log n] ≥ C ′′ 4 log n} + o(n -ε
) and conclude the proof with the same arguments we used to treat (4.5).

To get (4.7), observe first that

P {T ↑ j -b j ≥ C ′′ 4 log n} = P {T hn ≥ C ′′ 4 log n | H 0 ≥ h n } ≤ P { C ′′ 4 log n ≤ T hn < ∞}/P {H 0 ≥ h n }.
Furthermore, Lemma 1 yields

P { C ′′ 4 log n ≤ T hn < ∞} ≤ k≥ C ′′ 4 log n P {V (k) ≥ h n } ≤ k≥ C ′′ 4 log n e -k I( hn k ) ≤ k≥ C ′′ 4 log n e -k I(0) ≤ C n C ′′ 4 I(0)
, the third inequality being a consequence of the fact that the convex rate function I(•) is an increasing function on (m, +∞). Using (3.2), we get, for all large n,

P {T ↑ j -b j ≥ C ′′ 4 log n} ≤ C n C ′′ 4 I(0)-(1-ε)
, which yields (4.7), by choosing C ′′ large enough such that

C ′′ > 4 I(0) . For (4.8), observe first that ((V (k -b j )-V (b j )) a j ≤k≤b j , a j , b j ) has the same distribu- tion as ((V (k)) a -≤k≤0 , a -, 0) under P {•|V (k) ≥ 0, a -≤ k ≤ 0}, where a -:= sup{k ≤ 0 : V (k) ≥ D n }. Then, since P {V (k) ≥ 0, k ≤ 0} > 0 and since (V (-k), k ≥ 0) has the same distribution as (-V (k), k ≥ 0), we obtain P {b j -a j ≥ C ′′ 4 log n} ≤ CP {T (-∞,-Dn] > C ′′ 4 log n} ≤ CP {V ( C ′′ 4 log n) > -D n }.
Now, the arguments are the same as in the proof of (4.5).

Defining A(n

) := A 1 (n)∩A 2 (n)∩A 3 (n)∩A 4 (n)
, a consequence of Lemma 2, Lemma 3, Lemma 4 and Lemma 5, is that

P {A(n)} → 1. (4.9)
The following lemma tells us that the * -valleys coincide with the sequence of deep valleys with an overwhelming probability when n goes to infinity.

Lemma 6. If A * (n) := {K n = K * n ; (a j , b j , c j , d j ) = (a * j , b * j , c * j , d * j ), 1 ≤ j ≤ K n }
, then we have that the probability P {A * (n)} converges to 1, when n goes to infinity.

Proof. Since, by definition, the * -valleys constitute a subsequence of the deep valleys, Lemma 6 is a consequence of Lemma 4 together with Lemma 5.

Remark 5. Another meaning of this result is that, with probability tending to 1, two deep valleys are necessarily disjoint.

Preparatory lemmas.

In this subsection, we develop some technical tools allowing us to improve our understanding of the random walk's behavior. In Lemma 8, we prove that, after exiting a deep valley, the random walk will not come back to another deep valley it has already visited, with probability tending to one. Moreover, Lemma 9 specifies that the random walk typically exits from a * -valley on the right, while Lemma 10 shows that the time spent between two deep valleys is negligible. Lemma 11 states that the first valley coincides with the first * -valley with probability 1 -o(n -ε ), when n goes to infinity.

Preliminary estimates for inter-arrival times. Let us introduce

T ↑ (h) := min{x ≥ 0 : V ↑ (x) ≥ h}, h > 0, T ↓ (h) := min{x ≥ 0 : V ↓ (x) ≤ -h}, h > 0.
Lemma 7. Under assumptions of Theorem 1, we have, for h large enough,

E |0 [τ h ] ≤ C e h ,
where E |0 denotes the expectation under the law P |0 of the random walk in the random environment ω (under P ) reflected at 0 and τ h := τ (T ↑ (h) -1).

Proof. Using (Zeitouni [START_REF] Zeitouni | Random Walks in Random Environment, XXXI summer school in probability[END_REF], formula (2.1.14)), we obtain that E |0 [τ h ] is bounded from above by E 0≤i≤j<T ↑ (h) e V (j)-V (i) . Therefore, since T ↑ (h) ≤ T ↑ (h) • θ i for any i ≥ 0 (where θ denotes the shift operator for the environment), we obtain

E |0 [τ h ] ≤ i≥0 E 1 {i<T ↑ (h)} i≤j<T ↑ (h) e V (j)-V (i) ≤ β 1 (h) β 2 (h), (4.10)
where

β 1 (h) := E T ↑ (h) , β 2 (h) := E 0≤j<T ↑ (h) e V (j) .
To bound β 1 (h), let us introduce the number N of complete excursions before T ↑ (h), defined by N = N(h) := sup{i ≥ 0 : e i < T ↑ (h)}. Then, we can write

β 1 (h) = E[ N -1 i=0 (e i -e i-1 ) + (T ↑ (h) -e N )].
Observe that the definition of T ↑ (h) implies that N is a geometrical random variable with parameter q = q(h) := P {H ≥ h} and recall that, by (3.2), we have q ∼ C I e -κh , h → ∞. Therefore, we get, for h large enough,

β 1 (h) ≤ k≥0 (1 -q) k q kE[e 1 |H < h] + E[T h |H ≥ h] ≤ C k≥0 (1 -q) k q kE[e 1 ] + E[T h |H ≥ h] ,
the second inequality being a consequence of the fact that E[e 1 ] < ∞ (see (3.1)) together with P {H < h} → 1, h → ∞, by (3.2). By obvious calculations, this yields

β 1 (h) ≤ C(1 -q)q -1 E[e 1 ] + E[T h |H ≥ h], which implies with (3.2) that β 1 (h) ≤ Ce κh + E[T h |H ≥ h]. ( 4
E[T h |H ≥ h] ≤ Ce κh k≥0 (k + 1)E[1 {0<V (k)<h} e -κ(h-V (k)) ] ≤ C k≥0 (k + 1) ⌊h⌋ j=0 e κ(j+1) P {V (k) ≥ j}.
By Lemma 1, we have P {V (k) ≥ j} ≤ e -kI( j k ) . Now, the fact that I(•) is an increasing function on R + along with (4.2) imply

P {V (k) ≥ j} ≤ e -k 2 I( j k ) e -k 2 I( j k ) ≤ e -k I(0)
2 e -κ j 2 . Since I(0) > 0, this yields that there exists C > 0 such that, for all large h,

E[T h |H ≥ h] ≤ Ce κ 2 h . (4.12)
Combining together (4.11) and (4.12), we obtain for h large enough,

β 1 (h) ≤ Ce κh . (4.13)
Let us now bound β 2 (h). We introduce first E k := {max 0≤j≤k-1 H j < h ; H k ≥ h} and write

