Nathanaël Enriquez
email: nenriquez@u-paris10.fr

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

Olivier Zindy
email: olivier.zindy@u-paris10.fr

Ë L Nathana

Christophe Enriquez

Olivier Sabot

Zindy

Aging and quenched localization for one-dimensional random walks in random environment in

Keywords: 2000 Mathematics Subject Classification. primary 60K37; secondary 60G50, 60J45, 82D30 Random walk in random environment, aging, quenched localization

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Introduction

One-dimensional random walks in random environment have been the subject of constant interest in physics and mathematics for the last thirty years since they naturally appear in a great variety of situations in physics and biology.

In 1975, Solomon gave, in a seminal work [START_REF] Solomon | Random walks in a random environment[END_REF], a criterion of transience-recurrence for such walks moving to the nearest neighbours, 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 recall 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. Namely, 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. The potential itself is a usual random walk when the transition probabilities at each site are independent and identically distributed (i.i.d.).

The proof by Sinai of an annealed limit law in the recurrent case is based on a quenched localization result. Namely, a notion of valley of the potential is introduced, as well as an order on the set of valleys. It is then proved that the walk is localized at time t, with a probability converging to 1, around the bottom of the smallest valley of depth bigger than log t surrounding the origin. An annealed convergence in law of this site normalized by (log t) 2 implies the annealed limiting law for the walk.

In the case of transient random walks in random environment with zero asymptotic speed, the proof of the limiting law by Kesten, Kozlov and Spitzer [START_REF] Kesten | A limit law for random walk in a random environment[END_REF] does not follow this scheme. Therefore an analogous result to Sinai's localization in the quenched setting was missing. As we will see, the answer to this question is more complicated than in the recurrent case but still very explicit.

In the setting of sub-ballistic transient random walks, the valleys we introduce are, like in [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] and [START_REF] Peterson | Quenched limits for transient, zero speed one-dimensional random walk in random environment[END_REF], related to the excursions of the potential above its past minimum. Now, the key observation is that with a probability converging to 1, the particle at time t is located at the foot of a valley having depth and width of order log t. Therefore, since the walk spends a random time of order t inside a valley of depth log t, it is not surprising that this random walk exhibits an aging phenomenon.

What is usually called aging is a dynamical out-of-equilibrium physical phenomenon observed in disordered systems like spin-glasses at low temperature, defined by the existence of a limit of a given two-time correlation function of the system as both times diverge keeping a fixed ratio between them; the limit should be a non-trivial function of the ratio. It has been extensively studied in the physics literature, see [START_REF] Bouchaud | Out of equilibrium dynamics in spin-glasses and other glassy systems[END_REF] and therein references.

More precisely, in our setting, Theorem 1 expresses that, for each given ratio h > 1, the probability that the particle remains confined within the same valley during the time interval [t, th]. This probability is expressed in terms of the generalized Arcsine law, which confirms the status of universality ascribed to this law by Ben Arous and Černý in their study of aging phenomena arising in trap models [START_REF] Ben Arous | Dynamics of trap models, Ecole d' Éte de Physique des Houches[END_REF].

Recall that the trap model is a model of random walk that was first proposed by Bouchaud and Dean [START_REF] Bouchaud | Weak ergodicity breaking and aging in disordered systems[END_REF][START_REF] Bouchaud | Aging on Parisi's tree[END_REF] as a toy model for studying this aging phenomenon. In the mathematics litterature, much attention has recently been given to the trap model, and many aging result were derived from it, on Z in [START_REF] Fontes | Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension[END_REF] and [START_REF] Ben Arous | Bouchaud's model exhibits two aging regimes in dimension one[END_REF], on Z 2 in [START_REF] Ben Arous | Aging for Bouchaud's model in dimension two[END_REF], on Z d (d ≥ 3) in [START_REF] Ben Arous | Scaling limit for trap models on Z d[END_REF], or on the hypercube in [START_REF] Ben Arous | Glauber dynamics of the random energy model. I. Metastable motion on the extreme states[END_REF][START_REF] Ben Arous | Glauber dynamics of the random energy model. II. Aging below the critical temperature[END_REF]. A comprehensive approach to obtaining aging results for the trap model in various settings was later developed in [START_REF] Ben Arous | The arcsine law as a universal aging scheme for trap models[END_REF].

Let us finally mention that Theorem 1 generalizes the aging result obtained by heuristical methods of renormalization by Le Doussal, Fisher and Monthus in [START_REF] Doussal | Random walkers in one-dimensional random environments: Exact renormalization group analysis[END_REF] in the limit case when the bias of the random walk defining the potential tends to 0 (the case when this bias is 0 corresponding to the recurrent regime for the random walk in random environment). The recurrent case, which also leads to aging phenomenon, was treated in the same article and rigorous arguments were later presented by Dembo, Guionnet and Zeitouni in [START_REF] Dembo | Aging properties of Sinai's model of random walk in random environment[END_REF].

The second aspect of our work concerns localization properties of the walk and can be considered as the analog of Sinai's localization result in the transient setting. Unlike the recurrent case, the random walk is not localized near the bottom of a single valley. Nevertheless, if one introduces a confidence threshold α, one can say that, asymptotically, at time t, with a probability converging to 1 on the environment, the walk is localized with probability bigger than α around the bottoms of a finite number of valleys having depth of order log t. This number depends on t and on the environment, but is not converging to infinity with t. Moreover, in Theorem 2 and Corollary 1 we sharply estimate the probability for the walk of being at time t in each of these valleys.

Notation 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 = (X n , n ≥ 0) on Z N 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 , P ω (X n+1 = x -1 | X n = x) = 1 -ω 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. Let us introduce

ρ i := 1 -ω i ω i , i ∈ Z.
Our first main result is the following theorem which shows aging phenomenon in the transient sub-ballistic regime.

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, for all h > 1 and all η > 0, we have Let us now recall some basic result about X n : under the same assumptions (a)-(b), Kesten, Kozlov and Spitzer [START_REF] Kesten | A limit law for random walk in a random environment[END_REF] proved that X n /n κ converges in law to C κ ( 1 S ca κ ) κ where C κ is a positive parameter and S ca κ is the normalized positive stable law of index κ, i.e. with Laplace transform

lim t→∞ P(|X th -X t | ≤ η log t) = sin(κπ) π 1/h 0 y κ-1 (1 -y) -κ dy.
E[e -λS ca κ ] = e -λ κ , ∀λ > 0.
In [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF][START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] we gave a different proof of this result and we were able to give an explicit expression for the constant C κ .

The proof was based on a precise analysis of the potential associated with the environment, as it was defined by Sinai for its analysis of the recurrent case, see [START_REF] Sinai | The limiting behavior of a one-dimensional random walk in a random medium[END_REF].

In this paper, we use the techniques developed in [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF][START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] to prove Theorem 1. The potential, denoted by V = (V (x), x ∈ Z), is a function of the environment ω. It 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 the sequence (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 ] < ∞. (2.1)
Now, observe that the sequence ((e i , e i+1 ]) i≥0 stands 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 )) , i ≥ 0. (2.2)
Note that the (H i ) i≥0 's are i.i.d. random variables.

For t ∈ N, we introduce the critical height (2.3) h t := log tlog log t.

As in [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] we define the deep valleys from the excursions which are higher than the critical height h t . Let (σ(j)) j≥1 be the successive indexes of excursions, whose heights are greater than h t . More precisely, σ(1) := inf{i ≥ 0 :

H i ≥ h t }, σ(j) := inf{i > σ(j -1) : H i ≥ h t }, j ≥ 2.
We consider now some random variables depending only on the environment, which define the deep valleys.

Definition 1. For all j ≥ 1, let us introduce

b j := e σ(j) , a j := sup{k ≤ b j : V (k) -V (b j ) ≥ D t }, T ↑ j := inf{k ≥ b j : V (k) -V (b j ) ≥ h t }, 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 t }.
where D t := (1 + κ) log t. 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.

Moreover, let us introduce the first hitting time of x, denoted by τ (x) := inf{n ≥ 1 :

X n = x}, x ∈ Z,
and the index of the last visited deep valley at time t, defined by

ℓ t := sup{n ≥ 0 : τ (b n ) ≤ t}.
Before stating the quenched localization result, recall that X is defined on the sample probability space Z N . Then, let us introduce e = (e i , i ≥ 1) a sequence of i.i.d. exponential random variables with parameter 1, independent of X. We define e on a probability space Ξ and denote its law by P (e) . In order to express the independence between X and e, we consider for each environment ω, the probability space (Z N × Ξ, P ω × P (e) ) on which we define (X, e).

Furthermore, let us define the weight of the k-th deep valley by

W k (ω) := 2 a k ≤m≤n b k ≤n≤d k e Vω(n)-Vω(m) .
Moreover, let us introduce the following integer, for any t ≥ 0, ℓ

t,ω := sup i ≥ 0 :

i k=1 W k (ω)e k ≤ t .
We are now able to state our second main result. (ii) for all δ > 0, lim

t→∞ P d T V (ℓ t , ℓ (e)
t,ω + 1) > δ = 0, where d T V denotes the distance in total variation. Remark 2. The statement of Theorem 2 could be improved in the following way: the choice of the critical height h t is in some way arbitrary and we could take for h t any positive function such that lim t→∞ h t = ∞ and e ht = o(t). The meaning is that at time t the RWRE is localized at the bottom of a deep valley, deep meaning that its height H is such that e H is of order t. Furthermore, as in Theorem 1, the size of the localization window η log t could be replaced by any positive function a(t) such that lim t→∞ a(t) = ∞.

