Olivier Garet
email: olivier.garet@univ-lorraine.fr

R Égine Marchand
email: regine.marchand@univ-lorraine.fr

LARGE DEVIATIONS FOR THE CONTACT PROCESS IN RANDOM ENVIRONMENT

Keywords: 2000 Mathematics Subject Classification. 60K35, 82B43 Random growth, contact process, random environment, shape theorem, large deviation inequalities

The asymptotic shape theorem for the contact process in random environment gives the existence of a norm µ on R d such that the hitting time t(x) is asymptotically equivalent to µ(x) when the contact process survives. We provide here exponential upper bounds for the probability of the event { t(x) µ(x) ∈ [1ε, 1 + ε]}; these bounds are optimal for independent random environment. As a special case, this gives the large deviation inequality for the contact process in a deterministic environment, which, as far as we know, has not been established yet.

Introduction

Durrett and Griffeath [START_REF] Durrett | Contact processes in several dimensions[END_REF] proved that when the contact process on Z d starting from the origin survives, the set of sites occupied before time t satisfies an asymptotic shape theorem, as in first-passage percolation. In [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF], we extended this result to the case of the contact process in a random environment.

The random environment is given by a collection (λ e ) e∈E d of positive random variables indexed by the set of edges of the grid Z d . Given a realization λ of this environment, the contact process (ξ 0 t ) t≥0 in the environment λ is a homogeneous Markov process taking its values in the set P(Z d ) of subsets of Z d . If ξ 0 t (z) = 1, we say that z is occupied at time t, while if ξ 0 t (z) = 0, we say that z is empty at time t. The initial value of the process is {0} and the process evolves as follows:

• an occupied site becomes empty at rate 1,

• an empty site z becomes occupied at rate:

z-z ′ 1=1 ξ 0 t (z ′ )λ {z,z ′ } ,
all these evolutions being independent. We study then the hitting time t(x) of a site x: t(x) = inf{t ≥ 0 : x ∈ ξ 0 t }. In [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF], we proved that under good assumptions on the random environment, there exists a norm µ on R d such that for almost every environment, the family (t(x)) x∈Z d satisfies, when x goes to +∞, t(x) ∼ µ(x) on the event "the process survives".

We focus here on the large deviations of the hitting time t(x) for the contact process in random environment. As far as we know, such inequalities for the classical contact process have not been studied yet, they will be contained in our results.

The assumptions we will require on the random environment are the ones we already needed in [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF]. We denote by λ c (Z d ) the critical intensity of the classical contact process on Z d , we fix λ min and λ max such that λ c (Z d ) < λ min ≤ λ max and we set Λ = [λ min , λ max ] E d .

Assumptions (E). The support of the law ν of the random environment is included in Λ = [λ min , λ max ] E d ; the law ν is stationary, and if Erg(ν) denotes the set of x ∈ Z d \{0} such that the translation along vector x is ergodic for ν, then the cone generated by Erg(ν) is dense in R d . This last condition is obviously fulfilled if Erg(ν) = Z d \{0}. We will sometimes require the stronger following assumptions:

Assumptions (E'). The law ν of the random environment is a product measure:

ν = ν ⊗E d 0
, where ν 0 is some probability measure on [λ min , λ max ].

By taking for ν the Dirac mass (δ λ ) ⊗E d , with λ > λ c (Z d ), which clearly fullfills these assumptions, we recover the case of the classical contact process in a deterministic environment.

For λ ∈ Λ, we denote by P λ the (quenched) law of the contact process in environment λ, and by P λ the (quenched) law of the contact process in environment λ conditioned to survive. We define then the annealed probability measures P and P: P(.) = Λ P λ (.) dν(λ) and P(.) = Λ P λ (.) dν(λ).

We will study separately the probabilities of the "upper large deviations" and the "lower large deviations", i.e. respectively of the events {t(x) ≥ (1 + ε)µ(x)} and {t(x) ≤ (1 -ε)µ(x)}.

The most general result concerns the quenched "upper large deviations" for the hitting time t(x) and the coupling time t ′ (x) = inf{T ≥ 0 : ∀t ≥ T ξ 0 t (x) = ξ Z d t (x)}, where (ξ Z d t ) t≥0 is the contact process starting from Z d , and for the set of hit points H t and the coupled region K ′ t :

H t = {x ∈ Z d : t(x) ≤ t}, Ht = H t + [0, 1] d K ′ t = {x ∈ Z d : t ′ (x) ≤ t}, K′ t = K ′ t + [0, 1] d .
We only require here Assumptions (E).

Theorem 1.1. Let ν be an environment law satisfying Assumptions (E). For every ε > 0, there exist B > 0 and a random variable A(λ) such that for ν almost every environment λ, for every x ∈ Z d ,

P λ (t(x) ≥ µ(x)(1 + ε)) ≤ A(λ)e -B x , (1)
P λ (t ′ (x) ≥ µ(x)(1 + ε)) ≤ A(λ)e -B x , (2)
P λ ∀t ≥ T (1 -ε)tA µ ⊂ K′ t ∩ Ht ≥ 1 -A(λ)e -BT . (3)
We can note that the random variable A(λ) is almost surely finite, but that it could often be large. This question will be studied in a forecoming paper about annealed upper large deviations [START_REF] Garet | Annealed upper large deviations for the asymptotic shape of the contact process in Z d[END_REF]. The key point of the proof of Theorem 1.1, interesting on its own, is to control the times s when a site x is occupied and has infinite progeny. We will denote this event by {(0, 0) → (x, s) → ∞} by analogy with percolation.

Theorem 1.2. There exist C, θ, A, B > 0 such that ∀λ ∈ Λ ∀x ∈ Z d ∀t ≥ C x P λ (Leb{s ∈ [0, t] : (0, 0) → (x, s) → ∞} ≤ θt) ≤ A exp(-Bt).

For the "lower large deviations", the subadditivity gives a nice setting and allows to state a large deviations principle in the spirit of Hammersley [START_REF] Hammersley | Postulates for subadditive processes[END_REF]. Theorem 1.3. Let ν be an environment law satisfying Assumptions (E). Let x ∈ Z d . There exist a convex function Ψ x and a concave function K x taking their values in R + such that for ν almost every λ,

∀u > 0 lim n→+∞ - 1 n log P λ (t(nx) ≤ nu) = Ψ x (u); ∀θ ≥ 0 lim n→+∞ - 1 n log E λ [e -θt(nx) ] = K x (θ).
The functions Ψ x and K x moreover satisfy the reciprocity relations:

∀u > 0 ∀θ ≥ 0 Ψ x (u) = sup θ≥0 {K x (θ) -θu} and K x (θ) = inf u>0 {Ψ x (u) + θu}.
To obtain effective large deviation inequalities, we moreover have to prove that Ψ x (u) > 0 if u < µ(x). More precisely, Theorem 1.4. Let ν be an environment law satisfying Assumptions (E ′ ). For every ε > 0, there exist A, B > 0 such that for every x ∈ Z d , for every t ≥ 0, P(t(x) ≤ (1 -ε)µ(x)) ≤ A exp(-B x ), (4) P(∀s ≥ t H s ⊂ (1 + ε)sA µ ) ≥ 1 -A exp(-Bt). [START_REF] Couronné | Sur les grands clusters en percolation[END_REF] The annealed large deviations inequalities imply the quenched ones: setting

A(λ) = x∈Z d
exp(B x /2)P λ (t(x) ≤ (1 -ε)µ(x)) , we see that A(λ) is integrable with respect to ν, and thus is ν-almost surely finite.

So ∀x ∈ Z d P λ (t(x) ≤ (1 -ε)µ(x)) ≤ A(λ) exp(-B/2 x
). Unfortunately, we do not have a complete large deviation principle as Theorem 1.3 for the upper large deviations. However, we will see in Section 5 that when the environment is i.i.d, the exponential order given by these inequalities is optimal.

Asymptotic shape results for growth models are generally proved thanks to the subadditive processes theory initiated by Hammersley and Welsh [START_REF] Hammersley | First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory[END_REF], and especially with Kingman's subadditive ergodic theorem [START_REF] Kingman | Subadditive ergodic theory[END_REF] and its extensions. Since Hammersley [START_REF] Hammersley | Postulates for subadditive processes[END_REF], we know that subadditive properties offer a proper setting to study the large deviation inequalities. See also the survey by Grimmett [START_REF] Grimmett | Large deviations in subadditive processes and first-passage percolation[END_REF] and the Saint-Flour course by Kingman [START_REF] Kingman | Subadditive processes[END_REF]. However, as noted by Seppäläinen and Yukich [START_REF] Seppäläinen | Large deviation principles for Euclidean functionals and other nearly additive processes[END_REF], the general theory of large deviations for subadditive processes is patchy. The best known case is first-passage percolation, studied by Grimmett and Kesten in 1984 [14]. This paper introduced some lines of proof for the large deviations of growth processes, that have been reused later, for instance in the study of the large deviations for the chemical distance in Bernoulli percolation [START_REF] Garet | Large deviations for the chemical distance in supercritical Bernoulli percolation[END_REF]. For more recent results concerning first-passage percolation, see Chow-Zhang [START_REF] Chow | Large deviations in first-passage percolation[END_REF], Cranston-Gauthier-Mountford [START_REF] Cranston | On large deviation regimes for random media models[END_REF], and Théret et al [START_REF] Théret | On the small maximal flows in first passage percolation[END_REF][START_REF] Théret | Upper large deviations for the maximal flow in first-passage percolation[END_REF][START_REF] Rossignol | Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation[END_REF][START_REF] Rossignol | Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation[END_REF][START_REF] Cerf | Lower large deviations for the maximal flow through a domain of R d in first passage percolation[END_REF][START_REF] Cerf | Upper large deviations for the maximal flow through a domain of R d in first passage percolation[END_REF][START_REF] Cerf | Law of large numbers for the maximal flow through a domain of R d in first passage percolation[END_REF][START_REF] Rossignol | Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation[END_REF].

The renormalization techniques used by Grimmett and Kesten are well-known now: static renormalization for "upper large deviations" (control of a too slow growth), dynamic renormalization for "lower large deviations" (control of a too fast growth). However, the possibility for the contact process to die gives rise to extra difficulties that do not appear in the case of first-passage percolation or even of Bernoulli percolation. To our knowledge, the only growth process with possible extinction for which large deviations inequalities have been established is oriented percolation in dimension 2 (see Durrett [START_REF] Durrett | Oriented percolation in two dimensions[END_REF]). Note also that Proposition 20.1 in the PhD thesis of Couronné [START_REF] Couronné | Sur les grands clusters en percolation[END_REF] rules out the possibility of a too fast growth for oriented percolation in dimension d.

In Section 2, we construct the model, give the notation and state previous results, mainly from [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF]. Section 3 is devoted to the proof of the upper large deviation inequalities, Theorem 1.1, while lower large deviations -Theorems 1.3 and 1.4are proved in Section 4. Finally, the optimality of the exponential decrease given by these results is briefly discussed in Section 5.

Preliminaries

2.1. Definition of the model. Let λ min and λ max be fixed such that λ c (Z d ) < λ min ≤ λ max , where λ c (Z d ) is the critical parameter for the survival of the classical contact process on Z d . In the following, we restrict ourselves to the study of the contact process in random environment with birth rates λ = (λ e )

e∈E d in Λ = [λ min , λ max ] E d . An environment is thus a collection λ = (λ e ) e∈E d ∈ Λ.
Let λ ∈ Λ be fixed. The contact process (ξ t ) t≥0 in the environment λ is a homogeneous Markov process taking its values in the set P(Z d ) of subsets of Z d , that we sometimes identify with {0, 1} Z d : for z ∈ Z d we also use the random variable ξ t (z) = 1 1 {z∈ξt} . If ξ t (z) = 1, we say that z is occupied or infected, while if ξ t (z) = 0, we say that z is empty or healthy. The evolution of the process is as follows:

• an occupied site becomes empty at rate 1,

• an empty site z becomes occupied at rate

z-z ′ 1=1 ξ t (z ′ )λ {z,z ′ } ,
each of these evolutions being independent from the others. In the following, we denote by D the set of càdlàg functions from R + to P(Z d ): it is the set of trajectories for Markov processes with state space P(Z d ).