β 2 (h) = k≥0 E 1 E k 0≤j<T ↑ (h) e V (j) = k≥0 k-1 i=0 E 1 E k e V (e j ) J i + E 1 E k e V (e k ) J k ,
where J i := e i+1 j=e i e V (j)-V (e i ) for i ≥ 0 and J k := T ↑ (h)-1

j=e k e V (j)-V (e k ) which is well defined on E k . Observe that E k = {N(h) = k} and recall that N(h) is a geometrical random variable with parameter q = q(h) = P {H ≥ h}. Then, the Markov property applied at times (e j ) 1≤j≤k yields that β 2 (h) is less or equal than

k≥0 (1 -q) k q E[J 0 |H 0 < h] k-1 j=0 E[e V (e 1 ) |H 0 < h] j + E[J 0 |H 0 ≥ h]E[e V (e 1 ) |H 0 < h] k , which implies that β 2 (h) is bounded from above by 1 1 -E[e V (e 1 ) |H 0 < h] E[J 0 |H 0 < h] + q 1 -(1 -q)E[e V (e 1 ) |H 0 < h] E[J 0 |H 0 ≥ h]. Now, since V is transient to -∞, then H 0 is almost surely finite and E[e V (e 1 ) |H 0 < h] → E[e V (e 1 ) ] < 1, when h → ∞. Recalling that q = q(h) → 0, h → ∞, it follows that β 2 (h) ≤ C E[J 0 |H 0 < h] + qE[J 0 |H 0 ≥ h] , (4.14) for h large enough. Let us first bound E[J 0 |H 0 ≥ h].
Recall that if µ denotes the law of ρ 0 , thanks to assumption (a) of Theorem 1 we can define the law μ = ρ κ 0 µ, and the law P = μ⊗Z which is the law of a sequence of i.i.d. random variables with law μ. The definition of κ implies that log ρ μ(dρ) > 0. Then, using the explicit form of the Radon-Nykodym derivative between P and P , we can write

E[J 0 |H 0 ≥ h] ≤ Ce κh Ẽ[e -κV (T h ) J 0 1 {H 0 ≥h} ] ≤ C Ẽ e -κ(V (T h )-h) T h -1 k=0 e V (k) 1 {H 0 ≥h} ≤ C Ẽ T h -1 k=0 e V (k) 1 {min 0<k<T h V (k)>0} ≤ C Ẽ k≥0 ⌊h⌋ p=0 e V (k) 1 {p≤V (k)<p+1} ≤ C ⌊h⌋ p=0 e p+1 Ẽ k≥0 1 {p≤V (k)<p+1} .
Moreover, by Markov property, we have Ẽ

[ k≥0 1 {p≤V (k)<p+1} ] ≤ Ẽ[ k≥0 1 {0≤V (k)<1} ],
which is finite since (V (k)) k≥0 has a positive drift under P .

Therefore, recalling (4.14) and (3.2), we get

β 2 (h) ≤ C(E[J 0 |H 0 < h] + e (1-κ)h ) (4.
I [R|H < h] ≥ E[R 1 {H=S<h} ] ≥ E[J 0 1 {H=S<h} ].
Furthermore, since J 0 depends only on (V (k) ; 0 ≤ k ≤ e 1 ) and since P {V (k) ≤ 0 ; k ≥ 0} > 0, we get, by applying the strong Markov property at time

e 1 , that E[J 0 1 {H<h} ] ≤ CE I [R|H < h], which implies E[J 0 |H < h] ≤ CE I [R|H < h].
Therefore, we only have to prove that E I [R|H < h] ≤ Ce (1-κ)h . To this aim, we recall first that Corollary 4.1 in [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF] implies that, P I -almost surely,

E I [R|⌊H⌋] ≤ Ce ⌊H⌋ . (4.16) Now, observe that E I [R|H < h] ≤ CE I [R 1 {H<h} ]
and let us write

E I [R 1 {H<h} ] ≤ ⌊h⌋ k=0 E I 1 {⌊H⌋=k} E I [R|⌊H⌋ = k] ≤ C ⌊h⌋ k=0 E I 1 {⌊H⌋=k} e ⌊H⌋ ≤ C ⌊h⌋ k=0 e k P I {⌊H⌋ = k} ≤ C ⌊h⌋ k=0 e (1-κ)k ≤ Ce (1-κ)h , (4.17)
the second inequality is a consequence of (4.16) and the fourth inequality due to the fact that P I {⌊H⌋ = k} ≤ ce -κk for some positive constant c. Now assembling (4.10), (4.13), (4.15) and (4.17) concludes the proof of Lemma 7.

Important preliminary results.

Before establishing the announced lemmas, we introduce, for any x, y ∈ Z, τ (x, y) := inf{k ≥ 0 : X τ (x)+k = y}.

Recall that A(n) = A 1 (n) ∩ A 2 (n) ∩ A 3 (n) ∩ A 4 (n)
, where the events (A i (n)) 1≤i≤4 are defined at the beginning of Subsection 4.1. Then, we have the following results.

Lemma 8. Defining DT (n) := A(n) ∩ Kn j=1 τ (d j , b j+1 ) < τ (d j , d j ) , we have P {DT (n)} → 1, n → ∞.
Proof. Recalling (4.9), we only have to prove that

E 1 A(n) Kn j=1 P d j ω {τ (b j+1 ) > τ (d j )} → 0. (4.18)
By (Zeitouni [23], formula (2.1.4)), we get, for 1 ≤ j ≤ K n and for all ω in A(n) :

P d j ω τ (b j+1 ) > τ (d j ) = b j+1 -1 k=d j e V (k) b j+1 -1 k=d j e V (k) ≤ (b j+1 -d j )e V (d j )-V (d j )+hn .
Now, let us explain why b Kn+1 -d Kn ≤ 2n with probability tending to 1. Observe first that b Kn+1 -

d Kn ≤ n + T ↑ (h n ) • θ n if d Kn ≤ n and b Kn+1 -d Kn ≤ T ↑ (h n ) • θ n if d Kn > n.
Therefore it is sufficient to prove that P {T ↑ (h n ) ≥ n} → 0. But using Markov's inequality together with (4.13), we get P {T ↑ (h n ) ≥ n} ≤ Cn -1 e κhn → 0, when n → ∞.

Moreover we have b

j+1 -d j ≤ e n ≤ C ′ n on A 1 (n) for 1 ≤ j ≤ K n -1 and by definition V (d j ) -V (d j ) ≤ -D n for 1 ≤ j ≤ K n . Therefore, we get E 1 A(n) Kn j=1 P d j ω {τ (b j+1 ) > τ (d j )} ≤ C nE[K n ]e -Dn+hn .
Recalling that

D n = (1 + 1 κ ) log n, h n = 1-ε κ log n and since E[K n ] ≤ C n ε (K n has a binomial distribution with parameter (n, q n )), we obtain E 1 A(n) Kn j=1 P d j ω {τ (b j+1 ) > τ (d j )} ≤ C e ε(1-1/κ) log n ,
which implies (4.18).

