Olivier Zindy

Upper limits of Sinai's walk in random scenery by

Keywords: Random walk in random environment, random scenery, localization, concentration property. 2000 Mathematics Subject Classification. 60K37, 60G50, 60J55

We consider Sinai's walk in i.i.d. random scenery and focus our attention on a conjecture of Révész concerning the upper limits of Sinai's walk in random scenery when the scenery is bounded from above. A close study of the competition between the concentration property for Sinai's walk and negative values for the scenery enables us to prove that the conjecture is true if the scenery has "thin" negative tails and is false otherwise.

Introduction 1.Random walk in random environment

Problems involving random environments arise in different domains of physics and biology. Originally, one-dimensional random walk in random environment appeared as a simple model for DNA transcription. In the following, we consider the elementary model of one-dimensional Random Walk in Random Environment (RWRE), defined as follows. Let ω := (ω i , i ∈ Z) be a family of independent and identically distributed (i.i.d.) random variables 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 RWRE (X n , n ≥ 0) as a nearest-neighbor random walk on Z with transition probabilities given by ω: (X n , n ≥ 0) 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), by E ω the corresponding expectation, and 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 random walks in random environment.

Throughout the paper, we make the following assumptions on ω:

∃ δ ∈ (0, 1/2) :

P {δ ≤ ω 0 ≤ 1 -δ} = 1, (1.1) E[ log( 1 -ω 0 ω 0 ) ] = 0, (1.2)
σ 2 := Var[ log( 1 -ω 0 ω 0 ) ] > 0. (1.3)
Assumption (1.1) implies that | log( 1-ω 0 ω 0 )| is, P -a.s., bounded by the constant L := log( 1-δ δ ). It is a technical assumption, which can be replaced by an exponential moment of log( 1-ω 0 ω 0 ). According to a recurrence-transience result due to Solomon [START_REF] Solomon | Random walks in a random environment[END_REF], assumption (1.2) ensures that (X n ) n≥0 is P-almost surely recurrent, i.e., the random walk hits any site infinitely often.

Assumption (1.3) excludes the case of deterministic environment, which corresponds to the homogeneous symmetric random walk.

Under assumptions (1.1)-(1.3), the RWRE is referred to as Sinai's walk. Sinai [START_REF] Sinai | The limiting behavior of a one-dimensional random walk in a random medium[END_REF] proves that X n /(log n) 2 converges in law, under P, toward a non-degenerate random variable, whose distribution is explicitly computed 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]. This result contrasts with the usual central limit theorem which gives the convergence in law of

X n / √ n. Let L(n, x) := # {0 ≤ i ≤ n : X i = x} , n ≥ 0, x ∈ Z, (1.4) L(n, A) := x∈A L(n, x), n ≥ 0, A ⊂ Z.
In words, the quantity L(n, A) measures the number of visits to the set A by the walk in the first n steps.

The maximum of local time is studied by Révész ([20], p. 337) and Shi [START_REF] Shi | A local time curiosity in random environment[END_REF]: under assumptions (1.1)- (1.3), there exists c 0 > 0 such that lim sup n→∞ max x∈Z L(n, x) n ≥ c 0 , P-a.s.

It means that the walk spends, infinitely often, a positive part of its life on a single site. The liminf behavior is analyzed by Dembo, Gantert, Peres and Shi [START_REF] Dembo | Valleys and the maximum local time for random walk in random environment[END_REF], who prove that lim inf n→∞ max x∈Z L(n, x) n/ log log log n = c ′ 0 , P-a.s.,

for some c ′ 0 ∈ (0, ∞). A concentration property is obtained by Theorem 1.3 of Andreoletti [START_REF] Andreoletti | Almost sure estimates for the concentration neighborhood of Sinai's walk. To appear in Stochastic Processes and their Applications[END_REF], which says that, under assumptions (1.1)-(1.3) and for any 0 < β < 1, there exists ℓ(β) > 0 such that lim sup n→∞ sup x∈Z L (n, [xℓ(β), x + ℓ(β)]) n ≥ β, P-a.s. (1.5) In words, for any β close to 1, it is possible to find a length ℓ(β) such that, P-almost surely, the particle spends, infinitely often, more than a β-fraction of its life in an interval of length 2ℓ(β).

Random walk in random scenery

Random Walk in Random Scenery (RWRS) is a simple model of diffusion in disordered media, with long-range correlations. It is a class of stationary random processes exhibiting rich behavior. It can be described as follows: given a Markov chain on a state space, there may be a random field indexed by the state space, called a random scenery. As the random walk moves on this state space, he observes the scenery at his location. For a survey of recent results about RWRS, we refer to den Hollander and Steif [START_REF] Den Hollander | Random walk in random scenery: A survey of some recent results. Dynamics and Stochastics: Festschrift in Honor of Michael Keane[END_REF], and to Asselah and Castell [START_REF] Asselah | Random walk in random scenery and self-intersection local times in dimensions d ≥ 5[END_REF] for large deviations results in dimension d ≥ 5.

Let us now define the model of one-dimensional RWRS: consider S = (S n , n ≥ 0) a random walk on Z and ξ := (ξ(x), x ∈ Z) = (ξ x , x ∈ Z), a family of i.i.d. random variables defined on a probability space Ξ. We refer to ξ as the random scenery and denote by Q its law. Then, define the process (Y n , n ≥ 0) by

Y n := n i=0 ξ(S i ),
called RWRS or the Kesten-Spitzer Random Walk in Random Scenery. An interpretation is the following: if a random walker has to pay ξ x each time he visits x, then Y n stands for the total amount he has paid in the time interval [0, n].

The model is introduced and studied by Kesten and Spitzer [START_REF] Kesten | A limit theorem related to a new class of self-similar processes[END_REF] in dimensions d = 1

and d ≥ 3. They prove in dimension d = 1 that, when S and ξ belong to the domains of attraction of stable laws of indice α and β respectively, then there exists δ, depending on α and β, such that n -δ Y ⌊nt⌋ converges weakly. In the simple case where α = β = 2, they show

that n -3/4 Y ⌊nt⌋ ; 0 ≤ t ≤ 1 law -→ (Λ(t); 0 ≤ t ≤ 1) ,
where " law -→ " stands for weak convergence in law (in some functional space; for example in the space of bounded functions on [0, 1] endowed with the uniform topology). The process (Λ(t), t ≥ 0), called Brownian motion in Brownian scenery, is defined by Λ(0) = 0 and Λ(t) := R ℓ(t, x) dW (x) for t > 0, where (W (x); x ∈ R) denotes a two-sided Brownian motion and (ℓ(t, x), t ≥ 0, x ∈ R) denotes the jointly continuous version of the local time process of a Brownian motion (B(t), t ≥ 0), independent of (W (x); x ∈ R). Independently, Borodin analyzes the case of one-dimensional nearest-neighbor random walk in random scenery, see [START_REF] Borodin | Limit theorems for sums of independent random variables defined on a transient random walk. Investigations in the theory of probability distributions[END_REF] and [START_REF] Borodin | A limit theorem for sums of independent random variables defined on a recurrent random walk[END_REF]. Bolthausen [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF] studies the case d = 2. He proves that, if S is a recurrent random walk and ξ 0 has zero expectation and finite variance, then (n log n) -1/2 Y ⌊nt⌋ satisfies a functional central limit theorem.

Random environment and random scenery

In this paper, we consider Sinai's Walk in Random Scenery. Problems combining random environment and random scenery have been examined for more general models. Replacing Z by a more general countable state space, Lyons and Schramm [START_REF] Lyons | Stationary measures for random walks in a random environment with random scenery[END_REF] exhibit, under certain conditions, a stationary measure for Random Walks in a Random Environment with Random Scenery (RWRERS) from the viewpoint of the random walker. Häggström [START_REF] Häggström | Infinite clusters in dependent automorphism invariant percolation on trees[END_REF], Häggström and Peres [START_REF] Häggström | Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously[END_REF] treat the case where the scenery arises from percolation on a graph. In this particular case, the scenery determines the random environment of the associated RWRE, which is used by the authors to obtain information about the scenery.

Let us first describe the model of Sinai's walk in random scenery. We consider Sinai's walk (X n , n ≥ 0) under assumptions (1.1)-(1.3), and recall that the environment ω is defined on (Ω, P ). For the scenery, we consider a family of i.i.d. random variables ξ := (ξ(x), x ∈ Z) = (ξ x , x ∈ Z), defined on (Ξ, Q), independent of ω and (X n , n ≥ 0). To translate independence between ω and ξ, we consider the probability space (Ω×Ξ, P ⊗Q), on which we define (ω, ξ). Moreover, we denote by P ⊗ Q the law of (ω, (X n ) n≥0 , ξ). Then we define as Sinai's walk in random scenery the process (Z n , n ≥ 0):

Z n := n i=0 ξ(X i ).
Observe that Z n can be written using local time notation:

Z n = x∈Z ξ(x) L(n, x), n ≥ 0, (1.6)
where L(n, x) stands for the local time of the random walk at site x until time n, see (1.4).

We are interested in the upper limit of Z n in the case where a := ess sup ξ 0 is finite. We consider the concentration property of order β for Sinai's walk with β close to 1 (see ( It turns out that the conjecture holds only under some additional assumptions on the distribution of the random scenery. It is interesting to note that this conjecture follows immediately from the result of Andreoletti [START_REF] Andreoletti | Almost sure estimates for the concentration neighborhood of Sinai's walk. To appear in Stochastic Processes and their Applications[END_REF] mentioned earlier if ess inf ξ 0 is larger than -∞. In the general case, a close study of the competition between the concentration property for Sinai's walk and negative values for the scenery enables us to obtain the following theorem, which gives a solution to this problem, depending on the tail decay of ξ - 0 := max{-ξ 0 , 0}.

Theorem 1.1 Assume (1.1)-(1.
3) and a := ess sup ξ 0 < ∞.

(i) If Q{ξ - 0 > λ} ≤ 1 (log λ) 2+ε
, for some ε > 0 and all large λ, then

P ⊗ Q lim sup n→∞ Z n n = a = 1. (ii) If Q{ξ - 0 > λ} ≥ 1 (log λ) 2-ε
, for some ε > 0 and all large λ, then

P ⊗ Q lim n→∞ Z n n = -∞ = 1.
It is possible to give more precision in the case (ii), see Remark 3.5. On the other hand, the case ε = 0 is still open.

We mention that, under (1.1)-(1.3) and a := ess sup ξ 0 < ∞, it is possible to prove that P ⊗ Q{lim sup n→∞ Zn n = c}, satisfies a 0 -1 law, for any c ∈ [-∞, ∞]. The proof follows the lines of [START_REF] Gantert | Many visits to a single site by a transient random walk in random environment[END_REF], except that we need an additional argument saying that modifying a finite number of random variables in the scenery does not change the behavior of lim sup n→∞ Zn n . The latter can be done by means of Theorem 1 (see Example 1) in [START_REF] Hu | The local time of simple random walk in random environment[END_REF], which implies, for

any x ∈ Z, that L(n, x) = o(n), n → ∞, P-almost surely.
In general, we do not know whether lim sup n→∞ Zn n ∈ {-∞, a}, P-almost surely. The paper is organized as follows: in Section 2, we present some key results for the environment and for Sinai's walk when the environment is fixed (i.e., quenched results). In Section 3, we define precisely the notion of "good" environment-scenery and prove Theorem 1.1 by accepting two intermediate propositions. The first one, proved in Section 4, is devoted to the study of the RWRE within the "good" environment-scenery. The second one, proved in Section 5, does not concern the RWRE, but only the environment-scenery. We show that, P ⊗ Q-almost surely, (ω, ξ) is a "good" environment-scenery.

In the following, we use c i (1 ≤ i ≤ 33) to denote finite and positive constants.

Preliminaries

In this section, we collect some basic properties of random walk in random environment that will be useful in the forthcoming sections.

