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

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

GROWTH OF A POPULATION OF BACTERIA IN A DYNAMICAL HOSTILE ENVIRONMENT

Keywords: 2000 Mathematics Subject Classification. 60K35, 82B43 contact process, directed percolation, renormalization, stochastic domination, random environment, block construction, interacting particle system

We study the growth of a population of bacteria in a dynamical hostile environment corresponding to the immune system of the colonised organism. The immune cells evolve as subcritical open clusters of oriented percolation and are perpetually reinforced by an immigration process, while the bacteria try to grow as a supercritical oriented percolation in the remaining empty space. We prove that the population of bacteria grows linearly as soon as it survives. In this perspective, we build general tools to study dependent percolation models issued from renormalization processes.

A growth model in dynamical hostile environment

We consider the following discrete time interacting particle system: at time n = 0, a particularly fertile bacterium (represented here by a type 1 particle) is submerged in a population of immune cells (type 2 particles) that are going to impede its development. The immune cells are not very fertile but benefit from a constant immigration process. Our aim is to find conditions that ensure, when the bacteria survive, that their growth is linear.

Our system is described by a discrete time Markov chain taking its values in {0, 1, 2} Z d , depending on 3 parameters p, q, α ∈ (0, 1). The time is indexed by N = {0, 1, 2, . . . } and we also note N * = {1, 2, 3, . . . }. The transition between two states is in two steps. First, between time n and time n + 1/2, each particle tries to colonize its neighbor sites: it succeeds with probability p if it is a type 1 particle, and with probability q if it is a type 2 particle. All events are independent, and in case of conflict, the type 2 particle wins. Next, between time n + 1/2 and time n + 1, the immigration of type 2 particles occurs: on each site, a type 2 particle appears with probability α > 0, possibly taking the place of the particle previously occupying the site. Once again, all events are independent.

In the degenerate case where q = 0 and α = 0, we recover independent oriented percolation with parameter p, which provides a simple model for the spread of an infection. By classical arguments, there exists a critical probability -→ p c alt (d + 1) for the possibility for independent oriented percolation on Z d × N to grow infinitely. Of course, we choose p > -→ p c alt (d + 1) to avoid the almost sure extinction of the bacteria in the absence of immune cells. Hence, if q = 0 and α = 0, we know that the bacteria survive with positive probability, and when they survive, their growth is linear. These results have been proved for the supercritical contact process by Bezuidenhout-Grimmett [START_REF] Bezuidenhout | The critical contact process dies out[END_REF] and Durrett [START_REF] Durrett | The contact process, 1974-1989[END_REF], and can readily be transposed for supercritical independent oriented percolation.

On the contrary, we take q < -→ p c alt (d+1), which corresponds to the poor virulence of type 2 particles. However, the constant immigration rate α guarantees that type 2 particles are always present in the organism.

Let us now describe the model more formally. 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 set Ω = {0, 1}

-→

E d × {0, 1}
-→ E d × {0, 1} Z d and we endow the set Ω = ΩN * with its Borel σ-algebra for the product topology. We consider the probability P = P p,q,α = ν ⊗N * , where

ν = ν p,q,α = B(p) ⊗ - → E d ⊗ B(q) ⊗ - → E d ⊗ B(α) ⊗Z d
and where B(p) stands for the Bernoulli law with parameter p.

Starting from the initial configuration x ∈ {0, 1, 2} Z d , we define the Markov chain (η x n ) n≥0 taking its values in {0, 1, 2} Z d by η x 0 = x and η x n+1 = f (η x n , ω n+1 )

where f : {0, 1, 2} Z d × Ω → {0, 1, 2} Z d is defined as follows:

f (x, ((ω e 1 ) e∈ - →

E d , (ω e 2 ) e∈ - → E d , (ω k 3 ) k∈Z d )) =   max      2ω k 3 , 2 max(ω (i,k) 2 : i -k 1 ≤ 1, x i = 2), max(ω (i,k) 1 : i -k 1 ≤ 1, x i = 1)         k∈Z d .
Note that type 2 particles do not see type 1 particles in their evolution, which explains why type 2 particles are assimilated to an environment. Considering two disjoint subsets E 1 , E 2 of Z d that represent the initial sets occupied by type 1 and type 2 particles, we also use the notation η E1,E2 n = η

1 1E 1 +21 1E 2 n
. We denote by η E1,E2 1,n (resp. η E2 2,n ) the set of sites occupied by type 1 particles (resp. by type 2 particles) at time n, and we consider the evolution of the bacteria population (η {0},∅ 1,n ) n≥0 : can this process survive ? Does it grow linearly when it survives ? We naturally introduce the following extinction time and hitting times:

τ E1,E2 1 = inf{n ≥ 0 : η E1,E2 1,n = ∅}; ∀y ∈ Z d t E1,E2 1 (y) = inf{n ≥ 0 : y ∈ η E1,E2 1,n }.
Note that α → P p,q,α (τ E1,E2 1 = +∞) is non-increasing and exhibits a phase transition. We first prove that this phase transition does not depend on the initial configuration E 2 of the environment: Theorem 1.1. For every p > -→ p c alt (d + 1) and every q < -→ p c alt (d + 1), ∀α ∈ [0, 1] P p,q,α (τ 0,Z d \{0} 1

= +∞) > 0 ⇐⇒ P p,q,α (τ 0,∅ 1 = +∞) > 0.

We thus define α c (p, q) = sup{α ≥ 0 : P p,q,α (τ 0,∅ 1 = +∞) > 0}. Our main result is the following: Theorem 1.2. For every p > -→ p c alt (d + 1) and every q < -→ p c alt (d + 1), 0 < α c (p, q) < 1.

Moreover, for every α < α c (p, q), there exist positive constants A, B, C such that for every E ⊂ Z d \{0}, x ∈ Z d and t > 0, P p,q,α (τ 0,E 1 = +∞) > 0, (1)

P p,q,α (τ 0,E 1 = +∞, t 0,E 1 (x) ≥ C x 1 + t) ≤ A exp(-Bt), (2)
P p,q,α (t < τ 0,E 1 < +∞) ≤ A exp(-Bt). [START_REF] Broman | Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment[END_REF] We thus prove that if the immigration of type 2 particles is not too important, the bacteria population survives with positive probability, and, when it survives, grows linearly, as it happens in the absence of immune cells. We can also explain this model in terms of dependent oriented percolation: on the oriented graph Z d × N, we erase for each site (z, n) ∈ Z d × N, with probability α, the finite cluster of oriented percolation with parameter q starting from (z, n). The remaining random oriented graph is then given to the type 1 particle, which tries to develop as an oriented percolation with parameter p. Thus the growth of type 1 particles can be seen as a dependent oriented percolation model, with an unbounded but exponentially fast decreasing dependence. Our result ensures the linear growth of this oriented percolation when it percolates.

A natural question is then the existence of an asymptotic shape result:

Conjecture 1.3. For every p > -→ p c alt (d + 1), for every q < -→ p c alt (d + 1) and every α ∈ (0, α c (p, q)), there exists a norm µ on R d such that for any two disjoints subsets E 1 and E 2 in Z d with E 1 = ∅, we have for ε > 0: P p,q,α (.|τ E1,E2 1 = +∞) almost surely, for every large enough t,

(1 -ε)B µ (0, 1) ⊂ 1 t B t ⊂ (1 + ε)B µ (0, 1)
,

where B t = {x ∈ Z d : t E1,E2 1 (x) ≤ t} + [-1/2, 1/2] d .
We think that this result can be proved with subadditive methods similar to the ones we used in the case of the contact process in a random environment -see Garet-Marchand [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF].

We can find a certain number of similar competition mechanisms in the literature under the name of hierarchical competition (see Durrett-Møller [START_REF] Durrett | Complete convergence theorem for a competition model[END_REF]), of contact process (or oriented percolation) in a dynamical random environment (see Broman [START_REF] Broman | Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment[END_REF], Luo [START_REF] Luo | The Richardson model in a random environment[END_REF], Remenik [START_REF] Remenik | The contact process in a dynamic random environment[END_REF], Steif-Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF]), or without any specific denomination (see Durrett-Swindle [START_REF] Durrett | Are there bushes in a forest? Stochastic Process[END_REF], Durrett-Schinazi [START_REF] Durrett | Asymptotic critical value for a competition model[END_REF]). The common characteristic of these models is that one type of particles (here type 2 particles) evolves in a Markovian way, and that the second type evolves as a contact process or an oriented percolation in the remaining empty space.

In our paper, we are going to use renormalization techniques. This is not surprising: the efficiency of such techniques in the study of particle systems has been known for long, see for instance Bramson-Durrett [START_REF] Bramson | A simple proof of the stability criterion of Gray and Griffeath[END_REF], or Durrett [START_REF] Durrett | Multicolor particle systems with large threshold and range[END_REF][START_REF] Durrett | Ten lectures on particle systems[END_REF], and the use of renormalization is usual to prove that survival occurs with positive probability. However, studying the system conditioned to survive can be subtle. Indeed, the renormalization procedures tend to destroy the independence properties given by the Markovianity and the tried and tested restart arguments described in Durrett [START_REF] Durrett | Oriented percolation in two dimensions[END_REF] must be adapted with some care. While the general idea remains simple, the implementation is quite technical and, for the moment, there are no ready-made tools for this kind of situation. In the perspective of future works, we build tools in the spirit of the theorem of Liggett-Schonmann-Stacey [START_REF] Liggett | Domination by product measures[END_REF] but in the context of dependent oriented percolations resulting from renormalization procedures -see Theorem 2.5.