To define the contact process in the environment λ ∈ Λ, we use Harris' construction [START_REF] Harris | Additive set-valued Markov processes and graphical methods[END_REF]. It allows to make a coupling between contact processes starting from distinct initial configurations by building them from a single collection of Poisson measures on R + .

Graphical construction. We endow R + with the Borel σ-algebra B(R + ), and we denote by M the set of locally finite counting measures m = +∞ i=0 δ ti . We endow this set with the σ-algebra M generated by the maps m → m(B), where B describes the set of Borel sets in R + .

We then define the measurable space (Ω, F ) by setting

Ω = M E d × M Z d and F = M ⊗E d ⊗ M ⊗Z d .
On this space, we consider the family (P λ ) λ∈Λ of probability measures defined as follows: for every λ = (λ e ) e∈E d ∈ Λ,

P λ =   e∈E d P λe   ⊗ P ⊗Z d 1 ,
where, for every λ ∈ R + , P λ is the law of a Poisson point process on R + with intensity λ. If λ ∈ R + , we write P λ (rather than P (λ) e∈E d ) for the law in deterministic environment with constant infection rate λ.

For every t ≥ 0, we denote by F t the σ-algebra generated by the maps ω → ω e (B) and ω → ω z (B), where e ranges over all edges in E d , z ranges over all points in Z d , and B ranges over all Borel sets in [0, t].

We build the contact process in environment λ ∈ Λ from this family of Poisson process, as detailed in Harris [START_REF] Harris | Additive set-valued Markov processes and graphical methods[END_REF] for the classical contact process and in [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF] for the random environment case. Note especially that the process is attractive

(A ⊂ B) ⇒ (∀t ≥ 0 ξ A t ⊂ ξ B t
), and Fellerian; then it enjoys the strong Markov property. Time translations. For t ≥ 0, we define the translation operator θ t on a locally finite counting measure m = +∞ i=1 δ ti on R + by setting

θ t m = +∞ i=1 1 1 {ti≥t} δ ti-t .
The translation θ t induces an operator on Ω, still denoted by θ t : for every ω ∈ Ω, we set θ t ω = ((θ t ω e ) e∈E d , (θ t ω z ) z∈Z d ).

Spatial translations. The group Z d can act on the process and on the environment. The action on the process changes the observer's point of view: for x ∈ Z d , we define the translation operator T x by

∀ω ∈ Ω T x ω = ((ω x+e ) e∈E d , (ω x+z ) z∈Z d ),
where x + e the edge e translated by vector x.

Besides, we can consider the translated environment x.λ defined by (x.λ) e = λ x+e . These actions are dual in the sense that for every λ ∈ Λ, for every x ∈ Z d ,

∀A ∈ F P λ (T x ω ∈ A) = P x.λ (ω ∈ A). (6)
Consequently, the law of ξ x under P λ coincides with the law of ξ 0 under P x.λ .

Essential hitting times and associated translations. For a set A ⊂ Z d , we define the lifetime τ A of the process starting from A by τ A = inf{t ≥ 0 : ξ A t = ∅}. For A ⊂ Z d and x ∈ Z d , we also define the first infection time t A (x) of the site x from the set A by t A (x) = inf{t ≥ 0 : x ∈ ξ A t }. If y ∈ Z d , we write t y (x) instead of t {y} (x). Similarly, we simply write t(x) for t 0 (x).

In our previous paper [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF], we introduced a new quantity σ(x): it is a time when the site x is infected from the origin 0 and also has an infinite lifetime. This essential hitting time is defined from a family of stopping times as follows: we set u 0 (x) = v 0 (x) = 0 and we define recursively two increasing sequences of stopping times (u n (x)) n≥0 and (v n (x)) n≥0 with u 0 (x) = v 0 (x) ≤ u 1 (x) ≤ v 1 (x) ≤ u 2 (x) . . . as follows:

• Assume that v k (x) is defined. We set

u k+1 (x) = inf{t ≥ v k (x) : x ∈ ξ 0 t }. If v k (x) < +∞, then u k+1 (x)
is the first time after v k (x) where site x is once again infected; otherwise, u k+1 (x) = +∞.

• Assume that u k (x) is defined, with k ≥ 1. We set v k (x) = u k (x)+τ x •θ u k (x) . If u k (x) < +∞, the time τ x • θ u k (x)
is the lifetime of the contact process starting from x at time u k (x); otherwise, v k (x) = +∞.

We then set ( 7)

K(x) = min{n ≥ 0 : v n (x) = +∞ or u n+1 (x) = +∞}.
This quantity represents the number of steps before the success of this process: either we stop because we have just found an infinite v n (x), which corresponds to a time u n (x) when x is occupied and has infinite progeny, or we stop because we have just found an infinite u n+1 (x), which says that after v n (x), site x is nevermore infected.

We proved that K(x) is almost surely finite, which allows to define the essential hitting time σ(x) by setting σ(x) = u K(x) . It is of course larger than the hitting time t(x) and can been seen as a regeneration time.

Note however that σ(x) is not necessary the first time when x is occupied and has infinite progeny: for instance, such an event can occur between u 1 (x) and v 1 (x), being ignored by the recursive construction.

At the same time, we define the operator θx on Ω by:

θx = T x • θ σ(x) if σ(x) < +∞, T x otherwise,
or, more explicitly,

( θx )(ω) = T x (θ σ(x)(ω) ω) if σ(x)(ω) < +∞, T x (ω) otherwise.
We will mainly deal with the essential hitting time σ(x) that enjoys, unlike t(x), some good invariance properties in the survival-conditioned environment. Moreover, the difference between σ(x) and t(x) was controlled in [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF]; this will allow us to transpose to t(x) the results obtained for σ(x).

Contact process in the survival-conditioned environment. For λ ∈ Λ, we define the probability measure P λ on (Ω, F ) by

∀E ∈ F P λ (E) = P λ (E|τ 0 = +∞).
It is thus the law of the family of Poisson point processes, conditioned to the survival of the contact process starting from 0. Let then ν be a probability measure on the set of environments Λ. On the same space (Ω, F ), we define the corresponding annealed probabilities P and P by setting

∀E ∈ F P(E) = Λ P λ (E) dν(λ) and P(E) = Λ P λ (E) dν(λ).
2.2. Previous results. We recall here the results established in [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF] for the contact process in random environment.

Proposition 2.1 (Lemma 8 and Corollary 9 in [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF]). Let x, y ∈ Z d \{0}, λ ∈ Λ, A in the σ-algebra generated by σ(x), and B ∈ F . Then ∀λ ∈ Λ P λ (A ∩ ( θx ) -1 (B)) = P λ (A)P x.λ (B).

As consequences we have:

• The probability measure P is invariant under the translation θx .

• Under P λ , σ(y)• θx and σ(x) are independent. Moreover, the law of σ(y)• θx under P λ is the same as the law of σ(y) under P x.λ . • The random variables (σ(x) • ( θx ) j ) j≥0 are independent under P λ . Proposition 2.2 (Corollaries 20 and 21 in [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF]). There exist A, B, C > 0 and, for every p ≥ 1, a constant C p > 0 such that for every x ∈ Z d and every λ ∈ Λ,

E λ [σ(x) p ] ≤ C p (1 + x ) p , ( 8
) ∀t ≥ 0 ( x ≤ t) =⇒ P λ (σ(x) > Ct) ≤ A exp(-Bt 1/2 ) . (9)
Proposition 2.3 (Theorem 2 in [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF]). For every x ∈ Erg(ν), the measure-preserving dynamical system (Ω, F , P, θx ) is ergodic.

We then proved that P almost surely, for every x ∈ Z d , σ(nx) n converges to a deterministic real number µ(x). The function x → µ(x) can be extended to a norm on R d , that characterizes the asymptotic shape. Let A µ be the unit ball for µ. We define

H t = {x ∈ Z d : t(x) ≤ t}, G t = {x ∈ Z d : σ(x) ≤ t}, K ′ t = {x ∈ Z d : ∀s ≥ t ξ 0 s (x) = ξ Z d s (x)},
and we denote by Ht , Gt , K′ t their "fattened" versions:

Ht = H t + [0, 1] d , Gt = G t + [0, 1] d and K′ t = K ′ t + [0, 1] d .
We can now state the asymptotic shape result:

Proposition 2.4 (Theorem 3 in [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF]). For every ε > 0, P -a.s., for every t large enough,

(10) (1 -ε)A µ ⊂ K′ t ∩ Gt t ⊂ Gt t ⊂ Ht t ⊂ (1 + ε)A µ .
In order to prove the asymptotic shape theorem, we established exponential controls uniform in λ ∈ Λ. We set

B x r = {y ∈ Z d : y -x ∞ ≤ r},
and we write B r instead of B 0 r .

Proposition 2.5 (Proposition 5 in [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF]). There exist A, B, M, c, ρ > 0 such that for every λ ∈ Λ, for every y ∈ Z d , for every t ≥ 0

P λ (τ 0 = +∞) ≥ ρ, (11)
P λ (H 0 t ⊂ B Mt ) ≤ A exp(-Bt), (12)
P λ (t < τ 0 < +∞) ≤ A exp(-Bt), ( 13
)
P λ t 0 (y) ≥ y c + t, τ 0 = +∞ ≤ A exp(-Bt), (14)
P λ (0 ∈ K ′ t , τ 0 = +∞) ≤ A exp(-Bt). ( 15
)
Lemma 2.6. There exist A, B, C > 0 such that for every x ∈ Z d and every λ ∈ Λ, ( 16)

∀t ≥ 0 ( x ≤ t) =⇒ P λ (t ′ (x) > Ct) ≤ A exp(-Bt 1/2 ) .
Proof. For every λ ∈ Λ, for every x ∈ Z d ,

P λ (t ′ (x) > σ(x) + s) = P λ (x ∈ K ′ σ(x)+s ∩ G σ(x)+s ) = P λ (x ∈ K ′ σ(x)+s ) ≤ P λ (x ∈ x + (K ′ s ) • θx ) = P x.λ (0 ∈ K ′ s ) ≤ A exp(-Bs), (17)
with ( 11) and [START_REF] Hammersley | Postulates for subadditive processes[END_REF]. With [START_REF] Garet | Large deviations for the chemical distance in supercritical Bernoulli percolation[END_REF], this estimate gives the announced result.

2.

3. An abstract restart procedure. We formalize here the restart procedure for Markov chains. Let E be the state space where our Markov chains (X x n ) n≥0 evolve, x ∈ E being the starting point of the chain. We suppose that we have on our disposal a set Ω, an update function f : E × Ω → E, and a probability measure ν on Ω such that on the probability space (Ω, F , P) = ( ΩN * , B( ΩN * ), ν ⊗N * ), endowed with the natural filtering (F n ) n≥0 given by F n = σ(ω → ω k : k ≤ n), the chains (X x n ) n≥0 starting from the different states enjoy the following representation:

X x 0 (ω) = x X x n+1 (ω) = f (X x n (ω), ω n+1
). As usual, we define θ : Ω → Ω which maps ω = (ω n ) n≥1 to θω = (ω n+1 ) n≥1 . We assume that for each x ∈ E, we have defined a (F n ) n≥0 -adapted stopping time T x , a F T x -measurable function G x and a F -measurable function F x . Now, we are interested in the following quantities:

T x 0 = 0 and T x k+1 = +∞ if T x k = +∞ T x k + T x k (θ T x k ) with x k = X x θ T x k otherwise; K x = inf{k ≥ 0 : T x k+1 = +∞}; M x = K x -1 k=0 G x k (θ T x k ) + F X x K (θ T x K ).
We wish to control the exponential moments of the M x 's with the help of exponential bounds for G x and F x . In numerous applications to directed percolation or to the contact process, T x is the extinction time of the process (or of some embedded process) starting from the smallest point (in lexicographic order) in the configuration x.