We remark that we can easily deduce the following quenched localization in probability result by assembling part (i) and part (ii) of Theorem 2. We precise that our quenched localization result is in probability because one should not expect an almost sure result here, since no almost sure quenched limit results are expected to hold, see [START_REF] Peterson | Quenched limits for transient, zero speed one-dimensional random walk in random environment[END_REF]. For y < x, we denote by E x ω the expectation associated with the law P x ω of the particle in the environment ω, started at x. Corollary 1. Under assumptions (a)-(b) of Theorem 1, we have, for all δ, η > 0, that

P i≥1 P 0,ω (|X t -b i | ≤ η log t) -P (e) i-1 k=1 W k (ω)e k ≤ t < i k=1 W k (ω)e k > δ
converges to 0, when t tends to ∞.

The content of this result is twofold. It first says that, with a probability converging to 1, the process at time t is concentrated near the bottom of a valley of depth of order log t. It also determines, for each of these valleys, the probability that, at time t, the particle lies at the bottom of it. This probability is driven by a renewal Poisson process which is skewed by the weights of each of these valleys. This result may be of interest when trying to get information on the environment on the basis of the observation of a sample of trajectories of the particle. See [START_REF] Cocco | Reconstructing a random potential from its random walks[END_REF] for a recent example of this in a paper on DNA reconstruction.

Notation

A result of Iglehart [START_REF] Iglehart | Extreme values in the GI/G/1 queue[END_REF] which will be of constant use, says that, under assumptions (a)-(b) of Theorem 1, the tail of the height H i of an excursion above its past minimum is given by

(3.1) P (H 1 > h) ∼ C I e -κh , h → ∞,
for a positive constant C I (we will not need its explicit value).

The analysis done in [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF][START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] shows that on the interval [0, t], t ∈ N, the walk X n spends asymptotically all its time trying to climb excursions of height of order log t + C for a real C. Let us now introduce the integer n t := ⌊t κ log log t⌋.

The integer n t will be use to bound the number of excursions the walk can cross before time t. The strategy will be to show that we can neglect the time spent between two excursions of size smaller than h t , and to show that at time t the walk X t is close to the foot of an excursion of height larger than h t .

The deep valleys. Let us define the number of deep valleys in the n t first excursions by

K t := sup{j ≥ 0 : σ(j) ≤ n t }, which is the number of excursions higher than the critical height h t in the n t first excursions. [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] with n = n t , but with a different critical height. In [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] the critical height was h n = 1-ε κ log n, for ε such that 0 < ε < 1. Here, we see that h nt would be equal to (1 -ε) log t + 1-ε κ log log log t which is smaller than our critical height h t = log tlog log t. This means that the deep valleys are higher and less numerous in the present paper than in [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF]. We will see that this choice makes possible the control of the localization of the particle in any neighborhood of size η log t around the bottom of the last visited valley (recall Part (i) of Theorem 2).

Remark 3. This definition corresponds to the definition of deep valleys introduced in

3.2.

The * -valleys. Let us first 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.
The deep valleys defined above are not necessarily made of disjoint portions of the environment. To overcome this difficulty we defined another type of valleys, called * -valleys, which form a subsequence of the previous valleys. By construction, the * -valleys are made of disjoint portions of environment and will coincide with high probability with the previous valleys on the portion of the environment visited by the walk before time t.

γ * 1 := inf{k ≥ 0 : V (k) ≤ -D t }, T * 1 := inf{k ≥ γ * 1 : V ↑ (γ * 1 , k) ≥ h t }, 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 t }, 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 t }.
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 * t the number of such * -valleys before e nt , i.e. K * t := sup{j ≥ 0 :

T * j ≤ e nt }.
The * -valleys will be made of independent and identically distributed portions of potential (up to some translation).

Preliminary estimates

4.1. Good environments. We define in this subsection the good environments in the same manner as we did in [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] to give a complete characterization of the limit law. Since the critical height considered here is not the same (see Remark 3), the following results are not taken from [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] but proved with the same ideas, that we recall in this subsection. Let us introduce the following series of events, which will occur with high probability when t tends to infinity.

A 1 (t) := {e nt ≤ C ′ n t } , A 2 (t) := K t ≤ (log t) 1+κ 2 , A 3 (t) := ∩ Kt j=0 σ(j + 1) -σ(j) ≥ t κ/2 , A 4 (t) := ∩ Kt+1
j=1 {d j -a j ≤ C ′′ log t} , where σ(0) := 0 (for convenience of notation) and C ′ , C ′′ stand for positive constants (large enough) which will be specified below. In words, A 1 (t) bounds the total length of the first n t excursions. The event A 2 (t) gives a control on the number of deep valleys while A 3 (t) ensures that they are well separated and A 4 (t) bounds finely the length of each of them.

Lemma 1. Let A(t) := A 1 (t) ∩ A 2 (t) ∩ A 3 (t) ∩ A 4 (t), then lim t→∞ P (A(t)) = 1.
Proof. The fact that P (A 1 (t)) → 1 is a consequence of the law of large numbers. Concerning A 2 (t) and A 3 (t), we know that the number of excursions higher than h t in the first n t excursions is a binomial random variable with parameter (n t , q t ) where q t := P (H 1 ≥ h t ), from which we can easily deduce that P (A 2 (t) ∩ A 3 (t)) → 1. For example, since (3.1) implies q t ∼ C I e -κht , t → ∞, we have that E [K t ] = n t q t ∼ C I log log t(log t) κ . Using the Markov inequality we get that P (A 2 (t)) tends to 1, when t tends to infinity.

The proof for A 4 (t) requires a bit more explanations. Since K t ≤ (log t) with probability tending to one, we only have to prove, for j ≥ 1 that P (d j -a j ≥ C ′′ log t) = o((log t) -1+κ

2 ). Furthermore, observe that we can write d j -a j = (d j -

d j ) + (d j -T ↑ j ) + (T ↑ j -b j ) + (b j -a j )
. Therefore, the proof boils down to showing that, for each term in the previous sum, the probability that it is larger than C ′′ 4 log t is a o((log t) -1+κ

2 ). Here, we only prove that (4.1)

P (T ↑ j -b j ≥ C ′′ 4 log t) = o((log t) -1+κ 2 ), t → ∞,
the arguments for the other terms being similar and the results more intuitive.

Let us first introduce T h := inf{x ≥ 0 : V (x) ≥ h} for any h > 0. Then, recalling (3.1), we can write (4.2)

P (T ↑ j -b j ≥ C ′′ 4 log t) ≤ Ce κht P ( C ′′ 4 log t ≤ T ht < ∞).
Denoting by I(•) the convex rate function associated with the potential, we apply Chebychev's inequality in the same manner as is done in the proof of the upper bound in Cramer's theorem (see [START_REF] Hollander | Large Deviations[END_REF]) and obtain that the probability on the right-hand side in (4.2) is bounded above by

k≥ C ′′ 4 log t P (V (k) ≥ h t ) ≤ k≥ C ′′ 4 log t e -k I( h t k ) ≤ k≥ C ′′ 4 log t e -k I(0) ≤ Ct -C ′′ 4 I(0) . (4.3)
Now, let us recall that h t ≤ log t by definition. Morever, observe that the assumption (a) implies that E [ρ κ 0 ] = 1, which yields I(0) > 0. Then, assembling (4.2) and (4.3) yields (4.1) by choosing C ′′ larger than 4κ/I(0), which concludes the proof of Lemma 1.

The following lemma tells us that the * -valleys, which are i.i.d., coincide with the sequence of deep valleys with an overwhelming probability when t goes to infinity.

Lemma 2. If A * (t) := {K t = K * t ; (a j , b j , c j , d j ) = (a * j , b * j , c * j , d * j ), 1 ≤ j ≤ K * t }
, then we have that the probability P (A * (t)) converges to 1, when t goes to infinity.

Proof. By definition, the * -valleys constitute a subsequence of the deep valleys, and A * (t) occurs as soon as the valleys (a j , b j , c j , d j ) are disjoint for 1 ≤ j ≤ K t . Hence, we see that A 3 (t) ∩ A 4 (t) ⊂ A * (t). Then, Lemma 2 is a consequence of Lemma 1.

Directed traps.

Let us first recall that it is well-known (see for example [START_REF] Zeitouni | Random Walks in Random Environment, XXXI summer school in probability[END_REF], formula (2.1.4)) that for r < x < s, (4.4)

P x ω (τ (r) < τ (s)) = s-1 j=x e V (j) s-1 j=r e V (j) -1
.

Moreover, we introduce here the inter-arrival times, defined, for any x, y ∈ Z, by

τ (x, y) := inf{k ≥ 0 : X τ (x)+k = y}.
With the two following lemmas, we prove that the particle never backtracks before a j after reaching the bottom b j of the j-th valley, uniformly in 1 ≤ j ≤ K t , and that it visits each of them only once.

Lemma 3. Defining DT (t) := A(t) ∩ Kt j=1 τ (d j , b j+1 ) < τ (d j , d j ) , we have lim t→∞ P(DT (t)) = 1. Proof. Recalling that K t ≤ (log t) 1+κ 2
with probability tending to one, we have to prove, for j ≥ 1, that

E[1 A(t)∩{j≤Kt} P ω (τ (d j , b j+1 ) > τ (d j , d j ))] = o((log t) -1+κ
2 ), when t tends to infinity. Therefore, applying the strong Markov property at τ (d j ), we need to prove that

E[1 A(t)∩{j≤Kt} P d j ω (τ (b j+1 ) > τ (d j ))] = o((log t) -1+κ 2 ), t → ∞. (4.5)
By (4.4) we get that P

d j ω τ (b j+1 ) > τ (d j ) is bounded by (b j+1 -d j )e V (d j )-V (d j )+ht . Observe first that b j+1 -d j ≤ e nt ≤ C ′ n t on A(t) ∩ {j ≤ K t }. Then, recalling that V (d j ) -V (d j ) ≤ -D t
by definition (where D t = (1 + κ) log t) together with h t ≤ log t yields (4.5) and concludes the proof of Lemma 3.