Comparison and coupling results

While the setting of static renormalization can be defined quite formally, there are other types of renormalization that are harder to classify: they all have in common to consider local events that cannot be defined in an absolute way, but rather depend on a local component and also on the past of the renormalization process. This past can be associated to a time line as in Bezuidenhout-Grimmett [START_REF] Bezuidenhout | The critical contact process dies out[END_REF] and Durrett [START_REF] Durrett | The contact process, 1974-1989[END_REF], or to a sequence of spatial boxes as in Grimmett-Marstrand [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF].

After renormalization, we are led to study a dependent oriented percolation process. The fact that this process survives with positive probability can be proved quite directly from the comparison result of Liggett-Schonmann-Stacey [START_REF] Liggett | Domination by product measures[END_REF]. However, when one wants to study the oriented percolation process conditioned to survive, things are more intricate: our Theorem 2.5 gives thus a general setting to ensure that "conditioned on its survival, the oriented percolation process on Z d × N built from the renormalization process stochastically dominates an independent oriented percolation process with parameter as large as we want". The aim is of course to transfer the properties of the supercritical independent percolation process to the dependent percolation process.

We work on the graph Z d × N, as defined in the introduction. We consider Ω = {0, 1} -→ E d+1 alt endowed with its Borel σ-algebra and the probability

P p = B(p) ⊗ - → E d+1 alt ;
the edges such that ω e = 1 are said to be open, the other ones are closed. For two sites v, w in Z d × N, we denote by v → w the existence of an open oriented path from v to w. The critical probability is denoted by -→ p c alt (d + 1). The time translations θ n on Ω are defined by θ n ((ω (e,k)

) e∈ - → E d ,k≥1 ) = (ω (e,k+n) ) e∈ - → E d ,k≥1 . We set, for n ∈ N and x ∈ Z d , ξ x n = {y ∈ Z d : (x, 0) → (y, n)}, ξ Z d n = ∪ x∈Z d ξ x n , τ x = min{n ∈ N : ξ x n = ∅}, H x n = ∪ 0≤k≤n ξ x k , K x n = (ξ x n ∆ξ Z d n ) c = ξ x n ∪ (Z d \ξ Z d n ).
As for the contact process, (H x n ) n≥0 and (K x n ∩ H x n ) n≥0 grow linearly in case of survival: Lemma 2.1. We consider independent oriented percolation on Z d × N. For every p > -→ p c alt (d + 1), there exist strictly positive constants A, B, C such that for every x ∈ Z d , for every L, n > 0:

P p (τ x = +∞, [-L, L] d ⊂ K x CL+n ) ≤ Ae -Bn P p (τ x = +∞, [-L, L] d ⊂ H x CL+n ) ≤ Ae -Bn .
Proof. For the contact process, Durrett [START_REF] Durrett | The contact process, 1974-1989[END_REF] showed how to deduce an analogous result from the construction of Bezuidenhout-Grimmett [START_REF] Bezuidenhout | The critical contact process dies out[END_REF]. As explained in [START_REF] Bezuidenhout | The critical contact process dies out[END_REF], the proofs remain valid for oriented percolation, which is the discrete-time analogous of the contact process.

We now recall the comparison theorem of Liggett-Schonmann-Stacey [START_REF] Liggett | Domination by product measures[END_REF]. In the following, for two edges e and f in -→ E d , we denote by d(e, f ) the distance for . 1 between the middles of the edges e and f . Proposition 2.2. Let d ≥ 1 be fixed. For every M ≥ 1, there exists a function g M from [0, 1] to [0, 1] with lim q→1 g M (q) = 1 and such that if µ is a probability measure

on Ω = {0, 1} - → E d and q ∈ [0, 1] satisfying: for every e ∈ -→ E d , µ(ω e = 1|ω f , d(e, f ) ≥ M ) ≥ q, then µ stochastically dominates a product of Bernoulli law with parameter g M (q): µ B(g M (q)) ⊗ - → E d .
Relying on this theorem, we are going to prove analogous results for a certain class of dependent oriented percolations: Definition 2.3. 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 .
First, we give a stochastic comparison between fields in C d (M, q) and Bernoulli product measures: Lemma 2.4. Let d, M ≥ 1 be positive integers and q ∈ (0, 1).

If the distribution of (W n e ) e∈ - → E d ,n≥1 belongs to C d (M, q), then for each n, the distribution of the field

(W n+k e ) e∈ - → E d ,k≥1 conditioned by G n stochastically dominates B(g M (q)) ⊗ - → E d+1
alt , where the function g M has been defined in Proposition 2.2.

In other words, for each n ≥ 0, for each A ∈ G n , and each non-decreasing bounded function f , we have

E W [1 1 A (f • θ n )] ≥ P(A) {0,1} - → E d+1 alt f dB(g M (q)) ⊗ - → E d+1 alt ,
where θ n is the translation operator on Ω that has been defined previously.

Proof. Let E = {0, 1}

-→ E d , q ′ = g M (q) and fix n ≥ 1. We will show that for each nonnegative integer k, for every non-decreasing bounded function f that only depends on the k first time coordinates, we have

E[1 1 A f (W n+1 , W n+2 , . . . , W n+k )] ≥ P(A) f dB(q ′ ) ⊗ - → E d+1 alt .
When k = 0, f is constant and the result is obvious. Suppose the result holds for k and let us prove it for k + 1. Let h be a non-decreasing bounded function on E k+1 and consider A ∈ G n . Since we work on a Polish space, we can disintegrate P with respect to the σ-field G n+k (see e.g. Stroock [START_REF] Stroock | Probability theory, an analytic view[END_REF]). Then, we have, with the notation of Stroock [START_REF] Stroock | Probability theory, an analytic view[END_REF]:

E[1 1 A h(W n+1 , . . . , W n+k+1 )] = E[1 1 A E[h(W n+1 , . . . , W n+k+1 )|G n+k ] = A Ω h(W n+1 (ω ′ ), . . . , W n+k (ω ′ ), W n+k+1 (ω ′ )) dP G n+k ω (ω ′ ) dP(ω) = A Ω h(W n+1 (ω), . . . , W n+k (ω), W n+k+1 (ω ′ )) dP G n+k ω (ω ′ ) dP(ω).
Since we supposed that the distribution of (W n e ) e∈ - → E d ,n≥1 belongs to C d (M, q), the distribution of (W n+k+1 e ) e∈ - → E d under P G n+k ω satisfies, for every fixed ω, the assumptions of the Liggett-Schonmann-Stacey comparison Theorem (Theorem 2.2). Thus, it stochastically dominates B(q

′ ) ⊗ - → E d , which gives Ω h(W n+1 (ω), . . . , W n+k (ω), W n+k+1 (ω ′ )) dP G n+k ω (ω ′ ) ≥ E h(W n+1 (ω), . . . , W n+k (ω), x) dB(q ′ ) ⊗ - → E d (x) = f (W n+1 (ω), . . . , W n+k (ω)),
where f is defined by ( 4)

f (y 1 , . . . , y k ) = E h(y 1 , . . . , y k , x) dB(q ′ ) ⊗ - → E d (x).
Thus we obtain

E[1 1 A h(W n+1 , . . . , W n+k )] ≥ A f (W n+1 , . . . , W n+k ) dP.

But by the induction assumption,

A f (W n+1 , . . . , W n+k ) dP ≥ P(A)

E k f (y 1 , . . . , y k ) d(B(q ′ ) ⊗ - → E d ) ⊗k ,
which, from Definition (4), gives the desired result.

Then, we associate to every {0, 1}-valued random field (W n e ) e∈ - → E d ,n≥1 an oriented percolation process (ξ 0 n (W )) n≥1 = (ξ 0 n ) n≥1 starting from (0 Z d , 0) and defined in the usual way:

ξ 0 0 = {0} ξ 0 n+1 = {x ∈ Z d : ∃y ∈ ξ 0 n W n+1 (y,x) = 1}.
For simplicity, we will often say "oriented percolation in C d (M, q)" instead of "oriented percolation associated to a field χ ∈ C d (M, q)".

We define the extinction time of the oriented percolation associated to W and starting from (0 Z d , 0):

τ 0 (W ) = τ 0 = inf{n ≥ 1 : ξ 0 n = ∅}.
The following result allows a coupling between surviving dependent percolation in C d (M, q) and supercritical Bernoulli percolation: Theorem 2.5. Let d, M ≥ 1 be fixed positive integers and let q ∈ (0, 1) be such that g M (q) > -→ p c alt (d + 1). There exist positive constants β, γ such that for each field χ ∈ C d (M, q), we can find a probability space where live a field W = (W

(W ) = ∞) > 0; • T = τ 0 (W ) on the event {τ 0 (W ) < +∞}; • Conditioning by {τ 0 (W ) = +∞}, the open cluster issued from (0 Z d , 0) of the field (W ′n e ) e∈ - → E d ,n≥1
has the same distribution as the open cluster issued from (0 Z d , 0) conditioned on survival in independent oriented percolation with parameter g M (q); moreover, on {τ 0 (W ) = +∞}, we have ∀n ≥ 0 ξ 0 T +n (W ) ⊃ D + ξ 0 n (W ′ ). In fact, this theorem contains two results

• it ensures the existence of an embedded independent infinite cluster in the dependent infinite cluster, and controls its position. • when the dependent cluster is finite, it also controls its height.

Proof. Define q ′ = g M (q).
Let E 1 , . . . , E n be finite subsets of Z d . We define E = (E 1 , . . . , E n ) and |E| = n. The event

A E = |E| ∩ i=1 {ξ 0 i = E i }
is in G n ; on this event, the history of the directed percolation process starting from (0 Z d , 0) up to time n is characterized by E. We call that E an history.