Lemma 2.7 (Lemma 4.1 in [START_REF] Garet | Growth of a population of bacteria in a dynamical hostile environment[END_REF]). We suppose that there exist real numbers

A > 0, c < 1, p > 0, β > 0 such that the real-valued functions (G x ) x∈E , (F x ) x∈E defined above satisfy ∀x ∈ E    G(x) = E[exp(βG x )1 1 {T x <+∞} ] ≤ c; F(x) = E[1 1 {T x =+∞} exp(βF x )] ≤ A; T(x) = P(T x = +∞) ≥ p.
Then, for each x ∈ E, K x is P-almost surely finite and

E[exp(βM x )] ≤ A 1 -c < +∞.
2.4. Oriented percolation. We work, for d ≥ 1, on the following graph:

• The set of sites is

V d+1 = {(z, n) ∈ Z d × N}. • We put an oriented edge from (z 1 , n 1 ) to (z 2 , n 2 ) if and only if n 2 = n 1 + 1 and z 2 -z 1 1 ≤ 1; the set of these edges is denoted by -→ E d+1 alt . Define -→ E d in the following way: in -→ E d , there is an oriented edge between two points z 1 and z 2 in Z d if and only if z 1 -z 2 1 ≤ 1. The oriented edge in -→ E d+1 alt from (z 1 , n 1 ) to (z 2 , n 2 ) can be identified with the couple ((z 1 , z 2 ), n 2 ) ∈ -→ E d × N * . Thus, we identify -→ E d+1 alt and -→ E d × N * . We consider Ω = {0, 1} - → E d+1
alt endowed with its Borel σ-algebra: the edges e such that ω e = 1 are said to be open, the other ones are closed. For v, w in Z d × N, we denote by v → w the existence of an oriented path from v to w composed of open edges. We denote by -→ p c alt (d + 1) the critical parameter for the Bernoulli oriented percolation on this graph (i.e. all edges are independently open with probability p). We set, for n ∈ N and (x, 0) ∈ V d+1 , ξx

n = {y ∈ Z : (x, 0) → (y, n)}, τ x = max{n ∈ N : ξx n = ∅}.
We recall results from [START_REF] Garet | Growth of a population of bacteria in a dynamical hostile environment[END_REF] for a class C d (M, q) of dependent oriented percolation models on this graph. The parameter M controls the range of the dependence while the parameter q controls the probability for an edge to be open. Definition 2.8 (Class C d (M, q)). Let d ≥ 1 be fixed. Let M be a positive integer and q ∈ (0, 1).

Let (Ω, F , P) be a probability space endowed with a filtration (G n ) n≥0 . We assume that, on this probability space, a random field

(W n e ) e∈ - → E d ,n≥1 taking its values in {0, 1} is defined.

This field gives the states -open or closed -of the edges in

-→ E d+1 alt . We say that the law of the field

(W n e ) e∈ - → E d ,n≥1 is in C d (M, q) if it satisfies the two following conditions. • ∀n ≥ 1, ∀e ∈ -→ E d W n e ∈ G n ; • ∀n ≥ 0, ∀e ∈ -→ E d P[W n+1 e = 1|G n ∨ σ(W n+1 f , d(e, f ) ≥ M )] ≥ q, where σ(W n+1 f , d(e, f ) ≥ M ) is the σ-field generated by the random variables W n+1 f , with d(e, f ) ≥ M . Note that if 0 ≤ q ≤ q ′ ≤ 1, we have C d (M, q ′ ) ⊂ C d (M, q).
We can control the probability of survival and also the lifetime for these dependent oriented percolations. Proposition 2.9 (Corollary 3.1 in [START_REF] Garet | Growth of a population of bacteria in a dynamical hostile environment[END_REF]). Let ε > 0 and M > 1. There exist β > 0 and q < 1 such that for each χ

∈ C d (M, q), ∀x ∈ Z d E χ [1 1 {τ x <+∞} exp(β τ x )] ≤ ε and χ(τ x = +∞) ≥ 1 -ε. A point (y, k) ∈ Z d × N such that (x, 0) → (y, k) → ∞ is called an immortal descendant of x.
We will need estimates on the density of immortal descendants of x above some given point y in oriented dependent percolation. So we define

Ḡ(x, y) = {k ∈ N (x, 0) → (y, k) → ∞}, γ(θ, x, y) = inf{n ∈ N : ∀k ≥ n |{0, . . . , k} ∩ Ḡ(x, y)| ≥ θk}.
Proposition 2.10 (Corollary 3.3 in [START_REF] Garet | Growth of a population of bacteria in a dynamical hostile environment[END_REF]). Let M > 1. There exist q 0 < 1 and positive constants A, B, θ, α such that for each χ ∈ C d (M, q 0 ), we have

∀x, y ∈ Z d ∀n ≥ 0 χ(+∞ > γ(θ, x, y) > α x -y 1 + n) ≤ Ae -Bn .

Quenched upper large deviations

The aim is now to prove the quenched upper large deviations of Theorem 1.1. In order to exploit the subadditivity, we show that σ(x) admits exponential moments uniformly in λ ∈ Λ:

Theorem 3.1. There exist positive constants γ 1 , β 1 such that (18) ∀x ∈ Z d ∀λ ∈ Λ E λ (e γ1σ(x) ) ≤ e β1 x 1 .
As an immediate consequence, we get Corollary 3.2. There exist positive constants A, B, c, such that for each λ ∈ Λ, each x ∈ Z d and every t ≥ 0

P λ t ′ (x) ≥ x c + t ≤ A exp(-Bt).
Proof.

P λ t ′ (x) ≥ x c + t ≤ P λ σ(x) ≥ x c + t/2 + P λ (t ′ (x) -σ(x) ≥ t/2).
The second term is controlled by Inequality [START_REF] Harris | Additive set-valued Markov processes and graphical methods[END_REF] and Theorem 3.1 gives the desired result with c = γ1 β1 . The rest of this section is organized as follows. We first prove how the subadditive properties and the existence of exponential moments for σ given by Theorem 3.1 imply the large deviations inequalities of Theorem 1.1. Next we show how Theorem 1.2 gives Theorem 3.1. Finally, the last (and most important) part will be devoted to the proof of Theorem 1.2.

3.1. Proof of Theorem 1.1 from Theorem 3.1. Let ε > 0. Let β 1 and γ 1 be the constants given by [START_REF] Kesten | Aspects of first passage percolation[END_REF], and let [START_REF] Kingman | Subadditive ergodic theory[END_REF] C > 2β 1 /γ 1 .

Theorem 2.4 gives the almost sure convergence of σ(x)/µ(x) to 1 when x tends to +∞, and Proposition 2.2 ensures that the family (σ(x)/µ(x)) x∈Z d is bounded in L 2 (P), therefore uniformly integrable: then the convergence also holds in L 1 (P).

Let then M 0 be such that [START_REF] Kingman | Subadditive processes[END_REF] (µ(x)

≥ M 0 ) ⇒ E(σ(x)) µ(x) ≤ 1 + ε/8.
We assumed that {ay :

a ∈ R + , y ∈ Erg(ν)} is dense in R d . Its range by x → x µ(x) is therefore dense in {x ∈ R d : µ(x) = 1}, thus the set { y µ(y) : y ∈ Erg(ν), µ(y) ≥ M 0 } is also dense in {x ∈ R d : µ(x) = 1}. By a compactness argument, one can find a finite subset F in { y µ(y) : y ∈ Erg(ν), µ(y) ≥ M 0 } such that ∀x ∈ R d such that µ(x) = 1 ∃y ∈ F, y µ(y) -x 1 ≤ ε/C.
We let M = max{µ(y) : y ∈ F }.

For y ∈ F , note σ(y) = σ(y) -(1 + ε 4 )µ(y). Since, with [START_REF] Kesten | Aspects of first passage percolation[END_REF], σ(y) admits exponential moments, the asymptotics E[e tσ(y) ] = 1 + tE[σ(y)] + o(t) holds in the neighborood of 0. Since E[σ(y)] < 0, we have E[e tσ(y) ] < 1 when t is small enough. Since F is finite, we can find some constants α > 0 and c α < 1 such that

∀y ∈ F E[exp(α(σ(y) -(1 + ε/4)))] ≤ c α .
Let x ∈ Z d . We associate to x a point y ∈ F and an integer n such that

(21) x µ(x) - y µ(y) 1 ≤ ε C and n - µ(x) µ(y) ≤ 1.
By the definition of t(x), for each λ ∈ Λ, we have

P λ (t(x) ≥ (1 + ε)µ(x)) ≤ P λ n-1 i=0 σ(y) • θi y + σ(x -ny) • θn y ≥ (1 + ε)µ(x) ≤ P λ n-1 i=0 σ(y) • θi y ≥ 1 + ε 2 µ(x) + P λ σ(x -ny) • θn y ≥ ε 2 µ(x) . (22)
Let first consider the first term in [START_REF] Rossignol | Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation[END_REF]. With Proposition 2.1 and estimate [START_REF] Kesten | Aspects of first passage percolation[END_REF], it follows that

P λ σ(x -ny) • θn y ≥ ε 2 µ(x) = P ny.λ σ(x -ny) ≥ ε 2 µ(x) ≤ exp - γ 1 εµ(x) 2 E ny.λ (exp(γ 1 σ(x -ny))) ≤ exp - γ 1 εµ(x) 2 exp(β 1 x -ny 1 ).
Our choices [START_REF] Liggett | Domination by product measures[END_REF] for y and n and the definition of M ensure that

x -ny 1 ≤ x - µ(x) µ(y) y 1 + µ(x) µ(y) -n y 1 ≤ εµ(x) C + M.
Our choice [START_REF] Kingman | Subadditive ergodic theory[END_REF] for C gives then the existence of two positive constants A 1 and B 1 such that for each λ ∈ Λ and each x ∈ Z d ,

P λ σ(x -ny) ≥ ε 2 µ(x) ≤ A 1 exp(-B 1 x ).
Let us move to the first term of [START_REF] Rossignol | Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation[END_REF]. Our choices [START_REF] Liggett | Domination by product measures[END_REF] for y and n ensure that

µ(x) nµ(y) -1 ≤ 1 n ≤ µ(x) M -1 -1
.

Then, we can find T sufficiently large to have, for µ(x) ≥ T , that

µ(x) nµ(y) ≥ 1 + ε/4 1 + ε/2 .
Suppose now that µ(x) ≥ T . Proposition 2.1 ensures that the variables σ(y) • θi y are independent under P λ and moreover that the law of σ(y)• θi y under P λ coincides with the law of σ(y) under P iy.λ : thus

P λ n-1 i=0 σ(y) • θi y ≥ 1 + ε 2 µ(x) ≤ P λ n-1 i=0 σ(y) • θi y ≥ (1 + ε 4 )nµ(y) ≤ P λ n-1 i=0 exp α[σ(y) • θi y -(1 + ε 4 )µ(y)] ≥ 1 ≤ n-1 i=0 E iy.λ exp α(σ(y) -(1 + ε 4 )µ(y)) .
Applying the Ergodic Theorem to the system (Λ, B(Λ), ν, y.) and to the function λ → log E λ (exp[α(σ(y) -(1 + ε/4)µ(y))]), we get that for ν-almost every λ and for each y ∈ F ,

lim n→+∞ 1 n log P λ 1 nµ(y) n-1 i=0 σ(y) • θi y ≥ 1 + ε/4 ≤ Λ log E λ (exp[α(σ(y) -(1 + ε/4)µ(y))]) dν(λ) ≤ log Λ E λ (exp[α(σ(y) -(1 + ε/4)µ(y))]) dν(λ) ≤ log c α < 0.
Using the norm equivalence theorem and noting that the choices [START_REF] Liggett | Domination by product measures[END_REF] for n and y ensure that

n µ(x) ≤ 1 M + 1 T , we deduce that lim x →+∞ log P λ (t(x) ≥ µ(x)(1 + ε)) x ≤ -C ε , with C ε = max(-log c α , B 1
(t ′ (x) > T + εµ(x)) ≤ P x.λ (0 ∈ K ′ εµ(x) ) ≤ A exp(-Bµ(x)). Thus, since P λ (t ′ (x) > (1 + 2ε)µ(x)) ≤ P λ (T > (1 + ε)µ(x)) + P λ (t ′ (x) > T + εµ(x))
and T has already been controlled, inequality (2) follows.