Lemma 4. Defining DT * (t) := K * t j=1 τ (b * j , d * j ) < τ (b * j , a * j ) , we have lim t→∞ P(DT * (t)) = 1.
Proof. We omit the details here since the arguments are very similar to the proof of Lemma 3.

Finally, we need to know that the time spent between the deep valleys is small. Let us first recall the following technical result proved in Lemma 7 of [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF]. Lemma 5. Let T ↑ be defined by T ↑ (h) := inf{x ≥ 0 : V ↑ (x) ≥ h}, for any h ≥ 0. Then, there exists C > 0 such that, for all h,

E |0 τ (T ↑ (h) -1) ≤ Ce h ,
where E |0 denotes the expectation under the annealed law P |0 associated with the random walk in random environment reflected at 0. Now, we can prove that the time spent by the particle between the K t first deep valleys is negligible with respect to t with an overwhelming probability when t goes to infinity, which is the statement of the following lemma. Lemma 6. Let us introduce the following event

IA(t) := A(t) ∩ τ (b 1 ) + Kt j=1 τ (d j , b j+1 ) < t log log t .
Then, we have

lim t→∞ P(IA(t)) = 1.
Proof. Recalling Lemma 1, Lemma 3 and using the Markov inequality, we only need to prove that

E[1 A(t)∩DT (t) (τ (b 1 ) + Kt j=1 τ (d j , b j+1 ))] is o( t log log t
), when t goes to infinity. For y < x, let us denote by E x ω,|y the expectation associated with the law P x ω,|y of the particle in the environment ω, started at x and reflected at site y. Then, applying the strong Markov property at times τ (d Kt ), . . . , τ (d 1 ), we get that the above expectation is smaller than

E[1 A(t)∩DT (t) τ (b 1 )] + E 1 A(t)∩DT (t) Kt j=1 E d j ω,|d j [τ (b j+1 )] , (4.6)
since (X τ (d j )+n ) n≥0 under P ω has the same law as (X n ) n≥0 under P d j ω,|d j on A(t)∩DT (t). Concerning the second term of (4.6), we apply the strong Markov property for the potential at times d Kt , . . . , d 1 , such that we get

E 1 A(t)∩DT (t) Kt j=1 E d j ω,|d j [τ (b j+1 )] ≤ (log t) 1+κ 2 E |0 τ (T ↑ (h t ) -1) ≤ C(log t) 1+κ 2 e ht ≤ Ct(log t) -1-κ
2 , the second inequality being a consequence of Lemma 5. Now, let us mention that the bound Ce ht can be obtained in a similar way for the first term of (4.6), which yields that the expression in (4.6) is a o( t log log t ), when t tends to infinity and concludes the proof of Lemma 6.

4.3.

Localization in deep traps. In a first step, we state a technical result which ensures that the potential does not have excessive fluctuations in a typical box and which will be very useful to control the localization of the particle in a valley.

Lemma 7. If F γ (t) := ∩ Kt j=1 max{V ↑ (a j , b j ) ; -V ↓ (b j , c j ) ; V ↑ (c j , d j )} ≤ γ log t , then we have, for any γ > 0, lim t→∞ P(F γ (t)) = 1.
Proof. Observe first that Lemma 14 in [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] implies that, for all ε > 0, the valleys with height larger that (1 -ε) log t + 1-ε κ log log log t have fluctuations bounded by γ log t, with a probability tending to one, for any γ > ε/κ. Now, since h t is larger than (1 -ε) log t + 1-ε κ log log log t for any ε > 0 (see Remark 3), the deep valleys considered here are included in the valleys treated by Lemma 14 in [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] for any ε > 0, which concludes the proof of Lemma 7.

For each deep valley, let us introduce the position c i defined by

c i := inf{n ≥ c i : V (n) ≤ V (c i ) -h t /3}.
We first need to know that during its sojourn time inside a deep valley, the random walk spends almost all its time inside the interval (a i , c i ). This is a consequence of the following lemma.

Lemma 8. Let LT (t) be the event

LT (t) := Kt i=1 τ (c i , d i ) ≤ t log t .
Then,

lim t→∞ P(LT (t)) = 1.
This result just means that at the time scale t, if the walk reaches c i , then soon after it exits the deep valley (a i , d i ).

Proof. Recalling Lemma 1 and Lemma 7, we only have to prove that

P τ (c j , d j ) > t log t ; A 4 (t) ; F γ (t) ; j ≤ K t = o((log t) -1+κ 2 ), t → ∞,
for any j ≥ 1. Now, applying the strong Markov property at τ (c j ), we get that the previous probability is bounded by

E 1 A 4 (t)∩Fγ (t)∩{j≤Kt} P c j ω,|c j (τ (d j ) > t/ log t) + P c j ω (τ (c j ) < τ (d j )) .
Concerning the first term, we use the fact that E c j ω,|c j [τ (d j )] ≤ c j ≤u≤v≤d j e V (v)-V (u) (see (A1) in [START_REF] Golosov | Localization of random walks in one-dimensional random environments[END_REF]) and Chebychev inequality, such that we obtain (4.7)

P c j ω,|c j (τ (d j ) > t/ log t) ≤ log t t c j ≤u≤v≤d j e V (v)-V (u) ≤ C ′′ (log t) 2 t e γ log t , on A 4 (t)∩F γ (t)∩{j ≤ K t }.
For the second term, by (4.4) we obtain that the probability P

c j ω (τ (c j ) < τ (d j )) is less than (4.8) d j -1 k=c j e V (k)   d j -1 k=c j e V (k)   -1 ≤ (d j -c j ) e V (c j )+γ log t-V (c j ) ≤ C ′′ (log t) e γ log t-h t 3 , on A 4 (t) ∩ F γ (t) ∩ {j ≤ K t }.
O i := [a i + 1, c i -1] \ (b i -η log t + 1, b i + η log t -1), i ∈ N,
and the event

A 5 (t, η) := Kt i=1 min k∈O i ∩Z (V (k) -V (b i )) ≥ C ′′′ η log t ,
for a constant C ′′′ (small enough and independent of η) to be defined later. Then, we have the following result.

Lemma 9. For all η > 0, lim t→∞

P (A 5 (t, η)) = 1.
Proof. Observe first that if η > C ′′ , then the sets (O i , 1 ≤ i ≤ K t ) are empty on A 4 (t). Therefore, Lemma 9 is a consequence of Lemma 1.

Now, let us assume η ≤ C ′′ . The definition of c i implies that min

c i ≤k<c i (V (k) - V (b i )) ≥ 2 3 h t .
Then, choosing C ′′′ such that C ′′′ C ′′ < 2/3 implies that C ′′′ η log t < 2 3 h t for all large t, which yields

P Kt i=1 min c i ≤k<c i (V (k) -V (b i )) ≥ C ′′′ η log t = 1, (4.9)
for all large t. Then, let us introduce the sets

O ′ i := O i ∩ [b i , c i ], O ′′ i := O i ∩ [a i , b i ], i ∈ Z,
and the events

A ′ 5 (t, η) := Kt i=1 min k∈O ′ i ∩Z (V (k) -V (b i )) ≥ C ′′′ η log t , A ′′ 5 (t, η) := Kt i=1 min k∈O ′′ i ∩Z (V (k) -V (b i )) ≥ C ′′′ η log t .
Now, recalling (4.9), the proof of Lemma 9 boils down to showing that lim t→∞ P (A ′ 5 (t, η)) = 1, (4.10)

lim t→∞ P (A ′′ 5 (t, η)) = 1. (4.11)
Let us first prove (4.10). Recalling Lemma 1 and Lemma 7, we only need to prove that it is possible to choose C ′′′ small enough such that for some γ > 0

P min k∈O ′ 1 ∩Z (V (k) -V (b 1 )) < C ′′′ η log t ; F γ (t) = o((log t) - 1+κ
2 ), (4.12) when t → ∞. Now recalling assumption (a) of Theorem 1 and denoting by µ the law of log ρ 0 , 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. Now, let us simplify the notation by writing

H := H 0
(where H 0 is the height of the first excursion defined by H 0 := max 0≤k≤e 1 V (k)) and define the hitting time of level h for the potential by

T h := min{x ≥ 0 : V (x) ≥ h}, h > 0.
Then, introducing Fγ (t) := -V ↓ (0, T H ) ≤ γ log t , we can write that the probability term in (4.12) is smaller than

P min ⌊η log t⌋≤k≤T H V (k) < C ′′′ η log t ; Fγ (t) | H ≥ h t ≤ Ce κht P min ⌊η log t⌋≤k≤T H V (k) < C ′′′ η log t ; Fγ (t) ; H ≥ h t = C Ẽ e -κ(V (T H )-ht) 1 {min ⌊η log t⌋≤k≤T H V (k)<C ′′′ η log t ; Fγ (t) ; H≥ht} ≤ C P min ⌊η log t⌋≤k≤T H V (k) < C ′′′ η log t ; Fγ (t) ; H ≥ h t , (4.13)
the first inequality being a consequence of (3.1) and the equality deduced from Girsanov property. Now, let us introduce α = α(η) := cη with c satisfying 0 < c < min{ Ẽ [V (1)] ; 1/C ′′ } and γ = γ(η) := cη/2. Observe that α log t < h t for all large t, so that T α log t ≤ T ht ≤ T H < ∞ on {H ≥ h t }. Now since c < Ẽ [V (1)] , we use Chebychev's inequality in the same manner as is done in the proof of the upper bound in Cramer's theorem, see [START_REF] Hollander | Large Deviations[END_REF], and obtain that P (V (⌊η log t⌋) < α log t) ≤ C exp{-η Ĩ(c) log t} = o((log t) -1+κ

2 ), where Ĩ(•) denotes the convex rate function associated with V under P . This yields P (T α log t ≤ ⌊η log t⌋) = 1 -o((log t) -1+κ

2 ), when t tends to infinity. Therefore, we get P min Furthermore, observe that on Fγ (t), we have min T α log t ≤k≤T H V (k) ≥ (α-γ) log t, which yields min T α log t ≤k≤T H V (k) ≥ C ′′′ η log t, if we choose C ′′′ smaller than c/2. Therefore, for C ′′′ small enough (independently of η ≤ C ′′ ), we get that the probability term in (4.14) is null for all large t. Now, assembling (4.13) and (4.14) concludes the proof of (4.10).