Lemma 9. Defining DT * (n) := K * n j=1 τ (b * j , d * j ) < τ (b * j , γ * j ) , we have P {DT * (n)} → 1, n → ∞. Proof. Recall that A * (n) = {K n = K * n ; (a j , b j , c j , d j ) = (a * j , b * j , c * j , d * j ), 1 ≤ j ≤ K n }. Then, let us consider A † (n) := A * (n) ∩ A 3 (n) ∩ A * 4 (n) to control the * -valleys, where A * 4 (n) is defined by A * 4 (n) := ∩ K * n j=1 γ * j+1 -a * j ≤ C ′′ log n ∩ {γ * 1 ≤ C ′′ log n} .
Using the same arguments as in the proof of Lemma 5, we can prove that P {A * 4 (n)} → 1, n → ∞, for C ′′ large enough. Then, recalling that Lemma 4 and Lemma 6 imply

P {A * (n) ∩ A 3 (n)} → 1, n → ∞, it remains only to prove that E 1 A † (n) Kn j=1 P b j ω {τ (d j ) > τ (γ * j )} → 0. (4.19)
Observe that by (Zeitouni [23], formula (2.1.4)) we get, for 1

≤ j ≤ K n , P b j ω {τ (d j ) > τ (γ * j )} ≤ (d j -b j )e H (j) -(V (γ * j )-V (b j )) ≤ C log n e H (j) -(V (γ * j )-V (b j )) ,
the second inequality being a consequence of ω ∈ A * (n) ∩ A * 4 (n). Then, to bound e H (j) -(V (γ * j )-V (b j )) from above, observe that (3.2) implies P {H (j) > (1+ε ′ )

κ log n} ∼ n -(ε ′ +ε) = o(n -ε ), n → ∞, for any ε ′ > 0, which yields that P { Kn j=1 {H (j) < (1+ε ′ ) κ
log n}} tends to 1, when n tends to ∞. Therefore, recalling (4.19), we only have to prove that

C(log n)n (1+ε ′ ) κ E 1 A † (n) Kn j=1 e -(V (γ * j )-V (b j )) → 0. (4.20) Since γ * j -b j-1 ≤ C ′′ log n on A * 4 (n) and b j -b j-1 ≥ n 1-3ε on A 3 (n), we get b j -γ * j ≥ 1 2 n 1-3ε for 2 ≤ j ≤ K n on A † (n), for all large n. Similarly, γ * 0 ≤ C ′′ log n on A * 4 (n) and b 1 ≥ n 1-3ε on A 3 (n) yield b 1 -γ * 1 ≥ 1 2 n 1-3ε on A † (n).
Therefore, recalling the definition of b j , we can use Lemma 1 and obtain

P {A † (n) ; V (b j ) -V (γ * j ) ≥ -n 1-3ε 2 } ≤ P {V ( 1 2 n 1-3ε ) ≥ -n 1-3ε 2 } ≤ e -n 1-3ε 2 I -2n -1-3ε 2 = o(n -ε ),
for any 1 ≤ j ≤ K n , since I(0) > 0. This result implies that the term on the lefthand side in (4.20) is bounded from above by C log n n

(1+ε ′ ) κ E[K n ]e -n 1-3ε 2 . Then, since E[K n ] ≤ C n ε , this concludes the proof of Lemma 9.
Lemma 10. For any 0 < η < ε( 1κ -1), let us introduce the following event IA(n

) := A(n) ∩ Kn j=1 τ (d j , b j+1 ) < n 1/κ-η . Then, we have P {IA(n)} → 1, n → ∞.
Proof. Recalling that P {K n ≥ 2C I n ε } → 0, n → ∞, and that Lemma 8 implies that P {DT (n)} → 1, n → ∞, it only remains to prove

P DT (n) ∩ ⌊2C I n ε ⌋ j=1 τ (d j , b j+1 ) ≥ n 1/κ-η → 0, n → ∞.
Using Markov inequality, we have to prove that

E 1 DT (n) ⌊2C I n ε ⌋ j=1 τ (d j , b j+1 ) = o 1 n 1/κ-η , n → ∞. (4.21)
Furthermore, by definition of the event DT (see Lemma 8), we get

E 1 DT (n) ⌊2C I n ε ⌋ j=1 τ (d j , b j+1 ) ≤ E 1 A(n) ⌊2C I n ε ⌋ j=1 E d j ω,|d j [τ (b j+1 )] ≤ E 1 A(n) ⌊2C I n ε ⌋ j=1 E d j ω,|d j [τ (b j+1 )] .
Applying successively the strong Markov property at

d ⌊2C I n ε ⌋ , . . . , d 2 , d 1 , this implies E 1 DT (n) ⌊2C I n ε ⌋ j=1 τ (d j , b j+1 ) ≤ 2C I n ε E |0 [τ (T ↑ (h n ) -1)].
Therefore, Lemma 7 implies

E 1 DT (n) ⌊2C I n ε ⌋ j=1 τ (d j , b j+1 ) ≤ Cn ε e hn ≤ Cn 1 κ -ε( 1 κ -1) ,
which yields (4.21) and concludes the proof, since 0 < η < ε( 1κ -1).

Lemma 11. We have

P {(a 1 , b 1 , c 1 , d 1 ) = (a * 1 , b * 1 , c * 1 , d * 1 )} = o(n -ε ), n → ∞.
Proof. Since γ * 1 is a negative record for the potential V, it is sufficient to prove that there is no excursion higher than h n before γ * 1 . In a first step, we prove that for C large enough

P {γ * 1 ≥ C log n} = o(n -ε ), n → ∞. (4.22)
Indeed, applying Lemma 1, we get

P {γ * 1 ≥ C log n} ≤ P {V (C log n) ≥ -D n } ≤ exp -CI( 1 + κ -1 C ) log n = o(n -ε ),
by choosing C so that CI( 1+κ -1 C ) > ε, which is possible since I(0) > 0. In a second step, we prove that the probability that there is an excursion higher than h n before C log n is a o(n -ε ). Since the number of excursions before C log n is bounded by C log n, we will prove that

P max 0≤i≤C log n H i ≥ h n = o(n -ε ), n → ∞. (4.23)
But this result is obvious. Indeed, using (3.2) we obtain that the probability term in .