Let us prove inequality (3) of Theorem 1.1. Since t → K ′ t ∩ H t is non-decreasing, it is sufficient to prove that there exist constants A, B > 0 such thay Bn). The proof of the last inequality is classic. For points that have a small norm, we use inequality [START_REF] Grimmett | First-passage percolation, network flows and electrical resistances[END_REF] and Corollary 3.2; for the other ones, we use inequalities ( 1) and (2).

∀n ∈ N P((1 -ε)nA µ ⊂ K ′ n ∩ H n ) ≤ A exp(-

3.2.

Proof of Theorem 3.1 from Theorem 1.2. Theorem 1.2 ensures that with a probability exceeding 1 -A exp(-Bt), the Lebesgue measure of the times

s ≤ C x + t when (0, 0) → (x, s) → ∞ is at least θt. If σ(x) ≥ C x ∞ + t,
it means that all these times are ignored by the recursive construction of σ(x):

those times necessarily belong to

K(x)-1 ∪ i=1 [u k (x), v k (x)]
. Thus, we choose θ, C as in Theorem 1.2 and get

P λ (σ(x) ≥ C x + t) ≤ P λ {s ≤ C x + t : (0, 0) → (x, s) → ∞} ⊂ K(x)-1 ∪ i=1 [u k (x), v k (x)] ≤ P λ (Leb({s ≤ C x + t : (0, 0) → (x, s) → ∞}) ≤ θt) +P λ K(x)-1 i=1 (v k (x) -u k (x)) > θt .
Lemma 1.2 allows to control the first term. To control the second one with a Markov inequality, it is sufficient to prove the existence of exponential moments for

K(x)-1 i=1 (v k (x) -u k (x)).
To do so, we apply the abstract restart Lemma 2.7. We define, for each subset B in Z d , F B = 0 and

T B = inf{t > τ x : x ∈ ξ B t }, G B = τ x .
Estimate [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF] ensures that for each λ ∈ Λ,

P λ (T B = +∞) ≥ P λ (τ x = +∞) ≥ ρ > 0,
and estimate [START_REF] Grimmett | First-passage percolation, network flows and electrical resistances[END_REF] ensures the existence of α > 0 and c < 1 -that do not depend on B -such that for each λ ∈ Λ,

E λ [exp(αG B )1 1 {T B <+∞} ] ≤ E λ [exp(ατ x )1 1 {τ x <+∞} ] = E x.λ [exp(ατ 0 )1 1 {τ 0 <+∞} ] ≤ c.
Then, with the notation of Lemma 2.7, we have

E λ exp α K(x)-1 i=1 (v k (x) -u k (x)) = E λ exp α K(x)-1 i=0 τ x • T k ≤ 1 1 -c .
To conclude, we note, using [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF], that E λ (.) ≤ E λ (.)/ρ.

3.3.

Proof of Theorem 1.2. We will include in the contact process a block event percolation: sites will correspond to large blocks in Z d × [0, ∞), and the opening of the bonds will depend of the occuring of good events that we define now.

A( n0 , u, x 0 , x 1 ) = A C1,M1 I,L,δ ( n0 , u, x 0 , x 1 ) =          ∃t ∈ [0, C 1 L -δ] 2L n0 + x 1 ∈ ξ 2L n0+x0+[-I,I] d t ω 2L n0+x1 ([t, t + δ]) = 0 ∃s ∈ 2L( n0 + u) + [-L, L] d s + [-I, I] d ⊂ ξ 2L n0+x1 C1L-t • θ t t∈[0,C1L] ξ 2L n0+[-L-I,I+L] d t ⊂ 2L n0 + [-M 1 L, M 1 L] d          .
We let then T = C 1 L. When the event A( n0 , u, x 0 , x 1 ) occurs, we denote by s( n0 , u, x 0 , x 1 ) a point s satisfying the last condition that defines the event. Else, we let s( n0 , u, x 0 , x 1 ) = ∞.

If this event occurs, then:

• Starting from an area of size I centered at a starting point 2L n0 + x 0 in the box with spatial coordinate n0 , the process at time T colonizes an area of size I centered around the exit point 2L( n0 + u) + s( n0 , u, x 0 , x 1 ) in the box with spatial coordinate n0 + u. • Moreover, the point 2L n0 + x 1 is occupied between time 0 and time T in a time interval with duration at least δ. For each ε > 0, we can choose, in that specific order, two integers I ≤ L large enough and δ > 0 small enough such that for every λ ∈ Λ, n0 ∈ Z d , and each

u ∈ Z d with u 1 ≤ 1, ∀x 0 , x 1 ∈ [-L, L[ d P λ (A( n0 , u, x 0 , x 1 )) ≥ 1 -ε.
Moreover, as soon as n0 -n′ 0 ∞ ≥ 2M 1 + 1, for every u, u ′ , x 0 , x ′ 0 , x 1 , the events A( n0 , u, x 0 , x 1 ) and A( n0 ′ , u ′ , x ′ 0 , x 1 ) are independent. Proof. Let us first note that P λ (A( n0 , u, x 0 , x 1 )) = P 2L n0.λ (A(0, u, x 0 , x 1 )), which permits to assume that n0 = 0. Let ε > 0 be fixed. We first choose I large enough to have [START_REF] Rossignol | Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation[END_REF] ∀x Let then L 1 be such that

P λmin (∃n ∈ [0, L]; ∀y ∈ ne 1 + [-I, I] d , τ y = +∞) > 1 - ε ′ 12 P λmin (τ 0 = +∞).
By a time-reversal argument, we have for each t > 0,

P λmin (∃n ∈ [0, L] : ne 1 + [-I, I] d ⊂ ξ Z d t ) = P λmin (∃n ∈ [0, L] : ∀y ∈ ne 1 + [-I, I] d , τ y ≥ t) > 1 - ε ′ 12 P λmin (τ 0 = +∞).
We have for each t ≥ 0 and each λ ∈ Λ:

P x1.λ (τ 0 = +∞, ∀n ∈ [0, L], 2Lu -x 1 + ne 1 + [-I, I] ⊂ ξ 0 t ) ≤ P x1.λ (∀n ∈ [0, L], 2Lu -x 1 + ne 1 + [-I, I] ⊂ ξ Z d t ) +P x1.λ (τ 0 = +∞, [-(I + 4L), (I + 4L)] d ⊂ K ′ t ) ≤ P λmin (∀n ∈ [0, L], ne 1 + [-I, I] ⊂ ξ Z d t ) +P x1.λ (τ 0 = +∞, [-(4L + I), (4L + I)] d ⊂ K ′ t )
. Let C > 0 be large enough to satisfy properties [START_REF] Garet | Large deviations for the chemical distance in supercritical Bernoulli percolation[END_REF] and [START_REF] Hammersley | First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory[END_REF]. Then, with [START_REF] Hammersley | First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory[END_REF], we can find L 2 ≥ L 1 such that for L ≥ L 2 and t ≥ 5CL, we have

P x1.λ (∃n ∈ [0, L]; 2Lu -x 1 + ne 1 + [-I, I] ⊂ ξ 0 t ) ≥ 1 -ε ′ /6.
Let δ > 0 such that 1 -e -δ ≤ P λmin (τ 0 = +∞)ε ′ /6 and δ < 5CL: if we let

F t = ω 0 ([0, δ]) = 0; ∃n ∈ [0, L], 2Lu -x 1 + ne 1 + [-I, I] ⊂ ξ 0 t ,
we also have, for each λ ∈ Λ and each t ≥ 5CL, that

P x1.λ (F t ) ≥ 1 -ε ′ /3.
Then, with Proposition 2.1, one deduces that if y ∈ x + [-I, I] d , then

P y.λ (σ(x 1 -y) ≤ 4CL, θ-1 x1-y (F 9CL-σ(x1-y) )) ≥ P y.λ (σ(x 1 -y) ≤ 4CL)(1 -ε ′ /3).
Considering estimate (9), we can choose L 3 ≥ L 2 such that for L ≥ L 3 , we have

P y.λ (σ(x 1 -y) ≤ 4CL, θ-1 x1-y (F 9CL-σ(x1-y) )) ≥ 1 -ε ′ /2.
Let C 1 = 9C. With [START_REF] Rossignol | Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation[END_REF] and the definition of ε ′ , we get

P λ   ∃t ∈ [0, C 1 L -δ] : x 1 ∈ ξ x0+[-I,I] d t ω x1 ([t, t + δ]) = 0 ∃s ∈ 2Lu + [-L, L] d s + [-I, I] d ⊂ ξ x1 C1L-t • θ t   ≥ 1 -3ε/4.
Finally, one takes for M the constant given by equation ( 12) and lets M 1 = M C 1 + 2. With [START_REF] Garet | Growth of a population of bacteria in a dynamical hostile environment[END_REF], we can find L ≥ L 3 sufficiently large to have for each λ ∈ Λ:

(24) P λmax   0≤t≤C1L ξ [-L-I,L+I] d t ⊂ [-M 1 L, M 1 L] d   ≥ 1 -ε/4;
this fixes the integer L.

The local dependence of the events comes from the third condition in their definition. This concludes the proof of the lemma.

Dependent macroscopic percolation.

We fix C 1 , M 1 given by Lemma 3.3. We choose I ∈ N * , L ∈ N * and δ > 0 such that I ≤ L and δ < C 1 L and we let T = C 1 L.

Let x in Z d be fixed. We write x = 2L[x] + {x}, with {x} ∈ [-L, L[ d and [x] ∈ Z d . We will first, from the events defined in the preceding subsection, build a field ( x W n ( k,u) ) n≥0, k∈Z d , u 1≤1 . The idea is to construct a macroscopic oriented percolation on the bonds of -→ E d × N * , looking for the realizations, floor by floor, of translates of good events of type A(.). We start from an area centered at 0 in the box with coordinate 0; for each u such that u 1 ≤ 1, say that the bond between ( 0, 0) and (u, 1) is open if A( 0, u, 0, {x}) holds; in that case we obtain an infected square centered at the exit point s( 0, u, 0, {x}); all bonds in this floor that are issued from another point than 0 are open, with fictive exit points equal to ∞. Then we move to the upper floor: for a box (ȳ, 1), look if it contains exit points of bonds that were open at the preceding step. If it is the case, we choose one of these, denoted by d x 1 (ȳ), open the bond between (ȳ, 1) and (ȳ + u, 2) if A(ȳ, u, d x 1 (ȳ), {x}) • θ T happens and close it otherwise; in the other case we open all bonds issued from that box, and so on for every floor.

Precisely, we let d x 0 ( 0) = 0 and also d x 0 (ȳ) = +∞ for every ȳ ∈ Z d that differs from 0. Then, for each ȳ ∈ Z d , each u ∈ Z d such that u 1 ≤ 1 and for each n ≥ 0, we recursively define:

• If d x n (ȳ) = +∞, x W n (ȳ,u) = 1. • Otherwise, x W n (ȳ,u) = 1 1 A(ȳ,u,d x n (ȳ),{x}) • θ nT , d x n+1 (ȳ) = min{s(ȳ + u, -u, d x n (ȳ + u), {x}) • θ nT : u 1 ≤ 1, d x n (ȳ + u) = +∞}.
Recall that the definition of the function s has been given with the one of a good event in the preceding subsection. Then, d x n+1 (ȳ) represents the relative position of the entrance area for the x W n+1 (ȳ,u) 's, with u 1 ≤ 1. We may have several candidates, that are the relative positions of the exit areas of the x W n (ȳ+u,-u) 's; the min only plays the role of a choice function.