⌊η log t⌋≤k≤T H V (k) < C ′′′ η
The proof of (4.11) is similar but easier. Indeed, we do not have to use Girsanov property to study the potential on [a i , b i ].

Two versions of a Dynkin type renewal result

We define the sequence of random times (τ * i ) i≥1 as follows: conditioning on the environment ω, (τ * i ) i≥1 is defined as an independent sequence of random variables with the law of τ (d

:= sup{n ≥ 0 : τ * 1 + • • • + τ * n ≤ t}. For all 0 ≤ x 1 < x 2 ≤ 1, we have lim t→∞ P(t(1 -x 2 ) ≤ τ * 1 + • • • + τ * ℓ * t ≤ t(1 -x 1 )) = sin(κπ) π x 2 x 1 (1 -x) κ-1 x -κ dx.
For all 0 ≤ x 1 < x 2 , we have

lim t→∞ P(t(1 + x 1 ) ≤ τ * 1 + • • • + τ * ℓ * t +1 ≤ t(1 + x 2 )) = sin(κπ) π x 2 x 1 dx x κ (1 + x)
.

Observe that the result would exactly be Dynkin's theorem (cf e.g. Feller, vol II, [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF], p. 472) if the sequence (τ * i ) i≥1 was an independent sequence of random variables in the domain of attraction of a stable law with index κ. Here, the sequence (τ * i ) i≥1 implicitly depends on the time t, since the * -valleys are defined from the critical height h t . We will use the main intermediate result of [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] which gives an estimate of the Laplace transform of τ * 1 at 0. We deduce from Corollary 2 and Remark 7 of [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] the following lemma.

Lemma 10. We have

E 1 -e -λ τ * 1 t ∼ 2 κ πκ sin(πκ) C U t κ P (H ≥ h t ) λ κ , t → ∞,
for all λ > 0.

Proof. We apply Corollary 2 of [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] to n = ⌊t κ ⌋ and h n = h t = log tlog log t which satisfies the condition of Remark 7 of [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF]. The constant C U was made explicit in [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF] but we will not need this value here.

For the convenience of the reader, we give a brief idea of the arguments of the proof of this formula. Let us simply write (a, b, c, d)

for (a * 1 , b * 1 , c * 1 , d * 1
). The time it takes to cross the valley can be decomposed in a geometric number of unsuccessful attempts and a successful attempt, hence we can write

τ * 1 = τ (b, d) = F 1 + • • • + F N + S,
where N is a geometric random variable with parameter

1 -p(ω) := P b ω (τ (d) < τ + (b)) = ω b e V (b) d-1 x=b e V (x)
, where τ + (b) := inf{n > 0 : X n = b}. The random variables (F i ) i≥1 are i.i.d. and distributed as τ + (b) under

P b ω ( • |τ + (b) < τ (d)) and S is distributed as τ (d) under P b ω ( • |τ (d) < τ + (b))
. The first step is to prove that the successful attempt S can be neglected (this is done in [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] using some estimates on h-processes). Thus, we can write

E e -λ τ * 1 t ∼ E 1 -p(ω) 1 -p(ω)E ω e -λ t F 1 , t → ∞.
The second step is to linearize E ω [e -λ t F 1 ], i.e. to show that it can be replaced by (1

-λ t E ω [F 1 ]
) ( using again estimates on h-processes). This leads to

E e -λ τ * 1 t ∼ E 1 1 + λ t p(ω) 1-p(ω) E ω [F 1 ] , t → ∞.
Then we prove that p(ω)

1-p(ω) E ω [F 1 ] is of order Z = 2e H M 1 M 2 ,
where M 1 and M 2 are defined by M 1 := c k=a e -(V (k)-V (b)) and M 2 := d k=b e V (k)-V (c) . Then, we use the main result of [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF], where the tail estimate of Z is obtained (see Theorem 2.2).

Proof of Proposition 1. The arguments are essentially the same as in [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]. Let us introduce S * 0 = 0 and S * n := n i=1 τ * i , for n ≥ 1. Then, the inequality t(1

-x 2 ) ≤ τ * 1 + • • • + τ ℓ * t ≤ t(1 -x 1
) occurs iff S * n = ty and τ * n+1 > t(1 -y) for some combination n, y such that 1 -x 2 < y < 1 -x 1 . Summing over all n and possible y we get

P(t(1 -x 2 ) ≤ S * ℓ * t ≤ t(1 -x 1 )) = 1-x 1 1-x 2 G t (1 -y) P (H ≥ h t ) U t {dy}, (5.1)
where G t (x) := P (H ≥ h t )P(t -1 τ * 1 ≥ x), and U t {dx} denotes the measure associated with U t (x) := n≥0 P(t -1 S * n ≤ x). We introduce the measure dH t (u) such that

∞ x dH t (u) = G t (x)
, for all x ≥ 0. Lemma 11. For any x > 0, we have

lim t→∞ x κ t κ G t (x) = 2 κ Γ(1 + κ)C U . (5.2)
Moreover, the convergence is uniform on any compact set.

Proof. In a first step, observe that

E[1 -e -λ τ * 1 t ] = P (H ≥ h t ) -1 ∞ 0 (1 -e -λu ) dH t (u). Recalling Lemma 10, we obtain lim t→∞ t κ ∞ 0 (1 -e -λu ) dH t (u) = 2 κ Γ(1 + κ)C U Γ(1 -κ)λ κ . Since Γ(1 -κ)λ κ = λ ∞ 0 e -λu u -κ du, this implies lim t→∞ t κ ∞ 0 (1 -e -λu ) dH t (u) = 2 κ Γ(1 + κ)C U λ ∞ 0 e -λu u -κ du. (5.3)
On the other hand, integrating by parts, we get, for any t ≥ 0,

∞ 0 (1 -e -λu ) dH t (u) = λ ∞ 0 e -λu G t (u) du.
(5.4) Combining (5.3) and (5.4) implies that the measure t κ G t (u) du tends to the measure with density 2 κ Γ(1 + κ)C U u -κ . Therefore, we have for all x ≥ 0,

lim t→∞ t κ x 0 G t (u) du = 2 κ Γ(1 + κ)C U x 1-κ 1 -κ , (5.5) which yields lim ε→0 lim t→∞ (1+ε)x x G t (u) du ε x 0 G t (u) du = 1 -κ. (5.6)
Moreover, observe that the monotonicity of G t (•) implies

xG t ((1 + ε)x) x 0 G t (u) du ≤ (1+ε)x x G t (u) du ε x 0 G t (u) du ≤ xG t (x) x 0 G t (u) du . (5.7)
Now, combining (5.6) and (5.7), we obtain lim inf

t→∞ xG t (x) x 0 G t (u) du ≥ 1 -κ. Recalling (5.5), this yields lim inf t→∞ x κ t κ G t (x) ≥ 2 κ Γ(1 + κ)C U . (5.8)
Similarly, we obtain, for any ε > 0, lim sup

t→∞ x κ t κ G t ((1 + ε)x) ≤ 2 κ Γ(1 + κ)C U .
(5.9) Assembling (5.8) and (5.9) and letting ε → 0 conclude the proof of (5.2). Furthermore, observe that the uniform convergence on any compact set is a consequence of the monotonicity of x → G t (x), the continuity of the limit and Dini's theorem.

Lemma 12. The measure P (H≥ht) -1 t κ U t {dx} converges vaguely to the measure

1 Γ(κ)Γ(1 + κ)Γ(1 -κ)2 κ C U x κ-1 dx.
Proof. Observe first that the Laplace transform

U t (λ) := ∞ 0 e -λu U t {du} satisfies U t (λ) = n≥0 E[e -λ S * n t ] = (1 -E[e -λ τ * 1 t ]) -1 . Therefore, Lemma 10 yields lim t→∞ P (H ≥ h t ) -1 t κ U t (λ) = λ -κ Γ(1 + κ)Γ(1 -κ)2 κ C U .
Furthermore, since Γ(κ)λ -κ = ∞ 0 e -λu u κ-1 du, we deduce the vague convergence of the measure from the pointwise convergence of the Laplace transforms. Now, recalling (5.1), we observe that Lemma 11 together with Lemma 12 imply lim t→∞

P(t(1 -x 2 ) ≤ S * ℓ * t ≤ t(1 -x 1 )) = 1 Γ(κ)Γ(1 -κ) 1-x 1 1-x 2
(1 -y) -κ y κ-1 dy, = sin(κπ) π

x 2

x 1

(1 -y) κ-1 y -κ dy.