Let us first recall that Lemma 8 and Lemma 10 imply that

P {DT (n) ∩ IA(n)} → 1, n → ∞.
Then, we get that the lower bound in the previous interval is equal to

E |0 1 DT (n)∩IA(n) e -λn(τ (b 1 )+ P Kn j=1 {τ (b j ,d j )+τ (d j ,b j+1 )}) + o(1) = E |0 1 DT (n)∩IA(n) e -λn P Kn j=1 τ (b j ,d j ) + o(1) = E |0 e -λn P Kn j=1 τ (b j ,d j ) + o(1).
Then, applying the strong Markov property for the random walk successively at τ (b Kn ), τ (b Kn-1 ), . . . , τ (b 2 ) and τ (b 1 ) we get

E |0 e -λn P Kn j=1 τ (b j ,d j ) = E Kn j=1 E b j ω,|0 e -λnτ (d j ) = E 1 A * (n) K * n j=1 E b * j ω,|0 e -λnτ (d * j ) + o(1) = E K * n j=1 E b * j ω,|0 e -λnτ (d * j ) + o(1),
the second equality being a consequence of Lemma 6. Then, since Lemma 9 implies P{DT * (n)} → 1, we have

E |0 e -λn P Kn j=1 τ (b j ,d j ) = E K * n j=1 E b * j ω,|0 1 DT * (n) e -λnτ (d * j ) + o(1) = E K * n j=1 E b * j ω,|γ * j 1 DT * (n) e -λnτ (d * j ) + o(1) = E K * n j=1 E b * j ω,|γ * j e -λnτ (d * j ) + o(1), Since P{K n = K * n } → 1, and P{K n ≤ K n } → 1, with K n = ⌈nq n (1 + n -ε/4 )⌉,
E |0 e -λn τ (en) ≥ E E b * 1 ω,|γ * 1 e -λnτ (d * 1 ) K n + o(1).
Using Lemma 11 and recalling that

K n = ⌈nq n (1 + n -ε/4 )⌉ = O(n ε ), n → ∞, the strong Markov property applied at γ * 1 yields E |0 e -λn τ (en) ≥ E E b 1 ω,|0 e -λnτ (d 1 ) Kn + o(1).
Using similar arguments for the upper bound in the aforementioned interval, we get

E |0 e -λn τ (en) ∈ E E b 1 ω,|0 e -λnτ (d 1 ) K n + o(1) , E E b 1 ω,|0 e -λnτ (d 1 ) K n + o(1) . with K n := ⌊nq n (1-n -ε/4 )⌋. Furthermore, observe that we have E E b 1 ω,|0 e -λnτ (d 1 ) = E E b 1 ω,|a 1 e -λnτ (d 1 ) + o(n -ε
). This is a consequence of Lemma 5, definition of a and the fact that (3.2) implies P {H (1)

> (1+ε ′ ) κ log n} ∼ n -(ε ′ +ε) = o(n -ε ), n → ∞, for any ε ′ > 0, which gives E P b 1 ω {τ (a 1 ) < τ (d 1 )} ≤ C log n e (1+ε ′ ) κ log n-Dn = o(n -ε ).
This concludes the proof of Proposition 1.

Annealed Laplace transform for the exit time from a deep valley

This section is devoted to the proof of the linearization. It involves h-processes theory and "sculpture" of a typical deep valley. To ease notations, we shall use a, b, c, and d instead of a 1 , b 1 , c 1 and d 1 . Moreover, let us introduce, for any random variable Z ≥ 0, the functional

R n (λ, Z) := E 1 1 + λ n 1/κ Z , (5.1)
and the two important random variables given by

M 1 := d-1 x=a+1 e -( b V (x)-b V (b)) , (5.2)
M 2 := d-1 x=b e V (x)-V (c) , (5.3)
where V is defined below in (5.5). Then, the result can be expressed in the following way.

Proposition 2. For any ξ > 0, we have, for all large n, R n (e ξ λ, 2e

H (1) M 1 M 2 )+o(n -ε ) ≤ E E b ω,|a [e -λnτ (d) ] ≤ R n (e -ξ λ, 2e H (1) M 1 M 2 )+o(n -ε ).
5.1. Two h-processes. In order to estimate E b ω,|a e -λnτ (d) , we decompose the passage from b to d into the sum of a random geometrically distributed number, denoted by N, of unsuccessful attempts to reach d from b (i.e. excursions of the particle from b to b which do not hit d), followed by a successful attempt. More precisely, since N is a geometrically distributed random variable with parameter 1 -p satisfying (see [START_REF] Zeitouni | Random Walks in Random Environment, XXXI summer school in probability[END_REF], formula (2.1.4))

1 -p = ω b e V (b) d-1 x=b e V (x) , (5.4)
we can write τ (d) = N i=1 F i + G, where the F i 's are the successive i.i.d. failures and G the first success. The accurate estimation of the time spent by each (successful and unsuccessful) attempt leads us to consider two h-processes where the random walker evolves in two modified potentials, one corresponding to the conditioning on a failure (see the potential V and Lemma 12) and the other to the conditioning on a success (see the potential V and Lemma 13).

5.1.1. The failure case: the h-potential V . Let us fix a realization of ω. To introduce the h-potential V , we consider the valley a < b < c < d and define h(x)

:= P x ω {τ (b) < τ (d)}. For any b < x < d, we introduce ω x := ω x h(x+1) h(x) . Since h is a harmonic function, we have 1 -ω x = (1 -ω x ) h(x-1) h(x) . Now, V can be defined for x ≥ b by V (x) := V (b) + x i=b+1 log 1 -ω i ω i . (5.5)
We obtain for any b ≤ x < y < d,

V (y) -V (x) = (V (y) -V (x)) + log h(x) h(x + 1) h(y) h(y + 1) . (5.6) Since h(x) is a decreasing function of x by definition, we get h(x) h(x + 1) h(y) h(y + 1) ≥ 1. (5.7)
Thus we obtain for any b ≤ x < y ≤ c,

(5.8) V (y) -V (x) ≥ V (y) -V (x).
Lemma 12. For any environment ω, we have

(5.9) E ω [F 1 ] = 2 ω b b-1 i=a+1 e -(V (i)-V (b)) + d-1 i=b e -( b V (i)-b V (b)) ,

and

(5.10)

E ω F 2 1 = 4ω b R + + 4(1 -ω b ) R -,
where

R + := d-1 i=b+1 1 + 2 i-2 j=b e b V (j)-b V (i-1)
e -( b

V (i-1)-b V (b)) + 2 d-1 j=i+1 e -( b V (j-1)-b V (b)) , R -:= b-1 i=a+1 1 + 2 b j=i+2 e V (j)-V (i+1) e -(V (i+1)-V (b)) + 2 i-1 j=a+1 e -(V (j+1)-V (b)) .
Remark 6. Alili [START_REF] Alili | Asymptotic behavior for random walks in random environments[END_REF] and Goldsheid [START_REF] Goldsheid | Simple transient random walks in one-dimensional random environment: the central limit theorem[END_REF] prove a similar result for a non-conditioned hitting time. Here we give the proof in order to be self-contained.