We thus obtain an oriented percolation process. Among open bonds, only those corresponding to the realization of good events are relevant for the underlying contact process. Let us note however that the percolation cluster starting at 0 only contains bonds that correspond to the propagation of the contact process. Lemma 3.4. Again, we work with C 1 , M 1 given by Lemma 3.3. For each q < 1, we can choose parameters I, L, δ such that for each λ ∈ Λ, and each x ∈ Z d , the law of

( x W n e ) (e,n)∈ - → E d ×N * under P λ is in C(2M 1 + 1, q).
Proof. For each n ∈ N, let G n = F nT , with T = C 1 L. Let us note that, for each x, k ∈ Z d and n ≥ 1, the quantity d x n (k) is G n -mesurable, and so does x W n (k,u) . Lemma 3.3 ensures that the events A( k, u, x 0 , {x}) and A( l, v, x ′ 0 , {x}) are independent as soon as kl 1 ≥ 2M 1 + 1; so we deduce that, conditionally to G n , the random variables x W n+1 (k,u) and x W n+1 (l,v) are independent as soon as k-l 1 ≥ 2M 1 +1. Let now x, k ∈ Z d , n ≥ 0 and u ∈ Z d such that u 1 ≤ 1:

E λ [ x W n+1 (k,u) |G n ∨ σ( x W n+1 (l,v) , v 1 ≤ 1, l -k 1 ≥ 2M 1 + 1)] = E λ [ x W n+1 (k,u) |G n ] = 1 1 {d x n (k)=+∞} + 1 1 {d x n (k)<+∞} P λ [ x W n+1 (k,u) = 1|d x n (k) < +∞] = 1 1 {d x n (k)=+∞} + 1 1 {d x n (k)<+∞} P λ [A(k, u, d x n (k), {x})]
. With Lemma 3.3, we can choose integers I < L and δ > 0 in such a way that

E λ [ x W n+1 (k,u) |G n ∨ σ( x W n+1 (l,v) , v 1 ≤ 1, l -k 1 ≥ 2M 1 + 1)] ≥ q.
This concludes the proof of the lemma.

For this percolation process, we denote by τ k and γ(θ, k, l) the lifetime starting from k and the essential hitting times of l starting from k in the dependent oriented percolation induced by the Bernoulli random field ( x W n e ) (e,n)∈ - → E d ×N * . Lemma 3.5. We can choose the parameters I, L, δ such that the following holds:

• ∀λ ∈ Λ P λ (τ 0 = +∞) ≥ 1 2 . • ∀λ ∈ Λ ϕ(λ) = E λ [e ατ 0 1 1 {τ 0 <+∞} ] ≤ 1/2
• there exist strictly positive constants α 0 > 0, C such that for every x, y ∈ Z d ∀α ∈ [0, α 0 ] ∀λ ∈ Λ ℓ(λ, α, x, y) = E λ [1 1 {τ x =+∞} e αγ(θ,x,y)) ] ≤ 2e Cα x-y .

Proof. By Lemma 2.9, we know that there exist q < 1 and α > 0 such that we have

E[e ατ 0 1 1 {τ 0<+∞} ] ≤ 1/2
for each field in C(2M 1 + 1, q). By Lemma 3.4, we can choose I, L, δ such that

( x W n e ) (e,n)∈ - → E d ×N * ∈ C(2M 1 + 1, q)
, which gives the two first points. Then, from Lemma 2.10, we get constants A, B, C such that for every x, y ∈ Z d , every n ≥ 0 and each λ ∈ Λ, we have

P λ (+∞ > γ(θ, x, y) > C x -y 1 + n) ≤ Ae -Bn .
We can then find B ′ > 0 independent from x and λ such that the Exponential law with parameter B ′ dominates (γ(θ, x, y) -C x -y 1 )1 1 {γ(θ,x,y)<+∞} . Let then α ≤ B ′ /2: we have ℓ(λ, α, x, y) = e αC x-y 1 E λ [1 1 {τ x =+∞} e α((γ(θ,x,y)-C y-x 1)) ] ≤ e αC x-y 1 B ′ B ′ -α ≤ 2e αC x-y 1 .

Proof of Theorem 1.2.

We first choose I, L, δ in order to satisfy the inequalities of Lemma 3.5, and we let T = C 1 L.

We use a restart argument. The idea is as follows: fix λ ∈ Λ and x ∈ Z d ; if the lifetime τ 0 of the contact process in random environment is infinite, then one can find by the restart procedure an instant T K such that • ξ 0 TK contains an area z + [-2L, 2L] d , which allows to activate a block oriented percolation, as defined in the previous subsection, from some z0

∈ Z d such that 2z 0 L + [-L, L] d ⊂ z + [-2L, 2L] d ,
• the block oriented percolation issued from z0 infinitely survives. Then, with Lemma 2.10, we give a lower bound for the proportion of time when x0 = [x] is occupied by descendents having themselves infinite progeny. By the definition of good events, this will allow to bound from below the measure of {t ≥ 0; (0, 0) → (x, t) → ∞} in the contact process. Indeed, recall that the definition of the event A(x 0 , u, x 0 , {x}) targets {x} and ensures that each time the site x0 = [x] is occupied in the macroscopic oriented percolation, then the contact process occupies the site 2Lx 0 + {x} = x during δ units of time.

Definition of the restart procedure. We define the following stopping times: for each non-empty subset A ⊂ Z d ,

U A =    T if ∀z ∈ Z d z + [-2L, 2L] d ⊂ ξ A T , T (1 + τ 0 • T 2x A L • θ T ) otherwise with xA = inf{ m ∈ Z d : 2 mL + [-L, L] d ⊂ ξ A T } and U ∅ = +∞.
In other words, starting from a set A, we ask if the contact process contains an area in the form 2 mL + [-L, L] d at time T , : if the answer is no, we stop, otherwise we consider the lifetime of the macroscopic percolation issued from the macroscopic site corresponding to that area. Particularly, if A = ∅ and U A = +∞, then there exists, at time T , in the contact process issued from A, an area 2x A L + [-L, L] d which is fully occupied, and such that the macroscopic oriented percolation issued from thae macroscopic site xA percolates. We then search in that infinite cluster not too large a time when the proportion of individuals living at x0 = [x] and having infinite progeny becomes sufficiently large: if A = ∅ and U A = +∞, we note

R A = R A (x) = T (1 + γ(θ, xA , x0 )) if A = ∅ and U A = +∞; 0 otherwise .
Thus, when U A = +∞, the variable R A represents the first time (in the scale of the contact process, not that of the macroscopic oriented percolation) when the proportion of individuals living at x0 = [x] and having infinite progeny becomes sufficiently large.

Estimates for the restart procedure.

Lemma 3.6. There exist constants α > 0, q > 0, c < 1, A ′ , h > 0 such that for each λ ∈ Λ, each A ⊂ Z d , and each x ∈ Z d ,

P λ (U A = +∞) ≥ q; (25) E λ [exp(αU A )1 1 {U A <+∞} ] ≤ c; (26) E λ [exp(αR A (x))1 1 {U A =+∞} ] ≤ A ′ e αh( x0 ∞ + A ∞) . ( 27
)
Proof. We easily get (25) from a stochastic comparison: for each λ ∈ Λ and each non-empty A,

P λ (U A = +∞) ≥ P λmin ([-2L, 2L] d ⊂ ξ 0 T )P(τ 0 = +∞) = q > 0. Now, if α > 0 , A ⊂ Z d is
non-empty and λ ∈ Λ, we have with Lemma 3.5,

E λ [exp(αU A )1 1 {U A <+∞} ] = e αT 1 -P λ (∃z ∈ Z d , z + [-2L, 2L] d ⊂ ξ A T ) 1 -E[e αT τ 0 1 1 {τ 0 <+∞} ] ≤ e αT 1 - 1 2 P λ (∃z ∈ Z d , z + [-2L, 2L] d ⊂ ξ A T ) ≤ e αT 1 - 1 2 P λmin (∃z ∈ Z d , z + [-2L, 2L] d ⊂ ξ A T ) = c < 1
provided that α > 0 is small enough; this proves [START_REF] Théret | Upper large deviations for the maximal flow in first-passage percolation[END_REF]. By the strong Markov property and Lemma 3.5, if we choose α > 0 small enough, then for each λ ∈ Λ,

E λ [exp(αR A )1 1 {U A =+∞} |F T ] = 1 1 {∃z∈Z d , z+[-2L,2L] d ⊂ξ A T } e αT E[exp(αT γ(θ, xA , x0 ))1 1 {τ xA =+∞} ] ≤ 2e αT exp(CαT xA -x0 ∞ ) ≤ 2e αT (1+C x0 ∞ ) exp(CαT ξ A T ∞ ). ( 28
)
We use the comparison with Richardson's model to bound the mean of the last term: let us choose the positive constants M, β such that ∀s, t ≥ 0 P λmax ( ξ 0 s ∞ ≥ M s + t) ≤ e -βt . Then, for each non-empty finite set A, each t > 0, and each λ ∈ Λ,

P λ ( ξ A T ∞ ≥ 2 A ∞ + M T + t) ≤ P λmax (max a∈A ξ a T -a ∞ ≥ A ∞ + M T + t) ≤ |A|P λmax ( ξ 0 T ∞ ≥ M T + A ∞ + t) ≤ A d ∞ e -β( A ∞+t) ≤ α ′ exp(-βt). Then, for α small enough, E λ [exp(CαT ξ A T ∞ )] ≤ e CαT (2 A ∞ +MT ) 1 + CαT α ′ β -CαT ≤ 2e CαT (2 A ∞ +MT ) . ( 29
)
Inequality [START_REF] Théret | On the small maximal flows in first passage percolation[END_REF] immediately follows from (28) and (29).

Application of the restart lemma 2.7. Let

T 0 = 0 and T k+1 = +∞ if T k = +∞ T k + U ξ T k 0 • θ T k otherwise; K = inf{k ≥ 0 : T k+1 = +∞}.
The restart lemma, applied with T . = G . = U . and F . = 0, ensures that

E λ [exp(αT K )] ≤ A ′ 1 -c .
Applying now the restart lemma with G . = 0 and F . = R . , we get that

E λ [exp(α(R ξ T K 0 • θ TK -(h x0 ∞ + ξ TK 0 ∞ )))] ≤ A ′ 1 -c .
Particularly, it holds that for each s > 0 and t > 0,

P λ (T K > s) ≤ A ′ 1 -c exp(-αs); (30) P λ R ξ T K 0 •θT K ≥ t/2, T K ≤ s, H 0 s ⊂ B 0 Ms ≤ A ′ 1 -c exp(α(h( x0 ∞ + M s) -t/2)). ( 31
)
On the event {τ 0 = +∞}, one can be sure that the contact process is non-empty at each step of the restart procedure : the restart Lemma ensures that at time T K + T , one can find some area from which the directed block percolation percolates, and, by construction, that for every

t ≥ T K + R ξ T K 0 •θT K , Leb({s ∈ [T K + T, t] : (0, 0) → (x, s) → ∞}) ≥ δθInt( t -(T K + T ) T ) ≥ δθ 2T t as soon as T K ≤ t/2 -1. Let C = 2h L . Let now be x ∈ Z d , and t ≥ C x ∞ . P λ τ 0 = +∞, Leb({s ∈ [0, t] : (0, 0) → (x, s) → ∞}) < δθ 2T t ≤ P λ (T K > t/2 -1) + P λ (T K ≤ t/2 -1, t < T K + R ξ T K 0 • θ TK ) ≤ P λ (T K > t/2 -1) + P λ (R ξ T K 0 • θ TK > t/2).
We control the first term with (30). For the second one, we take s = t 8hM :

P λ (R ξ T K 0 • θ TK > t/2) ≤ P λ (R ξ T K 0 •θT K > t/2, T K ≤ s, H 0 s ⊂ B 0 Ms ) + P λ (T K > s) + P λ (H 0 s ⊂ B 0
Ms ). We control the last two terms with (30) and [START_REF] Garet | Growth of a population of bacteria in a dynamical hostile environment[END_REF]; for the first one, we use (31):

since x0 ∞ ≤ 1 2L x ∞ + 1, P λ R ξ T K 0 •θT K > t/2, T K ≤ s, H 0 s ⊂ B 0 Ms ≤ A ′ 1 -c exp(α(h( x0 ∞ + M s) -t/2)) ≤ A ′ e αh 1 -c exp α h 2L x ∞ - t 4 - t 8
≤ A ′ e αh 1 -c exp(-αt/8),
which concludes the proof.