This concludes the proof of the first part of Proposition 1. The second part of Proposition 1 is obtained using similar arguments.

Recall Lemma 6 which tells that the inter-arrival times are negligible. Now, we will prove that the results of Proposition 1 are still true if we consider, in addition, these inter-arrival times. Let δ 1 := τ (b 1 ), τ 1 := τ (b 1 , d 1 ) and

δ k := τ (d k-1 , b k ), τ k := τ (b k , d k ), k ≥ 2.
Moreover, we set

T k := δ 1 + τ 1 + • • • + τ k-1 + δ k , k ≥ 1, the entering time in the k-th deep valley. Proposition 2. Recall ℓ t = sup{n ≥ 0 : τ (b n ) ≤ t}. Then, we have P(T ℓt ≤ t < T ℓt + τ ℓt ) → 1, t → ∞.
For all 0 ≤ x 1 < x 2 ≤ 1, we have

lim t→∞ P(t(1 -x 2 ) ≤ T ℓt ≤ t(1 -x 1 )) = sin(κπ) π x 2 x 1 (1 -x) κ-1 x -κ dx.
For all 0 ≤ x 1 < x 2 , we have

lim t→∞ P(t(1 + x 1 ) ≤ T ℓt+1 ≤ t(1 + x 2 )) = sin(κπ) π x 2 x 1 dx x κ (1 + x) .
Proof. On the event A(t) ∩ DT * (t), we know that the random times (τ i ) 1≤i≤K * t have the same law as the random times (τ * i ) 1≤i≤K * t defined in Section 5. If we define lt := sup{n ≥ 0 : τ 1 + • • • + τ n ≤ t}, then, using Proposition 1 and Lemma 3, we get that the result of Proposition 1 is true with τ and lt in place of τ * and ℓ * t . Now, using Lemma 6 we see that

lim inf t→∞ P( lt = ℓ t -1 ; T ℓt ≤ t < T ℓt + τ ℓt ) ≥ lim inf t→∞ P(IA(t) ; |t -(τ 1 + • • • + τ lt )| ≥ ξt),
for all ξ > 0. Thus, using Proposition 1 (for lt and τ i ) and letting ξ tends to 0, we get that lim

t→∞ P( lt = ℓ t -1 ; T ℓt ≤ t < T ℓt + τ ℓt ) = 1.
We conclude the proof by the same type of arguments.

Proof of part (i) of Theorem 2: a localization result

We follow the strategy developed by Sinai for the recurrent case. For each valley we denote by π i the invariant measure of the random walk on [a i , c i ] in environment ω, reflected at a i and c i and normalized so that π i (b i ) = 1. Clearly, π i is the reversible measure given, for k ∈

[b i + 1, c i -1], by π i (k) = ω b i 1 -ω b i +1 • • • ω k-1 1 -ω k = ω b i ρ -1 b i +1 • • • ρ -1 k-1 (ρ -1 k + 1) ≤ e -(V (k)-V (b i )) + e -(V (k-1)-V (b i )) . Similarly, π i (k) ≤ e -(V (k)-V (b i )) + e -(V (k+1)-V (b i )) for k ∈ [a i + 1, b i -1]
. Since the walk is reflected at a i and c i , we have π i (a i ) = e -(V (a i +1)-V (b i )) and π i (c i ) = e -(V (c i -1)-V (b i )) . Hence on the event A 5 (t, η) we have

sup{π i (k) ; k ∈ [a i , c i ] \ (b i -η log t, b i + η log t)} ≤ Ce -C ′′′ η log t = Ct -C ′′′ η .
Moreover, since π i is an invariant measure and since π i (b i ) = 1, we have, for all k ≥ 0,

P b i ω,|a i ,c i | (X k = x) ≤ π i (x)
. Hence, on the event A(t) ∩ A 5 (t, η) we have, for all k ≥ 0,

P b i ω,|a i ,c i | (|X k -b i | > η log t) ≤ C(log t)t -C ′′′ η . (6.1)
Let ξ be a positive real, 0 < ξ < 1. Then, let us write

lim inf t→∞ P(|X t -b ℓt | ≤ η log t) ≥ lim inf t→∞ P(|X t -b ℓt | ≤ η log t ; ℓ t = ℓ t(1+ξ) ) ≥ lim inf t→∞ P(ℓ t = ℓ t(1+ξ) ) -lim sup t→∞ P(|X t -b ℓt | > η log t ; ℓ t = ℓ t(1+ξ) ).
Considering the first term, we get by using Proposition 2, lim inf t→∞

P(ℓ t = ℓ t(1+ξ) ) = lim inf t→∞ P(T ℓt+1 > t(1 + ξ)) = sin(κπ) π ∞ ξ dx x κ (1 + x) . (6.2)
In order to estimate the second term, let us introduce the event

T T (t) := A(t) ∩ A 5 (t, η) ∩ DT (t) ∩ DT * (t) ∩ A * (t) ∩ IA(t) ∩ LT (t) ∩ IT (t),
where IT (t) := {T ℓt ≤ t < T ℓt + τ ℓt }. Observe that the preliminary results obtained in Section 4 together with Proposition 2 imply that P(T T (t)) → 1, when t → ∞. Then, we have lim sup

t→∞ P(|X t -b ℓt | > η log t ; ℓ t = ℓ t(1+ξ) ) ≤ lim sup t→∞ P(T T (t) ; |X t -b ℓt | > η log t ; ℓ t = ℓ t(1+ξ) ) ≤ lim sup t→∞ E 1 T T (t) Kt i=1 1 {|Xt-b i |>η log t ; ℓt=ℓ t(1+ξ) =i} .
But on the event T T (t) ∩ {ℓ t = ℓ t(1+ξ) = i} we know that for all k ∈ [T i , t] the walk X k is in the interval [a i , c i -1]. Indeed, on the event LT (t) ∩ DT (t) ∩ IA(t) we know that once the position c i is reached then within a time t/ log t the position b i+1 is reached which would contradict the fact that ℓ t(1+ξ) = i. Hence, we obtain, for all i ∈ N,

P T T (t) ; i ≤ K t ; |X t -b i | > η log t ; ℓ t = ℓ t(1+ξ) = i ≤ E 1 {i≤Kt} 1 A(t)∩A 5 (t,η) sup k∈[0,t] P b i ω,|a i ,c i | (|X k -b i | > η log t) ≤ C(log t)t -C ′′′ η ,
where we used the estimate (6.1) on the event A(t) ∩ A 5 (t, η). Considering now that, on the event A(t), the number K(t) of deep valleys is smaller than (log t)

i := - 1 log(p i (ω)) e i ,
where 1 -p i (ω) denotes the probability for the random walk starting at b i to go to d i before returning to b i , which is equal to ω b e V (b i ) P d i -1

x=b i e V (x) . The parameter of this geometric law is now clearly equal to 1 -p i (ω). Now one can introduce like in [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] two random variables F (i) (resp. S (i) ) whose law are given by the time it takes for the random walk reflected at a i , starting at b i , to return to b i (resp. to hit d i ) conditional on the event that d i (resp. b i ) is not reached in between.

We introduce now a sequence of independent copies of F (i) we denote by (F i) which is going now to be compared with E ω [τ * i ]e i . Let us now estimate, for a given ξ > 0 (small enough),

(i) n ) n≥0 . The law of τ * i is clearly the same as F (i) 1 + • • • + F (i) N i + S (
P ∀i ≤ K t , (1 -ξ)(F (i) 1 + • • • + F (i) N i + S (i) ) ≤ E ω [τ * i ]e i < (1 + ξ)(F (i) 1 + • • • + F (i) N i + S (i) ) ≥ P ∀i ≤ K t , (1 -ξ 2 )(F (i) 1 + • • • + F (i) N i ) ≤ E ω [τ * i ]e i < (1 + ξ 2 )(F (i) 1 + • • • + F (i) N i ) -P ∃i ≤ K t , S (i) > ξ 3 (F (i) 1 + • • • + F (i) N i ) . (7.1)
Let us first treat the second quantity of the rhs of (7.1). For this purpose, we need an upper bound for E ω [S (i) ] which is obtained exactly like in Lemma 13 of [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] and can be estimated by controlling the size of the falls (resp. rises) of the potential during its rises from V (b i ) to V (c i ) (resp. falls from V (c i ) to V (d i )), see Lemma 7. Indeed, the random variable S (i) concerns actually the random walk which is conditioned to hit d i before b i . Therefore, this involves an h-process which can be viewed as a random walk in a modified potential between b i and d i . This modified potential has a decreasing trend (which encourages the particle to go to the right), and the main contribution to S (i) comes from the small risings of this modified potential along its global fall. More precisely, the particle starting at b i which is conditioned to hit d i before returning to b i moves like a particle in the modified random potential V (i) defined as follows: for all b i ≤ x < y ≤ d i ,

V (i) (y) -V (i) (x) = (V (y) -V (x)) + log g (i) (x) g (i) (x + 1) g (i) (y)g (i) (y + 1) , (7.2)
where g (i) (x) := P x ω (τ (d i ) < τ (b i )). The expectation of S (i) is given by the usual formula (see [START_REF] Zeitouni | Random Walks in Random Environment, XXXI summer school in probability[END_REF]), so that

E ω [S (i) ] ≤ 1 + d i k=b i +1 d i l=k e V (i) (l)-V (i) (k) .
We are therefore concerned by the largest rise of V (i) inside the interval [b i , d i ]. We first notice that, by standard arguments, for any b

i ≤ x < y ≤ d i , g (i) (x) g (i) (x + 1) g (i) (y) g (i) (y + 1) = x-1 j=b i e V (j) x j=b i e V (j)
y-1

j=b i e V (j) y j=b i e V (j) ≤ 1. (7.3) Therefore, we obtain for any b

i ≤ x < y ≤ d i V (i) (y) -V (i) (x) ≤ V (y) -V (x). (7.4)
This allows to bound the largest rise of V (i) on the interval [c i , d i ] by the largest rise of V on this interval.