From now on, we only consider histories satisfying χ(A E ) > 0; for such an history, we define a probability measure m E on {0, 1}

- → E d ×N * by m E (B) = χ((W |E|+k ) k≥1 ∈ B|A E );
we call it the law of the dependent oriented percolation with history E. Thanks to Lemma 2.4, the probability measure m E stochastically dominates B(q

′ ) ⊗ - → E d ×N * .
Thus, Strassen's Theorem ( [START_REF] Strassen | The existence of probability measures with given marginals[END_REF], see also Lindvall [START_REF] Lindvall | On Strassen's theorem on stochastic domination[END_REF]) allows to build a law ν E on ({0, 1}

- → E d ×N * ) 2 with marginals m E and B(q ′ ) ⊗ - → E d ×N * and is concentrated on {x ≥ y}, with ∀(x, y) ∈ ({0, 1} - → E d ×N * ) 2 x ≥ y ⇔ ∀e ∈ -→ E d × N * x e ≥ y e .
For every history E, the law ν E allows to make a coupling between the state of the bonds of dependent and of independent oriented percolations with common history E. Now, we can construct on the same probability space (Ω, F , P) a family of ({0, 1}

-→

E d ×N * ) 2 -valued independent processes ( E η, E η ′ ) E
, that are indexed by the collection of all histories E, in such a way that for every history E,

E η n e , E η ′n e e∈ - → E d ,n≥1 law = ν E .
We denote by E τ x the time where the independent directed percolation related to E η ′ and starting from x (and not from the whole history E) dies. We write ξ n ( E η) to denote the state at time n of the dependent percolation process with history E; thus, ξ 0 ( E η) = E |E| . We also denote by E N x = (ξ 1 ( E η)), . . . , ξE τ x ( E η)) the sequence of the configurations occupied up to time E τ x by the dependent percolation process associated to η with history E and denote by E L x its terminal configuration.

In words, an history E and a point x being given, we run the coupling between the independent percolation associated to E η and the dependent percolation associated to E η ′ up to time E τ x when the cluster issued from x in the independent one dies out. We then store the new history of the dependent percolation in E N x and its final state in E L x . Note that • the percolation fields both have history E;

• we define the whole percolation fields, and not only the clusters issued from a specific set; • we run the coupling until time E τ x , where open cluster issued from x in the independent percolation dies out; • if the terminal configuration E L x of the dependent percolation is empty, then, by stochastic comparison, E τ x is also the lifetime of the dependent percolation after history E.

Then we build three sequences: a sequence of sites (x n ), a sequence of times (t n ) and a sequence of compatible histories (ε n ). Denote by ∆ a cemetery point added to Z d . Then, we put ε 0 = {0}, t 0 = 0, x 0 = 0 and recursively define

• if x i = ∆, then t i+1 = +∞, x i+1 = ∆ and ε i+1 = ε i . • if x i = ∆ (and thus t i < +∞), then t i+1 = t i + εi τ xi ; if moreover εi τ xi < +∞ and εi L xi = ∅, then x i+1 = min εi L xi and ε i+1 = (ε i , εi N xi ),
where the min is for the lexical order on Z d . Otherwise, we set x i+1 = ∆ and ε i+1 = ε i .

Then, we define

K = min{k ≥ 1 : t k+1 = +∞}, T = t K , and D = x K . For i ≤ K and e ∈ -→ E d , we put W n e = εi η n-ti e for n ∈ [t i , t i+1 [. Finally, for each n ≥ 1 and each e ∈ -→ E d , we define W ′n e = εK η ′n e-xK .
This procedure, close to the classical so-called "restart argument" can be described as follows: starting from 0, we exhibit with {0} ν a coupling between dependent and independent percolations up to time t 1 = {0} τ 0 when independent percolation dies. Then, we record the history of the dependent percolation in ε 1 , and pick some point x 1 occupied by the dependent percolation process in the terminal configuration. We then construct another coupling ε1 ν between the dependent percolation and some new independent percolation process starting from x 1 , following this coupling until time t 2 when the new independent percolation also dies. We can complement the history of the dependent percolation and get ε 2 , then choose x 2 occupied by the dependent percolation process in the terminal configuration, and so on.

We will soon see that K is almost surely finite; hence t K < +∞ and t K+1 = +∞: this can occurs for two reasons:

• either εK τ xK = +∞, which means that the independent oriented percolation starting from x K at time t K lives for ever (and so does the dependent oriented percolation by stochastic domination); • or εK τ xK < +∞ and εK L xK = ∅, which means that the dependent oriented percolation died exactly at the same time as the independent oriented percolation starting from x K at time t K This procedure stops either because we find a time t K when our Kth independent percolation process survives, or because the dependent percolation process died together with the independent one. Let us denote by T n the σ-field generated by the ( E η, E η ′ ) |E|≤n . We have, for α > 0,

E[exp(α εn τ xn )1 1 {K>n} |T tn ] = E[exp(α εn τ xn )1 1 {tn+1<+∞} |T tn ] = E[exp(α εn τ xn )1 1 {tn<+∞, εn L xn =∅, εn τ xn <+∞} |T tn ] ≤ 1 1 {K>n-1} 1 1 {τ 0 <+∞} exp(ατ 0 ) dB(q ′ ) ⊗ - → E d ×N * . Thus, since q ′ > -→ p c alt (d + 1), if we put r = 1 1 {τ 0 <+∞} exp(ατ 0 ) dB(q ′ ) ⊗ - → E d ×N * , we can choose α > 0 small enough to have r < 1; then E[exp(αt n+1 )1 1 {K=n+1} ] ≤ E[exp(α( ε0 τ x0 + • • • + εn τ xn ))1 1 {K>n} ] ≤ rE[exp(αt n )1 1 {K>n-1} ] ≤ r n+1 , then E[exp(αT )] ≤ +∞ i=0 r i+1 = r 1 -r .
Particularly, since K ≤ T , we get the existence of exponential moments for K, and the fact that K is almost surely finite.

Stacking the conditional laws up, we can check that the field W has the desired distribution.

Assume that τ 0 (W ) < +∞ and K = k. Then t k < +∞ and t k+1 = +∞. For each n ∈ [t k , +∞[, we have by construction (W n e ) e∈ - →

E d = ( ε k η n-t k e ) e∈ - → E d . If ε k L x k = ∅, then t k+1 = t k + ε k τ x k = +∞, which implies that ε k τ x k = +∞, which can not happen because τ 0 (W ) < +∞. Thus ε k L x k = ∅, so τ 0 (W ) ≤ t k = T .
The inequality τ 0 (W ) ≥ t k directly follows from the inclusion between independent and dependent percolations. Finally, if τ 0 (W ) < +∞, then T = τ 0 (W ).

On the event {τ 0 (W ) = +∞}, we have by construction D ∈ ξ 0 T (W ), so the inclusion property gives ∀n ≥ 0 ξ 0 T +n (W )

⊃ D + ξ 0 n (W ′ ). Let B be any Borel set B in {0, 1} - → E d+1 alt and define, for x ∈ Z d , x.B = {(η n e+x ) e∈ - → E d ,n≥1 : η ∈ B}. Noting that {τ 0 (W ) = +∞, K = n, ε n = E, x n = x} ⊂ { E τ x =
+∞}, we get by independence that

P(τ 0 (W ) = +∞, K = n, ε n = E, x n = x, W ′ ∈ B) = P(τ 0 (W ) = +∞, K = n, ε n = E, x n = x, E η ′ ∈ (-x).B) = P(τ 0 (W ) = +∞, K ≥ n, ε n = E, x n = x, E η ′ ∈ (-x).B, E τ x = +∞) = P(τ 0 (W ) = +∞, K ≥ n, ε n = E, x n = x)P( E τ x = +∞, E η ′ ∈ (-x).B) = P(τ 0 (W ) = +∞, K ≥ n, ε n = E, x n = x)P q ′ (τ 0 = +∞, B)
Summing on all possible values for E, n, x, we obtain the existence of c such that

∀B ∈ B({0, 1} - → E d+1 alt ) P(τ 0 (W ) = +∞, W ′ ∈ B) = cP q ′ (τ 0 = +∞, B).
The constant c is identified by taking B = Ω, so we get P(W ′ ∈ B|τ 0 (W ) = +∞) = P q ′ (B|τ 0 = +∞).

Some properties of dependent oriented percolation

The coupling Theorem 2.5 permits to transfer some properties from supercritical independent oriented percolations to dependent oriented percolations in C d (M, q) for q close to 1. Practically, those processes often arise after the use of a dynamical renormalization scheme.

As a by-product of the proof of Theorem 2.5, we can get information on the exponential moments for extinction times. For oriented Bernoulli percolation, a Peierls-like argument shows that [START_REF] Durrett | Complete convergence theorem for a competition model[END_REF] lim

p→1 inf β>0 1 1 {τ 0 <+∞} exp(βτ 0 ) dP p = 0,
which can be transposed to the dependent fields of C d (M, q) as follows:

Corollary 3.1. Let ε > 0 and M > 1. There exist β > 0 and q < 1 such that for each χ ∈ C d (M, q),

E χ [1 1 {τ 0 <+∞} exp(βτ 0 )] ≤ ε.
Proof. We observed in the proof of Theorem 2.5 that T = τ 0 when {τ 0 < +∞}.