About the environment

In the study of one-dimensional RWRE, an important role is played by a function of the environment ω, called the potential. This process, noted V = (V (x), x ∈ Z), is defined on (Ω, P ) by:

V (x) :=    x i=1 log( 1-ω i ω i ) if x ≥ 1, 0 if x = 0, -0 i=x+1 log( 1-ω i ω i ) if x ≤ -1.
(2.1)

By (1.1), we observe that |V (x) -V (x -1)| ≤ L for any x ∈ Z. Moreover, we define P z {•} := P { • | V (0) = z},
for any z ∈ R; thus P = P 0 . (Strictly speaking, we should be working in a canonical space for V , with P z defined as the image measure of P under translation.)

Let us define, for any Borel set A ⊂ R,

ν + (A) := min {n ≥ 0 : V (n) ∈ A} .
We recall the following result, whose proof is given by a simple martingale argument.

Lemma 2.1 For any x < y < z, we have

y -x z -x + L ≤ P y {ν + ([z, ∞)) < ν + ((-∞, x])} ≤ y -x + L z -x .
Proof. Since (1.1) and (1.2) imply that (V (n); n ≥ 0) is a martingale with bounded jumps, we apply the optional stopping theorem ( [START_REF] Durrett | Probability: Theory and Examples[END_REF], p. 270

) at ν + ([z, ∞)) ∧ ν + ((-∞, x]) to get y = E y [X 0 ] = E y [X ν + ([z,∞)) ; ν + ([z, ∞)) < ν + ((-∞, x])] +E y [X ν + ((-∞,x]) ; ν + ([z, ∞)) > ν + ((-∞, x])]. Since X ν + ([z,∞)) ∈ [z, z + L] and X ν + ((-∞,x]) ∈ [x -L, x
] by ellipticity, we obtain

y ≥ zP y {ν + ([z, ∞)) < ν + ((-∞, x])} + (x -L)(1 -P y {ν + ([z, ∞)) < ν + ((-∞, x])}),
which yields the right inequality. The left inequality is obtained by similar arguments.

Moreover, we recall a result of Hirsch [START_REF] Hirsch | A strong law for the maximum cumulative sum of independent random variables[END_REF], which, under assumptions (1.1)-(1.3), takes the following simplified form: for any 0 < ε ′ < 1 34 , there exists c 1 > 0 such that

P { max 0≤x≤N V (x) < N 1 2 -ε ′ } ∼ c 1 N -ε ′ , N → ∞. (2.2)

Quenched results

We define, for any x ∈ Z,

τ (x) := min {n ≥ 1 : X n = x} , min ∅ := ∞.
(Note in particular that when X 0 = x, then τ (x) is the first return time to x.) Throughout the paper, we write

P x ω {•} := P ω { • | X 0 = x} (thus P 0 ω = P ω
) and denote by E x ω the expectation with respect to P x ω .

Recalling that

ω i /(1 -ω i ) = e -(V (i)-V (i-1
)) , we get, for any r < x < s,

P x ω {τ (r) < τ (s)} = s-1 j=x e V (j) s-1 j=r e V (j) -1 . (2.3)
This result is proved in [START_REF] Zeitouni | Random Walks in Random Environment, XXXI summer school in probability[END_REF], see formula (2.1.4).

The next result, which gives a simple bound for the expectation of τ (r) ∧ τ (s) when the walk starts from a site x ∈ (r, s), is essentially contained in Golosov [START_REF] Golosov | Localization of random walks in one-dimensional random environments[END_REF]; its proof can be found in [START_REF] Shi | A weakness in strong localization for Sinai's walk[END_REF]. For any integers r < s, we have

max x∈(r, s)∩Z E x ω [τ (s) 1 {τ (s)<τ (r)} ] ≤ (s -r) 2 exp max r≤i≤j≤s (V (j) -V (i)) . (2.4)
We will also use the following estimate borrowed from Lemma 7 of Golosov [START_REF] Golosov | Localization of random walks in one-dimensional random environments[END_REF]: for ℓ ≥ 1 and x < y,

P x ω {τ (y) < ℓ} ≤ ℓ exp -max x≤i<y [V (y -1) -V (i)] . (2.5)
Looking at the environment backwards, we get: for ℓ ≥ 1 and w < x,

P x ω {τ (w) < ℓ} ≤ ℓ exp -max w<i≤x [V (w + 1) -V (i)] . (2.6)
Finally we quote an important result about excursions of Sinai's walk (for detailed discussions, see Section 3 of [START_REF] Dembo | Valleys and the maximum local time for random walk in random environment[END_REF]). Let b ∈ Z and x ∈ Z, and consider L(τ (b), x) under P b ω . In words, we look at the number of visits to the site x by the random walk (starting from b) until the first return to b. Then there exist constants c 2 and c 3 such that

c 2 e -[V (x)-V (b)] ≤ E b ω [L(τ (b), x)] ≤ c 3 e -[V (x)-V (b)] . (2.7)
3 Good environment-scenery and proof of Theorem 1.1

For any j ∈ N * , we define

d + (j) := min {n ≥ 0 : V (n) ≥ j} , b + (j) := min n ≥ 0 : V (n) = min 0≤x≤d + (j) V (x) .
These variables enable us to consider the valley (0, b + (j), d + (j)). Similarly, we define

d -(j) := max {n ≤ 0 : V (n) ≥ j} , b -(j) := max n ≤ 0 : V (n) = min d -(j)≤x≤0 V (x) .
In the next sections, we will be frequently using the following elementary estimates. Lemma 3.1 For any ε ′ > 0, we have, P -almost surely for all large j,

j 2-ε ′ ≤ |b ± (j)| < |d ± (j)| ≤ j 2+ε ′ .
Proof. Fix ε ′ > 0. Let us consider the sequence (j p ) p≥1 defined by j p := p 12/ε ′ for all p ≥ 1. Using (2.2), we obtain

p≥1 P {d + (j p ) > 1 3 j 2+ε ′ p } < ∞. Therefore, Borel-Cantelli lemma implies that, P -almost surely, d + (j p ) ≤ 1 3 j 2+ε ′ p
for all large p, say p ≥ p 0 . We fix a realization of ω and consider j p ≤ j ≤ j p+1 with p ≥ p 0 . Since d + (j) ≤ d + (j p+1 ), we get

d + (j) ≤ 1 3 j 2+ε ′ p+1 ≤ j 2+ε ′ 1 3 j p+1 j 2+ε ′ ≤ j 2+ε ′ 1 3 j p+1 j p 2+ε ′ = j 2+ε ′ 1 3 (1 + p -1 ) 12(2+ε ′ ) ε ′
, which yields d + (j) ≤ j 2+ε ′ for all large j. In a similar way, we can prove that

j 2-ε ′ ≤ ν + ((-∞, -j 1-κ ]) ≤ d + (j)
for some κ > 0 and all large j, which implies j 2-ε ′ ≤ b + (j) for all large j. Moreover, the arguments are the same to prove that, P -almost surely,

j 2-ε ′ ≤ |b -(j)| < |d -(j)| ≤ j 2+ε ′ for all large j.
To introduce the announced "good" environment-scenery, we fix ε > 0 such that assumption of Part (i) of Theorem 1.1 holds. For α ∈ (0, 1) (which will depend on ε), 0 < c 4 < 1/6, and j ≥ 100, we define

γ 0 (j) := j, γ i (j) := j (1-α) i = (γ i-1 (j)) 1-α , i ≥ 1, (3.1) ε i (j) := exp -c 4 γ i+2 (j) , i ≥ 0. (3.2)
For convenience of notation we define ε -1 (j) := ε 0 (j). In words, (γ i (j)) i≥0 represents a decreasing sequence of distances, which enables us to classify the sites according to the value of V (x) -V (b + (j)).

Write log p for the p-th iterative logarithmic function. Fix ε ′ := min{1/35, ε/2} > 0, and introduce, for j ≥ 100,

M(j) := inf n ≥ 0 : γ n (j) ≤ (log 2 j) 1-α 2+ε ′ . (3.3)
By definition of M(j), we have

γ M (j)-1 (j) ∈ (log 2 j) 1-α 2+ε ′ , (log 2 j) 1 2+ε ′ .
Moreover, in view of (3.1) and since γ M (j) belongs to [(log 2 j)

(1-α) 2 2+ε ′ , (log 2 j) 1-α 2+ε ′ ], we get that M(j) ∼ 1 | log(1 -α)| log 2 j, j → ∞. (3.4)
Note that we choose α small enough such that

β := (1 -α) 2 (2 + ε) -(2 + ε ′ ) > 0, (3.5) β ′ := ε ′ 2 -α > 0. (3.6)
Then we introduce the set (the constant c 5 will be chosen small enough in (5.9))

Θ M (j)-1 (j) := b + (j) -c 5 (γ M (j)-1 (j)) 2+ε ′ , b + (j) + c 5 (γ M (j)-1 (j)) 2+ε ′ ,
and, for i = M(j) -2, ..., 1, 0, the sets (the constant c 6 ≥ 1 will be chosen large enough in (5.17))

Θ i (j) := b + (j) -c 6 (γ i (j)) 2+ε ′ , b + (j) + c 6 (γ i (j)) 2+ε ′ \ M (j)-1 p=i+1 Θ p (j).
Observe that the sets (Θ i (j)) 0≤i≤M (j)-1 form a partition of the interval [b + (j)-c 6 j 2+ε ′ , b + (j)+ c 6 j 2+ε ′ ]. The final sets we consider are given, for 0 ≤ i ≤ M(j) -1, by Θ i (j) := Θ i (j) ∩ I(j), where I(j) := [ν + ((-∞, -j]), d + (j)]. Note that ν + ((-∞, -j]) < d + (j) on A(j) which will be defined in (3.14). In this case, the sets (Θ i (j)) 0≤i≤M (j)-1 form a partition of I(j) into annuli (since c 6 ≥ 1). Loosely speaking, the set Θ i (j) contains the sites x satisfying

V (x) -V (b + (j)) ≈ γ i (j). To cover [d -(j), d + (j)], we define Θ -1 (j) := -j 2+ε ′ , j 2+ε ′ ∩ d -(j), ν + ((-∞, -j]) . (3.7)
Moreover, for the environment on Z + , we introduce the events

A env 1 (j) := -4j ≤ V (b + (j)) ≤ -3j , (3.8) A env 2 (j) := max 0≤x≤y≤b + (j) [V (y) -V (x)] ≤ j 4 . (3.9)
The first event ensures that the valley considered is "deep enough" and the second one that the particle reaches the bottom of the valley "fast enough". To control the time spent by the particle in different Θ i (j) during an excursion from b + (j) to b + (j), we define

A env ann (j) := M (j)-2 i=0    x∈Θ i (j) e -[V (x)-V (b + (j))] ≤ (ε i (j)) 2    =: M (j)-2 i=0
A env ann,i (j). (3.10) For the environment on Z -, let

B env (j) := V (b -(j)) ≤ - j 6 , max d -(j)≤x≤y≤0 [V (y) -V (x)] ≤ j 3 , (3.11)
which ensures that the particle will not spent too much time on Z -.

Recalling that ξ - x = max{-ξ x , 0}, we define for the scenery

A sce i (j) := max x∈Θ i (j) ξ - x < (ε i (j)) -1/2 , -1 ≤ i ≤ M(j) -2, (3.12)
which ensures that the scenery does not reach excessive negative value in each Θ i (j). In order to force the scenery in a neighborhood of the bottom (where the particle is concentrated), to be close to a = ess sup ξ 0 , we fix ρ ∈ (0, 1) and introduce

A sce M (j)-1 (j) := min x∈Θ M (j)-1 (j) ξ x ≥ a -ρ . (3.13)
We set

A env (j) := A env 1 (j) ∩ A env 2 (j) ∩ A env ann (j), A sce (j) := M (j)-1 i=-1
A sce i (j).

Moreover, we define

A(j) := A env (j) ∩ B env (j) ∩ A sce (j). (3.14)
A pair (ω, ξ) is a "good" environment-scenery if (ω, ξ) ∈ A(j) for infinitely many j ∈ N.

For future use, let us note that for ω ∈ B env (j) ∩ A env 2 (j), we have max

d -(j)≤x≤y≤b + (j) [V (y) -V (x)] ≤ 2j 3 . (3.15)
To prove Theorem 1.1, we need two propositions, whose proofs are respectively postponed until Sections 5 and 4. The first one ensures that almost all pair (ω, ξ) is a "good" environment-scenery, while the second one describes the behavior of the particle in a "good" environment.