Proof. Let us first introduce

N + i := ♯{k < τ (b) : X k = i -1, X k+1 = i}, i > b, N - i := ♯{k < τ (b) : X k = i + 1, X k+1 = i}, i < b,
and the quenched probability in the environment ω, denoted by P b ω . Then, observe that, under P b ω , for i > b and conditionally on N + i = x, N + i+1 is the sum of x independent geometrical random variables with parameter ω i ∈ (0, 1). It means that

E b ω [N + i+1 |N + i = x] = x b ρ i and Var b ω [N + i+1 |N + i = x] = x b ω i b ρ 2 i
. Similarly, under P ω , for i < b and conditionally on

N - i = x, N - i-1 is the sum of x independent geometrical ran- dom variables with parameter 1 -ω i . It means that E ω [N - i-1 |N - i = x] = xρ i and Var ω [N - i-1 |N - i = x] = xρ 2 i (1-ω i ) . Since E ω [F 1 ] = 2ω b E b ω [ d-1 b+1 N + i ] + 2(1 -ω b ) E ω [ b-1 a+1 N - i ],
an easy calculation yields (5.9).

To calculate E ω [F 2 1 ], observe first that

E ω [F 2 1 ] = 4ω b E b ω ( d-1 i=b+1 N + i ) 2 + 4(1 -ω b ) E ω ( b-1 i=a+1 N - i ) 2 .
Then, it remains to prove that

E b ω [( d-1 b+1 N + i ) 2 ] = R + and E ω [( b-1 a+1 N - i ) 2 ] = R -. We will only treat E b ω [( d-1 b+1 N + i ) 2 ], the case of E ω [( b-1 a+1 N - i ) 2 ] being similar. We get first E b ω ( d-1 b+1 N + i ) 2 = d-1 i=b+1 E b ω [(N + i ) 2 ] + 2 d-1 i=b+1 d-1 j=i+1 E b ω [N + i N + j ]. (5.11) Observe that E b ω N + i N + j = E b ω N + i E b ω N + j | N + i , . . . , N + j-1 = E b ω N + i N + j-1 b ρ j-1
, for i < j, so that we get, by iterating,

E b ω N + i N + j = E b ω (N + i ) 2 1 ρ j-1 . . . ρ i .
Recalling (5.11), this yields

E b ω ( d-1 b+1 N + i ) 2 = d-1 i=b+1 E b ω (N + i ) 2 1 + 2 d-1 j=i+1 1 ρ i . . . ρ j-1 = d-1 i=b+1 E b ω (N + i ) 2 1 + 2 d-1 j=i+1 e -( b V (j-1)-b V (i-1)) . (5.12) Now, observe that E b ω (N + i ) 2 = E b ω E b ω (N + i ) 2 |N + i-1
, which implies

E b ω (N + i ) 2 = E b ω k≥1 E b ω [G (i) 1 + • • • + G (i) k ]1 {N + i-1 =k} .
Since the G (i)

• 's are i.i.d., we get

E b ω [G (i) 1 + • • • + G (i) k ] = kVar b ω [G (i) 1 ] + k 2 E b ω [G (i) 1 ] 2 . Recalling that E b ω [G (i) 1 ] = 1 b ρ i-1 and Var b ω [G (i) 1 ] = 1 b ω i-1 b ρ 2 i-1
, this yields

E b ω (N + i ) 2 = E b ω N + i-1 ω i-1 ρ 2 i-1 + E b ω (N + i-1 ) 2 ρ 2 i-1 = 1 ω i-1 ρ b+1 . . . ρ i-2 ρ 2 i-1 + E b ω (N + i-1 ) 2 ρ 2 i-1 . (5.13) Denoting W b+1 := 1 and W i := ( ρ b+1 . . . ρ i-1 ) 2 E b ω (N + i ) 2 for b + 1 < i < d, (5.13) becomes W i -W i-1 = ρ b+1 . . . ρ i-1 ω i-1 = ρ b+1 . . . ρ i-1 + ρ b+1 . . . ρ i-2 ,
the second equality being a consequence of 1/ ω i-1 = ρ i-1 + 1. Therefore, we have W i = i b+2 (W j -W j-1 ) + W b+1 = ρ b+1 . . . ρ i-1 + 2(1 + i-2 b+1 ρ b+1 . . . ρ j ), which implies

E b ω (N + i ) 2 = 1 ρ b+1 . . . ρ i-1 + 2 i-2 j=b ρ b+1 . . . ρ j ( ρ b+1 . . . ρ i-1 ) 2 = e -( b V (i-1)-b V (b)) + 2 i-2 j=b e b V (j)-2 b V (i-1)+ b V (b) . (5.14)
Assembling (5.12) and (5.14) yields (5.10). g(x) . Since g is a harmonic function, we have 1ωx = (1 -ω x ) g(x-1) g(x) . Then, V can be defined for x ≥ b by

V (x) := V (b) + x i=b+1 log 1 -ωi ωi .
We have the following result for any b < x < y ≤ d, V (y) -V (x) = (V (y) -V (x)) + log g(x) g(x + 1) g(y)g(y + 1) . (5.15) Since g(x) is a increasing function of x by definition, we get g(x) g(x + 1) g(y) g(y + 1) ≤ 1. (5.16) Therefore, we obtain for any c ≤ x < y ≤ d, V (y) -V (x) ≤ V (y) -V (x). (5.17) Using the same arguments as in the failure case, we get the following result.

Lemma 13. For any environment ω, we have

(5.18) E ω [G] ≤ 1 + d i=b+1 d j=i e V (j)-V (i) .
5.2. Preparatory lemmas. The study of a typical deep valley involves the following event

A 5 (n) := max{V ↑ (a, b) ; -V ↓ (b, c) ; V ↑ (c, d)} ≤ δ log n ,
where δ > ε/κ. In words, A 5 (n) ensures that the potential does not have excessive fluctuations in a typical box. Moreover, we have the following result.

Lemma 14. For any δ > ε/κ,

P {A 5 (n)} = 1 -o(n -ε ), n → ∞.
Proof. We easily observe that the proof of Lemma 14 boils down to showing that

P {V ↑ (a, b) ≥ δ log n} = o(n -ε ), n → ∞, (5.19) P {-V ↓ (b, c) ≥ δ log n} = o(n -ε ), n → ∞, (5.20) P {V ↑ (c, d) ≥ δ log n} = o(n -ε ), n → ∞. (5.21)
In order to prove (5.21), let us first observe the following trivial inequality

P {V ↑ (c, d) ≥ δ log n} ≤ P {V ↑ (T ↑ 1 , d) ≥ δ log n}.
Looking at the proof of (4.6), we observe that P {d -T ↑ 1 ≥ C log n} = o(n -ε ′ ), for any ε ′ > 0, by choosing C large enough, depending on ε ′ . Therefore, we only have to prove that

P {V ↑ (T ↑ 1 , T ↑ 1 + C log n) ≥ δ log n} = o(n -ε ).
Then, applying the strong Markov property at time T ↑ 1 , we have to prove that P {V ↑ (0, C log n) ≥ δ log n} = o(n -ε ). Now, by Lemma 1 we get

P {V ↑ (0, C log n) ≥ δ log n} ≤ (C log n) 2 max 0≤k≤C log n P {V (k) ≥ δ log n} ≤ (C log n) 2 max 0≤k≤C log n e -kI( δ log n k ) ≤ (C log n) 2 exp{-κδ log n}.
Since δ > ε/κ, this yields (5.21).