Lower large deviations

4.1. Duality. We have seen that the hitting times σ(nx) enjoy superconvolutive properties. In a deterministic frame, Hammersley [START_REF] Hammersley | Postulates for subadditive processes[END_REF] has proved that the superconvolutive property allows to express the large deviation functional in terms of the moments generating function, as in Chernoff's Theorem. We will see that this property also holds in an ergodic random environment. The following proof is inspired by Kingman [START_REF] Kingman | Subadditive processes[END_REF].

Proof of Theorem 1.3. Since {t(x) ≤ t, τ x • θ t(x) = +∞} ⊂ {σ(x) ≤ t} ⊂ {t(x) ≤ t}, the Markov property ensures that

P λ (t(x) ≤ t)P λ (τ x = +∞) ≤ P λ (σ(x) ≤ t) ≤ P λ (t(x) ≤ t).
Thus, letting R = -log P λmin (τ 0 = +∞), we have

(32) -log(P λ (t(x) ≤ t)) ≤ -log(P λ (σ(x) ≤ t)) ≤ -log(P λ (t(x) ≤ t)) + R.
Similarly,

E λ [e -t(x) ] ≥ E λ [e -θσ(x) ] ≥ E λ [1 1 {τ x •θ t(x) =+∞} e -θt(x) ] = E λ [e -θt(x) ]P λ (τ x = +∞), which leads to (33) -log E λ [e -t(x) ] ≤ -log E λ [e -θσ(x) ] ≤ -log E λ [e -θt(x) ] + R.
Then, having a large deviation principle in mind, working with σ or t does not matter. We will work here with σ, which gives simpler relations. We know that

(34) t((n + p)x) ≤ σ(nx) + σ(px) • θnx ,
that σ(nx) and σ(px)• θnx are independent under P λ , and that the law of σ(px)• θnx under P λ is the law of σ(px) under P nx.λ (see Proposition 2.1). Then (35)

-log P λ (t((n + p)x) ≤ nu + pv) ≤ -log P λ (σ(nx) ≤ nu) -log P nx.λ (σ(px) ≤ pv). Let g x n (λ, u) = -log P λ (σ(nx) ≤ nu) + R and G x n (u) = Λ g x n (λ, u) dν(λ).
Inequalities (32) and (35) ensure that (36) g x n+p (λ, u) ≤ g x n (λ, u) + g x p (T n x λ, u). Since 0 ≤ g x 1 (λ, u) ≤ -log P λmin (σ(x) ≤ u) + R < +∞, Kingman's subadditive ergodic theorem ensures that

g x n (u,λ) n converges to Ψ x (u) = inf n≥1 1 n G x n (u) = lim n→+∞ 1 n G x n (u).
for ν-almost every λ. Note that (36) ensures that for every n, p ∈ N and every u, v > 0,

Ψ x nu + pv n + p ≤ 1 n + p G x n+p nu + pv n + p ≤ n n + p G x n (u) n + p n + p G x p (v) p . Let α ∈]0, 1[. Since Ψ x is non-increasing, considering some sequence n k , p k such n k n k +p k tends to α from above, we get Ψ x (αu + (1 -α)v) ≤ αΨ x (u) + (1 -α)Ψ x (v). So Ψ is convex. Similarly, let h x n (λ, θ) = -log E λ [e -θσ(nx) ] + R and H x n (θ) = h x n (λ, θ) dν(λ).
As previously, with (33) and the subadditive relation (34), we have

E λ [e -θσ((n+p)x) ] ≥ e -R E λ [e -θt((n+p)x) ] ≥ e -R E λ [e -θ(σ(nx)+σ(px)• θnx) ] = e -R E λ [e -θσ(nx) ]E λ [e -θσ(px) ],
and then the inequality (x) ] < +∞, Kingman's subadditive ergodic theorem ensures that for ν-almost every λ,

h x n+p (λ, θ) ≤ h x n (λ, θ) + h x p (T n x λ, θ). Since 0 ≤ h x 1 (λ, θ) ≤ -log E λmin [e -θσ
h x,θ n (λ,θ) n converges to K x (θ) = inf n≥1 1 n H x n (θ) = lim n→+∞ 1 n H x n (θ).
Let now θ ≥ 0 and u > 0. By the Markov inequality, we observe that

P λ (σ(nx) ≤ nu) ≤ e θnu E λ [e -θσ(nx) ], i.e. -g x n (., u) ≤ θnu -h x,θ n (., u), i.e. G x n (u) ≥ -θnu + H n (θ), i.e. Ψ x (u) ≥ -θu + K x (θ).
Thus, we easily get

∀u > 0 Ψ x (u) ≥ sup θ≥0 (K x (θ) -θu), (37)
∀θ > 0 K x (θ) ≤ inf u>0 (Ψ x (u) + θu). (38)
It remains to prove both reversed inequalities. Let us first prove

(39) ∀θ > 0 K x (θ) ≥ inf u>0 {Ψ x (u) + θu}.
Let θ > 0. Define M = inf u>0 {Ψ x (u) + θu} and note that for each u and each

integer n G x n (u) + nθu ≥ nΨ x (u) + nθu ≥ nM. Fix ε > 0. Define E n,ε = {λ : g x,u n (λ) ≥ G x n (u) -nε}.
We have

H x n (θ) ≥ En,ε h x n (θ) dν(λ) = En,ε (R -log E λ [e -θσ(nx) )] dν(λ) = En,ε -log +∞ 0
nθe -θnu e -R P λ (σ(nx) < nu) du dν(λ).

For every λ ∈ E n,ε and b > 0, one has

+∞ 0 nθe -θnu e -R P λ (σ(nx) < nu) du ≤ e -θnb + b 0 nθe -θnu e -R P λ (σ(nx) < nu) du ≤ e -θnb + b 0 nθe -θnu e -g x,u n (λ) du ≤ e -θnb + b 0 nθe -θnu e -G x n (u)+nε du ≤ e -θnb + nθbe -n(M-ε) ≤ (nM + 1)e -n(M-ε) with b = M/θ.
Finally,

H x n (θ) n ≥ ν(E n,ε ) - log(1 + nM ) n + M -ε .
Since ν(E n,ε ) tends to 1 when n goes to infinity, one deduces that

K x (θ) = lim 1 n H x n (θ) ≥ M -ε.
Letting ε tend to 0, we get (39).

Let us finally prove

(40) ∀u > 0 Ψ x (u) ≤ sup θ≥0 (K x (θ) -θu).
Let u > 0. It is sufficient to prove that there exists θ u ≥ 0, with Ψ x (u) ≤ -θ u u + K x (θ u ). Since Ψ x is convex and non-increasing, there exists a slope

-θ u ≤ 0 such that Ψ x (v) ≥ Ψ x (u) -θ u (v -u). Then K x (θ u ) = inf v>0 {Ψ x (v) + θ u v} ≥ inf v>0 {Ψ x (u) -θ u (v -u) + θ u v} ≥ Ψ x (u) + θ u u,
which completes the proof of (40) and of the reciprocity formulas. The function -K x (-θ) corresponds to Ψ x in the Fenchel-Legendre duality: therefore, it is convex. Particularly, the functions Ψ x and K x are continuous on ]0, +∞[. By the definition of Ψ x and K x , there exists Λ ′ ⊂ Λ with ν(Λ ′ ) = 1 and such that for each u ∈ Q ∩ (0, +∞) and each θ ∈ Q ∩ [0, +∞), we have

lim n→+∞ - 1 n log P λ (σ(nx) ≤ nu) = Ψ x (u),
and lim

n→+∞ - 1 n log E λ e -θσ(nx) = K x (θ).
Since the functions θ → h x,θ n and u → h x,u n are monotonic and their limits Ψ x and K x are continuous, it is easy to check that the convergences also hold for every λ ∈ Λ ′ , u > 0 and θ ≥ 0. 4.2. Lower large deviations. We prove here Theorem 1.4. Remember that P(.) = Λ P λ (.) dν(λ). The main step is actually to prove the following:

- µ , C + µ > 0 such that (41) ∀z ∈ R d C - µ z ∞ ≤ µ(z) ≤ C + µ z ∞ .
Let α, L, N, ε > 0. We define the following event, relatively to the space-time box

B N = B N (0, 0) = [-N, N ] d × [0, 2N ]: A α,L,N,ε = ∀(x 0 , t 0 ) ∈ B N ξ x0 αLN -t0 • θ t0 ⊂ x 0 + (1 + ε)(αLN -t 0 )A µ ∩ ∀(x 0 , t 0 ) ∈ B N ∪ 0≤s≤αLN -t0 ξ x0 s • θ t0 ⊂] -LN, LN [ d .
The first part of the event ensures that the descendants, at time αLN , of any point (x 0 , t 0 ) in the box B N are included in x 0 + (1 + ε)(αLN )A µ : it is a sharp control, requiring the asymptotic shape theorem. The second part ensures that the descendants, at all times in [0, αLN ], of the whole box B N are included in ] -LN, LN [ d : the bound is rough, only based on the (at most) linear growth of the process.

We say that the box B N is good if A α,L,N,ε occurs. We also define, for k ∈ Z d and n ∈ N, the event A α,L,N,ε (k, n) = A α,L,N,ε • T 2kN • θ 2nN and we say that the box B N (k, n) is good if the event A α,L,N,ε (k, n) occurs.

The proof of the lower large deviation inequalities is close to the one by Grimmett and Kesten [START_REF] Grimmett | First-passage percolation, network flows and electrical resistances[END_REF] for first passage-percolation. If a point (x, t) is infected too early, it means that its path of infection has "too fast" portions when compared with the speed given by the asymptotic shape theorem. For this path, we build a sequence of boxes associated with path portions, and the existence of a "too fast portion" forces the corresponding box to be bad. But we are going to see that we can choose the parameters to ensure that

• the probability under P for a box to be good is as close to 1 as we want,

• the events "B N (k, 0) is good" are only locally dependent.

We then conclude the proof by a comparison with independent percolation with the help of the Liggett-Schonmann-Stacey Lemma [START_REF] Liggett | Domination by product measures[END_REF] and a control of the number of possible sequences of boxes. Lemma 4.2. We have • The events ({B N (k, 0) is good}) k∈Z d are identically distributed under P.

• There exists α > 0 such that for every ε ∈ (0, 1), there exists an integer L (that can be taken as large as we want) such that lim

N →+∞ P(A α,L,N,ε ) = 1.
• If moreover ν = ν d 0 , then the events ({B N (k, 0) is good}) k∈Z d are (L+1)dependent under P.

Proof. The first and last points are clear. Let us prove the second point. The idea is to find a point (0, -k), with k large enough, such that

• the descendants of (0, -k) are infinitely many and behave correctly (without excessive speed) • the coupled region of (0, -k) contains a set of points that is necessarily crossed by any infection path starting from the box B N .