Concerning the largest rise of V (i) on the interval [b i , c i ], we notice, taking into account the small size of the fluctuations of V described in Lemma 7, (7.3) and (7.4), that for all η > 0, for all ω ∈ A 4 (t) ∩ F η (t), and for all i ≤ K t , the difference

V (i) (y) -V (i) (x) is less than or equal to [V (y) -max b i ≤j≤y V (j)] -[V (x) -max b i ≤j≤x V (j)] + O(log log t) ≤ η log t + O(log log t).
This reasoning yields for all η > 0 that, for all ω

∈ A 4 (t) ∩ F η (t), ∀i ≤ K t , E ω [S (i) ] ≤ t η .
This implies, by the Markov inequality, that, for all η > 0 and all ω ∈ A 4 (t) ∩ F η (t), ∀i ≤ K t , P ω (S (i) > t 2η ) < 1 t η . On the other hand, we have

P ω (F (i) 1 + • • • + F (i) N i < t 2η ) ≤ P ω (N i < t 2η ) = 1 -p i (ω) ⌊t 2η ⌋ = O t 2η log t t ,
the last equality coming from the definition of h t := log tlog(log t), which implies that 1 -p i (ω) is smaller than log t t . Hence, since A 2 (t) = {K t ≤ (log t) 1+κ 2 } satisfies P (A 2 (t)) → 1 (see Lemma 1), we obtain

lim t→+∞ P ∀i ≤ K t , P ω (F (i) 1 + • • • + F (i) N i < t 2η ) ≤ 1 t 1 2 -2η = 1.
Gathering these two informations on S (i) and F

(i) 1 + • • • + F (i) N i , we obtain lim t→+∞ P ∀i ≤ K t , S (i) < ξ 3 (F (i) 1 + • • • + F (i) N i ) = 1,
for all ξ > 0, which treats the second quantity of the rhs of (7.1).

The first quantity of the rhs of (7.1) is treated by going through

P (1 - ξ 4 )N i E ω [F (i) ] ≤ F (i) 1 + • • • + F (i) N i ≤ (1 + ξ 4 )N i E ω [F (i) ] ,
which, for all η > 0, is larger than

1-P F (i) 1 + • • • + F (i) N i N i -E ω [F (i) ] > ξ 4 E ω [F (i) ] ∩ {N i = 0} ∩ {E ω [(F (i) ) 2 ] ≤ t η } -P (E ω [(F (i) ) 2 ] ≥ t η ),
which is in turn, using the Bienaimé-Chebychev's inequality, larger than

1 -E E( t η N i 1 {N i =0} | N i ) 16 ξ 2 E ω [F ] 2 -P (E ω [(F (i) ) 2 ] ≥ t η ) ≥ 1 - 16t η ξ 2 E 1 N i 1 {N i =0} -P (E ω [(F (i) ) 2 ] ≥ t η ).
Now, we use again the reasoning based on h-processes to get an upper bound for E ω [(F (i) ) 2 ]. Like in the success case, the particle starting at b i which is conditioned to hit b i before returning to d i moves like a particle in the modified random potential V (i) defined as follows: for all a i ≤ x < y ≤ d i , V (i) (y) -V (i) (x) = (V (y) -V (x)) + log h (i) (x) h (i) (x + 1) h (i) (y)h (i) (y + 1) , (7.5) where h (i) (x) := P x ω (τ (b i ) < τ (d i )) (notice that V and V (i) coincide on the interval [a i , b i ]).

It happens now that E ω [(F (i) ) 2 ] can be computed explicitly in terms of V (i) (see Lemma 12 in [15]), and is bounded by a constant times (d (i) -a (i) ) 2 times the exponential of the maximum of the largest rise of V on [a i , b i ] and the largest fall of V (i) on [b i , d i ], which are treated in a similar way as the fluctuations of V (i) , above. So, we get ∀η > 0, P (E ω [(F (i) ) 2 ] ≥ t η ) = o 1 (log t) 2 . Moreover, we have

E 1 N i 1 {N i =0} = E - 1 -p i (ω) p i (ω) log(1 -p i (ω)) = O (log t) 2 t .
As a result, we obtain

P (1 - ξ 4 )N i E ω [F (i) ] ≤ F (i) 1 + • • • + F (i) N i ≤ (1 + ξ 4 )N i E ω [F (i) ] = 1 -o 1 (log t) 2 .
Now, the second step in the estimation of the first quantity of the rhs of (7.1) is the examination, for ξ > 0, of

P (1 - ξ 4 )N i E ω [F (i) ] ≤ E ω [τ i ]e i ≤ (1 + ξ 4 )N i E ω [F (i) ] ,
i.e.

P (1 - ξ 4 )N i E ω [F (i) ] ≤ (E ω [N i ]E ω [F (i) ] + E ω [S (i) ])e i ≤ (1 + ξ 4 )N i E ω [F (i) ] .
Neglecting again, like above, the contribution of S (i) we are back to prove that 2 ) → 1, when t → ∞, this concludes the proof that the rhs (and therefore the lhs) of (7.1) tends to 1 when t tends to infinity. Indeed, we obtain

P ∀i ≤ K t , (1-ξ)(F (i) 1 +• • •+F (i) N i +S (i) ) ≤ E ω [τ * i ]e i < (1+ξ)(F (i) 1 +• • •+F (i)
N i +S (i) ) → 1, from which we deduce

P ∀i ≤ K t , (1 -ξ)(τ * 1 + • • • + τ * i ) ≤ i k=1 E ω [τ * k ]e k < (1 + ξ)(τ * 1 + • • • + τ * i ) → 1.
Moreover, we use the fact that

E ω [τ * k ] = W * k -(d * k -b * k )
, where (cf for example [START_REF] Zeitouni | Random Walks in Random Environment, XXXI summer school in probability[END_REF]

P ∀i ≤ K t , (1 -ξ)(τ * 1 + • • • + τ * i ) ≤ i k=1 W * k e k < (1 + ξ)(τ * 1 + • • • + τ * i ) → 1.
Applying this, for i = ℓ * t and i = ℓ We conclude now the proof by reminding that lim ξ→0 P(ℓ * t = ℓ * t(1+ξ) ) = 1 as well as lim ξ→0 P(ℓ

Proof of Theorem 1

We fix h > 1 and η > 0 (η was used to define the event A 5 (t, η) before Lemma 9). Let us introduce the event

Theorem 2 .

 2 Under assumptions (a)-(b) of Theorem 1, we have, (i) for all η > 0, lim t→∞ P(|X t -b ℓt | ≤ η log t) = 1,

  1+κ

  we get respectively that, for all ξ > 0, ,ω → 1 and P(ℓ(e) t,ω ≤ ℓ * t(1+ξ) ) → 1.

  ),ω ) = 1.Proof of Part (ii) of Theorem 2. The passage from Proposition 3 to Part (ii) of Theorem 2 is of the same kind as the passage from Proposition 1 to Proposition 2.

1 ( 1 -

 11 y) -κ dy, which concludes the proof of Theorem 1.

  Now, we need to be sure that the bottom of the deep valleys are sharp. For η > 0, we introduce the following subsets of the deep valleys

	Then, assembling (4.7) and (4.8) yields

P τ (c j , d j ) > t log t ; A 4 (t) ; F γ (t) ≤ C(log t) e γ log t-h t 3 ,

which concludes the proof of Lemma 8 by choosing γ < 1/3.

  The first step in our proof is to derive the following result.

	Proposition 1. Let ℓ * t be the random integer defined by
		ℓ * t
	P at a * b * i ω,|a * i	b * i ω,|a * i is the law of the Markov chain in environment ω, starting from b * , where τ (d * i ) denotes the first hitting time of d * i and i and reflected

* i ) under P i . Hence, under the annealed law P, (τ * i ) i≥1 is an i.i.d. sequence since the * -valleys are independent and identically distributed.

  Part (ii) of Theorem 2: the quenched law of the last visited valleyIn order to prove the proximity of the distributions of ℓ t and ℓ Let us first associate to the exponential variable e i the following geometric random variable N

	Proof. The strategy is to build a coupling between ℓ * t and ℓ	(e) t,ω such that
	lim t→∞	P (P 0,ω (ℓ * t = ℓ	(e) t,ω ) > δ) = 0.
				κ+1 2	we get
	lim sup			C(log t)	3+κ 2 t -C ′′′ η
				t→∞
				= 0.
	Then, letting ξ tends to 0 in (6.2) concludes the proof of part (i) of Theorem 2.
	7. (e) t,ω , we go through ℓ * t = sup{n ≥ 0, τ * 1 +• • •+τ * n ≤ t} whose advantage is to involve independent random
	variables whose laws are clearly identified.
	Proposition 3. Under assumptions (a)-(b) of Theorem 1, we have, for all δ > 0,
	lim t→∞	P d T V (ℓ * t , ℓ	(e) t,ω ) > δ = 0,
	where d T V denotes the distance in total variation.

t→∞ P(|X t -b ℓt | > η log t ; ℓ t = ℓ t(1+ξ) ) ≤ lim sup

  Now, since P (K t ≤ (log t)

	P (1-	ξ 4	) (-	1 log(p i (ω))	)e i ≤	p i (ω) 1 -p i (ω)	e i ≤ (1+	ξ 4	) (-	1 log(p i (ω))	)e i = 1-o	1 (log t) 2 ,
	which is a direct consequence of 1 -p i (ω) ≤ log t t allied with
					P (e) e i >	log t t	= 1 -o	1 (log t) 2 .