We also have the bound

E χ [1 1 {τ 0 <+∞} exp(βτ 0 )] ≤ E χ [e βT ] ≤ r 1 -r ,
with r = 1 1 {τ 0 <+∞} exp(βτ 0 ) dP gM (q) ; the result then follows from [START_REF] Durrett | Complete convergence theorem for a competition model[END_REF].

As a direct application of the coupling Theorem 2.5, the linear growth of the set H n of points reached before time n, given in Lemma 2.1 for independent directed percolation, can be transposed to any dependent percolation in C d (M, q): Corollary 3.2. Let d, M ≥ 1 be fixed positive integers, and let q ∈ (0, 1) be such that g M (q) > -→ p c alt (d + 1). There exist positive constants β, D 1 , D 2 and random variables (S y ) y∈Z d such that

∀y ∈ Z d E[e βS y ] ≤ D 2 ,
and such that for each field χ ∈ C d (M, q), the directed percolation associated to χ satisfies: on the event {τ y = +∞},

∀n ∈ N y + [-D 1 n, D 1 n] d ⊂ H y S y +n .
Having in mind an accurate study of certain particle systems, it could be interesting to have estimates on the density of bi-infinite points in the dependent oriented percolation. Thus, we define

G(x, y) = {k ∈ N (x, 0) → (y, k) → ∞} γ(θ, x, y) = inf{n ∈ N : ∀k ≥ n |{0, . . . , k} ∩ G(x, y)| ≥ θk} Corollary 3.3. Let M > 1.
There exist q 0 < 1 and positive constants A, B, θ, β such that for each χ ∈ C d (M, q 0 ), we have

∀x, y ∈ Z d ∀n ≥ 0 P(+∞ > γ(θ, x, y) > β x -y 1 + n) ≤ Ae -Bn .
For instance, those estimates allow to study the large deviations of the asymptotic shape of the contact process [START_REF] Garet | Large deviations for the asymptotic shape of the contact process in Z d[END_REF]. Considering Theorem 2.5, Lemma 3.3 will easily follow from the independent case. We define

Ĩ∞ = {(x, n) ∈ Z d × N : Z d × {0} → (x, n) → ∞}.
One notes that if x ∈ K 0 k and (x, k) ∈ Ĩ∞ , then by the definition of the coupled region K 0 k , (0

Z d , 0) → (x, k) → ∞.
Lemma 3.4. Consider independent directed percolation on Z d × N. For each ρ ∈ (0, 1), there exists p 0 (ρ) < 1 such that for each p > p 0 (ρ),

∀ finite A ⊂ {0} × N P p (A ∩ Ĩ∞ = ∅) ≤ 16ρ |A|-2 .
Proof. Note first that by inclusion, it is sufficient to prove the lemma for d = 1; when d = 1, we can use contour arguments. The oriented graph we defined is not the classical graph for oriented percolation in dimension 2: our graph has more edges. But once again, by inclusion, it is sufficient to prove the lemma for the classical oriented percolation model in dimension 2 (see for example Durrett [START_REF] Durrett | Oriented percolation in two dimensions[END_REF]), for which the dual graph is particularly simple. So we consider i.i.d. percolation with parameter p on the following oriented graph L + :

• The set of sites is V = {(z, n) ∈ Z × Z : |z| + n is even}. • There is 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.
The critical probability for this model is denoted by -→ p c . For the need of the proof, we define L -by simply reversing the oriented edges of L + . The state -open or closed -of an edge is the same in the two graphs. We denote by → + , resp. → -, the event of being linked by an open oriented path in L + , resp. in L -. As before, we define for ε ∈ {+, -}:

ξ x ε,n = {y ∈ Z : (y, n) ∈ L ε (1), (x, 0) → ε (y, n)}, τ x ε = max{εn ∈ N : ξ x ε,n = ∅}, I ε ∞ = {(x, n) ∈ L ε ; τ x ε • θ εn = +∞}, I ∞ = I + ∞ ∩ I - ∞ .
As Ĩ∞ ⊃ I ∞ , it is sufficient to prove the lemma when we replace Ĩ∞ by I ∞ : Let A be a fixed finite subset of {0} × 2N and let n be the smallest integer larger than |A|/2:

P p (A ∩ I ∞ = ∅) ≤ P p (∃B ⊂ A; |B| = n; B ∩ I + ∞ = ∅) + P p (∃B ⊂ A; |B| = n; B ∩ I - ∞ = ∅) ≤ 2P p (∃B ⊂ A; |B| = n; B ∩ I + ∞ = ∅) ≤ 2 B⊂A;|B|=n P p (B ∩ I + ∞ = ∅)
We work from now on with the graph L + . We fix a finite set B ⊂ A. For v ∈ V, denote by C(v) the open cluster starting from v:

C(v) = {w ∈ V : v → + w}. We set C f (v) = C(v) if C(v) is finite and C f (v) = ∅ otherwise.
We set

C f (B) = ∪ v∈B C f (v).
If C ⊂ V is a finite set of vertices, we denote by ∂ e C the set of edges entering in or exiting from C and by ∂ * e C the union of the segment lines corresponding to the dual edges of ∂ e C: it is a union of circuits. Note that

|∂ e C| ≥ 2|C ∩ ({0} × 2Z)|. Thus, as B ⊂ A ⊂ {0} × 2Z, {B ∩ I + ∞ = ∅} ⊂ {B ⊂ C f (B)} ⊂ {|∂ e C f (B)| ≥ 2|B|} and so P p (B ∩ I + ∞ = ∅) ≤ i≥|B|/2 P(|∂ e C f (B))| = 4i) = i≥|A|/4 P(|∂ e C f (B)| = 4i).
Let i be a fixed integer and assume that |∂ e C f (B)| = 4i. Note first that all edges exiting from C f (B) must be closed. Looking on a "diagonal line", we see that there are at least as many edges exiting from C f (B) than edges entering in C f (B) (here, we count an edge which is both entering in C f (B) and exiting from C f (B) as an exiting edge), and thus at least half edges in C f (B) must be closed. Next, ∂ * e C f (B) is composed of at most i circuits. In C f (B), consider the set of minima for the order relation →: all edges entering B in these points are necessarily in ∂ e C f (B), which allows to root the circuits of ∂ * e C f (B) to some points in B. So,

P p (∂ e C f (B)) = 4i) ≤ |B| i 4 4i P 4i k=1 X k ≤ 2i ≤ 2 n 4 4i P 4i k=1 X k ≤ 2i ,
where (X k ) k≥1 are i.i.d random variable with Bernoulli law of parameter p. Now, large deviations inequalities imply that for every r ∈ (0, 1), there exists p(r) ∈ (0, 1) such that ∀p ≥ p(r),

P 4i k=1 X k ≤ 2i ≤ r 4i .
Let then ρ ∈ (0, 1) be fixed, and apply the previous estimate for r = ρ 4 √ 2 ∈ (0, 1). This gives, for every p ≥ p(r),

P p (A ∩ I ∞ = ∅) ≤ 2 B⊂A;|B|=n P p (B ∩ I + ∞ = ∅) ≤ 2 i≥|A|/4 P(∂ e C f (B)) = 4i) ≤ 4 × 2 |A|/2 i≥|A|/4 (4r) 4i ≤ 4 1 -4r (4 √ 2r) |A| ≤ 16ρ |A| .
Lemma 3.5. We consider independent directed percolation on Z d × N. There exist positive constants A, B, θ, β and p < 1 such that for every x, y ∈ Z d ,

(6) ∀n ∈ N P p (τ x = +∞, γ(θ, x, y) ≥ β y -x ∞ + n) ≤ Ae -Bn .
Proof. We actually prove the following simpler result: there exists p close to 1, positive constants A, B, C ′ , θ such that ∀x ∈ Z d ∀n ∈ N

P p τ 0 = +∞ |k ∈ {C ′ x ∞ , . . . , C ′ x ∞ + n : (0 Z d , 0) → (x, k) → ∞}| ≤ θn ≤ Ae -Bn . (7)
Let us show that ( 7) implies [START_REF] Durrett | Oriented percolation in two dimensions[END_REF]. We note that γ(θ, x, y) has the same distribution as γ(θ, 0, y -x), that θ < 1 and use [START_REF] Durrett | Asymptotic critical value for a competition model[END_REF]:

P p τ 0 = +∞, γ(θ, 0, x) ≥ C ′ θ x ∞ + n = P p τ 0 = +∞, ∃k ≥ C ′ θ x ∞ + n |{l ∈ {0..k} : (0 Z d , 0) → (x, l) → +∞}| ≤ θk ≤ P p τ 0 = +∞, ∃k ≥ n |{l ∈ {C ′ x ∞ , . . . , C ′ x ∞ + k} : (0 Z d , 0) → (x, l) → +∞}| ≤ θk ≤ k≥n P p τ 0 = +∞, |{l ∈ {C ′ x ∞ , . . . , C ′ x ∞ + k} : (0 Z d , 0) → (x, l) → +∞}| ≤ θk ≤ k≥n A exp(-Bk).
Taking β = C ′ /θ, this proves [START_REF] Durrett | Oriented percolation in two dimensions[END_REF].