Indeed, this will allow to find, for all the descendants of B N , a unique common ancestor, and thus to control the growth of all the descendants of B N by simply controlling the descendants of this ancestor. A control on a number of points of the order of the volume of B N will thus be replaced by a control on a single point. See Figure 2. Let ε > 0 be fixed. We first control the positions of the descendants of the box B N at time 4N . Let A, B, M be the constants given by Proposition 2.5. We recall that ω x , for x ∈ Z d , and ω e , for e ∈ E d are the Poisson point processes giving respectively the death times for x and the potential infection times through edge e. We define, for every integer N :

ÃN 1 = {H 0 4N ⊂ [-(4M + 1)N, (4M + 1)N ] d }, A N 1 =    x∈[-N,N ] d 1 1 { ÃN 1 •Tx•θt} d δ 0 + e∋x ω e (t) = 0    .
Note in particular that (42)

A N 1 ⊂ ∀(x 0 , t 0 ) ∈ B N ξ x0 4N -t0 • θ t0 ⊂ [-(4M + 1)N, (4M + 1)N ] d .
We have with [START_REF] Garet | Growth of a population of bacteria in a dynamical hostile environment[END_REF],

E   x∈[-N,N ] d e∋x 2N 0 1 1 { ÃN 1 •Tx•θt} d(δ 0 + ω e )(t)   ≤ (2N + 1) d 2d(1 + 2N λ max )P λmax ( ÃN 1 ) ≤ (2N + 1) d 2d(1 + 2N λ max )A exp(-4BN ),
and thus, with the Markov inequality, (43) lim

N →+∞ P(A N 1 ) = 1.
With (42), we deduce that with a large probability, if N is large enough, the descendants of B N at time 4N are included in [-(4M + 1)N, (4M + 1)N ] d . Now, we look for points with a good growth (we will look for the common ancestor of B N among these candidates):

Ãt 2 = {τ 0 = +∞, ∀s ≥ t K ′ s ⊃ (1 -ε)sA µ and ξ 0 s ⊂ (1 + ε/2)sA µ }, A t,N 2 = N -1 ∪ k=0 Ãt 2 • θ -k .
The first event says that the point (0, 0) lives forever and has a good growth after time t (at most linear growth, and at least linear growth for its coupled zone), while the second event says that there exists a point (0, -k) with a good growth and such that k ∈ [0..N -1]. Theorem 3 in Garet-Marchand [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF] ensures that So there exists t 2 such that P( Ãt2 2 ) > 0. As the time translation θ -1 is ergodic under P, we get (44) lim

N →+∞ P A t2,N 2 = lim n→+∞ P n-1 ∪ k=0 Ãt2 2 • θ -k = 1.
In other words, with a large probability, if N is large enough there exists k ∈ [0..N -1] such that the point (0, -k) has a good growth.

Take

L 1 = L 1 (ε) > 0 such that (45) ∀N ≥ 1 (L 1 + 1)N (1 -ε)A µ ⊃ [-(4M + 1)N, (4M + 1)N ] d .
Thus, if we find an integer k ≥ max(t 2 , L 1 N ) such that A t2 • θ -k occurs, then the descendants of the box B N at time 4N are in the coupled region of (0, -k).

Denote by ←τ y the life time of (y, 0) for the contact process when we reverse time. As the contact process is self-dual, ←τ y as the same law as τ y . Set

A N 3 = ∀y ∈ [-(4M + 1)N, (4M + 1)N ] d ← - τ y • θ 4N = +∞ or ← -τ y • θ 4N < 2N .
The control (13) of large lifetimes ensures that (46) lim

N →+∞ P(A N 3 ) = 1. -k 0 2N 4N y x 0 t 0 0 x t Figure 2. Coupling from the past Assume now that N ≥ t 2 /L 1 . Thus L 1 N ≥ t 2 . Let us see that on A N 1 ∩ (A t2,N 2 • θ -L1N ) ∩ A N 3 , we have (47) ∀t ≥ 4N ∪ (x0,t0)∈BN ξ x0 t-t0 • θ t0 ⊂ ((L 1 + 1)N + t)(1 + ε/2)A µ .
Indeed, let t ≥ 4N and x ∈ Z d be such that (x, t) is a descendant of (x 0 , t 0 ) ∈ B N . Let (y, 4N ) be an ancestor of (x, t) and a descendant of (x 0 , t 0 ). On the event A N 1 , the point y is in [-4M N, 4M N ] d . But, on A N 3 , the definition of y ensures that ←τ y • θ 4N = +∞: so (y, 4N ) has a living ancestor a time -k, for each k such that L

1 N ≤ k ≤ (L 1 + 1)N -1. But, on A t2,N 2 • θ -L1N , inclusion (
45) ensures that (y, 4N ) is in the coupled region of (0, -k) for one of these k, and so (y, 4N ) is a descendant of this (0, -k). Finally, (x, t) is also a descendant of (0, -k), and, always on

A t2,N 2 • θ -L1N , µ(x) ≤ (k + t)(1 + ε/2) ≤ ((L 1 + 1)N -1 + t)(1 + ε/2),
which proves (47).

We then choose α ∈ (0, 1) and an integer L such that

α < 2C - µ 3 ≤ C - µ 1 + ε/2 , L ≥ max 4 α , L 1 + 1 C - µ -α(1 + ε/2) , 4M + 1, 2 αε ((L 1 + 1)(1 + ε/2) + C + µ + 2 .
If N ≥ t 2 /L 1 , as αLN ≥ 4N , we can use (47) with t ∈ [4N, αLN ]; thus our choices for α, L and (42) ensure that on the event

A N 1 ∩ (A t2,N 2 • θ -L1N ) ∩ A N 3 , for every (x 0 , t 0 ) ∈ B N ∪ 4N ≤s≤αLN -t0 ξ x0 s • θ t0 ⊂ ((L 1 + 1 + αL)N )(1 + ε/2)A µ ⊂ [-LN, LN ] d , ∪ 0≤s≤α4N ξ x0 s • θ t0 ⊂ [-(4M + 1)N, (4M + 1)N ] d ⊂ [-LN, LN ] d , ξ x0 αLN -t0 • θ t0 ⊂ (L 1 + 1 + αL)N (1 + ε/2)A µ ⊂ x 0 + (1 + ε)(αLN -t 0 )A µ . Finally, if N ≥ t 2 /L 1 , A N 1 ∩ (A t2,N 2 • θ -L1N ) ∩ A N 3 ⊂ A α,L,N,ε
, and we conclude with (43), ( 44) and (46). We first prove the existence of C > 0 such that, with a large probability, the point (0, 0) can not give birth to more than Ct generations before time t: Lemma 4.3. There A, B, C > 0 such that for every λ ∈ [0, λ max ] E d , for every t, ℓ ≥ 0:

P λ ∃(x, s) ∈ Z d × [0, t]
and an infection path from (0, 0) to (x, s) with more than Ct + ℓ horizontal edges ≤ A exp(-Bℓ).

Proof. Let α > 0 be fixed. For every path γ in Z d starting from 0 and eventually self-intersecting, we set X γ = 1 1 {γ is the projection on Z d of an infection path starting from (0,0)} e -αt(γ) , where t(γ) is the time when the extremity is infected after visiting successively the previous points. More formally, if the sequence of points in γ is (0 = x 0 , . . . , x n ) and if we set T 0 = 0, and for k ∈ {0, . . . , n -1},

T k+1 = inf t > T k ; ω {x k ,x k+1 } ([T k , t]) = 1 and ω x k ([T k , t]) = 0 ,
we have t(γ) = T n . The random variable t(γ) is a stopping time (it is infinite if γ is not the projection of an infection path). Let γ be a path in Z d starting from 0 and let f be an edge at the extremity of γ. If we denote by γ.f the concatenation of γ with f , the strong Markov property at time t(γ) ensures that To conclude, we take α = 2dλ max , and then C such that ( 2d 2d+1 ) C = e -α .

E λ [X γ.f |F t(γ) ] ≤ X γ λ max α + λ max , and so E[X γ ] ≤ λ max α + λ max |γ| . Now, P λ ∃(x, s) ∈ Z d × [0, t]
• ϕ(0) = max{j ≥ 0 : ∀i ∈ [0..j] k i = 0};

• Assume we chose ϕ(i), then we take, if it is possible,

j 0 (i) = inf{j > ϕ(i) : k j = k ϕ(i) }, ϕ(i + 1) = max{j ≥ j 0 (i) : k j = k j0(i) }.
and we stop the extraction process otherwise.

Then, as in [START_REF] Grimmett | First-passage percolation, network flows and electrical resistances[END_REF] a

ϕ(ℓ) -x ∞ ≤ (L + 1)N, 0 ≤ t -t ϕ(ℓ) ≤ αLN, ∀i ∈ [0..ℓ -1] a ϕ(i)+1 -a ϕ(i+1) ∞ ≤ 2N, ∀i ∈ [0..ℓ -1] |t ϕ(i)+1 -t ϕ(i+1) | ≤ 2N.
Moreover, the upper bound (53) for ℓ ensures that

(54) 1 ≤ ℓ ≤ ℓ ≤ 2Ct (αL -1)N + 2.
On the other hand, as µ(x) -µ(a ϕ(ℓ) -x) ≤ µ(a ϕ(l) ), we have with (54):

(1 + ε)t -C + µ (L + 1)N ≤ µ   ℓ-1 i=0 a ϕ(i+1) -a ϕ(i)   ≤ ℓ-1 i=0 µ(a ϕ(i+1) -a ϕ(i)+1 ) + ℓ-1 i=0 µ(a ϕ(i)+1 -a ϕ(i) ) ≤ 2N C + µ ℓ + ℓ-1 i=0 µ(a ϕ(i)+1 -a ϕ(i) ) ≤ 2N C + µ 2Ct (αL -1)N + 2 + ℓ-1 i=0 µ(a ϕ(i)+1 -a ϕ(i) ).
This ensures, with the choice (50) we made for α, L, that

ℓ-1 i=0 µ(a ϕ(i)+1 -a ϕ(i) ) ≥ (1 + 2ε/3)t -2C + µ (L + 1)N. (55)
In other words, even after the extraction process, the sum of the lengths of the crossings remains of order (1 + 2ε/3)t. Let k ∈ Z d and n ∈ N. We say now that B N (k, n) is good if the event A α,L,N,ε/3 The first sum can be controlled with Theorem 4.1. For the second sum, the Markov property gives for any λ ∈ Λ,

(i)+1 -a ϕ(i) ) ≤ (1 + ε/3)(t ϕ(i)+1 - t ϕ(i) ); this ensures that µ   i: BN (k ϕ(i) ,n ϕ(i) ) good (a ϕ(i)+1 -a ϕ(i) )   ≤ i: BN (k ϕ(i) ,n ϕ(i) ) good µ(a ϕ(i)+1 -a ϕ(i) ) ≤ (1 + ε
P λ (∃t ∈ [0, 1] ξ 0 T +n ⊂ (1 + ε/2)(T + n)A µ , ξ 0 T +n+t ⊂ ((1 + ε)(T + n)A µ ) ≤ x∈(1+ε/2)(T +n)Aµ P x.λ (∃t ∈ [0, 1] ξ 0 t ⊂ (ε/2)(T + n)A µ ) ≤ |(1 + ε/2)(T + n)A µ |P(H 0 1 ⊂ (ε/2)(T + n)A µ ) ≤ A exp(-B(T + n))
, where the last upper bound comes from a comparison with the Richardson model. We conclude the proof of (56) by integrating with respect to λ.

Let us prove now the existence of A, B > 0 such that (57) ∀r > 0 P(H 0 r ⊂ (1 + ε)rA µ ) ≤ A exp(-Br). With [START_REF] Garet | Growth of a population of bacteria in a dynamical hostile environment[END_REF], we can find A 1 , B 1 > 0 and c < 1 such that P(H 0 cr ⊂ rA µ ) ≤ A 1 exp(-B 1 r). Now, P(H 0 r ⊂ (1 + ε)rA µ ) ≤ P(H 0 cr ⊂ rA µ ) + P(∃t ∈ (cr, r) ξ 0 t ⊂ (1 + ε)rA µ ) ≤ A 1 exp(-B 1 r) + P(∃t ≥ cr ξ 0 t ⊂ (1 + ε)tA µ ), and we conclude the proof of (57) with (56). To obtain (5), we just need to note that t → H t is non-decreasing.