  , formula (2.1.14)) Since on the event A 4 (t) we have for allk ≤ K t , d * k -b * k ≤ C ′′ log t, we see that on A 4 (t) we have (1 -C ′′ (log t) 2 * k ≤ E ω [τ * k ] ≤ W * k , since W *k ≥ e ht by definition. Hence, it implies that

	W * k = 2	e Vω(n)-Vω (m) .
	a * k b * k ≤n≤d * ≤m≤n k

t

)W

  T T (t, h):= T T (t) ∩ {X t -b ℓt ≤ η 2 log t} ∩ {X th -b ℓ th ≤ η 2 log t},whose probability tends to 1, when t tends to infinity (it is a consequence of Section 4 together with part (i) of Theorem 2). Then, we easily have({ℓ th = ℓ t } ∩ T T (t, h)) ⊂ ({|X th -X t | ≤ η log t} ∩ T T (t, h)) .Moreover, observe that on T T (t),ℓ th > ℓ t implies that |b ℓ th -b ℓt | ≥ t κ/2 (by definition of A 3 (t)). Therefore, we get for all large t. Thus, since Proposition 2 implies that lim t→∞ P(ℓ th = ℓ t ) exists, we obtain lim t→∞ P(|X th -X t | ≤ η log t) = lim

		t→∞	P(ℓ th = ℓ t )
	= lim t→∞	P(T ℓt+1 ≥ th)
	=	sin(κπ) π

({|X th -X t | ≤ η log t} ∩ T T (t, h)) ⊂ ({ℓ th = ℓ t } ∩ T T (t, h)) ,


José Orozco-Santiago

Armando Cuevas-Vallejo

Luc Trouche
email: luc.trouche@ens-lyon.fr

Instrumental orchestrations by a university teacher in an online linear algebra course

Keywords: Instrumental orchestration, linear algebra teaching and learning, university mathematics, online course

In this article, we present an instrumental orchestration for online math education. The case study is of a university teacher in a linear algebra course with engineering students in a remote learning setting. We used the instrumental orchestration approach as a theoretical framework for planning and organizing the artifacts involved in the environment (didactic configuration) and the modalities of their implementation (mode of exploitation). The activities were designed using virtual manipulators, guided exploration sheets, and video recordings of individual or paired work by the students. The results of the observation of five sessions are presented and the orchestrations of a pedagogical sequence to introduce the concepts of value and eigenvector are briefly discussed. This work suggests new instrumental orchestrations for the teaching of mathematics online.

Introduction.

In March 2020, the World Health Organization declared a global pandemic due to COVID-19 and recommended lockdowns and mobility restrictions, forcing educational institutions to migrate from their face-to-face teaching model to online education [START_REF] Engelbrecht | Will 2020 be remembered as the year in which education was changed?[END_REF]. Most institutions had to improvise online courses using various digital devices and internet platforms. Although remote education has been developing for years in various forms -distance learning, e-learning, and hybrid learning-to reach more students (Silverman & Hoyos, 2018), it had not fully permeated most educational institutions. [START_REF] Oktaç | Student discussions on a linear algebra problem in a distance-education course[END_REF] reports the implementation of a remote introductory linear algebra course in an asynchronous learning environment. Teaching linear algebra online is a challenge, as it is one of the first abstract math courses students encounter during their early years in college. At ICME13, there was a discussion group on teaching and learning linear algebra, which considered the ways a linear algebra course could be adapted to meet the needs of students from other disciplines, such as engineering, physics, and computer science. It also reflected on how technology should be used in teaching linear algebra (Stewart et al., 2018, p. ix). [START_REF] Gol Tabaghi | Using Dynamic Geometry Software to Explore Eigenvectors: The Emergence of Dynamic-Synthetic-Geometric Thinking[END_REF] studied the geometry of eigenvalues and real eigenvectors of a 2 × 2 matrix using dynamic geometry software (DGS), in which by dragging the vector 𝑥 onto the screen, the vector 𝐴𝑥 moves accordingly. They concluded that dynamic geometric representations in the sketch enabled students to understand the concepts of eigenvector and eigenvalue by identifying their invariant geometric properties and helped them develop dynamic-synthetic-geometric thinking.

On the other hand, and perhaps in anticipation of the current circumstances, [START_REF] Trouche | Managing the complexity of human/machine interactions in computerized learning environments: Guiding students' command process through instrumental orchestrations[END_REF] introduced the notion of instrumental orchestration for planning a lesson, defined as the didactic management of the different artifacts available in the classroom, the organization of space and time, and the way the artifacts are connected.

Theoretical background

An instrumental orchestration is "the intentional and systematic organization of the various artifacts available in a computerized learning environment by the teacher for a given mathematical situation, in order to guide students' instrumental genesis" (Drijvers & Trouche, 2008, p. 377). An instrumental orchestration is defined by a didactical configuration and the way in which this configuration is exploited. The Sherpa-student orchestration, which involves a student whose computer or calculator is displayed to the rest of the class on a projector, was for several years an emblematic didactic configuration. For a given configuration, there are several possible operating modes. As an instrumental orchestration is partly prepared in advance and partly created "on the spot" during teaching, Drijvers and colleagues ( 2010) add a third level called didactical performance to account for the multiple adjustments that the teacher and students may make. Drijvers and colleagues (2010) suggest considering the triplets model, a sort of jazz ensemble composed of beginners, advanced musicians, and the teacher as a conductor who prepares joint participation, but who is open to the pupil's improvisation and interpretation, on order to recognize the contributions of each of these levels. This metaphor became a reality in our experience because there was no precedent for online teaching of linear algebra, so we had to "improvise" ways and methods to teach both the mathematical task and the technological tool, for example, the forms of student-student and student-teacher through communication through the platform. Drijvers and his colleagues have extended the repertoire of instrumental orchestrations to include technical-demo, explain-the-screen, link-screen-board, discuss-the-screen, spot-and-show, Sherpa-at-work [START_REF] Drijvers | The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom[END_REF], board-instruction and guide-and-explain [START_REF] Drijvers | Digital resources inviting changes in mid-adopting teachers' practices and orchestrations[END_REF]. In the first three orchestrations is it mainly the teacher leading the communication, while in next three students have more opportunities for participation. In the technical-demo orchestration, the teacher explains the basic techniques of a tool in order to familiarize students with it.

In the explain-the-screen orchestration the teacher explains what is happening on the computer screen, but unlike the previous one, this orchestration involves mathematical content. Technical-support orchestration refers to the process of helping the learner with technical issues with the tool, such as software and/or hardware issues. Discuss-the-screen orchestration involves a group discussion about what is happening on the computer screen. In the link-screen-board orchestration, the teacher refers to the relationship between what the tool displays and the mathematical concept. In spot-and-show orchestration, the teacher chooses a student's work before class, either for the student to explain in class or for discussion with the whole class. For the Sherpa-at-work orchestration, the teacher asks a student (referred to as the Sherpa) to demonstrate or perform a certain process in the technological environment (for details, see [START_REF] Drijvers | Digital resources inviting changes in mid-adopting teachers' practices and orchestrations[END_REF]. Thus, our research question is: What instrumental orchestrations are applied by a university professor using technology to teach the concept of eigenvalue and eigenvector in a distance education setting?

Method

The study was developed under a design-based research methodology, which consists of three research phases: preparation and design, teaching experiment and, retrospective analysis [START_REF] Bakker | Design Research in Education A Practical Guide for Early Career Researchers[END_REF]. A first research cycle (Orozco-Santiago, 2020) was carried out in face-to-face teaching. For the design of our task sequence, we used a hypothetical learning trajectory [START_REF] Simon | Reconstructing Mathematics Pedagogy from a Constructivist Perspective[END_REF] but, due to the global public health emergency caused by COVID-19, circumstances compelled them to use the Zoom platform for the first time for online teaching. However, the teacher has extensive experience in the use of digital resources. The trajectory consists of seven activities. During the second cycle of research, the teaching experiment was carried out with a small group of ten students (6 women and 4 men) in their second year of engineering studies at a Mexican public university, over a period of two weeks. When designing the didactic path of our work, we were able to detect a priori the use of the following orchestrations for the teacher (Figure 1), and each activity for the pupils was accompanied by a didactic interactive virtual environment (DIVE) and a guided exploration sheet. The design of a configuration depends on the technological environment (Figure 2), and in this case the students had laptops or mobile devices (smartphones, tablets). However, the hardware and conditions available to them for remote study were less than ideal. Internet connections were quite weak, there was noise at home and outside, and due to pandemic conditions, students were prone to stress. Most students could not use their cameras or microphones during courses because these require more bandwidth, which many did not have. In the course, the following topics had already been taught: matrices, systems of linear equations, vector spaces, linear combination, linear independence, and linear transformations. The participants had not previously studied the concept of eigenvalue and eigenvector.