Proposition 3.2 Under assumptions (1.1)-(1.3), we have that P ⊗ Q-almost all (ω, ξ) is a "good" environment-scenery. More precisely, P ⊗ Q-almost surely, there exists a random sequence (m k ) k≥1 such that m k ≥ k 3k and (ω, ξ) is a good environment-scenery along (m k ) k≥1 , i.e., (ω, ξ) ∈ A(m k ), for all k ≥ 1.

In fact (m k ) k≥1 is constructed in the following way. Let us first introduce the sequence j p := p 3p for p ≥ 0. We define then (m k ) k≥1 by m 1 := inf{j p ≥ 0 : (ω, ξ) ∈ A(j p )} and

m k := inf{j p > m k-1 : (ω, ξ) ∈ A(j p )} for k ≥ 2. Then, Proposition 3.2 means that m k → ∞, k → ∞, P ⊗ Q-almost surely.
Before establishing the proposition about the behavior of the particle, we extract a random sequence (n k ) k≥1 from (m k ) k≥1 such that

k≥1 ε M (n k ) (n k ) < ∞. (3.16)
In fact, we consider the random sequence defined by

n 1 := inf{m p ≥ 1 : ε M (mp) (m p ) ≤ 1} and n k := inf{m p > n k-1 : ε M (mp) (m p ) ≤ 1 k 2 } for k ≥ 2.
To ease notations, we write throughout the paper,

d + k := d + (n k ), τ + k := τ (d + k ), b + k := b + (n k ) and d - k := d -(n k ), τ - k := τ (d - k )
. Moreover, we define, for all k ≥ 1,

t k := ⌊e n k ⌋. ( 3
.17) Proposition 3.3 For P ⊗ Q almost all (ω, ξ), we have that, P ω -a.s., for all large k,

L (t k , Θ -1 (n k )) ≤ ε -1 (n k ) t k , (3.18) L (t k , Θ i (n k )) ≤ ε i (n k ) t k , 0 ≤ i ≤ M(n k ) -2, (3.19) τ + k ∧ τ - k > t k . (3.20)
Remark 3.4 There is no measurability problem for the events described in Proposition 3.3, see the beginning of Section 4. Similar arguments apply to the forthcoming events.

Proof of Theorem 1.1. Proof of Part (i). For any δ > 0, we define ε (δ) (j) := M (j)-2 i=-1 ε δ i (j). Recalling (1.6), we use Proposition 3.3 and Lemma 3.1 to obtain, for P ⊗ Q-almost all realization of ω, ξ and (X j ) j≥0 ,

t k j=0 ξ(X j ) ≥ (1 -ε (1) (n k )) t k min x∈Θ M (n k )-1 (n k ) ξ x - M (n k )-2 i=-1 ε i (n k ) t k max x∈Θ i (n k ) ξ - x ,
for all large k. Then, Proposition 3.2 implies

t k j=0 ξ(X j ) ≥ (1 -ε (1) (n k )) t k (a -ρ) - M (n k )-2 i=-1 ε i (n k ) t k ≥ (1 -ε (1) (n k )) t k (a -ρ) -ε (1/2) (n k ) t k , (3.21)
for all large k. We claim that, for any δ > 0 and all large j,

ε (δ) (j) ≤ M (j) i=-1 ε δ i (j) ≤ 2 1 + 1 δ ε δ M (j) (j). (3.22)
To prove (3.22), we observe that

M (j) i=-1 ε δ i (j) ≤ 2 ε δ M (j) (j) + M (j)-1 i=0 ε i+1 (j) ε i (j) ε δ i (j) ε i+1 (j) -ε i (j) dx.
Recalling (3.2), we have that ε i+1 (j)-ε i (j) = ε i+1 (j) 1e -c 4 (γ i+2 (j)-γ i+3 (j)) . Recalling (3.1) we get that γ i+2 (j

) -γ i+3 (j) = γ i+2 (j)(1 -γ -α i+2 (j)). Since (3.1) and (3.3) imply γ i+2 (j) ≥ γ M (j)+2 (j) ≥ (log 2 j) (1-α) 4 2+ε ′
for 0 ≤ i ≤ M(j), we obtain that γ i+2 (j)γ i+3 (j) ≥ γ i+2 (j)/2, for all large j and for 0 ≤ i ≤ M(j). Therefore, we get ε i+1 (j)

-ε i (j) ≥ ε i+1 (j)/2, implying that ε (δ) (j) ≤ 2 ε δ M (j) (j) + 2 M (j)-1 i=0 ε i+1 (j) ε i (j) ε δ i (j) ε i+1 (j) dx.
Furthermore, M (j)-1 i=0 ε i+1 (j) ε i (j)

ε δ i (j) ε i+1 (j) dx ≤ M (j)-1 i=0 ε i+1 (j) ε i (j) x δ-1 dx = ε M (j) (j) ε 0 (j)
x δ-1 dx, which is less than ε δ M (j) (j)/δ. This implies (3.22). Combining (3.21) and (3.22) and recalling that ε δ M (j) (j) → 0 when j → ∞, we get lim sup

n→∞ 1 n n i=0 ξ(X i ) ≥ a -ρ, P ⊗ Q-a.s. (3.23)
To conclude the proof, it remains only to observe that (3.23) is true for all ρ > 0 and that the definition of a implies that P ⊗ Q-a.s.,

1 n n i=0 ξ(X i ) ≤ a, for all n ≥ 0.
Proof of Part (ii). Using Theorem 1.5 of [START_REF] Hu | The limits of Sinai's simple random walk in random environment[END_REF], we have that, for any ε ′′ > 0, P-almost surely, max 0≤i≤n X i ≥ (log n) 2-ε ′′ + 1, for all large n. This implies

n i=0 ξ(X i ) ≤ a n - max 0≤x≤⌈(log n) 2-ε ′′ ⌉ ξ - x . (3.24)
By assumption, there exists ε > 0 such that Q {ξ 0 < -λ} ≥ (log λ) -2+ε . Therefore, fixing ε ′′ < ε, we get for k ≥ 1 and all N ≥ 1,

Q max 0≤x≤N ξ - x < k a e N 1 2-ε ′′ ≤ exp -c 7 N δ , (3.25)
where δ := 1 -2-ε 2-ε ′′ > 0. We choose N p := ⌊(log p) T ⌋ for p ≥ 1 with T large enough such that T δ > 1. Therefore, (3.25) and the Borel-Cantelli lemma imply that, Q-almost surely, there exists p 0 (ξ) such that

max 0≤x≤Np ξ - x ≥ k a e N 1 2-ε ′′ p , (3.26)
for p ≥ p 0 (ξ). Fixing a realization of ξ, we define p(n) by

N p(n) ≤ ⌈(log n) 2-ε ′′ ⌉ ≤ N p(n)+1 , (3.27)
for all n such that p(n) ≥ p 0 (ξ). This yields

max 0≤x≤⌈(log n) 2-ε ′′ ⌉ ξ - x ≥ max 0≤x≤N p(n) ξ - x ≥ k a e N 1 2-ε ′′ p(n) ,
the last inequality being a consequence of (3.26). Therefore, we obtain max

0≤x≤⌈(log n) 2-ε ′′ ⌉ ξ - x ≥ ka exp ⌈(log n) 2-ε ′′ ⌉ 1 2-ε ′′ exp -⌈(log n) 2-ε ′′ ⌉ 1 2-ε ′′ -N 1 2-ε ′′ p(n) ≥ kan exp -N 1 2-ε ′′ p(n)+1 -N 1 2-ε ′′ p(n)
, the second inequality being a consequence of (3.27). Moreover, we easily get that N

1 2-ε ′′ p(n)+1 - N 1 2-ε ′′ p(n) → 0, when n → ∞, implying that for all large n, max 0≤x≤⌈(log n) 2-ε ′′ ⌉ ξ - x ≥ k 2 an. (3.28)
Assembling (3.24) and (3.28), we get that P ⊗ Q-almost surely, lim sup n→∞

1 n n i=0 ξ(X i ) ≤ a (1 -k 2 )
. We conclude the proof by sending k to infinity.

Remark 3.5 It is possible to give more precision in the case (ii). Indeed, using the same arguments, we can prove that if Q{ξ - 0 > λ} ≥ 1 (log λ) α , for some α < 2, then we have, for any

ε ′ > 0, that lim n→∞ n -2 α +ε ′ Z n = -∞, P ⊗ Q-almost surely.
L(t k , Θ i (n k )) = x∈Z L(t k , x)1 {x∈Θ i (n k )} .
Since the Θ i (n k )'s are measurable, so are the random variables

1 {x∈Θ i (n k )} .
To the other hand, the measurability of L(t k , x), for any x ∈ Z, is obvious, being the composition of the measurable applications ω → (t k ) k≥0 (ω) and t → L(t, x).

We now proceed to the proof of Proposition 3.3. To get (3.20), we observe that

P ω τ + k ∧ τ - k ≤ t k ≤ P ω τ + k ≤ t k + P ω τ - k ≤ t k .
Then using (2.5), (2.6) and (3.8), (3.11) we obtain

P ω τ + k ∧ τ - k ≤ t k ≤ t k (e -4n k + e -7n k /6 ) ≤ 2 e -n k /6 , Since n k ≥ k, this yields k≥0 P ω τ + k ∧ τ - k ≤ t k ≤ 2 k≥0 e -n k /6 < ∞.
We conclude by using the Borel-Cantelli lemma.

To prove (3.19), we apply the strong Markov property at τ (b + k ) and get for 0

≤ i ≤ M(n k ) -2, P ω {L (t k , Θ i (n k )) ≥ ε i (n k ) t k } ≤ P b + k ω {L (t k , Θ i (n k )) ≥ ε i (n k ) t k -λ k } + P ω λ k ≤ τ (b + k ) ≤ τ - k + P ω τ - k ≤ τ (b + k ) ,
for any λ k ≥ 0. By (2.3), (3.9) and Lemma 3.1, we get, for all large k,

P ω τ - k ≤ τ (b + k ) ≤ b + k e n k /3 e n k ≤ n 2+ε ′ k e -2n k /3 . Since P ω λ k ≤ τ (b + k ) ≤ τ - k ≤ λ -1 k E ω τ (b + k ) 1 {τ (b + k )≤τ - k } , (2.4
) and (3.15) yield

P ω λ k ≤ τ (b + k ) ≤ τ - k ≤ (b + k -d - k ) 2 λ k e 2n k /3 ≤ 2n 2 (2+ε ′ ) k λ k e 2n k /3 ,
for all large k, the second inequality being a consequence of Lemma 3.1. Choosing λ k := e 5n k /6 , we obtain, for all large k,

P ω λ k ≤ τ (b + k ) ≤ τ - k + P ω τ - k ≤ τ (b + k ) ≤ e -n k /7 . (4.1) To treat P k,i := P b + k ω {L (t k , Θ i (n k )) ≥ ε i (n k ) t k -λ k },
we observe that (3.17) implies λ k ≤ 2 e -n k /6 t k . Therefore, we obtain

P k,i ≤ P b + k ω L (t k , Θ i (n k )) ≥ ε i (n k ) -2 e -n k /6 t k .
Then, by Chebyshev's inequality, we get

P k,i ≤ 1 (ε i (n k ) -2 e -n k /6 ) t k E b + k ω [L (t k , Θ i (n k ))] .
Furthermore, observe that Sinai's walk can not make more than t k excursions from b + k to b + k before t k . Since these excursions are i.i.d., we obtain

P k,i ≤ t k (ε i (n k ) -2 e -n k /6 ) t k E b + k ω L τ (b + k ), Θ i (n k ) . Now we recall (2.7), which implies E b + k ω L τ (b + k ), Θ i (n k ) ≤ c 3 x∈Θ i (n k ) e -[V (x)-V (b + k )] , for all 0 ≤ i ≤ M(n k ) -2.
Moreover, by (3.10), we get for all large k and for 0

≤ i ≤ M(n k ) -2, P k,i ≤ c 3 (ε i (n k )) 2 (ε i (n k ) -2 e -n k /6 ) ≤ c 8 ε i (n k ),
for some c 8 > 0. The second inequality is a consequence of ε i (n k ) ≥ ε 0 (n k ) and the fact that c 4 < 1/6 implies e -n k /6 = o(ε 0 (n k )).

Summing from 0 to M(n k ) -2 and using (3.22), we get, with

c 9 := 2(1 + 1 δ ) c 8 , M (n k )-2 i=0 P k,i ≤ c 8 M (n k )-2 i=0 ε i (n k ) ≤ c 9 ε M (n k ) (n k ). (4.2)
Assembling (4.1), (4.2) and recalling (3.4), (3.16) we obtain

k≥1 M (n k )-2 i=0 P ω {L (t k , Θ i (n k )) ≥ ε i (n k ) t k } ≤ k≥1 c 9 ε M (n k ) (n k ) + M(n k ) e -n k /7 < ∞.
This implies (3.19) by an application of the Borel-Cantelli lemma.

We get (3.18) by an argument very similar to the one used in the proof of (3.19), the main ingredients being the facts that (3.8) and the definition of d + (j)), and that Θ -1 (n k ) contains less than 2 n 2+ε ′ k sites (by (3.7)). We feel free to omit the details.

V (x) -V (b + k ) ≥ 2 n k , for x ∈ Θ -1 (n k ) (which is a consequence of (3.11),

Proof of Proposition 3.2

We now prove that, for P ⊗ Q-almost all (ω, ξ), there exists a sequence (m k ) such that (ω, ξ) ∈ A(m k ), ∀k ≥ 1, where A(m k ) is defined in (3.14).

Let j k := k 3k (k ≥ 1) and F j k-1 := σ{V (x), ξ z , d -(j k-1 ) ≤ x, z ≤ d + (j k-1 )}.
In the following, we ease notations by using γ i , ε i and M instead of γ i (j k ), ε i (j k ) and M(j k ).

If we are able to show that

k P ⊗ Q A(j k ) | F j k-1 = ∞, P ⊗ Q-a.s., (5.1)
then Lévy's Borel-Cantelli lemma ( [START_REF] Durrett | Probability: Theory and Examples[END_REF], p. 237) will tell us that P ⊗ Q-almost surely there are infinitely many k such that (ω, ξ) ∈ A(j k ).

To bound P ⊗ Q{A(j k ) | F j k-1 } from below, we start with the trivial inequality A(j k ) ⊃ A(j k ) ∩ C(j k-1 ), for any set C(j k-1 ). We choose C(j k-1 ) := C env (j k-1 ) ∩ D env (j k-1 ) ∩ C sce (j k-1 ), where

C env (j k-1 ) := inf 0≤y≤d + (j k-1 ) V (y) ≥ -j k-1 log 2 j k-1 , D env (j k-1 ) := inf d -(j k-1 )≤y≤0 V (y) ≥ -j k-1 log 2 j k-1 , C sce (j k-1 ) := max d -(j k-1 )≤x≤d + (j k-1 ) ξ - x < (ε -1 (j k )) -1/2 . Clearly, C(j k-1 ) is F j k-1 -measurable. Moreover on C env (j k-1 ) ∩A env (j k ), we have d + (j k-1 ) ≤ ν + ((-∞, -j k ]) ≤ b + (j k ). Let E sce -1 (j k ) := max x∈Θ -1 \[d -(j k-1 ),d + (j k-1 )] ξ - x < (ε -1 (j k )) -1/2 ,
and consider

E sce (j k ) := M -1 i=0 A sce i (j k ) ∩ E sce -1 (j k ). Since C sce (j k-1 ) ∩ E sce -1 (j k ) ⊂ A sce -1 (j k ), it follows that P ⊗ Q A(j k ) | F j k-1 ≥ P ⊗ Q P ⊗ Q A env (j k ) , B env (j k ) , E sce (j k ) , C(j k-1 ) | F j k-1 , ω | F j k-1 ≥ P ⊗ Q 1 {A env (j k ) , B env (j k ) , C(j k-1 )} P ⊗ Q {E sce (j k ) | ω} | F j k-1 .
Now, we suppose for the moment that we are able to prove that there exists c 10 > 0 such that, for P -almost all ω,

P ⊗ Q {E sce (j k ) | ω} ≥ c 10 k 1/4 . (5.2)
We get

P ⊗ Q A(j k ) | F j k-1 ≥ c 10 k 1/4 P ⊗ Q A env (j k ) , B env (j k ) , C(j k-1 ) | F j k-1 ≥ c 10 k 1/4 P + k P - k 1 C sce (j k-1 ) , (5.3)
where we use the fact that (V (x), x ≥ 0) and (V (x), x < 0) are independent processes and introduce

P + k := P A env (j k ) , C env (j k-1 ) | F j k-1 , P - k := P B env (j k ) , D env (j k-1 ) | F j k-1 .
To bound P + k from below, we introduce

E env 2 (j k ) := max 0≤x≤y≤b + (j k ) [V (y) -V (x)] ≤ j k 4 -j k-1 log 2 j k-1 -j k-1 -L ,
and consider

E env (j k ) := A env 1 (j k ) ∩ A env ann (j k ) ∩ E env 2 (j k ).
Observe that C env (j k-1 ) ∩ {max d + (j k-1 )≤x≤y≤b

+ (j k ) [V (y) -V (x)] ≤ j k 4 -j k-1 log 2 j k-1 -j k-1 - L} ⊂ A env 2 (j k ). Thus, since V (d + (j k-1 )) ∈ I j k-1 := [j k-1 , j k-1 + L],
we have, by applying the strong Markov property at d + (j k-1 ),

P + k ≥ inf z∈I j k-1 P z {E env (j k )} 1 C env (j k-1 ) . (5.4)
To bound P - k from below, we observe the following inclusion

B env (j k ) ⊃ max d -(j k )≤x≤y≤d -(j k-1 ) [V (y) -V (x)] ≤ j k 3 ∩ D env (j k-1 ).
Then since V (d -(j k-1 )) belongs to I j k-1 , the strong Markov property applied at d -(j k-1 ) yields

P - k ≥ inf z∈I j k-1 P z {B env (j k )} 1 D env (j k-1 ) . (5.5)
Observe that an easy calculation yields 1 C(j k-1 ) = 1, P ⊗ Q-almost surely for all large k. Therefore, recalling (5.3), (5.4) and (5.5), the proof of (5.1) boils down to showing that lim inf k→∞ inf z∈I j k-1

P z {E env (j k )} > 0, (5.6) lim inf k→∞ inf z∈I j k-1 P z {B env (j k )} > 0. (5.7)
The rest of the section is devoted to the proof of (5.2) and (5.6), whereas (5.7) is an immediate consequence of Donsker's theorem.

Proof of (5.2)

Since the sets {Θ i } -1≤i≤M -1 are disjoint, the events E sce -1 (j k ) and {A sce i (j k )} 0≤i≤M -1 are mutually independent. We write

P ⊗ Q {E sce (j k ) | ω} = M -1 i=0 P ⊗ Q {A sce i (j k ) | ω} × P ⊗ Q E sce -1 (j k ) | ω .
Thus, (5.2) will be a consequence of the two following lemmas.

Lemma 5.1 For P -almost all ω, we have

P ⊗ Q A sce M -1 (j k ) | ω ≥ 1 k 1/4 .
Lemma 5.2 There exists c 11 > 0 such that, for P -almost all ω,

lim inf k→∞ M -2 i=0 P ⊗ Q {A sce i (j k ) | ω} × P ⊗ Q E sce -1 (j k ) | ω ≥ c 11 . (5.8)
Proof of Lemma 5.1. Recalling (3.13), (3.7) and (3.4), we get, P -almost surely,

P ⊗ Q A sce M -1 (j k ) | ω ≥ exp {c 5 log q log 2 j k } ,
where q := Q {ξ 0 ≥ a -ρ}. Note that the definition of a implies -∞ < log q < 0. Therefore, it remains only to observe that log 2 j k = log k + log 2 k + log 3 and to choose c 5 small enough such that c 5 log q > -1/5, (5.9) to conclude the proof.

Proof of Lemma 5.2. Recalling (3.12) and that (ξ - x ) x∈Z is a family of i.i.d. random variables, we get, P -almost surely, for 0 ≤ i ≤ M -2,

P ⊗ Q {A sce i (j k ) | ω} ≥ Q{ξ - 0 ≤ ε -1/2 i } 2c 6 γ 2+ε ′ i ≥ exp 2c 6 γ 2+ε ′ i log 1 -Q{ξ - 0 ≥ ε -1/2 i } .
Then, since Q{ξ -

0 ≥ ε -1/2 i
} tends to 0 when k tends to ∞ and using the fact that log(1-x) ≥ -c 12 x for x ∈ [0, 1/2) with c 12 := 2 log 2 > 0, it follows that

P ⊗ Q {A sce i (j k ) | ω} ≥ exp -c 13 γ 2+ε ′ i Q{ξ - 0 ≥ ε -1/2 i } ,
for all large k, with c 13 := 2 c 6 c 12 . Recalling that Q ξ - 0 ≥ λ ≤ (log λ) -(2+ε) for λ ≥ λ 0 > 0 and (3.2) we get for k large enough and uniformly in 0

≤ i ≤ M -2, P ⊗ Q {A sce i (j k ) | ω} ≥ exp -c 14 γ -β i , (5.10)
where . Similarly, since E sce -1 (j k ) ⊂ A sce -1 (j k ) and recalling (3.12), we obtain 5.11) for some c 15 > 0. Combining (5.10) and (5.11), we get

β := (1 -α) 2 (2 + ε) -(2 + ε ′ ) > 0 by (3.
P ⊗ Q E sce -1 (j k ) | ω ≥ exp -c 15 γ -β 0 , (
M -2 i=0 P ⊗ Q {A sce i (j k ) | ω} × P ⊗ Q E sce -1 (j k ) | ω ≥ exp{-c 16 σ β }, with c 16 := max{c 14 , c 15 } and σ β := γ -β 0 + M -2 i=0 γ -β i .
By the same way we proved (3.22), we obtain that, for any β > 0, there exists c 17 ≤ 1 + 2/β such that σ β ≤ c 17 γ -β M -1 . Recalling (3.4), it follows that σ β → 0 when k → ∞, which implies (5.8).

Proof of (5.6)

To prove (5.6), we need the following preliminary result. Lemma 5.3 For any δ > 0, k ≥ 1 and any 0 ≤ p ≤ M, we have

M i=p γ δ i ≤ 1 + 2 δ γ δ p .
Proof. Observe that we easily get

M i=p γ δ i ≤ γ δ p + M i=p+1 γ i-1 γ i γ δ i γ i-1 -γ i dx.
Recalling that γ i-1γ i ≥ γ i-1 /2, for all large j and for 1 ≤ i ≤ M, we get

M i=p γ δ i ≤ γ δ p + 2 M i=p+1 γ i-1 γ i γ δ i γ i-1 dx. Then, M i=p+1 γ i-1 γ i γ δ i γ i-1 dx ≤ M i=p+1 γ i-1 γ i x δ-1 dx = γp γ M x δ-1 dx ≤ γ δ p /δ yields Lemma 5.3.
We now proceed to prove (5.6). Let

a ℓ := -3j k -ℓ γ M , F env 1 (j k , ℓ) := a ℓ+1 ≤ V (b + (j k )) < a ℓ . Denoting θ k := ⌊ j k /γ M ⌋ -1, the inclusion θ k ℓ=0 F env 1 (j k , ℓ) ⊂ A env 1 (j k ) yields P z {E env (j k )} ≥ θ k ℓ=0 P z {F env 1 (j k , ℓ) , A env ann (j k ) , E env 2 (j k )} =: θ k ℓ=0 P + k,ℓ . (5.12)
To bound P + k,ℓ by below for 0 ≤ ℓ ≤ θ k , we define the following levels,

η i = η i (j k , ℓ) := a ℓ + γ i , 0 ≤ i ≤ M, (5.13) η M +1 = η M +1 (j k , ℓ) := a ℓ , (5.14) η M +2 = η M +2 (j k , ℓ) := a ℓ+1 . (5.15)
In the following, we introduce stopping times for the potential, which will enable us to consider a valley having "good" properties. Let us write

T = T (j k , ℓ) := ν + ((-∞, η M +1 ]), T = T (j k , ℓ) := ν + ((-∞, η M +2 ]). d + (j) T i T ′ M -i T b + (j) x V (x) j 0 η i a ℓ a ℓ+1 Figure 1: Example of ω ∈ G(j k , ℓ) ∩ G ′ (j k , ℓ) ∩ H(j k , ℓ) ∩ H ′ (j k , ℓ) then Lemma 5.3 yields [T i , T ′ M -i ] ⊂ [b + (j k ) -c 6 γ 2+ε ′ i , b + (j k ) + c 6 γ 2+ε ′ i ], 0 ≤ i ≤ M -2. (5.18)
Recall now definition of Θ i (j k ), so that, by assembling (5.16) and (5.18), we have on G

(j k , ℓ)∩ G ′ (j k , ℓ) ∩ H ′ (j k , ℓ), Θ i (j k ) ⊂ x ∈ Z : V (x) -V (b + (j k )) ≥ γ i+2 , 0 ≤ i ≤ M -2.

An easy calculation yields

x∈Θ i (j k ) exp{-[V (x) -V (b + (j k ))]} ≤ 2c 6 γ 2+ε ′ i e -γ i+2 , for all 0 ≤ i ≤ M -2, on G(j k , ℓ) ∩ G ′ (j k , ℓ) ∩ H ′ (j k , ℓ). On the other hand, since 6 c 4 < 1, we get 2c 6 γ 2+ε ′ i e -γ i+2 ≤ ε 2 i , for all large k and uniformly in 0 ≤ i ≤ M -2. This implies that G(j k , ℓ) ∩ G ′ (j k , ℓ) ∩ H ′ (j k , ℓ) ⊂ F env 1 (j k , ℓ) ∩ A env ann (j k ). We easily observe that G(j k , ℓ) ∩ H(j k , ℓ) ⊂ E env 2 (j k ), for all large k. Thus we obtain G(j k , ℓ) ∩ G ′ (j k , ℓ) ∩ H(j k , ℓ) ∩ H ′ (j k , ℓ) ⊂ F env 1 (j k , ℓ) ∩ A env ann (j k ) ∩ E env 2 (j k ).
Recalling (5.12), we get P + k,ℓ ≥ P z {G(j k , ℓ) , G ′ (j k , ℓ) , H(j k , ℓ) , H ′ (j k , ℓ)} .

To bound P z {G(j k , ℓ) , G ′ (j k , ℓ) , H(j k , ℓ) , H ′ (j k , ℓ)} by below, we will apply the strong Markov property several times. Since V (T ′ M ) ∈ I η 0 := [η 0 , η 0 + L], the strong Markov property applied at T ′ M implies, for z ∈ I j k-1 , P + k,ℓ ≥ P z G(j k , ℓ) , G ′ (j k , ℓ) , H(j k , ℓ) inf

y∈Iη 0 P y d + (j k ) ≤ ν + ((-∞, η 1 ]) .
To bound by below P y {• • •} on the right hand side, observe that P y {• • •} is greater than P η 0 {• • •}. Moreover since η 1 ≥ -4j k implies j kη 1 ≤ 5j k , Lemma 2.1 yields

P η 0 {d + (j k ) ≤ ν + ((-∞, η 1 ])} ≥ η 0 -η 1 5j k + L = η 0 (1 -η -α 0 ) 5j k + L ≥ c 18 ,
for all large k and some c 18 > 0, which implies P + k,ℓ ≥ c 18 P z {G(j k , ℓ) , G ′ (j k , ℓ) , H(j k , ℓ)} .

We now apply the strong Markov property successively at (T ′ M -i ) 1≤i≤M and T , such that P + k,ℓ ≥ c 18 P z {G(j k , ℓ) , H(j k , ℓ)} inf Recall that for any β ′′ > 0, there exists c 23 > 0 such that M p=1 γ -β ′′ M -p ≤ c 23 γ -β ′′ M . Then, recalling (5.19), this yields, for all large k, P + k,ℓ ≥ c 24 γ M γ 0 P z {G(j k , ℓ) , H(j k , ℓ)} , (5.20) with c 24 > 0. To bound P z {G(j k , ℓ) , H(j k , ℓ)} from below, we apply successively the strong Markov property at (T M -i ) 0≤i≤M such that P z {G(j k , ℓ) , H(j k , ℓ)} ≥ P z {H(j k , ℓ)} and M p=1 γ p ≤ c 29 γ 1 = o(j k ) for some c 28 , c 29 > 0, (5.21) yields P z {G(j k , ℓ) , H(j k , ℓ)} ≥ c 30 Q 0 P z {H(j k , ℓ)} , (5.22) for some c 30 > 0. Observe that Donsker's theorem implies that there exists c 31 > 0 such that min{P z {H(j k , ℓ)}, Q 0 } ≥ c 31 . Therefore, assembling (5.20) and (5.22), we get

P + k,ℓ ≥ c 32 γ M γ 0 ,

c 4 2+ε

 4 [START_REF] Borodin | A limit theorem for sums of independent random variables defined on a recurrent random walk[END_REF], and c 14 := c 13 2

  := P y d + (η M -p ) ≤ γ 2+ε ′ M -p , d + (η M -p ) < ν + ((-∞, η M -p+2 ]) , 0 ≤ p ≤ M. First, observe that inf y∈Iη M -p+1 Q ′ p,y ≥ Q ′ p,η M -p+1 =: Q ′ p , for all 1 ≤ p ≤ M and similarly inf y∈I η M +1 -L Q ′ 0,y ≥ Q ′ 0,η M +1 -L := Q ′ 0 . Therefore we only have to bound from below Q ′ p for 1 ≤ p ≤ M and Q ′ 0 . Recalling that P {A , B} ≥ P {A} -P {B c }, we get, for 1 ≤ p ≤ M, Q ′ p ≥ P η M -p+1 d + (η M -p ) < ν + ((-∞, η M -p+2 ]) -P η M -p+1 d + (η M -p ) ≥ γ 2+ε ′ M -p ,andQ ′ 0 ≥ P η M +1 -L d + (η M ) < ν + ((-∞, η M +2 ]) -P η M +1 -L d + (η M ) ≥ γ 2+ε ′ M .where c 22 := c 12 c 21 .

≥ 1 -

 1 whereQ p := P ηp ν + ((-∞, η p+1 ]) ≤ min d + (η p+1 + j k /5), γ 2+ε ′ p , 0 ≤ p ≤ M.Recall that P {A , B} ≥ P {A} -P {B c }. Then (2.2) yields, for 1 ≤ p ≤ M,P ηp ν + ((-∞, η p+1 ]) ≤ γ 2+ε ′ p c 25 γ -ε ′ /6 p ,with c 25 > 0. Moreover, using Lemma 2.1, we get, for 1 ≤ p ≤ M,P ηp d + (η p+1 + j k /5) ≤ ν + ((-∞, η p+1 ]) ≤ c 26 γ p j k ,with c 26 > 0. Therefore, observing that, for 1 ≤ p ≤ M, we have γpj k ≤ γ 1 j k = j -α k → 0, k → ∞,and using the fact that log(1x) ≥ -c 12 x, for x ∈ [0, 1/2), we get that M p=1 Q p ≥ exp -c 27 c 27 := c 12 max{c 25 , c 26 }.

  Let us first explain why the events described in Proposition 3.3 (more precisely in(3.18) and (3.19)) are measurable. Since the sequences (n k ) k≥0 and (t k ) k≥0 can be explicitly constructed, ω → (n k ) k≥0 (ω) and ω → (t k ) k≥0 (ω) are measurable. Moreover, this implies that Θ i (n k ) is measurable, for any -1 ≤ i ≤ M(n k ) -2. Now, let us write

	4 Proof of Proposition 3.3

Acknowledgements

I would like to thank my Ph.D. supervisor Zhan Shi for helpful discussions. Many thanks are due to two anonymous referees for careful reading of the original manuscript and for invaluable suggestions.

Moreover, we set

and

See Figure 1 for an example of

If we choose c 6 such that

and

Recalling (5.13) and (5.14), we bound by below

for all large k. The first inequality is a consequence of (1 + x) -1 ≥ 1x for any x ∈ (0, 1)

and the second one is a consequence of 0 < α < 1. Similarly, recalling (5.13), (5.14) and (5.15), we get, for all large k,

with c 18 > 0. Moreover, combining (2.2) and the fact that γ M -p ≤ γ

, for all large k and for some c 19 > 0. Therefore, we obtain Q ′ 0 ≥ c 20 for some c 20 > 0 and recalling (3.6) we get, for 1

where

, where c 32 := c 24 c 30 c 2 31 . Recalling (5.12) and θ k = ⌊ j k /γ M ⌋ -1, we get, uniformly in z ∈ I j k-1 , P z {E env (j k )} ≥ c 32 θ k γ M γ 0 ≥ c 33 , for all large k and for some c 33 > 0, which concludes the proof of (5.6).


Joseph Kamel

Reza Vosooghi
email: reza.vosooghi@irt-systemx.fr

Jakob Puchinger

Feirouz Ksontini

Göknur Sirin

Exploring the Impact of User Preferences on Shared Autonomous Vehicle Modal Split: A Multi-Agent Simulation Approach

Keywords: Multi-Agent Simulation, Shared Autonomous Vehicle, User Preferences

Shared autonomous vehicles are considered to have a transformative impact on future urban transportation and especially Shared Mobility. In order to assess the transport-related impact of this new mode, various models and simulations are under development. The majority of recently proposed simulations are based on activity-based multiagent approaches. Thanks to the disaggregated level of data in multi-agent simulation, the traveler decision making mechanisms might be individualized according the attributes. In this paper we try to take in advantage of this granularity in order to explore the impact of user preferences on the modal split of shared autonomous vehicles. To illustrate the proposed methodology the transport system of Paris is simulated by using an activity-based multi-agent tool called MATSim. The traveler preferences toward shared autonomous vehicles use are also summarized based on the literature review.

Introduction

The advent of Autonomous Vehicle (AV) technologies fosters new opportunities for transportation systems by evolving modal shift from private mobility to service-use. It is expected that the next revolution in Shared Mobility will be based on Shared Autonomous Vehicles (SAV). This new service seems to be of high importance for car producers given their recent investments in the technology. However the impact of such systems on traveler behavior, urban form, congestion, and the environment is unknown since no large-scale public SAV service exists today [START_REF] Stocker | Shared Automated Mobility: Early Exploration and Potential Impacts[END_REF]. Nevertheless, considerable efforts are being made nowadays for the planning of future SAV services, and the need for a reliable demand analysis and traffic simulation gains importance.

In order to predict the impact of SAVs on travel behavior and traffic state, new simulation-based approaches have recently been developed where mainly activity-based multi-agent simulations are used [START_REF] Vosooghi | A critical analysis of travel demand estimation for new one-way carsharing systems[END_REF]. The latter allow us to model and simulate future SAV systems at a fine level of detail. Actually a multi-agent simulation is developed in which travel demand emerges from the interactions of four types of agents in the transportation system: node, arc, traveler and vehicle [START_REF] Zheng | A Primer for Agent-Based Simulation and Modeling in: Transportation Applications[END_REF]. Different levels of granularity and interaction complexity between agent types and agents themselves can be supposed in the simulation. While the complex microscopic rules for traffic simulation are already well investigated in this approach, the traveler preferences to use different modes are not adequately reflected. However, multi-agent approaches allow us to consider more details with regards to travelers and to explore how these details affect predictions.

Despite the mentioned advantages of multi-agent approaches, most of the simulations relying on it are based on a homogeneous structure of behavior. This implies that all the travelers have similar preferences when they make mode choice or other decisions. This similarity renders the results less accurate especially considering that user preferences are one of the key components that determine SAV market share and its impact [START_REF] Stocker | Shared Automated Mobility: Early Exploration and Potential Impacts[END_REF]. This paper aims to: first, better understand how user preferences would impact the SAV modal split in a multi-modal transportation system; and second, to gain insight into how to integrate those factors in a multi-agent simulation approach. This study introduces the first large-scale multi-agent activity-based simulation of Paris area developed in MATSim, a multi-agent transport platform. The following sections describe related studies, the methodology of modeling and considering user preferences in a multi-agent simulation, the experiment, key findings, and study conclusions.

Literature review

According to the existing literature, large attention is nowadays given to behavioral experimental studies towards using AV services, especially on behavioral characteristics and perceptions, willingness-to-pay and level of awareness and attitudes [START_REF] Gkartzonikas | A Literature Review on Surveys for Autonomous Vehicles[END_REF]. Also there are a limited number of studies specifically addressed to the case of SAV. Some of them are just based on mode choice surveys, but in a few studies more advanced modeling and simulation approach have been conducted. [START_REF] Steck | How autonomous driving may affect the value of travel time savings for commuting[END_REF] addressed the impact of autonomous driving on the value of travel time savings (VTTS) and mode choices for commuting trips using stated choice experiments and mixed logit modelling. They compared privately owned autonomous vehicles and SAV to other modes. Their results show that mode-specific parameters such as in-vehicle time and cost play an important role for the mode choice, but the socio-demographic variables (gender, age and income) do not have a significant effect on modal split. The authors also estimated that using SAV will reduce the VTTS up to 10% for commuting trips. [START_REF] Liu | Tracking a system of shared autonomous vehicles across the Austin, Texas network using agent-based simulation[END_REF] simulated SAV mode requests using agent-based simulation through a stochastic process for four possible fare levels. In their study the out-of-pocket cost and the value of time variables are considered and the modal split of SAVs and conventional vehicles are compared.

In two above-mentioned studies the system-related attributes including travel and waiting time and cost are estimated or considered to have a strong impact on mode choice comparing to the individual-specific characteristics, while the latter have also significant effects according to other studies. [START_REF] Haboucha | User preferences regarding autonomous vehicles[END_REF] developed a model for autonomous vehicle long-term choice decisions and pointed from their panel data that travelers have a strong individual taste-variation for using SAV. What is interesting in their results is that although cost is an important variable in the SAV choice, 25% of travelers would refuse to use SAVs even if they would be completely free. Also their results show that gender, age, educational level and income are the most significant individual-related attributes that have an effect on SAV mode choice. [START_REF] Krueger | Preferences for shared autonomous vehicles[END_REF] selected a stated preferences (SP) mode choice survey to compare SAV with the currently chosen travel modes. In their study the respondents were asked to indicate whether they would switch to SAVs without or with ride-sharing on a recent trip. They further identified the characteristics of potential users by using a mixed logit model. Their results show that SAVs may be more attractive to young travelers and that their acceptance is very sensitive to service attributes. [START_REF] Bansal | Assessing public opinions of and interest in new vehicle technologies: An Austin perspective[END_REF] analysed individuals' stated frequencies to use SAV under different pricing scenarios and identified the characteristics of potential SAV adopters. In their study gender and age are found to have a significant effect on SAV use.

To the best of the author's knowledge only the above-mentioned studies have specifically dealt with user preferences and SAV mode choice. However, the last three reflect the importance of specific variables on decision mechanisms for choosing SAVs. The surveys investigating those variables and their effects on SAV mode choice from the literature are summarized in Table 1. According to these studies, the most important attributes impacting SAV use are gender, age, income and educational level. Notwithstanding, the consideration of user preferences according to the mentioned attributes in the simulation remains very limited.

Methodology

The work conducted in this research can be described in four major parts: 1) data preparation, 2) multi-agent simulation, 3) adding user preferences, and 4) model set up and calibration.

Data preparation

As mentioned above in multi-agent approaches, instead of getting a sample representing whole population of a given zone, the behavior of individuals and households is simulated. Thus, such an approach requires to have detailed data. As this microdata is not available for the whole population, a synthetic population is usually generated. This is done by drawing households and individuals from microdata samples on a zonal level. A typical synthetic population is composed of spatially located households which possess certain attributes, such as household size, income, car ownership, etc. Households consist of one or several individuals, who possess personal attributes, such as gender, age, main activity, etc. The preferences of travelers depending on the above-mentioned attributes can be considered in the mode choice decision mechanism in the simulation process.

In this research the synthetic population of around 12 million people for Greater Paris (Île-de-France) is generated from public use microdata (INSEE 2013) and regional transport survey (EGT 2010) relying on a simulation-based synthesis. The population is generated by randomly drawing the adequate number of households (defined by marginal data) from the microdata at a specific spatial zoning called IRIS (5261 zones), see Fig. 1a. The simulation is repeated until some minimum standardized root mean square errors (SRMSE) for other attributes (age ranges, socioprofessional categories, etc.) are reached as goodness-of-fit. The simulation-based synthesis has been selected for this study because compared to other popular methods such as iterative proportional fitting (IPF) or combinatorial optimization (CO), this approach is more flexible in terms of adding additional attributes from other sources [START_REF] Farooq | Simulation based population synthesis[END_REF]. Also microdata employed in this study consists of an acceptable sample rate (more than 35% of the whole population).

It should be noted that the area for the generation of the synthetic population has been chosen to be bigger than the case study area (i.e. Paris, with a population of around 2,200,000 inhabitants), see Fig. 1b, as this is necessary to capture all trips and activities starting or ending in Paris. Also all modes except cargo and taxi are simulated. For every individual in a household, activity chains and locations have also been associated to simulate their mobility in a given day, see Fig. 2. Activity chains for each socio-professional category of individuals were derived from a regional transport survey. The locations for home, study and work activities were already available in the bigger aggregated special zoning (commune, 1276 zones for Greater Paris) in the public use microdata. Based on these data, Origin-Destination (OD) matrices have been generated. For other trip purposes the OD matrix was derived from the regional transport survey. To geo-localize each activity inside the zone, facility data has been used. The activities in each zone are randomly distributed and the activity details have been derived from a regional transport survey analysis. In this analysis all the individuals were grouped by socio-professional attributes and for each group the probability of activity start times and durations have been extracted. Activity start times and durations have then been assigned to the synthesized individuals according to the estimated probability. For example, for the activity chain of home-study-home the mean and standard deviation values of start time and activity duration are calculated. Then these values are randomly assigned to each individual (who had the same activity according to their category). To avoid cumulative error at the end of the day, the start time windows for each activity are fixed.

Multi-agent simulation

Our contribution has been made by using the multi-agent simulation platform MATSim (Horni et al., 2016), which is currently the most widely applied tool for simulating new vehicle sharing systems [START_REF] Vosooghi | A critical analysis of travel demand estimation for new one-way carsharing systems[END_REF]. MATSim uses the population with an initial daily plan for each agent as input. The core of MATSim relies on a co-evolutionary algorithm for optimizing agent's plan. In fact, almost all agent-based transport simulations do not involve discrete choice models as they are used in conventional transport demand models, but these simulations are more based on stochastically finding the maximized utility for all agents. In each iteration agents choose or innovate a plan (a set of choices including mode, route and activity end-time) for their day. This plan is then executed with a traffic simulator. The simulation converges once the average performance measures for all agents stabilize. The performance measure in MATSim is presented by a standard scoring function based on the Charypar-Nagel scoring method [START_REF] Charypar | Generating complete all-day activity plans with genetic algorithms[END_REF]. This scoring function is divided into two parts: activity scoring and travel scoring: (1)

For each agent j, with N as the number of activities and trip q which is the trip that follows by activity q, the total score of an executed plan is calculated. The global concept is that the activity score is logarithmically increasing with the activity duration whilst the travel score is usually negative. These two scoring formulas contain various specific or individual-related parameters. The latter are usually set up as been the same for all the agents in the population. However, activity and travel scores can be modified in order to individualize scoring parameters. It is thus possible to integrate preferences to the simulation. Since the orientation of this paper is to study the impact of preferences on the mode choice, only the travel scoring has been modified. Nevertheless activity scoring is also modified in an indirect way by considering the marginal utility of traveling.

Adding user preferences

Each trip in MATSim is scored based on the mode, travel time and distance. We used in our approach only three major parts of the initial travel scoring function: , ( ) , , .

. . trav mode q mode(q) trav,mode(q) trav q m d,mode(q) trav q

S C t d       (2)
For each mode used in trip q the score is calculated from mode-specific constant 𝐶 𝑚𝑜𝑑𝑒 , marginal disutility of travel 𝛽 𝑡𝑟𝑎𝑣,𝑚𝑜𝑑𝑒 , travel time 𝑡 𝑡𝑟𝑎𝑣 , marginal utility of money 𝛽 𝑚 , monetary distance rate 𝛾 𝑑,𝑚𝑜𝑑𝑒 and 𝑑 𝑡𝑟𝑎𝑣,𝑞 which is the distance traveled between two activity locations. Instead of having the same coefficient values for all agents, we need to change the scoring according to the agent's attributes. In order to do this, preference factors are introduced. These factors are determined for three attributes: gender, age and income. We assumed that the travel scoring coefficients would vary according to the related factor with specific distributions, see Fig. 3.

As there are no data available on SAV user preferences for the Paris area, we use the study of [START_REF] Haboucha | User preferences regarding autonomous vehicles[END_REF] in order to set the variation of each factor. The distribution forms are defined according to our hypothetical assumptions. In general men are more likely to use a SAVs than women, as well as younger persons in comparison to an older ones and wealthy persons to poor ones. For the gender factor 𝜅 𝑠𝑒𝑥 two fix values are assumed. We also assume that the age preference factor 𝜅 𝑎𝑔𝑒 for young traveler changes linearly. For income, the preference factor 𝜅 𝑖𝑛𝑐𝑜𝑚𝑒 is assumed to grow logarithmically. The preference factor is then multiplied to corresponding coefficients in the SAV scoring function. It should be noted that since the factors are used two times in the calculation, the mean values of the coefficients distribution have been fixed to be 0.5: , 2 ( )

2 2. age sex age sex income income         (3)
So the modified travel scoring function for SAV is:

' , , , , , , . . . .
.

trav q sav sav age sex trav,sav trav q income m d sav trav q S C t d         (4)
In this function the income factor is multiplied to the monetary part of scoring only, while age and gender factors are multiplied to the second part. The mode-specific constant set during calibration process remains unchanged.

Model set up and calibration

To set up the travel scoring function, the marginal utility (and disutility) parameters as well as the monetary distance rate have been estimated based on the econometric analysis relying on transport regional surveys. The model is calibrated by mode specific constants which are set up during the simulation according to the overall modal share derived from regional transport statistics. The calibration process is done by trial and error; the goal has been to have a stable modal split that corresponds to reality for the basic scenario. SAVs have then been added to the simulation by means of the hypothetical constant and coefficients. Since the aim of the research is to compare two scenarios, these values are assumed to be hypotheticals, however those values could be set up once survey data for SAVs are available.

Experiments and results

Three scenarios have been simulated: 1) the transportation system of Paris area without SAV, 2) SAV added without considering user preferences, and 3) SAV added and user preferences are considered. SAV services are considered to be only car-share and not the ride-share. Four main modes for the basic scenario are considered (PT, private car, walk and bike). In the two last scenarios a fleet of 17000 SAVs with 20 initial distribution points are added to the simulation. The simulation is repeated 900 times and lasted a week (Bullx S6130 -CPU: Intel® Xeon® E7-88 threads -RAM: 1TBs DDR4). The modal splits of the three scenarios are shown in Table 2. SAV modal shares are estimated to be 5.3% without, and 3.8% with introducing preferences. While the overall modal split of the SAV after introduction of user preferences decreases by 28%, the use of this mode decreases by 38% among the travelers. The modal share of private car decreases up to 26-27% in the two last scenarios. Meanwhile the modal shares of PT increase. That can be because of the users who take SAV for PT access and egress. Also use of SAV results in the higher use of bike. It can be due to the fact that the previously car users are now more likely to use bike as a well-integrated mode to SAV or PT. In the contrary, walk modal shares decrease in the second and the third scenarios. The longer walking trips could potentially be replaced by SAV in these scenarios. The main difference between two scenarios with SAV is the PT modal split which unusually increases in the third scenario. It seems that this modal split grows because the eliminated SAV sub-trips are replaced by PT. The SAV use fluctuations for each attribute according to their variation before and after the introduction of preferences are also shown in Fig. 4. These changes for older people, females and persons with higher income are more significant. In order to have a more in-depth analysis of how user preferences (all attributes together) change the SAV modal split, we grouped the users with similar attributes (called profile). The profiling process in the current study is based on one of the attributes that we have generated in the synthetic population. Then the SAV uses by each profile are compared, Table 3. In Table 3, the second column presents population distribution by gender for each profile. The female rates for retired persons and for women or men at home are higher than other profiles. The fourth column shows the average household income. It should be noted that this value for the unemployed people is significant (even if is less than for other profiles) because of the unemployment benefits funded by local authority. Concerning SAV use, detailed analysis of the population according to the profiles shows bigger changes, especially for retired persons and for women or men at home. For instance, relative difference for the retired or pre-retired population is about two times bigger than for employees. That is because of the higher average age of retired persons as well as the bigger portion of female population in this profile. The SAV use for students older than 14 years and the persons younger than 14 years remains approximately unchanged. Also the changes for employees and unemployed is estimated to be almost the same. That is because of the similar attributes values. As the values of time for these two profiles are not the same, logically the changes should be more significant. The reason for it is related to the main drawback of today's surveys: the measurements for SAV preferences are related more to the classic socio-demographic attributes, while it seems that profiling could be more useful in this case.

Discussion and conclusion

In this article we explored the impact of user preferences on SAV modal share by simulating multimodal transportation system of Paris area applying a multi-agent simulation. The simulation results reveal that neglecting user preferences in classic multi-agent simulations can significantly change the outputs for future scenarios. Also, a detailed analysis of the population according to the profiles shows that demographic structure plays an important role regarding to the use of SAV. Today, despite the considerable number of studies related to exploring the influence of travelers' preferences on the demand of SAV, the investigation about introduction of these preferences in the simulation remains very limited. We started to bridge this gap, but further explorations are still required.

Future work will involve investigations on how to add further attributes to the mode choice mechanism of agents as well as how to integrate them to the activity-related performance measure of multi-agent simulation.

Fig. 1 .

 1 Fig. 1. (a) IRIS spatial zones of Paris and Greater Paris (Île de France); (b) road and transit network of case study area (Paris).

Fig. 2 .

 2 Fig. 2. Data preparation process in order to find synthetic households and individuals with their activity characteristics.

Fig. 4 .

 4 Fig. 4. Modal share (SAV use) changes for each attribute before and after the introduction of preferences.

Table 1 .

 1 Socio-demographic variables and their effects on SAV mode choice

	Author(s), year	Number of	Variables (positive: + , not significant: * )
		respondents
	Bansal et al. (2016)	347	Gender (male + ), age (younger + ),
	Haboucha et al. (2017)	721	Gender (male + ), age (younger + ), income (higher + ), educational level (higher + )
	Krueger et al. (2016)	435	Age (younger + )
	Steck et al. (2018)	172	Gender

* , age * , income *

Table 2 .

 2 Comparison of modal splits for three scenarios.

	Scenario			Modal split (%)
		PT	Private car	Walk	Bike	SAV
	W/O SAV	29.7	21.4	47.1	1.8	-
	W/ SAV W/O Preferences 30.1	15.7	45.5	3.4	5.3
	W/ SAV W/ Preferences	31.3	15.9	45.6	3.4	3.8

Table 3 .

 3 Comparison between SAV use by profile with and without preferences.

	Profile	Population,	Average Age	Average	W/O	W/	Relative
		female / male	(year)	Household	Preferences	Preferences	Difference
		(% of total)		Income (€)	(%)	(%)	(%)
	Employed workers	51.2 / 48.8	41.0	35 736	20.1	12.1	-40
	Unemployed	52.1 / 47.9	37.9	29 588	15.6	9.8	-37
	Retired or pre-retired	60.7 / 39.3	74.5	39 866	11.9	3.3	-72
	Students >14 years	54.9 / 45.1	20.3	45 042	19.0	19.2	+1
	< 14 years of age	50.6 / 49.4	9.2	45 891	15.9	16.0	+1
	Women or men at home	95.6 / 4.4	55.3	51 310	13.3	3.2	-76
	Other inactive	48.6 / 51.4	45.6	30 435	16.3	8.6	-47

Fig. 3. Distribution graph of preference factors for gender (sex), age range and income.

Acknowledgements

This research work has been carried out in the framework of IRT SystemX, Paris-Saclay, France, and therefore granted with public funds within the scope of the French Program "Investissements d'Avenir".


T Gloriant
email: thierry.gloriant@insa-rennes.fr

Y Yang

P Castany

M Cornen

F Prima

S J Li

Y L Hao

Characterization of the martensitic transformation in the superelastic Ti-24Nb-4Zr-8Sn alloy by in situ synchrotron X-ray diffraction and dynamic mechanical analysis

Keywords: titanium alloy, martensitic transformation, tensile tests, synchrotron diffraction, DMA

Characterization of the martensitic transformation in

the superelastic Ti-24Nb-4Zr-8Sn alloy by in situ synchrotron X-ray diffraction and dynamic mechanical analysis.

Introduction

In this study, the metastable β-type Ti-24Nb-4Zr-8Sn (wt.%, Ti2448 for short) alloy was investigated in order to characterize the martensitic transformation occurring in this alloy. Indeed, due to this martensitic transformation, the Ti2448 alloy shows good superelasticity at room temperature characterized by a large recoverable strain of 3.3% [1][2][3]. This recoverable strain is much higher than that observed in the conventional Ti-6Al-4V alloy and in other binary Ti-Nb superelastic alloys [4,5].

Thus, the good mechanical properties of the Ti-24Nb-4Zr-8Sn alloy, including high strength (>800 MPa), low modulus (approx. 50 GPa) and the superelasticity mentioned above enable it to be used as bone substitutes (implants or prostheses) for biomedical applications.

With this alloy, the superelastic (or pseudoelastic) behavior is obtained when the quenched microstructure is composed of the metastable β phase at room temperature.

In such cases, a stress-induced martensitic (SIM) transformation between the austenitic β phase (bcc) and the martensitic α" phase α" (orthorhombic) occurs and large elastic recovery (superelasticity) can be obtained once the mechanical stress is released due to the fact that this SIM transformation is fully reversible. By tensile tests, the superelastic effect can be highlighted by the presence of a stress plateau on the tensile curve presenting thus a characteristic double-yielding behavior.

In superelastic alloys, the study of the SIM transformation is challenging due to its reversibility after the release of the stress. That is why in situ experiments are needed to characterize the SIM transformation under loading conditions. For example, conventional in situ X-ray diffraction (XRD) studies were performed on binary Ti-26Nb or Ti-13Nb-4Mo alloys (at.%) and allowed the detection of the SIM α" phase transformation under loading conditions [4,6]. However, it is difficult to separate the main peaks belonging to β and α" phases because of the relatively longer wavelength and the presence of both K α1 and K α2 wavelengths of conventional X-ray sources used in laboratories. In order to avoid these limitations, synchrotron X-ray diffraction (SXRD) can be used because of the combination of a shorter wavelength and a better monochromaticity. Thus, SXRD appears to be a very efficient method to characterize such SIM transformation and was never used to study the Ti2448 alloy in which superelasticity is predominantly due to this mechanism. Furthermore, martensitic transformations are usually described using characteristic temperatures such as martensite start, martensite finish, austenite start and austenite finish temperatures, respectively denoted M s , M f , A s and A f [7,8]. These temperatures can be evaluated by dynamic mechanical analysis (DMA) for different applied static stresses, and the martensitic transformation is thus generally highlighted by a drop of the dynamic elastic modulus corresponding to an increase of the damping factor tan δ [9,10].

The objective of this study is to characterize the martensitic transformation occurring in the Ti2448 alloy by both in situ synchrotron radiation X-ray diffraction (SXRD) and dynamic mechanical analysis (DMA).

Materials and methods

Hot-forged Ti-24Nb-4Zr-8Sn (wt.%) cylinder ingot with a diameter of 55 mm was used as raw material in this work. The chemical composition of the alloy is listed in Table 1. Each slice-cut sample used was directly cold rolled without intermediate annealing into 0.5 mm final thickness sheet, which corresponds to a cold rolling reduction rate of 94%. Then, two thermal treatments were performed for comparison: on the one hand, cold-rolled specimens were solution treated at 900 °C under high vacuum for 30 min followed by water quenching (solution treated state, ST); on the other hand, cold-rolled specimens were flash treated at 700 °C for 3 min followed by air cooling (flash treated state, FT). After thermal treatments, all specimens were cleaned in an acid solution made of 50% HF and 50% HNO 3 (in volume) to remove any oxidation layer.

Microstructures were observed by optical microscopy (Leica microscope). To be observed, the specimens were mechanically prepared and finally "mirror" polished using a colloidal silica suspension solution. To reveal the microstructure, each specimen was etched in an acid solution composed of 8% HF, 15% HNO 3 and 77%

H 2 O (in volume).
The mechanical properties of the Ti2448 alloy (ST and FT states) were evaluated by conventional tensile tests until rupture using an INSTRON 3369 tensile machine with a strain control mode at a rate of 10 -4 s -1 . To check the reproducibility of the mechanical properties, conventional tensile tests were performed on 3 different samples for each state. The superelasticity was more precisely characterized by cyclic loading-unloading tensile tests using the same tensile machine (strain control mode at a rate of 10 -4 s -1 for both loading and unloading). With these kinds of tests, the strain was incremented by steps of 0.5% until an elongation of 5% and each step was followed by a total release of the stress. The tensile direction of the specimen was chosen parallel to the rolling direction. The tensile specimens used have a gauge width of 3 mm and a thickness of 0.5 mm.

In situ synchrotron X-ray diffraction (SXRD) under cyclic tensile tests was conducted on beam line ID31 at the European Synchrotron Radiation Facility (ESRF, Grenoble, France). For these in situ tensile tests, a 5 KN Gatan micro-tensile dispositive was used. The ID31 offers a high-resolution incident X-ray beam (λ = 0.040002106 nm).

Nine-channel multi-analyzer installed as detectors can collect scanning data over the angular range 2-14°, with a scanning step of 0.005°. These in situ cyclic tensile tests under synchrotron were realized on tensile specimens (gauge width of 3 mm and a thickness of 0.5 mm) with increments of 0.5% until 5.0% and then increments of 1.0% until 10%. SXRD scans were then obtained after each increment for both loading and unloading conditions. Dynamic mechanical analysis (DMA) was done with a Metravib DMA50 in tensile mode. Samples for DMA were cut with a low-speed diamond wire saw. The cooling/heating thermal cycle was chosen between -150 °C and 50 °C with a rate of 5 °C/min. The frequency of the sinusoidal dynamic mechanical solicitation was fixed at 1 Hz and the dynamic/static ratio was kept at σ dyn /σ stat = 1/4. The tests were performed with the tensile direction parallel to the rolling direction for different static stress values: 160, 200, 240 and 280 MPa. Two important characteristic parameters of the DMA analysis: storage modulus (E ' ) and damping factor (tanδ) as a function of temperature can be obtained by this method. The determination of characteristic transformation temperatures is based on intersection of tangents from the curves obtained.

Results

Microstructure and mechanical properties

The microstructures of the Ti2448 alloy observed by optical microscopy are shown in Fig. 1 for the solution treated alloy (ST, Fig. 1a) and for the flash treated alloy (FT, Fig. 1b), respectively. The grain sizes after ST and FT were evaluated to be about 50 µm and 7 µm on these micrographs, respectively. Fig. 2 displays examples of conventional and cyclic tensile curves for the ST (Fig. 2a) and FT (Fig. 2b) conditions. From all tensile tests, elongation at rupture, ultimate tensile strength and incipient Young's modulus were measured to be 10±2%, 870±20MPa and 56±6GPa for ST and 9±2%, 990±30MPa and 59±8GPa for FT, respectively. There is no obvious variation for both ductility and Young's modulus for the two thermal conditions. Only higher strength is obtained for the FT condition due to the reduced grain size observed. On these tensile curves, a double yielding phenomenon associated with the superelastic effect was observed, suggesting that the SIM transformation has occurred for both ST and FT conditions. The first yielding corresponds to the critical stress inducing the SIM transformation, which is experimentally determined at the end of the linear part on each tensile curve as shown in Fig. 2. Thus, the critical stress to induce the SIM transformation was measured to be 410 ± 10 MPa for ST and 440 ± 10 MPa for FT, respectively. The highest recoverable strain of 2.75% is obtained for the FT while 2.3% maximum is reached for the ST. The higher value of recoverable strain for the FT sample was attributed to grain refinement in a recent work [11].

In situ synchrotron X-ray diffraction

For obtaining a direct evidence of the SIM transformation and further the crystallographic information of both β and α" phases during all the deformation stages, in situ SXRD during cyclic loading/unloading tensile tests were conducted.

The whole angle range spectrums at 0% strain and after 10% strain are first displayed in Fig. 3a for the ST sample and Fig. 3b for the FT sample, respectively. Before any deformation, both ST and FT samples display only the β phase and its typical diffraction peaks: (110) β , (200) β , (211) β and (220) β as shown in Fig. 3a andb. But after being deformed up to 10%, the diffraction peaks (020) α" , (002) α" , (111) α" , (021) α" , (022) α" , (131) α" , (1 1 3) α" , (0 4 0) α" and (0 0 4) α" of the martensitic α" phase are well detected in both ST and FT samples, which coexist with residual β phase. Thus, stress-induced martensitic transformation was clearly evidenced for both ST and FT alloys.

In order to characterize more precisely the SIM transformation, Fig. 4 displays the SXRD patterns obtained during the in situ cyclic tensile tests: on the ST sample after loading (Fig. 4a) and after each unloading (Fig. 4b) and on the FT sample after loading (Fig. 4c) and after each unloading (Fig. 4d). The different strains reached are indicated beside the spectra. In these figures, only the partial spectra in the angle range 8-11.5° around the main (110) β peak, where the most intense peaks are present, are shown.

From Fig. 4a, the (020) α" , (111) α" , (021) α" peaks start to be visible from 1.5% of strain and their intensities continue to increase as the applied strain increases. At the same time, the (110) β peak intensity decreases meaning that the SIM transformation has occurred on loading in the ST alloy. After unloading (Fig. 4b), it can be clearly observed that all the martensitic α" phase could be transformed back to the β phase as the (110) β peak intensity is recovered while the α" peaks disappear.

The SIM transformation and its reversibility is also evidenced in the FT alloy (Fig. 4c andd) with the difference that in this case, the SIM transformation seems to start later as the α" peaks start to be visible at about 2% on loading. These values are in good agreement with the tensile curve results.

From the loading spectra (Fig. 4a andc), a clear shift to the lower angles of the α" and β peaks due to cell parameter variations can also be observed. Thus, from each SXRD spectrum, the angular positions of all peaks were determined to evaluate the different cell parameters and their variations under loading. The evolution of these cell parameters (a β , a α" , b α" and c α" ) were plotted as a function of the applied percentage strain for the ST alloy (Fig. 5a) and the FT alloy (Fig. 5b), respectively.

Dynamic mechanical analysis

Fig. 6 displays the evolution of the storage modulus (E ' ) and the damping factor (tan δ) upon cooling and heating as a function of temperature under various applied stresses for the Ti2448 alloy after ST (Fig. 6a) and after FT (Fig. 6b). On these figures, E ' curves are presented in black and tanδ curves in red. On cooling, E ' and tanδ curves are represented with solid circles, while on heating, they are represented with open circles. The applied static stress (σ stat ) was initially fixed at 160 MPa and then was subsequently increased by 40 MPa after each temperature cycle until 280 MPa on the same specimen. All the figures show that both the martensitic transformation on cooling and the austenitic transformation on heating are characterized by a decrease of the storage modulus and an increase of the damping factor. Thus, modulus drops and damping peaks are clearly observed.

In order to characterize the martensitic transformation in the present case, the martensite start (M s ) and austenite finish (A f ) temperatures can be evaluated on the E ' curves. The characteristic temperature of M s is the point at which the change of slope on the E ' curves occurs, and also corresponds exactly to the initial increasing point on tanδ curve upon cooling, while the A f is determined on E ' curves upon heating. Other detected characteristic temperatures are M max and A max , which correspond to the maximum SIM α" transformation.

Therefore, all the detected characteristic temperatures M s , M max , A f , and A max were plotted as a function of the applied stress in Fig. 7 for the ST alloy (Fig. 7a) and for the FT alloy (Fig. 7b). As shown, a quite good linear fitting can be obtained in good agreement with the Clausius-Clapeyron relationship. It can be found that the slope is 0.19 K/MPa for M s and M max , 0.32 K/MPa for A f and 0.24 K/MPa for A max for the ST sample. On the other hand, all the slopes were found to be 0.20-0.22 K/MPa for the FT sample.

Discussion

From the evolution of the different cell parameters determined by SXRD (Fig. 5), it can be observed firstly that the β phase is elastically deformed in the 0-2.0% strain range (Fig. 5a) and in the 0-2.5% strain range (Fig. 5b) for ST and FT alloys, respectively. As shown, the a β increases from 3.30 to 3.33 Å (in ST alloy) and from 3.30 to 3.34 Å (in FT alloy). On the other hand, once the α" phase appears, this phase is observed to be elastically deformed up to 4.5% for ST and 5% for FT, respectively. This elastic deformation is clearly observed to be accommodated by an increase of the b α" and c α" parameters and by a decrease of the a α" parameter. These observations are in agreement with the tensile curves. Consequently, from both SXRD analysis and tensile tests, a three-step deformation sequence can be established as follows: firstly, the β phase is elastically deformed up to around 1.5-2% of strain in the ST alloy (2-2.5% for FT alloy); secondly, the deformation is accommodated by the SIM α" transformation, which is elastically deformed up to 4.5% in ST (5% in FT); from this value, both residual β phase and α" phase start to be deformed plastically.

From the DMA experiments, the martensitic transformation is particularly well highlighted by the tanδ curve. This kind of damping peak was commonly observed in other shape memory alloys such as Ni-Ti [12,13], Ti-Nb based alloys [14,15] and Cu-Al-Ni alloys [16]. It is well known that the reversible SIM transformation, which is accompanied by the movements of coherent austenite/martensite interfaces, can induce mechanical energy dissipation and thus constitutes an important damping source [13]. Consequently, a clear damping peak is observed in the present case and its maximum (P max ) corresponds more or less to M max . The high value of damping factor Tanδ (0.2-0.25) due to the SIM α" transformation in the Ti2448 alloy (Fig. 6) shows a superior damping capacity by comparison with the other β-type Ti-Ta-Nb [14] and Ti-Nb-Al [17] alloys under the same tensile mode under 1 Hz frequency. On the other hand, higher damping capacity is observed for the ST sample by comparison with the FT sample. This is caused by the larger β grain size observed in ST that permits longer distance mobility of the α"/β interfaces.

It has to be mentioned that the M f and A s temperatures cannot be clearly detected in the present case. This is due to the fact that the damping peak includes also the contribution of martensite/martensite interface movement as it was explained in a recent work [14]. The hysteretic movement of twin boundaries between martensite variants is another important damping source to dissipate mechanical energy that can overlap the movements of the austenite/ martensite interfaces [13]. That is also for this reason that the position of P max does not correspond exactly to the M max, especially under high applied stress.

From the tensile curves obtained at room temperature (Fig. 2), the critical stress inducing the SIM α" transformation was found to be 410 ± 10 MPa for the ST sample and 440 ± 10 MPa for the FT sample, respectively. From the DMA analyses, the critical stress inducing the SIM α" transformation at room temperature can also be obtained by extrapolating the data plotted in Fig. 7 (dot-lines). Indeed, at 20 °C the critical stress corresponding to the M s value was found to be 415 MPa for the ST sample and 440 MPa for the FT sample, respectively, as it is indicated in Fig. 7a andb. These critical stress values coincide perfectly with those obtained by tensile tests meaning that DMA analysis and mechanical tensile tests are in very good agreement.

In this work, the SIM α" transformation could be clearly evidenced by tensile tests, SXRD and DMA. However, the SIM α" phase morphology could not be observed in the present study although great efforts have been done by using transmission electron microscopy (TEM). Indeed, only the β phase was observed by TEM. One reason can be due to the fact that it is well known that spontaneous strain relaxation occurs during the preparation of the thin TEM specimens and consequently the SIM α" phase could have been transformed back to the β phase [START_REF] Lütjering | Titanium[END_REF].

One another reason can be linked with very recent works concerning the presence of nanodomains in this kind of alloy [START_REF] Liu | [END_REF], which were first reported by Miyazaki et al. [20,21]. As reported, nanodomains consist of nanosized modulated domain structure distributed homogeneously and randomly in the β phase and acted as obstacles for the long-range martensitic transformation. Consequently, nanodomain observation by conventional TEM remains tricky and high resolution TEM investigations are required for such characterization.

Conclusions

In this study, the superelastic Ti-24Nb-4Zr-8Sn alloy was investigated by tensile tests, in situ SXRD and DMA in order to characterize the martensitic transformation.

From all the characterization methods used, the following conclusions can be made:

(1) By tensile tests, the Ti2448 alloy showed low Young's modulus, high strength and good superelasticity, which enable it to be used as bone substitutes (implants or prostheses) for biomedical applications.

(2) By in situ SXRD, the typical diffraction peaks of the β and α" phases during loading and unloading were clearly observed and the reversible transformation between the β phase and the SIM α" phase was evidenced for both ST and FT states.

On the other hand, variations of the β and α" cell parameters were characterized in both loading and unloading conditions.

(3) From both SXRD analysis and tensile tests, a three step deformation sequence could be established as follows: firstly, the β phase is elastically deformed up to around 1.5-2% of strain in the ST alloy (2-2.5% for FT alloy); secondly, the deformation is accommodated by the SIM α" transformation, which is elastically deformed up to 4.5% in ST (5% in FT); from this value, both residual β phase and α" phases start to be deformed plastically.

(4) From the DMA experiments, clear damping peaks were observed and the characteristic temperatures detected (M s , M max , A f , and A max ) were plotted as a function of the applied stress. A quite good linear fitting could be obtained in agreement with the Clausius-Clapeyron relationship.

(5) Very good accordance concerning the critical stress inducing the martensitic transformation was obtained by comparing the results obtained from SXRD, DMA and conventional tensile tests.

Fig. 1 .

 1 Fig. 1. Optical micrographs of the Ti2448 alloy microstructure after (a) solution treatment at 900°C for 30min and (b) flash treatment at 700°C for 3min.

Fig. 2 .

 2 Fig. 2. Conventional and cyclic tensile strain-stress curves for the Ti2448 alloy after (a) solution treatment at 900°C for 30min and (b) flash treatment at 700°C for 3min.

Fig. 3 .10%Fig. 4 .Fig. 5 .

 345 Fig. 3. In situ SXRD spectra of the Ti2448 alloy before deformation and after to be deformed at 10% of strain for ST (a) and FT (b).

Fig. 6 .

 6 Fig. 6. Storage modulus and damping factor (in black and red, respectively) evolutions as a function of the σ stat during cooling and heating (in solid and open circles, respectively) for the Ti2448 alloy after (a) solution treatment at 900°C for 30min and (b) flash treatment at 700°C for 3min.

Fig. 7 .

 7 Fig. 7. Characteristic temperatures of M s , M max , A f and A max as a function of the applied static stresses σ stat for the Ti2448 alloy after different thermal treatment: (a) solution treatment at 900°C for 30min and (b) flash treatment at 700°C for 3min.

Table 1 .

 1 Chemical composition of the hot forged Ti2448 alloy (wt%).

	Nb	Zr	Sn	O	Ti
	23.9	4.05	8.22	0.16	Bal.

Acknowledgement

This work was financially supported by China scholarship Council (CSC), with No. 2011-6031. We acknowledge the European Synchrotron Radiation Facility for provision of synchrotron radiation facilities and we would like to thank Yves Watier for assistance in using beamline ID31.