Let us now prove [START_REF] Durrett | Asymptotic critical value for a competition model[END_REF]. We define

Ĩ∞ = {(x, n) ∈ V d+1 : Z d × {0} → (x, n) → ∞}.
One notes that if x ∈ K 0 k and (x, k) ∈ Ĩ∞ , then by the definition of the coupled region, (0 Z d , 0) → (x, k) → ∞. We take C ′ = ⌈1/C⌉, where C comes from Lemma 2.1: we choose any θ with 0 < θ < 1/4 then

P p (τ 0 = +∞, |{k ∈ {C ′ x ∞ , . . . , C ′ x ∞ + n} : (0 Z d , 0) → (x, k) → ∞}| ≤ θn) ≤ P p (τ 0 = +∞, ∃k ≥ C ′ x ∞ + n 2 , K 0 k ⊃ [-CC ′ x ∞ , CC ′ x ∞ ] d ) +P p (|{k ∈ {C ′ x ∞ + n/2, . . . , C ′ x ∞ + n} : (x, k) ∈ Ĩ∞ }| ≤ θn)
For the first term, we use Lemma 2.1:

P p (τ 0 = +∞, ∃k ≥ C ′ x ∞ + n 2 , K 0 k ⊃ [-CC ′ x ∞ , CC ′ x ∞ ] d ) ≤ k≥n/2 P p (τ 0 = +∞, K C ′ x ∞+k ⊃ [-CC ′ x ∞ , CC ′ x ∞ ] d ) ≤ k≥n/2
A exp(-Bk).

To control the second term, we use Lemma 3.4. With its notation, we choose 0 < ρ < 1 such that 2ρ 1/2 < 1 and obtain, for p ≥ p 0 (ρ),

P p |k ∈ {C ′ x ∞ + n 2 , . . . , C ′ x ∞ + n} : (x, k) ∈ Ĩ∞ | ≤ θn ≤ 2 n/2+1 16ρ n/2-θn-3 .
This concludes the proof of ( 7), and therefore of the Lemma.

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, where x ∈ E denotes 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 in such a way 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 4.1. We suppose that there exist real numbers A > 0, c < 1, p > 0, β > 0, and that the real-valued functions (G x ) x∈E , (F x ) x∈E defined above are such that

∀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 < +∞.
Before the proof, we note that we could give a statement about Markov chains avoiding the use of an update function, by working directly with the trajectory space of the Markov chain rather than with the generic underlying space: in that way, P(T x = +∞) would be replaced by P x (T = +∞) and a lot of formulas would be simpler. However, the processes we plan to apply this lemma to are often built from a graphical construction (here, the Ω where the growth model lives) and the functions G . , H . we plan to apply the lemma to are defined from the graphical representation, and not from the Markov chain.

Proof. We can assume without loss of generality that β = 1.

Let x ∈ E be fixed. At first, we have for each n ≥ 0

P[K x > n|F T x n ] = P(T x n+1 < +∞|F T x n ) = P(T x n < +∞, T xn (θ T x n ) < +∞|F T x n ) = 1 1 {T x n <+∞} (1 -T(x n )) ≤ (1 -p)1 1 {T x n <+∞} = 1 1 {K x >n-1} (1 -p), then P(K x > n) ≤ (1 -p)P(K x > n -1)
, which ensures that K x is P-almost surely finite.

Let S x -1 = 1 and, for k ≥ 0, put

S x k = exp k i=0 G xi (θ T x i ) 1 1 {T x k+1 <+∞} . We note that S x k is F T x k+1 -measurable. For k ≥ 0, one has exp(M x )1 1 {K x =k} = S x k-1 1 1 {T x k •θ T x k =+∞} exp(F x k ), hence by the strong Markov property E[exp(M x )1 1 {K x =k} |F T x k ] = S x k-1 F(x k ), then E[exp(M x )1 1 {K x =k} ] ≤ AE[S x k-1 ]. For k ≥ 1, the strong Markov property gives again E[S x k+1 |F T x k+1 ] = S x k × G(x k+1 ), then E[S x k+1 ] ≤ cE[S x k ], and E[exp(M x )1 1 {K x =k} ] ≤ Ac k .
We conclude the proof by summing on k.

Application to the Model

5.1. Dependence to initial conditions. We first prove that the positivity of the probability of survival for the bacteria does not depend on the initial condition of the environment. We can note that Steif and Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF] have proved a similar result for the model introduced by Broman [START_REF] Broman | Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment[END_REF].

Proof of Theorem 1.1. Let p > -→ p c , q < -→ p c , α > 0 such that P p,q,α (τ 0,∅ 1 = +∞) > 0.

We want to show that P p,q,α (τ 0,Z d \{0} 1 = +∞) > 0. Let us denote by C n the event : "there exists x ∈ [-n, n] d such that Z d × {0} is linked to (x, n) by open bonds of directed oriented percolation with parameter q": by a time reversal argument, we get

P p,q,α (C n ) ≤ (2n + 1) d P q (T > n) ≤ A exp(-Bn),
where T is the extinction time of some subcritical oriented percolation process with parameter q. We conclude that, if

A N = ∩ k≥N C c k , lim N →+∞ P p,q,α (A N -1 • θ 1 ) = lim N →+∞ P p,q,α (A N -1 ) = 1,
whence lim

N →+∞ P p,q,α (τ 0,∅ 1 = +∞, A N -1 • θ 1 ) = P p,q,α (τ 0,∅ 1 = +∞).
In particular, there exists N such that P p,q,α (τ

0,∅ 1 = +∞, A N -1 • θ 1 ) > 0.
Let us denote by B the event: "all the oriented edges issued from [-3N, 3N ] d × {0} are closed for the percolation with parameter q". By independence, one has

P p,q,α (τ 0,∅ 1 = +∞, A N -1 • θ 1 , B) = P p,q,α (τ 0,∅ 1 = +∞, A N -1 • θ 1 )P p,q,α (B) > 0.
It remains to prove that τ 0,Z d \{0} 1

= +∞ holds on this event. It is sufficient to prove that the processes (η

0,Z d \{0} 1,n
) n≥0 and (η 0,∅ 1,n ) n≥0 coincide on this event; but because of the definition of the dynamics, it is sufficient to note that on the event

(A N -1 • θ 1 ) ∩ B, we have ∀n ≥ 1 η ∅ 2,n ∩ [-n, n] d = η Z d \{0} 2,n ∩ [-n, n] d ,
which ends the proof.

5.2.

Outline of the proof of Theorem 1.2. The idea of the proof is to define a local block event with probability close to 1, that expresses the fact that if the bacterium occupies a sufficiently large area at a given place, it will presumably extend itself a bit further. If the associated block process percolates, then the linear growth is ensured by Theorem 2.5. With a restart argument, we will find a point of the space-time, not too far from the origin, where the bacterium occupies a sufficiently large area and where the associated block process percolates, which will give the desired result. The statement in Theorem 1.2 actually contains two results that must be proved separately. On one side, there is the fact that α c > 0, on the other side the fact that for α < α c , the process, when surviving, linearly grows. We can find in the literature many examples of block events similar to the ones we will use. Most of these papers take inspiration from the Bezuidenhout-Grimmett article [START_REF] Bezuidenhout | The critical contact process dies out[END_REF]: The critical contact process dies out. We think that for this kind of dynamical renormalization schemes, the existence of a coupling between the dependent oriented percolation of blocks and a Bernoulli oriented percolation conditioned to survive is barely explained in the literature. This led us to write Theorem 2.5. In Subsection 5.3, we focus on the case where α is small and the renormalization event simpler. The construction for α < α c , technically more subtle, is explained in Subsection 5.4. Figure 1. A simulation with p = 0.7, q = 0.25, and α = 10 -3 . 5.3. Positivity of α c (the case of small α). We prove here that when α is small enough, P p,q,α (τ 0,Z d \{0} 1 = +∞) > 0 and the growth is linear on the event {τ 0,Z d \{0} 1 = +∞}.

The block event.

Let I, L ∈ N * with I < L. Recall that the constant C is given in Lemma 2.1. We let T = 6CL and J = 2(L + T ).

For

k ∈ Z d , x ∈ [-L, L[ d and u ∈ Z d such that u 1 ≤ 1,
we define the following event:

A(k, x, u) =              ∃s ∈ [-L, L[ d , 2L(k + u) + s + [-I, I] d ⊂ η 2Lk+x+[-I,I] d ,Z d \(2Lk+[-J,J] d ) 1,T , η 2Lk+x+[-I,I] d ,Z d \(2Lk+[-J,J] d ) 2,T ∩ (2L(k + u) + [-J, J] d ) = ∅, 2Lk + [-L, L] d ⊂ ∪ 0≤t≤T η 2Lk+x+[-I,I] d ,Z d \(2Lk+[-J,J] d ) 1,t .              .
If A(k, x, u) holds, we denote by s(k, x, u) an element s satisfying the condition above.

Let us briefly explain the signification of the event A(k, x, u): obviously, η A,B 1,T is non-decreasing with respect to A, non-increasing with respect to B, whereas η B 2,T is non-decreasing with respect to B. Thus, if A(k, x, u) holds and if one knows that at time 0, the block 2Lk + x + [-I, I] d is full of "1" and the block 2Lk + [-J, J] d contains no "2", then one knows that analogous conditions will be fulfilled around 2L(k + u) at time T . Of course, the idea is to follow a chain of such events in an oriented percolation and to draw a path ensuring the development of the bacteria.

Lemma 5.1. For each p > -→ p c alt (d + 1), each q < -→ p c alt (d + 1), and each ε > 0, we can find integers I < L large enough and α ∈ (0, 1) small enough such that for every

k ∈ Z d , x ∈ [-L, L[ d and u ∈ Z d such that u 1 ≤ 1, (8) P p,q,α (A(k, x, u)) ≥ 1 -ε.
Moreover, as soon as k -l 1 > 4 + 18C, for every x, y ∈ [-L, L] d , every u, v ∈ Z d such that u 1 ≤ 1 and v 1 ≤ 1, the events A(k, x, u) and A(l, y, v) are independent.