To get (5.20), observe first that

P {-V ↓ (b, c) ≥ δ log n} ≤ P {-V ↓ (b, T ↑ 1 ) ≥ δ log n} + P {-V ↓ (T ↑ 1 , c) ≥ δ log n}.
The first term on the right-hand side is equal to P {V ↓ (0, T ↑ (h n )) ≥ δ log n|H 0 > h n }.

Recalling that (3.2) implies P {H 0 > h n } ≤ Cn -(1-ε) for all large n and observing the trivial inclusion

V ↓ (0, T ↑ (h n )) ≥ δ log n ; H 0 > h n ⊂ T ↓ (δ log n) < T hn < T (-∞,0] , it follows that P {-V ↓ (b, T ↑ 1 ) ≥ δ log n} is less or equal than Cn 1-ε P {T ↓ (δ log n) < T hn < T (-∞,0] } ≤ Cn 1-ε ⌊hn⌋ p=⌊δ log n⌋ P {M δ ∈ [p, p + 1) ; T ↓ (δ log n) < T hn < T (-∞,0] },
where M δ := max{V (k); 0 ≤ k ≤ T ↓ (δ log n)}. Applying the strong Markov property at time T ↓ (δ log n) and recalling (3.4) we bound the term of the previous sum, for ⌊δ log n⌋ ≤ p ≤ ⌊h n ⌋ and all large n, by

P {S ≥ p} P {S ≥ h n -(p -δ log n)} ≤ Ce -κp e -κ(hn-p+δ log n)) ,
where S = sup{V (k); k ≥ 0}. Thus, we get P {-V ↓ (b, T ↑ 1 ) ≥ δ log n} ≤ C⌊h n ⌋n -κδ , for all large n, which yields

P {-V ↓ (b, T ↑ 1 ) ≥ δ log n} = o(n -ε ), n → ∞, since δ > ε/κ. Furthermore, applying the strong Markov property at T ↑ 1 , we obtain that P {-V ↓ (T ↑ 1 , c) ≥ δ log n} ≤ P {-V ↓ (0, V max ) ≥ δ log n}.
In a similar way we used before (but easier), we get, by applying the strong Markov property at T ↓ (δ log n), that P {-V ↓ (T ↑ 1 , c) ≥ δ log n} ≤ n -κδ for all large n. Since δ > ε/κ this yields (5.20). For (5.19), observe first that ((

V (k -b) -V (b)) a≤k≤b , a, b) has the same distribution as ((V (k)) a -≤k≤0 , a -, 0) under P {•|V (k) ≥ 0 , a -≤ k ≤ 0}, where a -:= sup{k ≤ 0 : V (k) ≥ D n }. Then, since P {V (k) ≥ 0 , k ≤ 0} > 0 and since (V (-k) , k ≥ 0) has the same distribution as (-V (k) , k ≥ 0), we obtain P {V ↑ (a, b) ≥ δ log n} ≤ CP {V ↑ (0, T (-∞,-Dn] ) ≥ δ log n}.
Now, the arguments are the same as in the proof of (5.21).

Proof of Proposition 2.

Recall that we can write τ (d) = N i=1 F i + G, where the F i 's are the successive i.i.d. failures and G the first success. Then, denoting F 1 by F, we have

E b ω,|a [e -λnτ (d) ] = E b ω,|a [e -λnG ] k≥0 E b ω,|a [e -λnF ] k (1 -p)p k = E b ω,|a [e -λnG ] 1 -p 1 -p E b ω,|a [e -λnF ] . (5.22) In order to replace E b ω,|a [e -λnF ] by 1 -λ n E b ω,|a [F ], we observe that 1 -λ n E b ω,|a [F ] ≤ E b ω,|a [e -λnF ] ≤ 1 -λ n E b ω,|a [F ] + λ 2 n 2 E b ω,|a [F 2 ], which implies that E[ 1-p 1-p E b ω,|a [e -λnF ] ] be- longs to E 1 -p 1 -p(1 -λ n E b ω,|a [F ]) ; E 1 -p 1 -p(1 -λ n E b ω,|a [F ] + λ 2 n 2 E b ω,|a [F 2 ])
.

Now
R + ≤ (d -b) 1 + 2(d -b)e -b V ↓ (b,d) 3(d -b) max b≤j≤d e -( b V (j)-b V (b)) . (5.23)
Recalling that the estimates (4.5)-(4.8) imply that P {d -a ≥ C ′′ log n} = o(n -ε ) and that Lemma 14 tells that P {A 5 (n)} = 1 -o(n -ε ), we are interested in the event A ‡ (n) := {d -a ≤ C ′′ log n} ∩ A 5 (n), whose probability is greater than 1 -o(n -ε ) for n large enough. It allows us to sculpt the deep valley (a, b, c, d), such that we can bound R + . We are going to show that the fluctuations of V are, in a sense, related to the fluctuations of V controlled by A 5 (n). Indeed, (5.8) yields V ↓ (b, c) ≥ V ↓ (b, c) ≥ -δ log n on A ‡ (n). Moreover, (5.6) together with (5.7) imply that V (y) -V (x) is greater than

[V (y) -max y≤j≤d-1 V (j)] -[V (x) -max x≤j≤d-1 V (j)] -O(log 2 n), for any c ≤ x ≤ y ≤ d, on A ‡ (n). Since V (x) -max x≤j≤d-1 V (j) ≤ 0 and V (y) - max y≤j≤d-1 V (j) ≥ -δ log n on A ‡ (n), this yields V ↓ (c, d) ≥ -δ log n -O(log 2 n).
Furthermore, since (5.6) and (5.7) imply that V (c) is larger than max

b≤j≤c V (j) - O(log 2 n), assembling V ↓ (b, c) ≥ -δ log n with V ↓ (c, d) ≥ -δ log n -O(log 2 n) yield V ↓ (b, d) ≥ -δ log n -O(log 2 n), (5.24)
on A ‡ (n). Therefore, we have, on A ‡ (n) and for all large n,

R + ≤ C(log n) 3 n δ max b≤j≤d e -( b V (j)-b V (b)) . (5.25) Since V (b) = V (b) and (5.7) implies V (x) ≥ V (x), for all b ≤ x ≤ c (in particular V (c) ≥ V (c)), it follows from (5.24) that V (j) -V (b) = ( V (j) -V (c)) + ( V (c) - V (b)) ≥ h n -δ log n -O(log 2
n), which is greater than 0 for n large enough whenever δ < (1 -ε)/κ (it is possible since δ > ε/κ and 0 < ε < 1/3). Therefore, recalling (5.25), we obtain, on A ‡ (n),

R + ≤ C(log n) 3 n δ . (5.26)
In a similar way, we prove that R

-≤ C(log n) 3 n δ , on A ‡ (n), which implies that λ n E b ω,|a [F 2 ] ≤ C(log n) 3 n δ-1 κ . Now, observe that, for any ξ > 0, {λ n E b ω,|a [F 2 ] ≤ 2(1 - e -ξ )} is included in A ‡ (n), so that λ n E b ω,|a [F 2 ] ≤ 2(1 -e -ξ )E b ω,|a [F ] with probability larger than 1 -o(n -ε ). Then, introducing R ′ n (λ) := E 1 1 + λ n 1/κ p 1-p E b ω,|a [F ] ,
we get, for n large enough, We observe first that (5.17) implies V ↑ (c, d) ≤ V ↑ (c, d), which large n, whenever δ < 1-ε 2κ , which is possible since δ > ε/κ and 0 < ε < 1/3. Therefore, we obtain, for n large enough, R n (e ξ λ, 2e

R ′ n (λ) + o(n -ε ) ≤ E 1 -p 1 -p E b ω,|a [e -λnF ] ≤ R ′ n (e -ξ λ) + o(n -ε ). (5.
H (1) M T 1 M 2 ) + o(n -ε ) ≤ R n (λ, 2e H (1) M 1 M 2 ) ≤ R n (λ, 2e H (1) M T 1 M 2 ).
Step 2: we replace M T 1 by M T Then, for any ξ > 0, we obtain that M T 1 ≥ e -ξ M T 1 , on A ‡ (n) and for all large n. This implies R n (λ, 2e

H (1) M T 1 M 2 ) ≤ R n (λ, 2e H (1) M T 1 M 2 ) ≤ R n (e -ξ λ, 2e H (1) M T 1 M 2 ) + o(n -ε
). Now, assembling Step 1 and Step 2, we get that, for any ξ > 0 and n large enough, R n (λ, 2e H (1) M 1 M 2 ) belongs to R n (e ξ λ, 2e H (1) M T 1 M 2 ) + o(n -ε ) ; R n (e -ξ λ, 2e H (1) M T 1 M 2 ) + o(n -ε ) . (6.1)

Step 3: the "good " conditioning.

Let us first observe that ((V (k -b) -V (b)) a≤k≤d , a, b, c, d) has the same law as ((V (k)) a -≤k≤d + , a -, 0, T H , d + ) under P {•|I ′ n }, where I ′ n := {H ≥ h n ; V ↑ (a -, 0) ≤ h n ; V (k) ≥ 0 , a -≤ k ≤ 0}. Moreover, we easily obtain that P {{V (k) ≥ 0 , a -≤ k ≤ 0} \ {V (k) ≥ 0 , k ≤ 0}} = O(n -(1+κ) ) = o(n -ε ), that P {{H ≥ h n } \ {H = S}} = O(n -2(1-ε) ) = o(n -ε ) and that P {V ↓ (a -, 0) > h n } ≤ P {V ↓ (a -, 0) > δ log n} = o(n -ε ), with the same arguments as in the proof of Lemma 14. Therefore, we have P {I ′ n △ I n } = o(n -ε ). Since 0 ≤ R n (λ, Y ) ≤ 1, for any λ > 0 and any positive random variable Y, this yields R n (λ, 2e H (1) M T 1 M 2 ) = R n (λ) + o(n -ε ). (6.2) Combining (6.1) and (6.2) together concludes the proof of Proposition 3.

Proof of Theorem 1

Observe first that R n (λ) can be written

R n (λ) = 1 -E 1 - 1 1 + 2λ n Z |I n .
Then, we can use Corollary A.1 and Remark A.1 in [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF], that together imply

E 1 - 1 1 + 2λ n Z I n ∼ 2 κ πκ sin(πκ) E[M κ ] 2 C I nP {H ≥ h n } λ κ , n → ∞.
where the random variable M defined by (7.1)

M := k<0 e -V ′ k + k≥0 e -V ′′ k ,
where (V ′ k ) k<0 is distributed as the potential under P {•|V k ≥ 0, ∀k < 0} while (V ′′ k ) k≥0 is independent of (V ′ k ) k<0 and is distributed as the potential under P {•|V k > 0, ∀k > 0}.

Therefore, combining together the results of Proposition 1, Proposition 2, Proposition 3 and recalling that q n := P {H ≥ h n }, we get that, for any ξ > 0, lim inf Now, one can be tempted to express the functional E[M κ ] in terms of the more usual constant C K , see (2.2). This is the content of Theorem 2.1 in [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF], which yields Hence, we obtain that the limit is the positive stable law with index κ and parameter 2 κ πκ 2 sin(πκ

C K = E[M κ ]C F = E[M κ ] ( 1
) C 2 K E[ρ κ 0 log ρ 0 ].
We can easily see that we can deduce from this proof the asymptotic of the Laplace transform of the time needed to cross the first * -valley.

Corollary 2. We have

E 1 -e -λ n 1/κ τ * 1 ∼ 2 κ πκ sin(πκ) C U nP (H ≥ h n ) λ κ ,
where C U = C I E[M κ ] is the constant which appears in the tail estimate of Z, in [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF].

Remark 7. This result would hold for a different choice of h n . Indeed, from the proof of Proposition 2 and Proposition 3 and from Corollary A.1 of [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF], we see that the result holds for any choice of h n such that e hn = o(n 1 κ ) and h n ≥ n 1-ǫ κ for some 0 < ǫ < 1/3 (this last condition comes from the technical assumption in (4.1) which is needed in the proof Proposition 2, see (5.26)).

Figure 1 .

 1 Figure 1. Potential and valleys.

Figure 2 .

 2 Figure 2. Zoom on the j-th valley.

  [START_REF] Kesten | The limit distribution of Sinai's random walk in random environment[END_REF] and only have to boundE[J 0 |H 0 < h]. Recall that R = k≥0 e V (k) and observe that J 0 ≤ R. Moreover, let us denote by E I [•] the expectation under P I {•} := P {•|I}, with I := {H = S}. Then, we first observe that E

  (4.23) is less than C log n e -κhn = o(n -ε ). Now assembling (4.22) and (4.23) concludes the proof of Lemma 11.

4. 3 .

 3 Proof of Proposition 1. Since the time spent on Z -is almost surely finite, we reduce our study to the random walk in random environment reflected at 0 and observe that E e -λn τ (en) = E |0 e -λn τ (en) + o(1), n → ∞, where E |0 denotes the expectation under the law P |0 of the random walk in the random environment ω (under P ) reflected at 0. Furthermore, by definition, τ (e n ) satisfies τ (b 1 ) + Kn-1 j=1 {τ (b j , d j ) + τ (d j , b j+1 )} ≤ τ (e n ) ≤ τ (b 1 ) + Kn j=1 {τ (b j , d j ) + τ (d j , b j+1 )}, such that we easily get that E |0 e -λn τ (en) belongs to E |0 e -λn (τ(b1)+ P Kn j=1 {τ (b j ,d j )+τ (d j ,b j+1 )}) , E |0 e -λn (τ(b1)+ P Kn-1 j=1 {τ (b j ,d j )+τ (d j ,b j+1 )})

  we get E |0 e -λn τ (en) ≥ E the strong Markov property (for the potential V ) successively at times γ * Kn , . . . , γ * 2 and observing that the E

5. 1 . 2 .

 12 The success case: the h-potential V . In a similar way, we introduce the hpotential V by considering the valley a < b < c < d and defining g(x) := P x ω {τ (d) < τ (b)}. For any b < x < d, we introduce ωx := ω x g(x+1)

  27)In order to bound E b ω,|a e -λnG by below, we observe that e -x ≥ 1 -x, for anyx ≥ 0, such that E b ω,|a [e -λnG ] ≥ 1 -λ n E b ω,|a [G].Therefore, we only have to bound E b ω,|a [G] from above. Recalling (5.18), we get E b ω,|a [G] ≤ (d -b) 2 e V ↑ (b,d) . Now, let us bound V ↑ (b, d).

1 .( hn 2 ) 1 ≤ M T 1 .,

 1211 Let us denote M T 1 := T k=a+1 e -(V (k)-V (b)) . Since T ( hn 2 ) ≤ c, (5.8) implies that M T Observe that (5.6) with (5.7) imply that V (y) -V (b) -(V (y) -V (b)) is less or equal than log for any b ≤ y ≤ d. Therefore, on A ‡ (n), we obtain V (y) -V (b) ≤ (V (y) -V (b)) + C log ne -hn 2 for any b ≤ y ≤ T ( hn2), which yields M T 1 ≥ exp{C log n e -hn 2 }M T 1 .

  n→∞ E[e -λn τ (en) ] ≥ exp -2 κ πκ sin(πκ) E[M κ ] 2 C I (e ξ λ) κ , lim sup n→∞ E[e -λn τ (en) ] ≤ exp -2 κ πκ sin(πκ) E[M κ ] 2 C I (e -ξ λ) κ .Since this result holds for any ξ > 0, we get,lim n→∞ E[e -λn τ (en) ] = exp -2 κ πκ sin(πκ) E[M κ ] 2 C I λ κ .

2 K C I C 2 F λ κ + o( 1 )

 221 -E[e κV (e 1 ) ]) κE[ρ κ 0 log ρ 0 ]E[e 1 ].Therefore, the Laplace transform of n -1/κ τ (e n ) isE[e -λ n 1/κ τ (en) ] = exp -2 κ πκ sin(πκ) C = exp -2 κ πκ 2 sin(πκ) C 2 K E[ρ κ 0 log ρ 0 ]E[e 1 ] λ κ + o(1).

Finally

  , since, by the law of large numbers, e n /n converges almost surely to E[e 1 ], we conclude thatE[e -λ n 1/κ τ (n) ] = exp -2 κ πκ 2 sin(πκ) C 2 K E[ρ κ 0 log ρ 0 ] λ κ + o(1).

  .11) Now, let us bound E[T h |H ≥ h]. For this purpose, we observe first that E[T h |H ≥ h] ≤ Ce κh k≥0 (k + 1)P {T h = k + 1 ; H ≥ h}. Then, applying the Markov property at time k, we get

Acknowledgements Many thanks are due to an anonymous referee for careful reading of the original manuscript and for helpful comments.

yields V ↑ (c, d) ≤ δ log n on A ‡ (n). Moreover, (5.15) together with (5.16) imply that V (y) -V (x) is less or equal than

for any b ≤ x ≤ y ≤ c, on A ‡ (n). Since V (y)max b≤j≤y V (j) ≤ 0 and V (x)max b≤j≤x V (j) ≥ -δ log n on A ‡ (n), this yields V ↑ (b, c) ≤ δ log n + O(log 2 n). Furthermore, (5.17) and the fact that

κ -ε, which is possible since δ > ε/κ and 0 < ε < 1/3. Therefore, recalling (5.27), we obtain

(5.28) Recalling (5.9) and (5.4), we get

where

Now, assembling (5.28) and (5.29) concludes the proof of Proposition 2.

Back to canonical meanders

Recall that S = max{V (k) ; k ≥ 0} and let us set

and introduce the random variable

we get the following result.

Proposition 3. For any ξ > 0, we have, for n large enough,

Proof.

Step 1: we replace

. Recalling (5.24), we observe that M 1 ≤ M T 1 +C ′′ log ne -hn 2 +δ log n on A ‡ (n). This implies that, for any ξ > 0, we have M 1 -M T 1 ≤ (e ξ -1) M T 1 for all

Proof of Corollary 1

We are in the case when the law of the environment satisfies

with α, β > 0 and B(α, β) :=

The assumption of Theorem 1 corresponds to the case where 0 < α -β < 1 and an easy computation leads to κ = α -β. Now, a classical argument of derivation under the sign integral shows that

Γ(z) . Furthermore, a work of Chamayou and Letac [START_REF] Chamayou | Explicit stationary distributions for compositions of random functions and products of random matrices[END_REF] shows that C K can be made explicit. Indeed, with the notations of [START_REF] Chamayou | Explicit stationary distributions for compositions of random functions and products of random matrices[END_REF], ρ 0 follows the law β (2) p,q (dx) := 1 B(p,q) x p-1 (1 + x) -p-q 1 R + (x)dx with p = β and q = α. Then, Example 9 of [START_REF] Chamayou | Explicit stationary distributions for compositions of random functions and products of random matrices[END_REF] says that k≥1 e V (k) follows the law of β

9. Toward the case κ = 1

We intend to treat the critical case κ = 1 between the transient ballistic and subballistic cases. This case turns out to be more delicate. Indeed, Lemma 7 is replaced by a weaker statement, which says that τ (e n ) reduces to the time spent by the walker to climb excursions which are higher than α log n for α arbitrarily small. Due to this reduced height, the new "high" excursions are much more numerous and are not anymore well separated. The definition of the valleys should then be adapted as well as the "linearization" argument, which is more difficult to carry out. Moreover, a result of Goldie [START_REF] Goldie | Implicit renewal theory and tails of solutions of random equations[END_REF] gives an explicit formula for the Kesten's renewal constant, namely C K = 1 E[ρ 0 log ρ 0 ] . As a result, we should obtain, as a consequence of a fluctuation result, the following result, which takes a remarkably simple form: X n /( n log n ) converges in probability to E[ρ 0 log ρ 0 ]/2.