Throughout the course, activity data was collected by the following: a) e-mail; b) WhatsApp; c) Google Drive; d) the Zoom platform; e) free software for video recording OBS Studio; d) digital capture of students' notebooks; e) video recordings of the researcher's classroom activities on Zoom; and f) video recording of student work and teamwork, in the DIVE and guided exploration sheets. Five of the lectures on the concepts of eigenvalues and eigenvectors were recorded on video. To answer the research question, after the end of the teaching experiment, the five video recordings of the class were observed and the data analysis presented in this article focused on the identification of the orchestrations proposed by Drijvers and colleagues (2013), in the first three video recordings. In the video recording of session 1 (S1), 12 different episodes were identified. In S2 19 and in S3, 21.

Results

The instrumental orchestrations reported by [START_REF] Trouche | Managing the complexity of human/machine interactions in computerized learning environments: Guiding students' command process through instrumental orchestrations[END_REF] and [START_REF] Drijvers | The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom[END_REF][START_REF] Drijvers | Digital resources inviting changes in mid-adopting teachers' practices and orchestrations[END_REF], were developed within the framework of face-to-face teaching where the teacher and the students are in a classroom. But because of lockdown requirements during the pandemic, this teaching experience was developed synchronously online, which allowed for the use of some orchestrations and required the improvisation of others. For example, the technical-demo orchestration was used for the students to familiarize themselves with the Zoom software, offering instructions on how to use meeting control.

Since in the virtual model, the Work-and-walk-by orchestration was not possible, and we wanted to observe the individual instrumental genesis developed by the students and at the same time discuss as a group the correctness or not of the answers and the work of each student, the modality that we called Student-shared-screen appeared, which allowed the teacher to accomplish the task, it should be noted that the name, the peculiarity, and the characteristic of this orchestration are made in the analysis after the experiment with the data obtained described in Figures 1 and2. As this orchestration was improvised, when we analyzed the data and using the taxonomy defined by Drijvers and colleagues, they mention that the didactic configuration for the Spot-and-show orchestration "includes access to the students' work in the technological environment during lesson preparation" and for Sherpa-at-work orchestration, "includes access to the technology and projecting facilities, preferably access to student work" (2013, p. 999), but to implement access to the pupil's screen, it is necessary to ask the student's permission. Not having found an orchestration for this type of activity which we believe is important in online education, we call it the new Student-shared-screen orchestration, which consists of a didactic configuration in which the student allows the whole class to access his work on his computer screen. As an exploitation mode, the teacher asks the student to share his screen and discuss his work with the whole group. As can be seen, this is an orchestration prior to Sherpa-at-work and spot-and-show.

To simulate chalk, the teacher chose to use the Zoom Annotation tool and use a blank PowerPoint slide or a DIVE as a whiteboard, on which the teacher and students could write in order to present, explain and solve the exercises. We will call this the Share-board orchestration, and it is a recurring didactical configuration very similar to traditional classroom teaching (see Figure 3). In the third task, students had to drag two vectors onto ℝ 2 in the DGS and observe their relationship to the columns of a 2 × 2 matrix. The teacher identified an inappropriate response from the video recordings: some students did not drag the vectors, but directly changed the values of the matrix in the algebraic view. Through to the student-share-screen and spot-and-show orchestration, the teacher decides to ask the student to explain to the whole class how he modified the matrix, Teacher:

The coordinates of the vector a: what relation does it have with the matrix 𝐴? Student1:

None. Teacher:

When you drag vector a1, what happens to matrix 𝐴? Student1:

Nothing happens to it.

Student2:

Changes the values a11 and a22 of the matrix [this answer is provided by the chat].

Student3:

The values in the first column change.

Students who did not drag the vectors and directly modified the values of the matrix failed to identify the relationship between the vectors and the columns of matrix 𝐴. This didactical performance was observed thanks to the design of the DGS.

For the fourth task, the teacher used the explain-the-screen and share-board orchestration, using the work of one of the students as a mode of exploitation. The selected student's work was the criteria for the first work to be uploaded to Google Drive (see Figure 4). [START_REF] Thomas | Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking[END_REF] point out that textbooks show 𝐴𝑥 ⃗ = 𝜆𝑥 ⃗, then 𝐴𝑥 ⃗ = 𝜆𝐼𝑥 ⃗, but do not explain that it is 𝜆𝐼𝑥 ⃗ -𝐼 is a matrix and not a scalar. Using the board-instruction-tech orchestration, the teacher explains how the equation 𝐴𝑥 ⃗ = 𝜆𝑥 ⃗ is transformed into (𝐴 -𝜆𝐼)𝑥 ⃗ = 0 ⃗⃗ with student participation (see Figure 3a). In another activity, the teacher began with explain-the-screen orchestration, providing students with a DIVE that was not considered in the learning trajectory. After having carried out the calculation of the eigenvalues and the eigenvectors on the screen, the teacher explained to the students that in the CAS view of the DGS, they could observe step by step the calculation of the characteristic polynomial, of the roots of the characteristic polynomial (eigenvalues), and the solution of the equation (𝐴 -𝜆𝐼)𝑥 ⃗ = 0 ⃗⃗ for each of the eigenvalues (if they exist). After each line of the CAS was explained, a didactical performance was presented. Here the students commented that on their computers, the IME given did not work like the one the teacher presented, and the CAS did not recognize the same commands as those of the teacher (see Figure 5a). Using student-shared-screen orchestration, a student split his screen to perform and follow the teacher's instructions, and the student's computer screen froze. The teacher had to ask another student to share his screen. To solve the command problem, the teacher was able to use his technological knowledge to search the GeoGebra Wiki and find another command which made the DIVE work correctly on the students' computers (see Figure 5b).

Discussion

For the student-share-screen orchestration, a didactical configuration for this orchestration is the sharing of the videoconference screen by a student, which allows the students to follow the discussion, using Sherpa-at-work or spot-and-show. As an exploitation mode, the teacher can ask the Sherpastudent to explain her work, or he can ask questions and ask the student-Sherpa to perform specific actions. In our case, when a student wanted to share their paper-and-pencil work, there were several exploitation modes. 1) A student with a weak internet connection could take a photo of their notebook using their smartphone, upload it to the WhatsApp group, their cloud folder, or conference chat, so that the teacher or another classmate could download and project their work while they explained it. This way we can define this orchestration as Shared-resource. But because of the weak internet connection, the gestures of the teacher and the students were overlooked.

The videoconferencing platform used allows for the creation of small work teams (Breakout Rooms) assigned either randomly using the app, or by the teacher. It also allows the teacher to move among the teams to observe their work and, on the same platform, the students in each teams can ask for help from the teacher (Ask for Help). A downside to this option is that if multiple teams ask for help with the same issue, only members of the team visited by the teacher will be able to hear and see the discussion.

Conclusion

Supporting mathematics teachers in their efforts to integrate technology into their teaching practice remains a challenge for the mathematics education community [START_REF] Trouche | Understanding the work of teachers through their interaction with their teaching resources -a history of trajectories[END_REF]. We believe that teachers need basic skills and knowledge in the use of technologies so that they can support their students' learning in these online environments.

Returning to our research question: What instrumental orchestrations are applied by a university professor using technology to teach the concept of eigenvalue and eigenvector in a distance education setting? As technology evolves, existing types of instrumental orchestrations [START_REF] Drijvers | The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom[END_REF][START_REF] Drijvers | Digital resources inviting changes in mid-adopting teachers' practices and orchestrations[END_REF][START_REF] Trouche | Managing the complexity of human/machine interactions in computerized learning environments: Guiding students' command process through instrumental orchestrations[END_REF] need to be re-examined to determine how they might be modified and expanded. We examine the teaching practice of a university engineering professor during the onset of the COVID-19 pandemic through online education, using the theoretical framework of instrumental orchestration. Students contributed through video recordings of their work and ideas, which allowed the teacher to access this work and decide between Sherpa-student or spot-and-show orchestrations from this work. Some orchestrations from traditional education were also observed. In this observation, we identified four new types of instrumental orchestration: student-share-screen, share-board, board-instruction-tech and shared-resource.

In this course, being able to observe the personal development on paper-and-pencil by the whole class made the students more willing to share their incomplete attempts at problem-solving.

Videoconferencing technologies offer means of communication that raise new questions regarding the organization of lessons. The pandemic has raised new questions about synchronous teaching: the use of computer screens and webcams instead of blackboards, the ability to carry out multiple activities such as Chat watching, student participation by raising their hand, muting and unmuting the microphone, solving technical problems on the fly. The activities developed in this study can be implemented in face-to-face classes, encouraging autonomous and collaborative work, and installed on a freely accessible server where students can learn at their own pace.

The role of the smartphone as a computer has changed the way we can experience mathematics [START_REF] Borba | The future of mathematics education since COVID-19: humans-with-media or humans-with-non-living-things[END_REF]. We believe that the impact of the pandemic has changed the traditional way of teaching and learning, which, together with the ever-increasing demand for study and the limitations of institutions, makes hybrid distance learning a necessity. It is, therefore, necessary to carry out studies on the design and implementation of educational software on digital devices and, of course, new orchestrations.

Figure 1 :

 1 Figure 1: Orchestration suggestions for teachers

Figure 2 :

 2 Figure 2: A remote configuration

Figure 3

 3 Figure 3: a) PowerPoint and b) DGE, both used as a whiteboard

Figure 4 :

 4 Figure 4: Search for collinearity of the u-vectors and the Au-vector

Figure 5 :

 5 Figure 5: Calculation of eigenvalues and eigenvectors in the CAS view of the DGE