Proof. First note that P p,q,α (A(k, x, u)) = P p,q,α (A(0, x, u)), which allows to consider only the case n = 0. Under P p,q,α , the collection of random variables

ω 1 = (ω e 1,n ) e∈ - → E d ,n∈N * -resp. ω 2 = (ω e 2,n ) e∈ - → E d ,n∈N * -
has the law of the bonds of an independent directed percolation with parameter presp. q. We realize these percolation structures on Ω, keeping the notation of the introduction: thus, (ξ A n (ω 1 )) n≥0 is under P p,q,α a directed Bernoulli percolation process with parameter p starting from the set A, and (τ x 1 (ω 2 )) n≥0 is under P p,q,α the extinction time for a directed Bernoulli percolation process with parameter q starting from x. Under P p,q,α , the collection of random variables ω 3 = (ω e 3,n ) e∈ - → E d ,n∈N * are independent Bernoulli with parameter α. They represent the immigration of immune cells.

Let ε > 0. We choose two integers I, L with I < L -their values will be fixed later. Define

B(x, u) =            ∃s ∈ [-L, L[ d 2Lu + s + [-I, I] d ⊂ ξ x+[-I,I] d T (ω 1 ), ∀(y, n) ∈ [-(4L + 2T ), (4L + 2T )] d × {1, . . . , T } ω y,n 3 = 0 and τ y 1 • θ n (ω 2 ) ≤ T /2, [-L, L] d ⊂ ∪ 0≤t≤T ξ x+[-I,I] d t (ω 1 ).            .
We will show that B(x, u) ⊂ A(0, x, u) and also that one can choose I and L in such a way that P p,q,α (B(x, u)) ≥ 1 -ε, which will give the desired result. The advantage of using B is that it does not deal with the competition process, using only the directed percolation and the immigration processes. Thus, it is easier to estimate its probability.

Step 1. Let us show that B(x, u) ⊂ A(0, x, u).

The existence of a convenient s for the condition of A(0, x, u) is given by B(x, u) for the oriented percolation with parameter p embedded in the model. We have now to verify that our event ensures that the type 2 particles can not disturb the progress of type 1 particles.

Note A = x + [-I, I] d and B = Z d \[-J, J] d . At time 0, the smallest distance between points in η A,B 1,0 and η B 2,0 is at least 2L + 2T -(L + I) > 2T . In the zone [-J, J] d , there is no immigration between time 0 and time T , so η A,B 1,t and η B 2,t get closer at a speed that does not exceed 2 per time unit; thus at time T , the type 2 particles could not disturb the move of type 1 particles yet.

It remains to see that η 2,T can not reach 2Lu + [-J, J] d . Remember that there is no immigration between time 0 and time T in the area [-(4L + 2T ), (4L + 2T )] d . Moreover, type 2 particles that are outside [-(4L + 2T ), (4L + 2T )] d at time 0 do not have enough time to reach 2Lu + [-J, J] d at time T , so only type 2 particles that were already inside [-(4L + 2T ), (4L + 2T )] d at time 0 must be considered. But these ones are all dead at time T /2. This completes the proof of the inclusion.

Step 2. Bounding the probability of B(x, u) from below. Remember that P = P p,q,α . We first choose an integer I large enough to have [START_REF] Durrett | Ten lectures on particle systems[END_REF] ∀x ∈ Z d P(τ

x+[-I,I] d 1 (ω 1 ) = +∞) ≥ 1 -ε/12.
By the FKG inequality, P(∀y ∈ [-I, I] d , τ y 1 (ω 1 ) = +∞) > 0. Translation invariance and ergodicity of P then give

lim L→+∞ P(∃n ∈ [0, L] : ∀y ∈ nu + [-I, I] d , τ y 1 (ω 1 ) = +∞) = 1.
Then, let L 1 > I be such that for each L ≥ L 1 ,

P(∃n ∈ [0, L] : ∀y ∈ nu + [-I, I] d , τ y 1 (ω 1 ) = +∞) > 1 - ε 12 .
Let L ≥ L 1 . By a time reversal argument, we have for each t > 0, (10)

P ∃n ∈ [0, L] : nu + [-I, I] d ⊂ ξ Z d t (ω 1 ) = P ∃n ∈ [0, L] : ∀y ∈ nu + [-I, I] d , τ y 1 (ω 1 ) ≥ n ≥ 1 - ε 12 .
Now, Lemma 2.1 gives the existence of some L 2 ≥ L 1 such that for each L ≥ L 2 , we have simultaneously

P(∃y ∈ [-2L, 2L] d : τ y 1 (ω 1 ) = +∞, Lu + [-2L, 2L] d ⊂ K y 6CL (ω 1 ))) ≤ (4L + 1) d P(τ 0 1 (ω 1 ) = +∞, [-5L, 5L] d ⊂ K 0 6CL (ω 1 )) ≤ ε/12, (11)
and

P(∃y ∈ [-2L, 2L] d : τ y 1 (ω 1 ) = +∞, [-L, L] d ⊂ H y 6CL (ω 1 )) ≤ (4L + 1) d P(τ 0 1 (ω 1 ) = +∞, [-3L, 3L] d ⊂ H y 6CL (ω 1 )) ≤ ε/12. (12)
With ( 10) and ( 11), we get

P(τ x+[-I,I] d 1 (ω 1 ) = +∞, ∀n ∈ [0, L] Lu + nu + [-I, I] d ⊂ ξ x+[-I,I] d T (ω 1 )) ≤ P(∃y ∈ x + [-I, I] d : τ y 1 (ω 1 ) = +∞, Lu + [-2L, 2L] d ⊂ K y T (ω 1 )) +P(∀n ∈ [0, L] Lu + nu + [-I, I] d ⊂ ξ Z d T (ω 1 )) ≤ ε 6 .
With ( 9) and ( 12), we conclude that for each

x ∈ [-L, L] d (13) P H x+[-I,I] T (ω 1 ) ⊃ [-L, L] d , ∃n ∈ [0, L] (L + n)u + [-I, I] d ⊂ ξ x+[-I,I] T (ω 1 ) ≥ 1 - ε 3 .
Since q < -→ p c alt (d + 1), there exist positive constants A, B such that for each L,

P(∃y ∈ [-(4L + 2T ), (4L + 2T )] d : τ y 1 (ω 2 ) > T /2) ≤ (8L + 4T + 1) d A exp(-BT /2
). One deduces that there exists some integer L 3 ≥ L 2 such that for each L ≥ L 3 , ( 14) P(∃y ∈ [-(4L + 2T ), (4L + 2T )] d : τ y 1 (ω 2 ) > T /2) ≤ ε/3. Now fix L ≥ L 3 and choose α > 0 small enough to have [START_REF] Liggett | Domination by product measures[END_REF] P(∃(y, n) ∈ [-(4L + 2T ), (4L + 2T )] d × {1, . . . , T } ω y 3,n = 1) ≤ ε/3. We conclude by putting ( 13), ( 14) and ( 15) together.

Block events percolation.

Let I < L be fixed integers. First, for each x ∈ Z d , we will build a field ( x W n (z,u) ) n≥1,z∈Z d , u 1≤1 from the events defined above. The random variable W n+1 (z,u) will give the state of the oriented bond between the macroscopic sites (z, n) and (z + u, n + 1); those sites correspond to the coordinates of the boxes (2Lz, nT

) + [-L, L] d × [1, T ] and (2L(z + u), (n + 1)T ) + [-L, L] d × [1, T ]. The field ( x W n (z,u) ) n≥1,z∈Z d , u 1 ≤1
then defines a macroscopic dynamical dependent oriented percolation.

For x ∈ Z d , we denote by [x] 2L ∈ Z d the unique integer such that

x ∈ 2L[x] 2L + [-L, L[ d and we set {x} 2L = x -2L[x] 2L ∈ [-L, L[ d .
We set

d x 0 ([x] 2L ) = {x} 2L and also d x 0 (k) = +∞ for every k ∈ Z d that is not equal to [x] 2L . Then, for each k ∈ Z d , each u ∈ Z d with u 1 ≤ 1 and each n ≥ 1, we recursively define: • If d x n (k) = +∞, x W n+1 (k,u) = 1. • Otherwise, x W n+1 (k,u) = 1 1 A(k,d x n (k),u) • θ nT , d x n+1 (k) = min{s(k -u, d x n (k -u), u) • θ nT : u 1 ≤ 1, d x n (k -u) = +∞}. Let G n = σ(ω e,k 1 , ω e,k 2 , ω x,k 3 , e ∈ -→ E d , x ∈ Z d , k ≤ nT ).
Note that conditionally to G n , the random variables x W n+1 (k,u) et x W n+1 (l,v) are independent as soon as k -l 1 > 4 + 18C. Then we take M = 5 + 18C, and prove the following lemma: Lemma 5.2. For each p > -→ p c alt (d + 1), q < -→ p c alt (d + 1), and q 0 < 1, we can find some integers I < L and a parameter α > 0 such that for each x ∈ Z d , the law of ( x W n e ) n≥0,e∈ - → E d under P p,q,α belongs to C(M, q 0 ). Proof. Note that for every x, k ∈ Z d , for each n ≥ 1, the variable

d x n (k) is G n - measurable, and so does x W n (k,u) . Let us now consider x, k ∈ Z d , n ≥ 0 and u ∈ Z d such that u 1 ≤ 1: Lemma 5.1 ensures that E p,q,α [ x W n+1 (k,u) |G n ∨ σ( x W n+1 (l,v) , v 1 ≤ 1, l -k 1 ≥ M )] = E p,q,α [ x W n+1 (k,u) |G n ] = 1 1 {d x n (k)=+∞} + 1 1 {d x n (k)<+∞} P p,q,α [ x W n+1 (k,u) = 1|d x n (k) < +∞] = 1 1 {d x n (k)=+∞} + 1 1 {d x n (k)<+∞} P p,q,α [A(k, d x n (k), u)].
With Lemma 5.1, one can find some integers I < L and a parameter α > 0 in such a way that

E p,q,α [ x W n+1 (k,u) |G n ∨ σ( x W n+1 (l,v) , v 1 ≤ 1, l -k 1 ≥ M )] ≥ q 0 .
This completes the proof of the Lemma.