Finally, for x ∈ Z d \{0}, P(t(x) ≤ (1 -ε)µ(x)) ≤ P(H 0 (1-ε)µ(x) ⊂ µ(x)A µ ). Applying (57), we end the proof of (4), and thus of Theorem 1.4.

About the order of the deviations

By Theorems 1.1 and 1.4, we have for ν-almost every λ and each ε > 0:

lim x→+∞ 1 x log P λ t(x) µ(x) ∈ [1 -ε, 1 + ε] < 0.
To see that the exponential decrease in x is optimal, we need too see that lim In fact, we will prove here that for every (s, t) with 0 < s < t, there exists a constant γ > 0 such that for each λ ∈ Λ and each x ∈ Z d , P λ (t(x) ∈ [s, t] x 1 ) ≥ exp(-γ x 1 ). Proof. Let s, t with 0 < s < t. For each u ∈ Z d such that u 1 = 1, we define T u = inf{t ≥ 0 : ξ 0 t = {u}, ∀s ∈ [0, t) ξ 0 s = {0}}. We are going to prove that ∃γ > 0 ∀λ ∈ Λ ∀u ∈ Z d , u 1 = 1 P λ (T u ∈ [s, t]) ≥ e -γ .

In order to ensure that T u ∈ [s, t], it it sufficient to satisfy

• The lifetime of the particle at (0, 0) is strictly between (s+t)/2 and t, which happens with probability e -(s+t)/2 -e -t under P λ ; • The first opening of the bond between 0 and u happens strictly between s and (s + t)/2, which happens with probability exp(-λ {0,u} s) -exp(-λ {0,u} (s + t)/2) ≥ exp(-λ max s)(1 -exp(-λ min (t -s)/2))

under P λ ; • There is no opening between time 0 and time t, on the set J constituted by the 4d -2 bonds that are neighour of 0 or u and differ from {0, u}, which happens under P λ with probability j∈J exp(-λ j t) ≥ exp(-(4d -2)λ max t);

• There is no death at site u between 0 and t, which happens under P λ with probability e -t . Then, using the independence of the Poisson processes, we get

P λ (T u ∈ [s, t])
≥ (e -(s+t)/2) -e -t )e -t e -(4d-2)λmaxt e -λmaxs (1 -e -λmin(t-s)/2 ) = e -γ . Moreover, T u is obviously a stopping time. Then, applying the strong Markov property x 1 times, we get, P λ (t(x) ∈ [s, t] x 1 ) ≥ exp(-γ x 1 ). This gives the good speed for both upper and lower large deviations.

Note that the order of the large deviations is the same for upper and lower deviations, as happens for the chemical distance in Bernoulli percolation (see ). Conversely, it is known that these orders may differ for first-passage percolation (see Kesten [START_REF] Kesten | Aspects of first passage percolation[END_REF] and Chow-Zhang [START_REF] Chow | Large deviations in first-passage percolation[END_REF]).

Figure 1 . 3 . 3 . 1 .

 1331 Figure 1. The good event A( n0 , u, x 0 , x 1 ). 3.3.1. Good events. Let C 1 > 0 and M 1 > 0 be fixed. Let I ∈ N * , L ∈ N * and δ > 0 such that I ≤ L and δ < C 1 L. For n0 ∈ Z d , x 0 , x 1 ∈ [-L, L[ d and u ∈ Z d such that u 1 ≤ 1, we define the following event:

•Lemma 3 . 3 .

 33 The realization of this event only depends on what happens in the spacetime box (2L n0 + [-M 1 L, M 1 L]) × [0, T ]. Let us give a summary of the different parameters: L spatial scale of the macroscopic boxes I size of the entrance area and of the exit area (I ≤ L) T temporal size of the macroscopic boxes (T = C 1 L) δ minimum duration for the infection of x 1 n0 macroscopical spatial coordinate (coordinate of the big box) u direction of move ( u 1 ≤ 1) x 0 relative position of the entrance area in the box (x 0 ∈ [-L, L[ d ) x 1 relative position of the target point (x 1 ∈ [-L, L[ d ) s( n0 , u, x 0 , x 1 ) relative position of the exit area in the box with coordinate( n0 + u) (s( n0 , u, x 0 , x 1 ) ∈ [-L, L[ d )We can find constants C 1 > 0 and M 1 > 0 such that we have the following property.

  ∈ Z d P λmin (τ x+[-I,I] d = +∞) ≥ 1 -ε/4. We let ε ′ = ε/(2I + 1) d . By the FKG Inequality, P λmin (∀y ∈ [-I, I] d , τ y = +∞) > 0. Translation invariance gives then lim L→+∞ P λmin (∃n ∈ [0, L] : ∀y ∈ ne 1 + [-I, I] d , τ y = +∞) = 1.

Theorem 4 . 1 .

 41 Assume that ν = ν ⊗E d 0 and that the support of ν 0 is included in [λ min , λ max ]. For every ε > 0, there exist A, B > 0 such that ∀t ≥ 0 P(ξ 0 t ⊂ (1 + ε)tA µ ) ≤ A exp(-Bt). Using the norm equivalence on R d , we introduce constants C

2 )

 2 dν(λ) = P λ ( Ãt 2 )P λ (τ 0 = +∞)dν(λ) ≥ P λ ( Ãt 2 )P λmin (τ 0 = +∞)dν(λ) ≥ P λmin (τ 0 = +∞)P( Ãt 2 ).

  is an infection path from (0, 0) to (x, s) with more than Ct + ℓ horizontal edges = P λ ∪ γ:|γ|≥Ct+ℓ {X γ ≥ e -αt } ≤ e αt γ:|γ|≥Ct+ℓ λ max α + λ max |γ| ≤ e αt n≥Ct+ℓ 2dλ max α + λ max n .

2 ℓ

 2 N (0, 0) good) = 1.By the extraction process, the spatial coordinates of the boxes in Γ are all distinct. With the comparison theorem by Liggett-Schonmann-Stacey[START_REF] Liggett | Domination by product measures[END_REF], we can, for any p 1 < 1, take N large enough to ensure that for any family n = (n k ) k∈Z d ∈ N Z d , the law of the field (η n k ) k∈Z d under P stochastically dominates a product on Z d of Bernoulli laws with parameter p 1 . Thus, if x is such that µ(x) ≥ (1 + ε)t, then P   there exists an infection path γ from (0, 0) to (x, t) with less than Ct horizontal edges and such that Γ = Γ(γ) has at leastεt 3C + µ (L+1)N -2 bad boxes Card{Γ; |Γ| = ℓ} A classical counting argument gives the existence of a constant K = K(d, α, L) independent of N such that ∀ℓ ≥ 1 Card{Γ; |Γ| = ℓ} ≤ K ℓ .We get then an upper bound for our probability of the formA t N (1 -p 1 ) ε 3C + µ (L+1) (2K) 2C αL-1 t/N, which leads to a bound of the form A 3 exp(-B 3 t) as soon as p 1 is close enough to 1. Summing over all x ∈ [-M t, M t] d , we have again an exponential bound. With this last upper bound, (48) and (49), we end the proof of Theorem 4.1.Proof of Theorem 1.4. We first prove there exist A, B > 0 such that (56) ∀T > 0 P(∃t ≥ T ξ 0 t ⊂ (1 + ε)tA µ ) ≤ A exp(-BT ). Indeed,P(∃t ≥ T ξ 0 t ⊂ (1 + ε)tA µ )) ≤ P(∃n ∈ N ξ 0 T +n ⊂ (1 + ε/2)(T + n)A µ ) +P(∃n ∈ N ∃t ∈ [0, 1] ξ 0 T +n ⊂ (1 + ε/2)(T + n)A µ , ξ 0 T +n+t ⊂ ((1 + ε)(T + n)A µ ) ≤ n≥0 P(ξ 0 T +n ⊂ (1 + ε/2)(T + n)A µ ) + n≥0 P(∃t ∈ [0, 1] ξ 0 T +n ⊂ (1 + ε/2)(T + n)A µ , ξ 0 T +n+t ⊂ ((1 + ε)(T + n)A µ ).

x→+∞ 1 x

 1 log P λ t(x) µ(x) ∈ [1 -ε, 1 + ε] > -∞.

  In other words, if t > 0, if x is such that µ(x) ≥ (1 + ε)t, if there exists an infection path γ from (0, 0) to (x, t) with less than Ct horizontal edges, the associated sequence Γ has a number of bad boxes proportional to t.Note that Lemma 4.2 says that for any deterministic family n= (n k ) k∈Z d ∈ N Z d , the field (η n k ) k∈Z d , defined by η n k = 1 1 {BN (k,n k )good} is locally dependent and that lim

				3	)	i: BN (k ϕ(i) ,n ϕ(i) ) good	(t ϕ(i)+1 -t ϕ(i) )
	≤ (1 +	ε 3	)t.
	With (55), it implies that
	µ(a ϕ(i)+1 -a ϕ(i) ) ≥	ε 3	t -2C + µ (L + 1)N,
	and then, with (51)
	|{i : B N (k ϕ(i) , n ϕ(i) ) bad}| ≥	εt µ (L + 1)N 3C +	-2.

i: BN (k ϕ(i) ,n ϕ(i) ) bad

Proof of Theorem 4.1. Let ε > 0 and t > 0 be fixed. Obviously

The second term is controlled by equation [START_REF] Garet | Growth of a population of bacteria in a dynamical hostile environment[END_REF] Assume that ξ 0 t ⊂ (1+ε)tA µ : let x ∈ ξ 0 t be such that µ(x) ≥ (1+ε)t, x ∞ ≤ M t and let γ be an infection path from (0, 0) to (x, t). With Lemma 4.3, we choose C > 1, A 2 , B 2 > 0 such that for every t ≥ 0, (49)

P

there exists an infection path from (0, 0) to Z d × {t} with more than Ct horizontal edges

With the last estimate, we can assume that γ has less than Ct horizontal edges. We take 0 < α < 1 and L = L(α, ε) large enough to apply Lemma 4.2 and such that

We fix an integer N and we cut the space-time Z d × R + space-time boxes:

We associate to the path γ a finite sequence Γ = (k i , n i , a i , t i ) 0≤i≤ℓ , where the (k i , n i ) ∈ Z d × N are the coordinates of space-time boxes and the (a i , t i ) are points in Z d × R + in the following manner:

• k 0 = 0, n 0 = 0, a 0 = 0 and t 0 = 0: B N (k 0 , n 0 ) is the box containing the starting point (a 0 , t 0 ) = (0, 0) of the path γ. • Assume we have chosen (k i , n i , a i , t i ), where (a i , t i ) is a point in γ and (k i , n i ) are the coordinates of the space-time box containing (a i , t i ). To the box B N (k i , n i ), we add the larger box (2N

), we take for (a i+1 , t i+1 ) the first point -if it exists -along γ after (a i , t i ) to be outside this large box, and we take for (k i+1 , n i+1 ) the coordinates of the space-time box that containing (a i+1 , t i+1 ). Otherwise, we stop the process. The idea is to extract from the path a sequence of large portions, i.e. the portions of γ between (a i , t i ) and (a i+1 , t i+1 ). We have the following estimates:

The two first estimates just say that -spatially for (51) and in time for (52)the point (a i+1 , t i+1 ) remains in the large box centered around B N (k i , n i ), which contains (a i , t i ). Now consider the third estimate. We note that a path can get out of a large box either with its time coordinate -and the number of such exits is smaller than t (αL-1)N + 1 -, or by the space coordinate -, and the number of such exits is smaller than Ct (L-1)N + 1. The last inequality comes from C > 1 and α < 1. To ensure that the space coordinates of the boxes associated to the path are all distinct, we extract a subsequence Γ = (k ϕ(i) ) 0≤i≤ℓ with the loop-removal process described by Grimmett-Kesten [START_REF] Grimmett | First-passage percolation, network flows and electrical resistances[END_REF]: