Olivier Haeberlé
email: o.haeberle@uha.frdu

Hiromitsu Furukawa

Koji Tenjimbayashi

Keywords: Electromagnetism, Optics Polarized confocal theta microscopy Microscopie de fluorescence, Microscopie en réflexion, Microscopie Theta

We propose a comprehensive treatment of theta microscopy based on dipole emission, which describes better the fluorescence emission than the isotropic emission model, as fluorescence emission is often polarized. Formulas describing the point spread function for polarized confocal fluorescence theta microscopy are given. Examples are given and some advantages of polarized theta fluorescence microscopy are presented.

Fluorescence microscopy, Reflection microscopy, Theta microscopie

Microscopie confocale theta polarisée

Résumé

Nous présentons un modèle pour la microscopie de fluorescence theta basé sur le rayonnement dipolaire, qui décrit mieux le phénomène de fluorescence que le modèle isotropique, car l'émission de fluorescence est souvent polarisée. Les formules décrivant la tache de diffraction pour la microscopie de fluorescence polarisée en montage theta sont données. Des exemples sont donnés, et certains avantages à utiliser la polarisation en microscopie de fluorescence theta sont présentés.

Version française abrégée

En microscopie classique et confocale [1,2], un compromis entre résolution et distance de travail doit être accepté, les objectifs à grande distance de travail ayant une faible ouverture numérique, et donc une faible résolution. Le microscope theta a été proposé pour pallier à cet inconvénient. Il consiste à utiliser deux objectifs de microscope pour l'excitation et la détection, dont les axes optiques sont perpendiculaires [3]. La Figure 1 en décrit le principe. L'objectif 1 est utilisé pour illuminer le specimen, et la détection se fait au travers de l'objectif 2.

Nous proposons un traitement détaillé du microscope theta, basé sur la théorie vectorielle de la diffraction pour les objectifs à grande ouverture numérique, et sur le modèle d'émission dipolaire.

La configuration du microscope theta oblige à utiliser des objectifs sans lamelle, la RIO d'excitation est donc calculée à l'aide du modèle de Richards et Wolf [1]. Les équations (1,2) donnent le champ électromagnétique au point focal. La RIO du microscope theta est alors souvent calculée à l'aide de l'équation (3), en considérant une émission de fluorescence isotrope et non-polarisée [3]. L'émission de fluorescence étant polarisée [4,5], il est plus rigoureux de considérer le champ émis par un dipole [6,7]. Le moment et le champ dipolaires sont alors donnés par les équations (4) et (5). Le champ est capté par l'objectif 2, et refocalisé sur le détecteur. Avec un objectif de refocalisation de faible ouverture numérique, le détecteur quadratique enregistre alors le carré de la transformée de Fourier du champ électromagnétique: équation (7), les équations (6a) et (6b) décrivant le champ, calculé à l'aide des intégrales (8). La formule (9) lie les angles entre l'objectif de détection 2 et la lentille de refocalisation [6,7].

Nous avons calculés les

Introduction

In classical and even confocal microscopy [1,2], a trade-off between resolution and working distance must always be done. Long working distance objectives have a low numerical aperture, and therefore a low resolution, especially along the optical axis. High numerical aperture objectives permit a high resolution, but at the price of a usually very short working distance. Theta microscopy has been proposed as an effective solution to combine high-resolution with long working distances. Theta microscopy therefore provides the unique feature of allowing the study of large biological sample while keeping a good resolution. In previous work, the Point Spread Function of theta microscopy was calculated using an isotropic emission model, with unpolarized excitation. However, fluorescence is known to be polarized. As a consequence, dipole emission describes better the fluorescence process. We propose a comprehensive treatment of polarized fluorescence microscopy based on the electromagnetic theory of high numerical aperture objective, taking into account dipole emission by the fluorescent dye.

Polarized Fluorescence Theta Microscopy

The idea of theta microscopy is to diminish the size of the Point Spread Function by using two microscope objectives with crossed optical axes [3]. Doing so, the lower resolution along the optical axis of the detection objective is compensated by the fact that this axis corresponds to a lateral axis for the illumination objective. Figure 1 describes the considered setup. Objective 1 is used for fluorescence excitation, and objective 2 is for confocal detection. This optical configuration does not allow to use glass slides, so, the Richard and Wolf model of diffraction describes well the illumination PSF formation [1]. Considering a plane wave, linearly polarized along the x-axis, being focused by objective 1, the electric field at point P with coordinates x=x',y=-z',z=y' (see Fig. 1: (x,y,z) coordinates are for detection arm and (x',y',z') coordinates are for illumination arm) in the focal region is written as:

E x = -i(I 0 + I 2 cos2φ) , E y = 2I 1 cosφ, E z = -i(I 2 sin2φ) (1)
with:

I 0 = a(θ) 0 α ∫ sinθ(1 + cosθ )J 0 (k x 2 + z 2 sinθ
)exp(-iky cosθ)dθ

I 1 = a(θ)sin 2 θ 0 α ∫ J 1 (k x 2 + z 2 sinθ )exp(-iky cosθ)dθ I 2 = a(θ) 0 α ∫ sinθ(1 -cosθ )J 2 (k x 2 + z 2 sinθ)exp(-iky cosθ)dθ (2)
The apodization function in Eqs. ( 2) is given by a(θ) = cos 1/2 θ, φ the angle between the direction of observation and the x-axis (see Figure 1), α is the objective lens aperture angle, J n is the Bessel function of first kind or order n, and k=k 0 .n with n being the immersion medium refraction index. The classical treatment of confocal or theta confocal microscopy consists in considering that the excited molecule acts as an isotropic radiator. The final Point Spread Function is simply calculated as the product of the illumination PSF by the detection PSF. For a theta setup, it is written as [3]:

PSF theta (x,y,z) = PSF (x, y, z) × PSF (-z, y, x) (3)
as the difference in wavelength between illumination and detection is often neglected for simplicity. In that approach, the illumination and detection PSF are computed using the Richards and Wolf model. However, this model is for a plane wave focused into the medium, thus valid to compute the illumination PSF, but calculating the detection PSF as an illumination PSF constitutes another approximation. Furthermore, fluorescence is known to be polarized [4,5]. A more rigorous approach therefore consists in considering a point scatterer whose far field emission is that of an electric dipole, which moment p is proportional to the illumination field [6,7]. Hence, the Cartesian components of the dipole moment are given by:

p x = -i(I 0 + I 2 cos2φ) , p y = 2I 1 cosφ, p z = -i(I 2 sin 2φ ) (4)
The electric field in the radiation zone being given by:

E dp = -r × (r × p) (5)
The dipole electric vector is collimated by the detection objective, then passed through an analyzer and refocused onto the detector. Considering a low aperture detector lens, a quadratic detector therefore records the intensity I = |E d | 2 , with E d being the Fourier transform of the scalar field [6]:

E dx = p x (I 0 + I 2 cos2φ p ) + p y I 2 sin2φ p -2ip z cosφ p (6a) E dy = p x I 2 sin2φ p + p y (I 0 -I 2 cos2φ p ) -2ip z sinφ p (6b)
and then one obtains the intensity PSF after the polarizer as [6]:

PSF (x, y, z ) = cos 2 γE x + sinγ cosγE y 2 + cosγ sinγE x + sin 2 γE y 2 (7)
with:

I 0 d = cosθ 1 0 α d ∫ sin 2θ d J 0 (kρsinθ d )(1 + cosθ 1 )exp(-ikz s cosθ 1 )dθ d I 1 d = cosθ 1 sin 2θ d 0 α d ∫ J 1 (kρsinθ d )sinθ 1 exp(-ik 1 z s cosθ 1 )dθ d I 2 d = cosθ 1 0 α d ∫ sin 2θ d J 2 (kρsinθ d )(1 -cosθ 1 )exp(-ik 1 z s cosθ 1 )dθ d (8)
γ being the angle between the analyzer axis and the x-axis. The angular aperture of the detector lens α d and the coordinate angle θ d are linked to the objective lens angular aperture α1 and the coordinate θ1 by the relation [6,7]:

(n 1 sinα 1 )/( n d sinα d ) = (n 1 sinθ 1 ) /( n d sinθ d ) = β ( 9
)
where β is the nominal magnification of the objective lens, and n 1 and n d are the indices of refraction of the immersion medium and the image space, respectively [6,7].

Exemples

Figure 2 shows computed point spread functions. An Olympus Universal Plan Fluorite 40x waterimmersion objective with N.A.=0.8 is considered. This objective has a working distance of 3.3 mm and would be well suited for a theta setup, after a slight modification of the enclosure [8]. The excitation PSF is computed at λ = 400 nm and we consider emission at λ = 450 nm (corresponding to excitation and emission of Cascade Blue from Molecular Probes). The specimen is scanned through the focus. For the sake of simplicity, a point-like incoherent detector is considered [6,7]. Figure 2(a) shows the PSF computed using the classical isotropic emission model, for unpolarized excitation and detection, using Eq. (3). Figure 2(b) shows the PSF using the dipole model for x-polarized excitation and x-polarized detection. It is shown that the classical model tends to slightly overestimate the x-lateral resolution, while simultaneously slightly understimates the y-lateral and z-longitudinal resolution. The dipole emission models predicts a more isotropic PSF, with better separation of the lateral lobes (more noticeable differences are expected if larger N.A. objectives may be used [6,7,9]). The x-resolution is 220 nm, the yand z-resolution being of 270 nm (versus 210 nm and 290 nm for the isotropic emission model).

An inherent problem of polarized detection in confocal microscopy is that the detection of molecules excited by light with z-axis polarization is limited by the fact that the excitation (and therefore the available signal) is much weaker than for polarization along the x-axis or the y-axis [9,10]. Typically the signal is 100 times lower [9]. A possible approach is to use phase filters to enhance the z polarized component of the excitation and detection [10].

Theta microscopy offers another possibility: as the illumination and detection axes are crossed, the optical axis for the detection arm corresponds to a lateral axis of the illumination arm, for which the component is expected to be of much higher intensity. As shown on Figure 1, choosing a x'-or y'-polarized illumination wave corresponds to enhancing detection for the x-or z-component, respectively. The equations giving the illumination field, therefore the dipole moment, are readily obtained from Eqs. (1-2) by a 90° rotation around the (z'=-y)-axis, thus permuting the roles of the variable x and -z. Figure 2(c) shows the PSF obtained for z-polarized excitation and x-polarized detection. Note that the maxima of the PSFs are the same in x-polarized or z-polarized excitation. Polarized theta microscopy may therefore constitute an interesting alternative to z-polarized confocal microscopy for large specimen [10].

Figure 3 shows PSFs computed for crossed polars, with y-polarized detection. Not only are the shapes different, but also the relative intensities of the PSFs, the detection for crossed-polarization being typically 400 times weaker. Figure 3(a) shows the computed PSF whith x-polarized excitation and ypolarized detection (in the z=0 µm and y=0 µm planes, respectively). It exhibits the characteristic fourleave pattern also observed in confocal polarized microscopy [6,7,9,11], but now in both planes. The pattern is however now asymetric along the z-axis, as can be seen on Figure 3(b), showing (x-z) cuts along the y=-0.18 µm and y=+0.18 µm, respectively. Figure 3(c) shows the PSF computed for z-polarized excitation and y-polarized detection (in the z=0 µm and y=0 µm planes, respectively). The PSF has now a very different shape, as can be seen on Figures 3(d) and 3(c), which show cuts along various y-and zplanes. Note that a central symmetry is conserved.

To favor the y-polarized component, one could either symetryze the detection and illumination arms, or use a second illumination objective, placed perpendicularly to the theta setup (along the x-axis), in a configuration similar to Multiple Objective Microscopy [12] or Multiple Imaging Axis Microscopy [START_REF] Stelzer | Multiple imaging axis microscopie (MIAM)[END_REF].

Conclusion

We have given a rigorous electromagnetic treatment of polarized confocal theta microscopy, using the dipole emission model to describe the fluorescence process. Formulas to compute Point Spread Functions for this type of microscopy are given as well as typical examples of PSF. The dipole emission model predicts a slightly more isotropic resolution than the isotropic emission one. The main advantage is the possibility of using strongly z-polarized excitation. Taking into account polarization properties of the fluorescence emission may permit to obtain additional information about the distribution and orientation of the dipoles in large specimen with a good spatial resolution.

This work was made possible with the financial support of the New Energy and Industrial Technology Development Organization (NEDO) of Japan under an Industrial Technology Research Grant Program.

  RIO pour un microscope theta utilisant des objectifs d'ouverture numérique N.A.=0.8 à immersion à eau [8], avec une excitation à λ = 400 nm et une détection λ = 450 nm (Cascade Blue from Molecular Probes). La Figure 2 montre les résultats obtenus. La Figure 2(a) montre la RIO calculée avec le modèle isotrope. Les Figures 2(b) et 2(c) montrent la RIO calculée avec le modèle dipolaire, pour une excitation x-polarisée et une détection x-polarisée et pour une excitation z-polarisée et une détection x-polarisée, respectivement. Le modèle isotrope a tendance à surestimer la résolution en x, et sous-estimer la résolution en z et en y. La Figure 3 montre les RIO pour une détection y-polarisée. Non seulement les formes des RIO sont très différentes, mais surtout l'intensité est environ 400x plus faible. Les RIOs montrent la forme caractéristique en feuille de trèfle aussi prévue en microscopie confocale polarisée [6,7,9,11] (Figures 3(a) et 3(c)). Notons cependant que les RIO sont maintenant asymétriques, comme le montrent les Figures 3(b), 3(d) et 3(e), qui présentent des coupes (x-y) et (x-z) à divers plans en profondeur et latéralement.

Figure 1 :

 1 Figure 1: Sketch of a confocal theta microscope. Objective 1 is used for sample excitation. Objective 2, with optical axis at 90° with respect to objective 1, is used for detecting the fluorescence signal.

Figure 1 :Figure 2 : 2 :

 122 Figure 1 : Schéma de principe d'un microscope confocal theta. L'objectif 1 sert à l'excitation du specimen. L'objectif 2, dont l'axe optique est à 90° de l'objectif 1, sert à la detection du signal de fluorescence

Figure 3 :

 3 Figure 3: Same as Figure 2, but for y-polarized detection. (a) x-polarized excitation, y-polarized detection. (b) same as (a)-bottom, but in the y=-0.18 µm and y=+0.18 µm plane respectively. Note the assymetry. (c) z-polarized excitation, y-polarized detection.(d-e) same as (c) but in the z=±0.15 µm and y=±0.15 µm plane. Note the complex shape of the PSF in that case.

Figure 3 :

 3 Figure 3 : Idem à Figure 2, mais pour une détection y-polarisée. (a) excitation x-polarisée, détection ypolarisée. (b) identique à (a)-bas dans les plans y=-0.18 µm et y=+0.18 µm. Notez l'assymétrie. (c) excitation z-polarisée, détection y-polarisée. (d-e) identique à (c) mais dans les plans z=±0.15 µm et y=±0.15 µm. Notez la forme complexe de la RIO dans ce cas.

Figure 2

 2 Figure 2

Figure 3

 3 Figure 3


Élodie Bouchet
email: bouchet@math.univ-lyon1.fr

AND Alejandro F Ramírez
email: aramirez@mat.puc.cl

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

SHARP ELLIPTICITY CONDITIONS FOR BALLISTIC BEHAVIOR OF RANDOM WALKS IN RANDOM ENVIRONMENT

Keywords: 2000 Mathematics Subject Classification. 60K37, 82D30 Random walk in random environment, Dirichlet distribution, Reinforced random walks, Max-flow min-cut theorem

We sharpen the ellipticity criteria for random walks in i.i.d. random environments introduced by Campos and Ramírez which ensure ballistic behavior. Furthermore, we construct new examples of random environments for which the walk satisfies the polynomial ballisticity criteria of Berger, Drewitz and Ramírez. As a corollary we can exhibit a new range of values for the parameters of Dirichlet random environments in dimension d = 2 under which the corresponding random walk is ballistic.

Introduction

We continue the study initiated in [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF] sharpening the ellipticity criteria which ensure ballistic behavior of random walks in random environment. Furthermore, we apply our results to exhibit a new class of ballistic random walks in Dirichlet random environments in dimensions d = 2.

For x ∈ R d , denote by |x| 1 and |x| 2 its L 1 and L 2 norm respectively. Call U := {e ∈ Z d : |e| 1 = 1} = {e 1 , . . . , e 2d } the canonical vectors with the convention that e d+i = -e i for 1 i d. We set P := {p(e) : p(e) 0, e∈U p(e) = 1}.

An environment is an element ω := {ω(x) : x ∈ Z d } of the environment space Ω := P Z d . We denote the components of ω(x) by ω(x, e).

The random walk in the environment ω starting from x is the Markov chain {X n : n 0} in Z d with law P x,ω defined by the condition P x,ω (X 0 = x) = 1 and the transition probabilities P x,ω (X n+1 = x + e|X n = x) = ω(x, e) for each x ∈ Z d and e ∈ U.

Let P be a probability measure defined on the environment space Ω endowed with its Borel σ-algebra. We choose {ω(x) : x ∈ Z d } i.i.d. under P. We call P x,ω the quenched law of the random walk in random environment (RWRE) starting from x, and P x := P x,ω dP the averaged or annealed law of the RWRE starting from x.

The law P is said to be elliptic if for every x ∈ Z d and e ∈ U, P(ω(x, e) > 0) = 1. We say that P is uniformly elliptic if there exists a constant γ > 0 such that for every x ∈ Z d and e ∈ U, P(ω(x, e) γ) = 1.

Given l ∈ S d-1 we say that the RWRE is transient in direction l if

P 0 (A l ) = 1, with A l := { lim n→∞ X n • l = ∞}.
Furthermore, it is ballistic in direction l if P 0 -a.s.

lim inf n→∞ X n • l n > 0.
Given Λ ⊂ Z d , we denote its outer boundary by ∂Λ := {x / ∈ Λ : |x -y| 1 = 1 for some y ∈ Λ}.

We denote any nearest neighbour path with n steps joining two points x, y ∈ Z d by (x 1 , x 2 , . . . , x n ), where x 1 = x and x n = y.

1.1. Polynomial condition, ellipticity condition. In [START_REF] Berger | Effective polynomial ballisticity condition for randow walk in random environment[END_REF], Berger, Drewitz and Ramírez introduced a polynomial ballisticity condition within the uniformly elliptic context, which was later extended to the elliptic case by Campos and Ramírez in [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF]. This condition will be of interest for our results. It is effective, in the sense that it can a priori be verified explicitly for a given environment.

To define it, we need for each L, L > 0 and l ∈ S d-1 to consider the box

B l,L, L := R (-L, L) × (-L, L) d-1 ∩ Z d ,
where R is a rotation of R d that verifies R(e 1 ) = l.

For each subset A ⊂ Z d we note the first exit time from the set A as

T A := inf{n 0 : X n / ∈ A}.
Definition 1. Given M 1 and L 2, we say that condition (P ) M in direction l is satisfied on a box of size L (also written as (P ) M |l) if there exists an L 70L 3 such that one has the following upper bound for the probability that the walk does not exit the box B l,L, L through its front side :

P 0 (X T B l,L, L • l < L) 1
L M . This condition has proven useful in the uniformly elliptic case. Indeed, (P ) M for M 15d + 5 implies ballisticity (see [START_REF] Berger | Effective polynomial ballisticity condition for randow walk in random environment[END_REF]).

For non uniformly elliptic environments in dimensions d 2, there exist elliptic random walks which are transient in a given direction but not ballistic in that direction (see for example Sabot-Tournier [START_REF] Sabot | Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment[END_REF], Bouchet [START_REF] Bouchet | Sub-ballistic random walk in Dirichlet environment[END_REF]). In [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF], Campos and Ramírez introduced ellipticity criteria on the law of the environment which ensure ballisticity if condition (P ) M is satisfied for M 15d + 5. In this article we will sharpen this ellipticity criteria.

Define η α := max e∈U E 1 ω(0,e) α , α := sup{α 0 : η α < ∞}. We need the constant c 0 := 2 3 3 120d 4 +3000d(log η α/2 ) 2 : throughout the rest of this paper, whenever we assume that the polynomial condition (P ) M is satisfied, it will be understood that this happens on a box of size L c 0 .

Let us first recall the ellipticity condition of [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF]. The polynomial condition (P ) M implies the existence of an asymptotic direction (see for example Simenhaus [START_REF] Simenhaus | Asymptotic direction for random walks in random environment[END_REF]): there exists v ∈ S d-1 such that P 0 -a.s.,

lim n→∞ X n |X n | 2 = v.
We call v the asymptotic direction. Under (P ) M we can then define the half space Furthermore, when v exists, we say that the ellipticity condition (E ′ ) β is satisfied towards the asymptotic direction if there exists an {α(e) : e ∈ U} satisfying (1.1) and (1.2) and such that there exists

H v := {l ∈ R d : l • v 0}.
α 1 > 0 that satisfies α(e) = α 1 for e ∈ H v ∩ U while α(e) α 1 for e ∈ U \ H v . Remark 1. Replacing (1.2) by E e e ′ α(e ′ ) log 1 ω(0,e ′ )
< ∞ gives an equivalent condition. The direct implication is straightforward. And since 1 e∈U 1 {ω(0,e) 1 2d } , we get E e e ′ α(e ′ ) log 1 ω(0,e ′ ) e∈U e α(e) log(2d) E e e ′ =e α(e ′ ) log 1 ω(0,e ′ ) . This gives the reverse implication.

Remark 2. The results of [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF] still hold (with no changes in the proofs) if we replace the previous definition of η α by

η α := max e∈H v E 1 ω(0, e) α .
It means that the constant c 0 in the definition of condition (P ) M depends only of the integrability towards the asymptotic direction.

But knowing the existence of v does not mean that we know its value. In most cases, v is found to be inaccessible. A notable exception is the result of Tournier ( [START_REF] Tournier | Asymptotic direction of random walks in Dirichlet environment[END_REF]) that gives the value of v in the case of random walks in Dirichlet environments.

1.2. Ballisticity results. Our main results are a generalization of theorems 1.2 and 1.3 of [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF] where we remove the "towards the asymptotic direction" condition of Theorems 1.2 and 1.3 of [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF].

Let τ v 1 be the first renewal time in the direction v, its precise definition is recalled in the next section. We prove the following tail estimate on renewal times, which improves proposition 5.1 of [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF].

Theorem 1. Let l ∈ S d-1 , β > 0 and M 15d + 5. Assume that (P ) M |l is satisfied and that (E ′ ) β holds (cf. (1.1), (1.2)). Then lim sup u→∞ (log u) -1 log P 0 (τ v 1 > u) -β.
The condition (E ′ β ) is sharp in a sense that is made precise in remark 3 below. Together with previous results of Sznitman, Zerner, Seppäläinen and Rassoul-Agha, cf [START_REF] Sznitman | A law of large numbers for random walks in random environment[END_REF][START_REF] Zerner | A non-ballistic law of large numbers for random walks in i.i.d. random environment[END_REF][START_REF] Sznitman | Slowdown estimates and central limit theorem for random walks in random environment[END_REF][START_REF] Rassoul-Agha | Almost sure functional central limit theorem for ballistic random walk in random environment[END_REF], it implies the following: a)

(Annealed central limit theorem) If (E ′ ) 2 is satisfied then ε 1/2 (X [ε -1 n] -[ε -1 n]v)
converges in law under P 0 as ε → 0 to a Brownian motion with non-degenerate covariance matrix. b)

(Quenched central limit theorem) If (E ′ ) 176d is satisfied, then P-a.s. we have that ε 1/2 (X [ε -1 n] -[ε -1 n]v
) converges in law under P 0,ω as ε → 0 to a Brownian motion with non-degenerate covariance matrix.

Removing the "towards the asymptotic direction" is a real improvement: in section 1.3.2, we will give some examples of environments (in the class of Dirichlet environments) that satisfy (E ′ β ) but not towards the asymptotic direction. For those environments, our new theorems allows to prove a LLN or CLT. Furthermore, our final goal would be to get a ballisticity condition that depends only locally on the environment (i.e. a condition that depends only on the law of the environment at one point). Condition (E ′ ) β is local, whereas (E ′ ) β towards the asymptotic dimension is not: removing the "towards the asymptotic direction" is then a first step in this direction. Ideally, we would also need to get rid of condition (P ) M |l, that is not local either. This is a much more difficult problem, not solved even in the uniformly elliptic case.

Remark 3. The condition of theorem 1 is sharp under the following assumption on the tail behavior of the environment at one site: there exists some (β e ) e∈U , β e 0, and a positive constant C > 1 such that for all e ∈ U C -e)) then E (τ v 1 ) β = ∞. Indeed, consider a direction e 0 which realizes the maximum in sup e∈U (β(e) + β(-e)) and set K = {0, e 0 }. We denote by ∂ + K the set of edges that exit the set K, which is composed of the edges {(0, e)} e =e 0 and {(e 0 , e)} e =-e 0 . For small t > 0, under the condition that ω(x, y) t for all (x, y) ∈ ∂ + K we have P 0,ω (T K n) (1 -(2d -1)t) n . Hence,

P 0 (T K n) (1 -(2d -1)/n) n P (ω(x, y) 1/n, ∀(x, y) ∈ ∂ + K) (1 -(2d -1)/n) n C -1 n -(( e ′ β(e ′ ))-(β(e 0 )+β(-e 0 )))
which implies that E 0 (T β K ) = ∞. Since T K is clearly a lower bound for the first renewal time it gives the result. Remark 4. Theorem 1.1 of [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF] states that for i.i.d. environments in dimensions d 2 satisfying the ellipticity condition (E ′ ) 0 , the polynomial condition (P ) M |l (for l ∈ S d-1 and M 15d + 5) is equivalent to Sznitman's condition (T ′ )|l (see for example [START_REF] Sznitman | On a class of transient random walks in random environment[END_REF] for the definition). We can therefore replace (P ) M |l by (T ′ )|l in the statements of Theorems 2 and 3.

1.3.

New examples of random walks satisfying the polynomial condition. In this article we also introduce new examples of RWRE in environments which are not uniformly elliptic and which satisfy the polynomial condition (P ) M for M 15d + 5. In subsection 1.3.1 we prove the polynomial condition for a subset of marginal nestling random walks, including a particular environment introduced by Campos and Ramírez in [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF]. In subsection 1.3.2, we prove the polynomial condition for a class of random walks in Dirichlet random environments which do not necessarily satisfy Kalikow's condition.

1.3.1. Example within the class of marginal nestling random walks. Following Sznitman [START_REF] Sznitman | Slowdown estimates and central limit theorem for random walks in random environment[END_REF], we say that a law P on Ω is marginal nestling if the convex hull K o of the support of the law of d(0, ω) := e∈U ω(0, e)e is such that 0 ∈ ∂K o . We will prove in section 4 that a certain subset of the marginal nestling laws satisfies the polynomial condition.

Theorem 4. Consider an elliptic law P under which {ω(x) : x ∈ Z d } are i.i.d. Assume that there exists an r > 1 such that ω(0, e 1 ) = rω(0, e 1+d ). Then the polynomial condition (P ) M | e 1 is satisfied for some M 15d + 5.

Remark 5. This theorem is valid for all i.i.d. elliptic environments satisfying ω(0, e 1 ) = rω(0, e 1+d ), including uniformly elliptic environments. However, the environments are marginal nestling only in the non-uniformly elliptic case.

The above result includes an example suggested in [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF], by Campos and Ramírez, of an environment which satisfies the polynomial condition and for which the random walk is directionally transient but not ballistic. They showed that on this environment, (E ′ ) α is satisfied for α smaller but arbitrarily close to 1, and that the walk is transient but not ballistic in a given direction. The proof that this environment satisfies the polynomial condition was left for a future work.

Let us define the environment introduced in [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF]. Let ϕ be any random variable taking values on the interval (0, 1/4) and such that the expected value of ϕ -1/2 is infinite, while for every ε > 0, the expected value of ϕ -(1/2-ε) is finite. Let X be a Bernoulli random variable of parameter 1/2. We now define ω(0, e 1 ) = 2ϕ, ω(0, -e

1 ) = ϕ, ω(0, e 2 ) = Xϕ + (1 -X)(1 -4ϕ) and ω(0, -e 2 ) = X(1 -4ϕ) + (1 -X)ϕ.
For every ε > 0 this environment satisfies (E ′ ) 1-ε : traps can appear because the random walk can get caught on two edges of the type (x, e 2 ), (x+e 2 , -e 2 ). Furthermore, it is transient in direction e 1 but not ballistic in that direction.

Examples within the class of Dirichlet random environments. Random Walks in

Dirichlet Environment (RWDE) are interesting because of the analytical simplifications they offer, and because of their link with reinforced random walks. Indeed, the annealed law of a RWDE corresponds to the law of a linearly directed-edge reinforced random walk ( [START_REF] Enriquez | Edge oriented reinforced random walks and RWRE[END_REF], [START_REF] Pemantle | Phase transition in reinforced random walk and RWRE on trees[END_REF]).

Given a family of positive weights (β 1 , . . . , β 2d ), a random i.i.d. Dirichlet environment is a law on Ω constructed by choosing independently at each site x ∈ Z d the values of (ω(x, e i )) i∈[ [1,2d]] according to a Dirichlet law with parameters (β 1 , . . . , β 2d ). That is, at each site we choose independently a law with density

Γ 2d i=1 β i 2d i=1 Γ (β i ) 2d i=1 x β i -1 i dx 1 . . . dx 2d-1 on the simplex {(x 1 , . . . , x 2d ) ∈]0, 1] 2d , 2d i=1 x i = 1}.
Here Γ denotes the Gamma function Γ(β) = ∞ 0 t β-1 e -t dt , and dx 1 . . . dx 2d-1 represents the image of the Lebesgue measure on R 2d-1 by the application

(x 1 , . . . , x 2d-1 ) → (x 1 , . . . , x 2d-1 , 1 -x 1 -• • • - x 2d-1
). Obviously, the law does not depend on the specific role of x 2d . Remark 6. Given a Dirichlet law of parameters (β 1 , . . . , β 2d ), the ellipticity condition

(E ′ ) β is satisfied if and only if κ ((β 1 , . . . , β 2d )) = 2 2d i=1 β i -max i=1,...,d (β i + β i+d ) > β.
As stated in remark 3, this ellipticity condition is optimal to get theorem 1 in the case of Dirichlet environments. Remark that for Dirichlet environments, (E ′ ) β is much sharper that (E ′ ) β towards the asymptotic direction. Indeed, (E ′ ) β is satisfied towards the asymptotic direction v if and only if we also have the existence of a constant c > 0 such that for all i, β i c, and for all i such that e i • v 0, β i = c. The result of Tournier ([16]) gives us the value of v in terms of the (β 1 , . . . , β 2d ). But there are many obvious cases when we do not need to find v to notice that (E ′ ) β is not satisfied toward the asymptotic direction, for example when we cannot find d of the β i with the same value.

In the case of RWDE, it has been proved that Kalikow's condition, and thus the (T ′ ) condition, is satisfied whenever

(1.3) max 1 i d |β i -β i+d | > 1
(see Enriquez and Sabot in [START_REF] Enriquez | Random walks in a Dirichlet environment[END_REF] and Tournier in [START_REF] Tournier | Integrability of exit times and ballisticity for random walks in Dirichlet environment[END_REF]). The characterization of Kalikow's condition in terms of the parameters of a RWDE remains an open question. On the other hand, we believe that for RWDE condition (T ′ ) is satisfied if and only if

max 1 i d |β i -β i+d | > 0.
Nevertheless, in this article we are able to prove the following result.

Theorem 5. Let β 1 , β 2 , . . . , β d , β d+2 , . . . , β 2d be fixed positive numbers. Then, there exists an ε ∈ (0, 1) depending on these numbers such that if β 1+d is chosen so that β 1+d ε, the Random Walk in Dirichlet Environment with parameters (β 1 , . . . , β 2d ) satisfies condition (P ) M |e 1 for M 15d + 5.

Theorem 5 gives as a corollary new examples of RWDE which are ballistic in dimension d = 2 since they do not correspond to ranges of the parameters satisfying condition 1.3 of Tournier [START_REF] Tournier | Integrability of exit times and ballisticity for random walks in Dirichlet environment[END_REF] and Sabot and Enriquez [START_REF] Enriquez | Random walks in a Dirichlet environment[END_REF]. Indeed, by Theorem 2, if

2d i=1 β i -sup 1 i d (β i + β i+d ) > 1
and one of the parameters {β i : 1 i d} is small enough, the walk is ballistic.

Remark 7. In dimension d

3, in [START_REF] Sabot | Random Dirichlet environment viewed from the particle in dimension d 3[END_REF][START_REF] Bouchet | Sub-ballistic random walk in Dirichlet environment[END_REF], precise conditions on the existence of an invariant measure viewed from the particle absolutely continuous with respect to the law have been given ; this allows to characterize parameters for which there is ballisticity, but it fails to give information on the (T') condition and on the tails of renewal times. It also fails to give a CLT.

Theorem 3 then gives us annealed CLTs for Dirichlet laws when the parameters (β 1 , . . . , β 2d ) satisfy 2d i=1 β isup 1 i d (β i + β i+d ) > 2 along with condition (1.3) or the hypothesis of theorem 5.

Remark 8. For the Dirichlet laws in dimension

d = 2 with parameters (β 1 , . . . , β 4 ) satisfying 4 i=1 β i -sup 1 i 2 (β i +β i+d ) > 1,
with one of the parameters {β i : 1 i 4} small enough, but for which there are no vector v and constant c such that for all i, β i c, and for all i such that e i • v 0, β i = c, our theorem 2 gives the ballisticity when the results of [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF] would not have been enough.

For the Dirichlet laws in dimension d 2 with parameters (β 1 , . . . , β 2d ) satisfying

2d i=1 β i -sup 1 i d (β i + β i+d ) > 2, with condition (1.
3) or the hypothesis of theorem 5, but for which there are no vector v and constant c such that for all i, β i c, and for all i such that e i • v 0, β i = c, our theorem 3 gives the annealed CLT when the results of [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF] would not have been enough.

This illustrates the relevance of having removed the "toward the asymptotic direction" hypothesis in theorem 1.

First tools for the proofs

In this section we will introduce some tools that will prove necessary for the proof of theorem 1.

2.1. Regeneration times. The proofs in [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF] are based on finding bounds on the regeneration times. We thus begin by giving the definition and some results about the regeneration times with respect to a direction l.

We define {θ n : n 1} as the canonical time shift on Z d N . For l ∈ S d-1 and u 0, we define the times T l u := inf{n 0 : X n • l u} and T l u := inf{n 0 :

X n • l u}. Set (2.1) a > 2 √ d and D l := min{n 0 : X n • l < X 0 • l}. We define S 0 := 0, M 0 := X 0 • l, S 1 := T l M 0 +a , R 1 := D l • θ S 1 + S 1 , M 1 := sup{X n • l : 0 n R 1 },
and recursively for k 1,

S k+1 := T l M k +a , R k+1 := D l • θ S k+1 + S k+1 , M k+1 := sup{X n • l : 0 n R k+1 }.
The first regeneration time is then defined as

τ 1 := min{k 1 : S k < ∞, R k = ∞}.
We can now define recursively in n the (n + 1)-th regeneration time τ n+1 as τ 1 (X • ) + τ n (X τ 1 +• -X τ 1 ). We will occasionally write τ l 1 , τ l 2 , . . . to emphasize the dependence on the chosen direction.

Remark 9. The condition (2.1) on a is only necessary to prove the non-degeneracy of the covariance matrix of part (a) of theorem 3.

It is a standard fact (see for example Sznitman and Zerner [START_REF] Sznitman | A law of large numbers for random walks in random environment[END_REF]) to show that the sequence ((τ 1 , X (τ 1 +•)∧τ 2 -X τ 1 ), (τ 2 -τ 1 , X (τ 2 +•)∧τ 3 -X τ 2 ), . . .) is independent and (except for its first term) i.i.d.. Its law is the same as the law of τ 1 with respect to the conditional probability measure P 0 (•|D l = ∞).

Those regeneration times are particularly useful to us because of the two following theorems :

Theorem 6 (Sznitman and Zerner [START_REF] Sznitman | A law of large numbers for random walks in random environment[END_REF], Zerner [START_REF] Zerner | A non-ballistic law of large numbers for random walks in i.i.d. random environment[END_REF], Sznitman [START_REF] Sznitman | Slowdown estimates and central limit theorem for random walks in random environment[END_REF]). Consider a RWRE in an elliptic i.i.d. environment. Let l ∈ S d-1 and assume that there is a neighbourhood V of l such that for every l ′ ∈ V the random walk is transient in the direction l ′ . Then there is a deterministic v such that P 0 -a.s.

lim n→∞ X n n = v.
Furthermore, the following are satisfied.

a) If E 0 (τ 1 ) < ∞, the walk is ballistic and v = 0. b) If E 0 (τ 2 1 ) < ∞, ε 1/2 X [ε -1 n] -[ε -1 n]v
converges in law under P 0 to a Brownian motion with non-degenerate covariance matrix.

Theorem 7 (Rassoul-Agha and Seppäläinen [START_REF] Rassoul-Agha | Almost sure functional central limit theorem for ballistic random walk in random environment[END_REF]). Consider a RWRE in an elliptic i.i.d. environment. Take l ∈ S d-1 and let τ 1 be the corresponding regeneration time.

Assume that E 0 (τ p 1 ) < ∞, for some p > 176d. Then P-a.s. we have that

ε 1/2 X [ε -1 n] -[ε -1 n]v
converges in law under P 0,ω to a Brownian motion with non-degenerate covariance matrix.

2.2.

Atypical Quenched Exit Estimate. The proof of theorem 1 is based on an atypical quenched exit estimate proved in [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF]. We will also need this result, and thus recall it in this section. Let us first introduce some notations.

Without loss of generality, we can assume that e 1 is contained in the open half-space defined by the asymptotic direction so that v • e 1 > 0.

We define the hyperplane :

H := {x ∈ R d : x • e 1 = 0}.
Let P := P v be the projection on the asymptotic direction along the hyperplane H defined for z ∈ Z d by

P (z) := z • e 1 v • e 1 v,
and Q := Q l be the projection of z on H along v so that

Q(z) := z -P (z).
Now, for x ∈ Z d , β > 0, ρ > 0 and L > 0, we define the tilted boxes with respect to the asymptotic direction v by :

(2.2) B β,L (x) := y ∈ Z d s.t. -L β < (y -x) • e 1 < L and Q(y -x) ∞ < ρL β .
and their front boundary by

∂ + B β,L (x) := {y ∈ ∂B β,L (x) s.t. (y -x) • e 1 = L}.
We have :

Proposition 8 (Atypical Quenched Exit Estimate, proposition 4.1 of [3]). Set α > 0 such that η α := sup e∈U E 1 ω(0,e) α < ∞. Take M 15d + 5 such that (P ) M |l is satisfied. Let β 0 ∈ (1/2, 1), β ∈ β 0 +1
2 , 1 and ζ ∈ (0, β 0 ). Then, for each γ > 0 we have that

lim sup L→∞ L -g(β 0 ,β,ζ) log P P 0,ω X T B β,L (0) ∈ ∂ + B β,L (0) e -γL β < 0, where g(β 0 , β, ζ) := min{β + ζ, 3β -2 + (d -1)(β -β 0 )}.
2.3. Some results on flows. The main tools that enables us to improve the results of [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF] is the use of flows and max-flow-min-cut theorems. We need some definitions and properties that we will detail in this section. In the following we consider a finite directed graph G = (V, E), where V is the set of vertices and E is the set of edges.

For all e ∈ E, we denote by e and e the vertices that are the head and tail of the edge e (the edge e goes from e to e).

Definition 3. We consider a finite directed graph G = (V, E). A flow from a set

A ⊂ V to a set Z ⊂ V is a non-negative function θ : E → R + such that : • ∀x ∈ (A ∪ Z) c , div θ(x) = 0. • ∀x ∈ A, div θ(x) 0. • ∀x ∈ Z, div θ(x) 0.
where the divergence operator is div :

R E → R V such that for all x ∈ V , div θ(x) = e∈E,e=x θ(e) - e∈E,e=x θ(e).
A unit flow from A to Z is a flow such that x∈A div θ(x) = 1. (Then we have also x∈Z div θ(x) = -1 ).

We will need the following generalized version of the max-flow-min-cut theorem : Proposition 9 (proposition 1 of [START_REF] Sabot | Random Dirichlet environment viewed from the particle in dimension d 3[END_REF]). Let G = (V, E) be a finite directed graph. Let (c(e)) e∈E be a set of non-negative reals (called capacities). Let x 0 be a vertex and (p x ) x∈V be a set of non-negative reals. There exists a non-negative function θ :

E → R + such that (2.3) div θ = x∈V p x (δ x 0 -δ x ), (2.4)
∀e ∈ E, θ(e) c(e), if and only if for all subset K ⊂ V containing x 0 we have

(2.5) c(∂ + K) x∈K c p x ,
where

∂ + K = {e ∈ E, e ∈ K, e ∈ K c } and c(∂ + K) = e∈∂ + K c(e).
The same is true if we restrict the condition (2.5) to the subsets K such that any y ∈ K can be reached from 0 following a directed path in K.

We will give here an idea of the proof, that explains why we call this result a generalized version of the classical max-flow-min-cut theorem. The complete proof can be found in [START_REF] Sabot | Random Dirichlet environment viewed from the particle in dimension d 3[END_REF]. The reversed implication is an easy consequence of the classical max-flow min-cut theorem on finite directed graphs (see for example [START_REF] Lyons | Probabilities on trees and networks[END_REF] section 3.1). If (c(e)) e∈E satisfies (2.5), we consider the new graph G = (V ∪ {δ}, Ẽ), where

Ẽ = E ∪ {(x, δ), x ∈ V }.
We define a new set of capacities (c(e)) e∈ Ẽ where c(e) = c(e) for e ∈ E and c((x, δ)) = p x . The strategy is to apply the max-flow min-cut theorem with capacities c and with source x 0 and sink δ. It gives a flow θ on G between x 0 and δ with strength x∈V p x and such that θ c. The function θ obtained by restriction of θ to E satisfies (2.4) and (2.3).

For the proof of theorem 1, we will consider the oriented graph (Z d , E Z d ) where E Z d := {(x, y) ∈ (Z d ) 2 s.t. |x -y| 1 = 1}. This graph is not finite, but we will only consider flows with compact support (θ(e) = 0 for all e except in a finite subset of E Z d ). We can then proceed as if the graph were finite, and use the previous definition and proposition.

Proof of theorem 1

Let l ∈ S d-1 , β > 0 and M 15d + 5. Assume that (P ) M |l is satisfied and that (E ′ ) β holds.

Let us take a rotation R such that R(e 1 ) = v. We fix β ′ ∈ 5 6 , 1 , M > 0 and for simplicity we will write τ 1 instead of τ v 1 . For u > 0, take

L = L(u) := 1 4M √ d 1 β ′ (log u) 1 β ′ , C L := x ∈ Z d : -L 2(v • e 1 ) x • R(e i ) L 2(v • e 1 )
, for 0 i 2d

Following the proof of proposition 5.1 in [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF], we write

P 0 (τ 1 > u) P 0 τ 1 > u, T C L(u) τ 1 + E F c 1 , P 0,ω T C L(u) > u + P(F 1 ),
with

F 1 := ω ∈ Ω : t ω C L(u) > u (log u) 1 β ′ and t ω (A) := inf n 0 : sup x P x,ω (T A > n) 1 2 .
As in [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF], the term P 0 τ 1 > u, T C L(u) τ 1 is bounded thanks to condition (P ) M |l, and the term E F c 1 , P 0,ω T C L(u) > u is bounded thanks to the strong Markov property. This part of the original proof is not modified, so we will not give more details here. It gives the existence for every γ ∈ (β ′ , 1) of a constant c > 0 such that :

P 0 (τ 1 > u) e -cL γ (u) c + 1 2
(log u)

1 β ′ + P(F 1 ).
It only remains to show that we can find a constant C > 0 such that P(F 1 ) Cu -β for u big enough.

For each ω ∈ Ω, still as in [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF], there exists x 0 ∈ C L(u) such that

P x 0 ,ω ( Hx 0 > T C L(u) ) 2|C L(u) | t ω (C L(u) )
where for y ∈ Z d , Hy = inf{n 1 : X n = y}. It gives

P(F 1 ) P ω ∈ Ω s.t. ∃x 0 ∈ C L(u) s.t. P x 0 ,ω ( Hx 0 > T C L(u) ) 2(log u) 1 β ′ u C L(u) .
We define for each point x ∈ C L(u) a point y x , closest from x + 2 L β ′ v•e 1 v. To bound P(F 1 ), we will need paths that go from x to y x with probability big enough and the atypical quenched exit estimate (proposition 8).

Define :

N := |v| log u 2M √ d(v • e 1 ) .
It is straightforward that

N -1 |y x -x| 1 N + 1.
The following of the proof will be developed in three parts : first we will construct unit flows θ i,x going from {x, x + e i } to {y x , y x + e i }, for all x ∈ C L(u) . Then we will construct paths with those flows, and use the atypical quenched exit estimate to bound P(F 1 ) in the case that those paths are big enough. We will conclude by bounding the probability that the paths are not big enough.

3.1. Construction of the flows θ i,x . We consider the oriented graph (Z d , E Z d ) where

E Z d := {(x, y) ∈ (Z d ) 2 s.t. |x -y| 1 = 1}.
We want to construct unit flows θ i,x going from {x, x + e i } to {y x , y x + e i }, for all x ∈ C L(u) . But there are additional constraints, as we will need them to construct paths that have a probability big enough. The aim of this section is to prove the following proposition : Proposition 10. For all x ∈ C L(u) , for all α 1 , . . . , α 2d positive constants, there exists 2d unit flows θ i,x : E Z d → R + , respectively going from {x, x + e i } to {y x , y x + e i }, such that :

(3.1) ∀e ∈ E Z d , θ i,x (e) α(e) κ i ,
where κ i := 2 2d j=1 α j -(α i + α i+d ), and α(e) := α i for e of the type (z, e i ). Furthermore, we can construct θ i,x with a compact support, and in a way that allows to find γ and S ⊂ E Z d , |S| independent of u, such that θ i,x (e)κ i γ < α(e) for all e ∈ S c . We will construct the θ i,x to prove their existences. For this we need three steps. Let B(x, R) be the box of Z d of center x and radius R, and B i (x, R) be the same box, where the vertices x and x + e i are merged (and we suppress the edge between them). We note 2 } the corresponding sets of edges. We will construct a unit flow in the graph

E B(x,R) := {(x, y) ∈ E Z d ∩ (B(x, R)) 2 } and E B i (x,R) := {(x, y) ∈ E Z d ∩ (B i (x, R))
(B i (x, R), E B i (x,R) ) from {x, x + e i } to B i (x, R) c , a unit flow in the graph (B i (y x , R), E B i (yx,R) ) from B i (y x , R) c to {y x ,
x + e i }, and then connect them. At each step, we will ensure that condition 3.1 is fulfilled.

First step : construction of a unit flow from {x, x + e i } to

B i (x, R) c : Lemma 1. Set x ∈ C L(u)
, and α 1 , . . . , α 2d positive constants. If R max i κ i min j α j , there exists 2d unit flows θ i,x :

E B i (x,R) → R + such that : div θ i,x = z∈∂B i (x,R) 1 |∂B i (x, R)| (δ x -δ z ) and ∀e ∈ E B i (x,R) , θ i,x (e) α(e) κ i where ∂B i (x, R) = {z ∈ B i (x, R) that has a neighbour in B i (x, R) c }.
The divergence condition ensures that the flow will be a unit flow, that it goes from x, and that it leaves B i (x, R) uniformly on the boundary of the box.

Proof. The result is a simple application of proposition 9. We fix x ∈ C L(u) and i between 1 and 2d. Define

p z = 1 |∂B i (x,R)| if z ∈ ∂B i (x, R), p z = 0 if z / ∈ ∂B i (x, R).
To prove the result we only have to check that ∀K ⊂ B i (x, R) containing x,

e∈∂ + K α(e) κ i z / ∈K p z , where ∂ + K = {e ∈ E B i (x,R) s.t.
e ∈ K and e / ∈ K}. We have two cases to examine:

• If K ∂B i (x, R) = ∅, z / ∈K p z = 1.
We then need e∈∂ + K α(e) κ i . For K = {x}, e∈∂ + K α(e) = κ i as we merged x and x + e i . For bigger K, we consider for all j = i the paths (x + ne j ) n∈N and for all j = i + d the paths (x + e i + ne j ) n∈N . They intersect the boundary of K in 2d + 1 different points, and the exit directions give us the corresponding α j , that sum to κ i . It gives that e∈∂ + K α(e) κ i .

• If K ∂B i (x, R) = ∅, z / ∈K p z < 1.
As K contains a path from x to ∂B i (x, R),

e∈∂ + K α(e) κ i R min j (α j +α j+d ) κ i
. It is bigger than 1 thanks to the hypothesis on R. It gives the result.

Second step : by the same way, we construct a flow θ i,x :

E B i (yx,R) → R + such that div θ i,x = z∈∂B i (yx,R) 1 |∂B i (y x , R)| (δ z -δ yx ). and ∀e ∈ E B i (yx,R) , θ i,x (e) α(e) κ i .
Third step : we will join the flows on E B i (x,R) and E B i (yx,R) with simple paths, to get a flow on E Z d . Take R max i κ i min j α j , and make sure that

1 |∂B(x,R)| < α(e)
κ i for all e ∈ E Z d (always possible by taking R big enough, R depends only on the α i and the dimension).

We can find ∂B(x, R) simple paths π j ⊂ E Z d satisfying :

• ∀j, π j connects a point of ∂B(x, R) to a point of ∂B(y x , R).

• ∀j, π j stays outside of B(x, R) and B(y x , R), except from the departure and arrival points. • If two paths intersect, they perform jumps in different direction after the intersection (no edge is used by two paths). If (x, e i ) is in a path, then (x + e i , -e i ) is not in any path. • The number of steps of each path is close to N : there exists constants K 1 and K 2 independent of u such that the length of π j is smaller than

K 1 N + K 2 .
(For example we can use the paths π (i,j) p45 of [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF], and make them exit the ball B(x, R) instead of {x, x + e i }). For all i, ∂B i (x, R) = ∂B(x, R) and ∂B i (y x , R) = ∂B(y x , R) as soon as R > 1. By construction,div θ i,x (z 1 ) = div θ i,x (z 2 ) = 1 |∂B(x,R)| for any z 1 ∈ ∂B i (x, R) and z 2 ∈ ∂B i (y x , R). We can then join the flows of the first two steps by defining a flow θ i,x (e) = 1 |∂B(yx,R)| for all e ∈ π j (and 0 on all the other edges of E Z d ).

We have thus constructed a unit flow θ i,x on E Z d , from {x, x + e i } to {y x , y x + e i }, satisfying (3.1) ( (3.1) is satisfied on E B i (x,R) and E B i (yx,R) as R max i κ i min j α j thanks to lemma 1, and outside those balls as

1 |∂B(x,R)| < α(e)
κ i for all e ∈ E Z d ). It concludes the proof of the first part of proposition 10.

As θ i,x (e) = 0 out of the finite set

E B i (x,R) ∪ E B i (yx,R) ∪ {e ∈ π j , 1 j ∂B(x, R
)}, the flow has a compact support. And as we made sure that

1 |∂B(x,R)| < α(e) κ i , we can take S = B(x, R) ∪ B(y x , R) and γ = κ i |∂B(x,R)| to conclude the proof.
3.2. Bounds for P(F 1 ). We apply proposition 10 for the α 1 , . . . , α 2d of the definition of (E ′ ) β (see (1.1) and (1.2)). It gives flows θ i,x on E Z d , constructed as in the previous section.

We can decompose a given θ i,x (for i and x fixed) in a finite set of weighted paths, each path starting from x or x + e i and arriving to y x or y x + e i . It suffices to choose a path σ where the flow is always positive, to give it a weight p σ := min e∈σ θ i,x (e) > 0 and to iterate with the new flow θ(e) := θ i,x (e)p σ 1 e∈σ .

The weigh p σ of a path σ then satisfies : for all e ∈ E Z d , θ i,x (e) = σ containing e p σ . As θ i,x is a unit flow we get σ path of θ i,x p σ = 1. We will use those weights in the next section, to prove that those paths are "big enough" with high probability.

We now introduce :

F 2,i = ω ∈ Ω s.t. ∀x ∈ C L(u) , ∀σ path of θ i,x , ω σ := e∈σ ω e u 1 M -1
and

F 2 = 2d i=1 F 2,i . Define F 3 := ω ∈ Ω s.t. ∃x 0 ∈ C L(u) s.t. P x 0 ,ω ( Hx 0 > T C L(u) )
2(log u)

1 β ′ u C L(u) , F 2 .
We get immediately :

P(F 1 ) P(F 3 ) + P(F c 2 )
. It gives two new terms to bound. We start by bounding P(F 3 ). For this we will use the same method as in [START_REF] Campos | Ellipticity criteria for ballistic behavior of random walks in random environment[END_REF] : on the event F 3 , for all 1 i 2d we can use a path σ of θ i,x to join x or x + e i to y x or y x + e i . It gives :

ω(x 0 , e i )u 1 M -1 inf z∈{yx 0 ,yx 0 +e i } P z,ω (T C L(u) < H x 0 ) P x 0 ,ω (T C L(u) < Hx 0 ) 2(log u) 1 β ′ u C L(u) ,
where the factor ω(x 0 , e i ) corresponds to the probability of jumping from x to x + e i , in the case where the path σ starts from x + e i . As 2d i=1 ω(x 0 , e i ) = 1, it gives

u 1 M -1 inf z∈V (yx 0 ) P z,ω (T C L(u) < H x 0 ) 4d(log u) 1 β ′ u C L(u) ,
where V (y x 0 ) := {y x 0 , (y x 0 + e i ) i=1,...,2d }.

In particular, on F 3 , we can see that for u large enough V (y x 0 ) ⊂ C L(u) . As a result, on F 3 , we have for u large enough

inf z∈V (yx 0 ) P z,ω (X Tz+U β ′ ,L • e 1 > z • e 1 ) inf z∈V (yx 0 ) P z,ω (T C L(u) < H x 0 ) 1 u 1 2M = e -2 √ dL(u) β ′ , where U β ′ ,L := {x ∈ Z d : -L β ′ < x • e 1 < L}.
From this and using the translation invariance of the measure P, we conclude that :

P ∃x 0 ∈ C L(u) s.t. P x 0 ,ω ( Hx 0 > T C L(u) ) 4d(log u) 1 β ′ u C L(u) , F 2 P ∃x 0 ∈ C L(u) s.t. inf z∈V (yx 0 ) P z,ω (X Tz+U β ′ ,L • e 1 > z • e 1 ) e -2 √ dL(u) β ′ (2d + 1) C L(u) P P 0,ω (X T U β ′ ,L(u) • e 1 > 0) e -2 √ dL(u) β ′ (2d + 1) C L(u) P P 0,ω (X T B β ′ ,L(u) • e 1 > 0) e -2 √ dL(u) β ′
, where the titled box B β ′ ,L(u) is defined as in (2.2).

We conclude with the atypical quenched exit estimate (proposition 8) : there exists a constant c > 0 such that for each β 0 ∈ ( 1 2 , 1) one has :

P(F 3 ) 1 c e -cL(u) g(β 0 ,β ′ ,ζ) ,
where g(β 0 , β ′ , ζ) is defined as in proposition 8. Note that for each β ′ ∈ 5 6 , 1 there exists a β 0 ∈ 1 2 , β such that for every ζ ∈ 0, 1 2 one has g(β 0 , β ′ , ζ) > β ′ . Therefore, replacing L by its value, we proved that there exists c > 0 such that : P(F 3 ) cu -β .

Bound for P(F c

2 ). To conclude the bound for P(F 1 ) and the proof of theorem 1, it only remains to control P(F c

2 ). It is in this section that we will use the conditions that were imposed on θ i,x during the construction of the flows.

P(F c

2 )

2d i=1 P(F c 2,i ) 2d i=1 x∈C L(u) P(∀σ path of θ i,x , ω σ u 1 M -1 ) As θ i,x is a unit flow, if ∀σ path of θ i,x , ω σ u 1 M -1 then : σ path of θ i,x p σ ω σ u 1 M -1 σ path of θ i,x p σ = u 1 M -1 .
Jensen's inequality then gives :

σ path of θ i,x ω pσ σ = e∈E d ω θ i,x (e) e u 1 M -1 .
It allows to write :

P(F c 2 ) 2d i=1 x∈C L(u) P   e∈E d ω θ i,x (e) e u 1 M -1   2d i=1 x∈C L(u) E e∈E d ω -κ i θ i,x (e) e u -κ i ( 1 M -1)
We will use the integrability given by the flows to bound the expectations. The independence of the environment gives (for i and x fixed) : is a finite constant independent on u.

E e ω -κ i θ i,
It remains to deal with the case of z / ∈ S. As we chose R to get θ i,x (e)κ i < γ for the edges outside S, and thanks to the bounds on the number of edges with positive flow (there is a finite number of paths, and each path has a bounded length), we have : , where c 1 and c 2 are positive constants, independent of u. Then, putting all of those bounds together,

P(F c 2 ) 2d i=1 x∈C L(u) C 1 C C 3 N 2 u κ i ( 1 M -1) 2d i=1 x∈C L(u) C 4 u C 5 +κ i M -κ i C 6 (log u) C 7 u C 8 M -min i κ i
where all the constants C i are positive and do not depend on u. As remark 4 tells us that we can choose M as large as we want, we can get C 8 M as small as we want. Then we can find a constant C > 0 such that P(F c

2 ) Cu -β for u big enough. It concludes the proof.

P 0 X T B e 1 ,L, L • e 1 < L 1
L M , for some M 15d + 5. We first decompose this probability according to whether the exit point of the random walk from the box B e 1 ,L, L is on the bottom or on one of the sides of the box, so that,

P 0 X T B e 1 ,L, L • e 1 < L = P 0 X T B e 1 ,L, L • e 1 = -L + P 0 X T B e 1 ,L, L • e 2 = L + P 0 X T B e 1 ,L, L • e 2 = -L .
We will first bound the probability to exit through the sides. We do the computations for P 0 X T B e 1 ,L, L • e 2 = L but the other term can be dealt with in the same way. Suppose that X T B e 1 ,L, L • e 2 = L, and define n 0 , . . . , n L-1 the finite hitting times of new levels in direction e 2 as follows :

n k := inf{n 0 s.t. X n • e 2 k}.
To simplify notation define ϕ(x) := ω(x, e 1+d ). We now choose a constant 1 > δ > 0, and we will call "good point" any x ∈ Z 2 such that ϕ(x) > δ. We define p := P(ϕ(x) > δ). Note that p does not depend on x since the environment is i.i.d., so that it depends only on δ and the law of ϕ.

We now introduce the event that a great number of the X n k are good points :

C 1 := X T B e 1 ,L, L • e 2 = L and at least p 2 L of the X n k , 1 k L -1, are good .
We get immediately

P 0 X T B e 1 ,L, L • e 2 = L = P 0 (C 1 ) + P 0 X T B e 1 ,L, L • e 2 = L ∩ (C c 1 )
. By construction of the X n k and independence of the environment, and with Z an independent random variable following a binomial law of parameters p and L, we can bound the second term of the sum :

P 0 X T B e 1 ,L, L • e 2 = L ∩ (C c 1 ) P Z p 2 L exp -2 (p L -p L/2) 2 L = exp - p 2 L 2
where the last inequality is Hoeffding's inequality. It only remains to bound P 0 (C 1 ). For that, we introduce the following new event

C 2 := C 1 and X n k +1 -X n k = e 1 for at least δp 4 L of the good X n k ,
that states that the walk goes often in direction e 1 just after reaching a X n k that is a good point. We can then write

P 0 (C 1 ) = P 0 (C 2 ) + P 0 (C 1 ∩ (C c 2 )). To bound P 0 (C 1 ∩ (C c 2
)), we use the uniform bound "ϕ(x) > δ" for good points that gives us that ω(x, e 1 ) > rδ on those points. And we get Z ′ an independent random variable following a binomial law of parameters rδ and p 2 L :

P 0 (C 1 ∩ (C c 2 )) P Z ′ δp 4 L exp -pδ 2 L r - 1 2 2 .
It only remains to bound P 0 (C 2 ). Set n + (respectively n -) the total number of jumps in direction e 1 (respectively -e 1 ) before exiting the box B e 1 ,L, L. We will need a third new event

C 3 := n + 1 + r r n -,
that allows us to write

P 0 (C 2 ) = P 0 (C 2 ∩ C 3 ) + P 0 (C 2 ∩ (C c 3
)). First notice that for L big enough, C 2 ∩C 3 = ∅. Indeed, C 1 implies that we exit the box B e 1 ,L, L by the side "x• e 2 = L". Now, since the vertical displacement of the walk before exiting the box B e 1 ,L, L is n +n -, on the event C 3 we know that this displacement is at least equal to 1 1+r n + . Therefore, since on C 2 the walk makes at least δp 4 L = 35δp 2 L 3 moves in the direction e 1 , on C 2 ∩ C 3 its vertical displacement before exiting the box But X T B e 1 ,L, L • e 1 = -L also gives that necessarily, n -L. Then 4.2. Proof of Theorem 5. It is classical to represent Dirichlet distributions with independent gamma random variables : if γ 1 , . . . , γ N are independent gamma random variables with parameters β 1 , . . . , β N , then γ 1 γ i , . . . , γ N γ i is a Dirichlet random variable with parameters (β 1 , . . . , β N ). We get a restriction property as an easy consequence of this representation (see [START_REF] Wilks | Mathematical statistics[END_REF], pages 179-182) : for J a non-empty subset of {1, . . . , N}, the random variable x j i∈J x i j∈J follows a Dirichlet law with parameters (β j ) j∈J and is independent of i∈J x i . This property will be useful in the following.

We consider the box B e 1 ,L, L for L = 70L 3 , and want to find some L > c 0 such that P 0 X T B e 1 ,L, L • e 1 < L 1 L M to prove (P ) M |e 1 . Let l i := {x ∈ Z d s.t. x • e 1 = i} and t i := inf{n 0 : X n ∈ l i , X n+1 / ∈ l i }. We first consider the events that, when the walk arrives on l i for the first time, it gets out of it by an edge in direction e 1 (the alternative being getting out by an edge in direction -e 1 ) :

G 1,i := {X t i +1 -X t i = e 1 }. At the point X t i , we know that the walk will go either to X t i + e 1 or to X t ie 1 . Thanks to the restriction property of the Dirichlet laws, we know that ω(Xt i ,e 1 ) ω(Xt i ,e 1 )+ω(Xt i ,-e 1 ) follows a beta law of parameters (β 1 , β 1 + β 1+d ) and is independent of the previous trajectory of the walk on l i . Then

P 0 (G 1,i ) = β 1 β 1 + β 1+d .

Now define

G 1 := L-1 i=0 G 1,i ,
and note that P 0 (G c 1 ) L β 1+d β 1 + β 1+d .

We can now write Then the walk has to get out by one of the 2d -2 "side boundaries" :

P 0 X T B e 1 ,L, L • e 1 < L ∩ G 1 = P 0 ∪ d j=2 X T B e 1 ,L, L • e j = ± L ∩ G 1 .
On the event ∪ d j=1 {X T B e 1 ,L, L • e j = ± L} define n 0 , . . . , n L-1 as the finite hitting times of new levels in any direction perpendicular to e 1 as follows : Putting the previous bounds together, we finally get for all L L 0 :

P 0 X T B e 1 ,L, L • e 1 < L L β 1+d β 1 + β 1+d + exp - p 2 2 L .
Let now L 1 be such that for all L L 1 exp -

p 2 2 L 1 2L M .
Take now L 2 := max{c 0 , L 0 , L 1 } and then choose β 1+d ( 1) so that

L 2 β 1+d β 1 + β 1+d ≤ 1 2L M 2 .
We then conclude that for this choice of β 1+d there exists an L c 0 such that

P 0 X T B e 1 ,L 2 , L • e 1 < L 1 L M .

Definition 2 .

 2 Let β > 0. We say that the law of the environment satisfies the ellipticity condition (E ′ ) β if there exists an {α(e) : e ∈ U} ∈ (0, ∞) 2d such that (1.1) κ ({α(e) : e ∈ U}) := 2 e ′ α(e ′ )sup e∈U (α(e) + α(-e)) > β and for every e ∈ U (1.2) E e e ′ =e α(e ′ ) log 1 ω(0,e ′ ) < ∞.

  Idea of the proof. If θ satisfies (2.3) and (2.4) then e, e∈K, e∈K c θ(e)e, e∈K, e∈K c θ(e) = x∈K div θ(x) = x∈K c p x . It implies (2.5) by (2.4) and positivity of θ.

  t. e=z ω -κ i θ i,x (e)

4 . 1 .

 41 New examples of random walks satisfying the polynomial condition 4.Proof of Theorem 4. Consider the box B e 1 ,L, L for L = 70L 3 . We want to find some L > c 0 such that

P 0 X 2 .

 02 T B e 1 ,L, L • e 1 = -L P 0 n + < n + + n -Therefore we can find L 4 L 3 such that for all L L 4 ,P 0 X T B e 1 ,L, L • e 1 = -L 1 2L M ,from where we conclude that for all L L 4 ,P 0 X T B e 1 ,L, L • e 1 < L 1 L M .

P 0 X

 0 T B e 1 ,L, L • e 1 < L P 0 X T B e 1 ,L, L • e 1 < L ∩ G 1 + P 0 (G c 1 ) P 0 X T B e 1 ,L, L • e 1 < L ∩ G 1 + L β 1+d β 1 + β 1+d ,and we only need to bound the first term of this sum. If G 1 is satisfied, the walk cannot get out of the box B e 1 ,L, L by the "lower boundary" {x ∈ Z d s.t. x • e 1 = -L}.

1 1+

 1 n k := inf{n 0 s.t. sup 2 j d |X n • e j | k}.Let now p = β i =1+d β i and consider the event

G 3 : 2 L 3 ) 1 35p.

 3231 = G 1 and X n k +1 -X n k = e 1 for at least p of the points X n k .Suppose β 1+d1, then p E(ω(0, e 1 )). Consider now a random variable Z with a binomial law of parameters p and L. Using Hoeffding's inequality, we see thatP (G c But clearly G 1 ∩ G 3 = ∅ for L L 0 := Therefore we have in this case P 0 ∪ d j=2 X T B e 1 ,L, L • e j = ± L ∩ G 1 exp -

  ′ ) t e ′ , ∀e ′ ∈ U, e ′ = e) C ′ ) e ′ ∈U \{e} , 0 t e ′ ≤ 1. Dirichlet environment (cf the next section) is a typical example of environment that satisfies this condition. Indeed, in this case we easily see that (E ′ ) β is satisfied if and only if β < 2 e ′ β(e ′ )sup e∈U (β(e) + β(-e)). On the other hand if β 2 e ′ β(e ′ )sup e∈U (β(e) + β(

	-1	e ′ ∈U,e ′ =e	t β e ′ e ′	P (ω(0, e e ′ ∈U,e ′ =e	t β e ′ e ′
	for all (t e

  As θ i,x satisfies (3.1), the ellipticity condition (E ′ ) β gives that each of the expecta-By construction |S| is finite and does not depend on u : z∈S E e s.t. e=z ω -κ i θ i,x (e) e

		e	x (e)	=	E	ω -κ i θ i,x (e) e
					z∈Z d	e s.t. e=z
					=	E	ω -κ i θ i,x (e) e	E	ω -κ i θ i,x (e)
					z∈S	e s.t. e=z	z / ∈S	e s.t. e=z
	tions E	e s.t. e=z ω	-κ i θ i,x (e) e	are finite.

e

where we recall that S = B(x, R) ∪ B(y x , R).

Acknowledgements. The authors would like to thank Alexander Drewitz for pointing out the ideas of the proof of Theorem 4.

This work was partially supported by Fondo Nacional de Desarrollo Científico y Tecnológico grant 1100746 and by the ANR project MEMEMO2.

is at least 35δp 2(1+r) L 3 . Since on C 2 ∩ C 3 the walk exits the box by the "x • e 2 = L" side we see that for L larger than L 1 := 2(1+r) 35δp the event C 2 ∩ C 3 is empty. We now want to bound P 0 (C 2 ∩ (C c 3 )).

Now note that whenever we go through a vertical edge from a point x, the law of the environment tells us that it is an edge (x, e 1 ) with probability r 1+r , and (x, -e 1 ) with probability 1 1+r . Then, defining Z ′′ as a random variable following a binomial law of parameters 1 1+r and pδ 4 L, we have the bound :

where we need

1+2r to apply Hoeffding's inequality in the last inequality. We can find L 2 such that this is true for L L 2 .

Choose M 15d + 5. By putting all of our previous bounds together, we finally get, for all L L 2 ,

where we recall that L = 70L 3 , δ > 0 and p = P(ϕ(x) > δ). Then, for any choice of δ, we can find L 3 max(c 0 , L 1 , L 2 ) such that for all L L 3 ,

We now only need to bound P 0 X T B e 1 ,L, L • e 1 = -L to prove (P ) M |e 1 . We will use again the notations n + (respectively n -) for the total number of jumps in direction e 1 (respectively -e 1 ) before exiting the box B e 1 ,L, L. Suppose that X T B e 1 ,L, L • e 1 = -L. Then necessarily n + < n -, which gives n + < n + +n -

2

. As n + conditioned to n + + n - follows a binomial law of parameters r 1+r and n + + n -, Hoeffding's inequality gives the bound :

2 .