5.3.3.

From macroscopic to microscopic scale.

Proof of Theorem 1.2 for small α. The inequality α c ≤ 1-1 2d+1 easily follows from a counting argument. Let p > -→ p c alt (d + 1), q < -→ p c alt (d + 1), and take M = 5 + 18C as previously. By Lemma 3.1, we can find q 0 < 1 with g M (q 0 ) > -→ p c alt (d + 1) and β 0 > 0 such that for each field χ ∈ C d (M, q 0 ):

(16) E χ [1 1 {τ 0 1 <+∞} exp(β 0 τ 0 1 )] ≤ 1 2 .
We choose I, L, α as determined by Lemma 5.2. We will prove that for this α, the survival of the bacteria is possible, as well as the other announced estimates. Let x ∈ {0, 1, 2} Z d be some configuration; we denote by E 1 (x) the set of sites occupied by type 1 particles in configuration x. If E 1 (x) = ∅, we denote by j(x) the smallest point in 1 (x) (in lexicographic order). Note that there exists c > 0 such that [START_REF] Luo | The Richardson model in a random environment[END_REF] ∀x ∈ {0, 1, 2}

Z d P p,q,α (η x 1,4dT ⊃ j(x) + [-4T, 4T ] d ) ≥ c. Indeed, it is sufficient to open for ω 1 every bond in B = (j(x), 0) + [-4dT -1, 4dT + 1] d × [0, 4T ],
to close for ω 2 every bond in B, and to forbid in ω 3 every birth of type 2 in B: all of this corresponds to fixing a finite number of coordinates in ω, which can be done with a positive probability.

If the event in (17) happens, we have at time 4dT a large box j(x) + [-4T, 4T ] d occupied by type 1 particles. From this box, we can start the macroscopic percolation by building the random field

x W = ( j(x) W n e • θ 4dT ) e∈ - → E d ,n≥0
. The choice we made for I, L, α and Lemma 5.2 ensure that x W belongs to C d (M, q 0 ). Since g M (q 0 ) > -→ p c alt (d + 1), [START_REF] Luo | The Richardson model in a random environment[END_REF] gives

P p,q,α (τ x 1 = +∞) ≥ P p,q,α (η x 1,4dT ⊃ j(x) + [-4T, 4T ] d )P gM (q0) (τ 0 1 = +∞) > 0,
which proves (1).

To show the exponential estimates, we will apply Lemma 4.1. If E 1 (x) = ∅, we let T x = +∞; otherwise, let

T x = 4dT if the event in (17) does not occur, 4dT + T × τ [j(x)]2L 1 • θ 4dT otherwise, where τ [j(x)]2L 1
represents the extinction time in the percolation x W starting from the macroscopic site [j(x)] 2L containing j(x).

For each x ∈ {0, 1, 2} Z d such that E 1 (x) = ∅, we have

P p,q,α (T x = +∞) ≥ cP gM (q0) (τ 0 1 = +∞).
We take G x = T x ; for 0 < β 1 < β 0 , Inequality [START_REF] Lindvall | On Strassen's theorem on stochastic domination[END_REF] gives:

E p,q,α [e β1T x 1 1 {T x <+∞} ] ≤ e β14dT sup χ∈C(M,q0) E χ [1 1 {τ 0 1 <+∞} exp(β 0 τ 0 1 )] ≤ e β14dT /2 ≤ 2/3
provided that β 1 is small enough. We take F ∅ = 0 and for x = ∅,

F x = T × S [j(x)]2L • θ 4dT
where S has been defined in Corollary 3.2. This corollary moreover gives the existence of exponential moments for S. Thus, the restart lemma ensures that the variable

M x = T x K + F η x T x K • θ T x K admits exponential moments.
Let us begin to work on the event {τ x 1 = +∞}. In that case, η x 1,T x K is non-empty and, at time T x K + 4dT , the bacteria occupy a large box j(η x T x K +4dT ) + [-4T, 4T ] d , from which the macroscopic percolation lives forever; moreover,

M x = T x K + T × S [j(η x T x K +4dT )]2L • θ T x K +4dT
. By the definition of the macroscopic percolation, if the bond

j(η x T x K +4dT ) W n k,u • θ T x K +4dT
is open, then every point in the box 2Lk+[-L, L] d is visited by the bacteria between time T x K +4dT +nT and time T x K +4dT +(n+1)T . In particular, using Corollary 3.2, it comes that

∀n ∈ N 2L[j(η x T x K +4dT )] 2L + [-2nD 1 L, 2nD 1 L] d ⊂ ∪ 0≤m≤n+M x +4dT η x 1,m ;
we then deduce Estimate (2) and the existence of exponential moments for M x and T x K . Finally, since {τ x 1 < +∞} ⊂ {τ x 1 ≤ M x }, Estimate (3) follows from the bound for the exponential moments of M x given by Lemma 4.1; this completes the proof of Theorem 1.2 for small α.

5.4. The case α < α c (p, q): the Bezuidenhout-Grimmett way. We fix p, q, α such that P p,q,α (τ

0,Z d \{0} 1 = +∞) = P(τ 0,Z d \{0} 1 = +∞) > 0,
or in other words such that α < α c (p, q). The proof for the linear growth of the bacteria conditioned to survive is, as in the case of a small α, based on a renormalization process leading to the construction of a d-dimensional supercritical oriented percolation.

In the previous case, when building the local block event, we could choose α small enough for our model to behave nearly as independent oriented percolation. This is no longer the case when α is close to α c . Instead, we adapt the strategy developed by Bezuidenhout-Grimmett [START_REF] Bezuidenhout | The critical contact process dies out[END_REF] for the supercritical contact process on Z d , which is also the one followed by Steif-Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF] in the case of a contact process where the death rate depends on a dynamical environment. We will closely follow the proofs exposed by Liggett in [START_REF] Thomas | Stochastic interacting systems: contact, voter and exclusion processes[END_REF] p 45-54 and by Steif-Warfheimer in [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF]. The key point is the following proposition (which corresponds to Proposition 2.22 in Liggett [START_REF] Thomas | Stochastic interacting systems: contact, voter and exclusion processes[END_REF] or Lemma 4.10 in Steif-Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF]). We denote by V the set of e ∈ Z d with e 1 ≤ 1. 5.4.1. The block event.

Proposition 5.3. Let ε > 0 and k ≥ 1 be fixed. There exists n, a, b with n < a such that for every u ∈ V , every n0 ∈ Z d , every x 0 ∈ [-a, a] d , and every t 0 ∈ [0, b], we can define random variables Y ( n0 , u, x 0 , t 0 ) ∈ Z d and S( n0 , u, x 0 , t 0 ) ∈ N ∪ {+∞} such that

• Y ( n0 , u, x 0 , t 0 ) ∈ [-a, a] d • S( n0 , u, x 0 , t 0 ) ∈ [5kb, (5k + 1)b] ∪ {+∞} • y + 2ka( n0 + u) + [-n, n] d ⊂ η x0+2ka n0+[-n,n] d ,Z d \(x0+2ka n0+[-n,n] d ) 1,s-t0
• θ t0 on the event {Y ( n0 , u, x 0 , t 0 ) = y, S( n0 , u, x 0 , t 0 ) = s} • P p,q,α (S( n0 , u, x 0 , t 0 ) < +∞) ≥ 1 -ε • The event {Y ( n0 , u, x 0 , t 0 ) = y, S( n0 , u, x 0 , t 0 ) = s} belongs to the σ-algebra generated by the background random variables related to the space-time area

k-1 ∪ j=0 ([-5a, 5a] d × [0, 6b]) + (2jau, 5jb) ∩ (Z d × [t 0 , s]).
Remark 5.4. Note that the event {Y ( n0 , u, x 0 , t 0 ) = y, S( n0 , u, x 0 , t 0 ) = s} belongs to the σ-algebra generated by the background random variables related to the (moderately) simpler space-time area

Z d × [0, 5kb] ∪ ([-7a + 2ka( n0 + u), 7a + 2ka( n0 + u)] × [5kb, 5kb + s]) .
The idea of this proposition is the following: starting from a fully occupied source square (x 0 + 2a n0 , t 0 ) + [-n, n] d × {0}, bacteria can with high probability colonize a (random) target square (Y ( n0 , u, x 0 , t 0 ) + 2a( n0 + u), S( n0 , u, x 0 , t 0 )) + ([-n, n] d × {0}), in a manner measurable with respect to the background random variables related to the space-time area

k-1 ∪ j=0 ([-5a, 5a] d × [0, 6b]) + (2jau, 5jb) ∩ (Z d × [t 0 , s]).
The occurence of this event will correspond to the opening of the macroscopic edge between the macroscopic sites ( n0 , 0) and ( n0 + u, 1), corresponding respectively to microscopic coordinates (2a n0 , 0) and (2a( n0 + u), 5b). Note that the source square and the target square are floating, in the sense that their respective centers (x 0 + 2a n0 , t 0 ) and (Y ( n0 , u, x 0 , t 0 ) + 2a( n0 + u), S( n0 , u, x 0 , t 0 )) are only known to be in the boxes (2a n0 ,

0) + [-2a, 2a] d × [0, b] and (2a( n0 + u), 5b) + [-2a, 2a] d × [0, b] .
The measurability properties of the concerned event will thus be crucial to control the dependence of the d-dimentional percolation process. The possibility for type 2 particles coming from outside a box to influence what happens inside the box leads us to state our proposition in a way that differs from Liggett and Steif-Warfheimer.

We split its proof in several lemmas. For each of these lemmas, we quote the corresponding results in Liggett [START_REF] Thomas | Stochastic interacting systems: contact, voter and exclusion processes[END_REF] and Steif-Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF].

First, as the bacteria survive, we can take a source square large enough to ensure that with high probability, bacteria starting from this square will survive whatever the configuration outside the square is: Lemma 5.5 (Proposition 2.1 in Liggett [START_REF] Thomas | Stochastic interacting systems: contact, voter and exclusion processes[END_REF], Lemma 4.1 in Steif-Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF]).

lim n→+∞ P ∀t ∈ N η [-n,n] d ,Z d \[-n,n] d 1,t = ∅ = 1.
Proof. Let n be fixed. By monotonicity,

P ∀t ∈ N η [-n,n] d ,Z d \[-n,n] d 1,t = ∅ ≥ P ∃x ∈ [-n, n] d ∀t ∈ N η {x},Z d \{x} 1,t = ∅ ;
this last terms converges, when n goes to infinity, to

P ∃x ∈ Z d ∀t ∈ N η {x},Z d \{x} 1,t = ∅ ,
which is the probability of a translation invariant event. By ergodicity, this probability is either null or full, and as α < α c (p, q), it is positive and thus equal to 1.

Then, to control the spatial dependence, we define a truncated process: for every positive integer L, for every finite

A ⊂ (-L, L) d ∩ Z d , the process ( L η A,Z d \A 1,t ) t∈N evolves as (η A,Z d \A 1,t
) t∈N , except that outside the space-time box (-L, L) d × N, all sites are in state 2 -which is the worst case from the bacteria point of view. Thus

P | L η A,Z d \A 1,t | ≥ N = P ∀t ∈ N η A,Z d \A 1,t = ∅ .
Proof. Let A be a fixed finite subset of Z d , and N be a fixed positive integer. Let us first note that

∀t ∈ N η A,Z d \A 1,t = L∈N L η A,Z d \A 1,t .
Indeed, the inclusion ⊃ follows from positivity, and if L ≥ A ∞ + 2t + 1, then for every s ≤ t, η

A,Z d \A 1,s = L η A,Z d \A 1,s . Thus |η A,Z d \A 1,t | ≥ N = L∈N | L η A,Z d \A 1,t | ≥ N ,
and thus lim

L→+∞ P | L η A,Z d \A 1,t | ≥ N = P |η A,Z d \A 1,t | ≥ N .
Now, for s ∈ N, denote by F s the σ-algebra generated by all the Bernoulli random variables indexed by a time coordinate smaller than or equal to s. By blocking the edges that allow the expansion of the 1's, we see that

P τ A,Z d \A 1 < +∞ | F s ≥ (1 -p) (2d+1)|η A,Z d \A 1,s | .
By the martingale convergence theorem, lim

s→+∞ P τ A,Z d \A 1 < +∞ | F s = 1 1 {τ A,Z d \A 1 <+∞} . So on the event {τ A,Z d \A 1 = +∞}, lim s→+∞ |η A,Z d \A 1,s | = +∞, which implies (by domi- nated convergence for instance) that lim s→+∞ P(τ A,Z d \A 1 = +∞, |η A,Z d \A 1,s | ≤ N -1) = 0. Finally, lim t→+∞ P |η A,Z d \A 1,t | ≥ N = P(τ A,Z d \A 1 = +∞).
Then, using the FKG inequality with a classical square root trick, we can ensure that the truncated process at time t contains many points in a given orthant of Z d : Lemma 5.7 (Proposition 2.6 in Liggett [START_REF] Thomas | Stochastic interacting systems: contact, voter and exclusion processes[END_REF], Proposition 4.5 in Steif-Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF]). For every positive integers n, N, t, for every integer L ≥ n,

P | L η [-n,n] d ,Z d \[-n,n] d 1,t ∩ [0, L) d | ≤ N 2 d ≤ P | L η [-n,n] d ,Z d \[-n,n] d 1,t | ≤ N 2 d .
Prescribing a given orthant will not be sufficient to ensure a strictly positive move of the bacteria between time 0 and time T (remember we want to build an open oriented macroscopic edge). We thus also work with the points on the lateral faces of the box

[-L, L] d × [0, T ] colonized from a subset A of [-L, L] d .
For every positive integers L, T and every finite A ⊂ [-L, L] d , we define N A,Z d \A (L, T )

as the maximal number of points (x, t)

such that t ∈ [0, T ], x ∞ = L, x ∈ L η A,Z d \A 1,t
and satisfying the following extra property: if (x, t) and (y, s) are two distinct points in this set, then |t -s| ≥ 2n and x -y ∞ ≥ 2n. The next point is to ensure that when the bacteria survive, they must colonize many points and on the top face and on the lateral faces of a large box: Lemma 5.8 (Proposition 2.8 in Liggett [START_REF] Thomas | Stochastic interacting systems: contact, voter and exclusion processes[END_REF], Lemma 4.4 in Steif-Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF]).

For any positive integers M, N , for every finite

A ⊂ Z d , lim L→+∞, T →+∞ P N A,Z d \A (L, T ) ≤ M P | L η [-n,n] d ,Z d \[-n,n] d 1,T | ≤ N ≤ P ∃t η A,Z d \A 1,t = ∅ .
Proof. For two integers L, T , let F L,T be the σ-algebra generated by the restriction of the graphical representation ω to the box [-L, L] d × [0, T ]. Let A be a finite subset of Z d , and M, N be fixed integer. Set k = M + N . Let (T j ) j and (L j ) j be two increasing sequences of integers.

H j = {N A,Z d \A (L j , T j )+| Lj η [-n,n] d ,Z d \[-n,n] d 1,Tj | ≤ k} and G = {∃t η A,Z d \A 1,t = ∅}.
Then, as H j ∈ F Lj,Tj ,

P(G|F Lj ,Tj ) ≥ [(1 -p) 2d+1 ] k 1 1 Hj .
By the martingale convergence theorem, P(G|F Lj ,Tj ) almost surely converges to 1 1 G , which implies that lim j→+∞ H j ⊂ G, and thus lim j→+∞ P(H j ) ≤ P lim j→+∞ H j ≤ P(G).

Using once again the FKG inequality, note that

P(H j ) = P(N A,Z d \A (L j , T j ) + | Lj η [-n,n] d ,Z d \[-n,n] d 1,Tj | ≤ M + N ) ≥ P N A,Z d \A (L j , T j ) ≤ M P | Lj η [-n,n] d ,Z d \[-n,n] d 1,Tj | ≤ N ,
which ends the proof.

Exactly as in Lemma 5.7, the FKG inequality and the symmetries of the process allow to control the number of colonized points in a prescribed orthant of a lateral face of the box [-L, L] d × [0, T ].

For every positive integers L, T and every finite A ⊂ Z d , we define N A,Z d \A + (L, T ) as the maximal number of points (x, t) such that t ∈ [0, T ],

x 1 = L, x i ≥ 0 for 2 ≤ i ≤ d, x ∈ L η A,Z d \A 1,t
and satisfying the following extra property: if (x, t) and (y, s) are two distinct points in this set, then |t -s| ≥ 2n and x -y ∞ ≥ 2n. Then Lemma 5.9 (Proposition 2.11 in Liggett [START_REF] Thomas | Stochastic interacting systems: contact, voter and exclusion processes[END_REF], Proposition 4.6 in Steif-Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF]). For every positive integers n, N, t, for every integer L ≥ n,

P N [-n,n] d ,Z d \[-n,n] d + (L, T ) ≤ M d2 d ≤ P N [-n,n] d ,Z d \[-n,n] d (L, T ) ≤ M d2 d .
With the previous lemmas in hand, we can now prove that starting from a fully occupied square, the bacteria colonize with high probability a similar square on the top face and a similar square on the lateral faces of a large box. The orthant can even be prescribed: Lemma 5.10 (Theorem 2.12 in Liggett [START_REF] Thomas | Stochastic interacting systems: contact, voter and exclusion processes[END_REF], Theorem 4.7 in Steif-Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF]). For every ε > 0, there exist positive integers n, L, T , with n ≤ N such that

P ∃x ∈ [0, L) d L+2n η [-n,n] d ,Z d \[-n,n] d 1,T ⊃ x + [-n, n] d ≥ 1 -ε; (18)
P ∃x ∈ {L + n} × [0, L) d-1 , ∃t ∈ [0, T ) L+2n η [-n,n] d ,Z d \[-n,n] d 1,t ⊃ x + [-n, n] d ≥ 1 -ε. (19)
Proof. Let ε > 0 and δ > 0 to be chosen later. With Lemma 5.5, we choose a positive integer n such that [START_REF] Strassen | The existence of probability measures with given marginals[END_REF] P ∀t ∈ N η

[-n,n] d ,Z d \[-n,n] d 1,t = ∅ > 1 -δ 2 .
Choose an integer N ′ such that

1 -P n η {0},Z d \{0} 1,2n+1 ⊃ [-n, n] d N ′ ≤ δ,
and then N such that every finite subset A of Z d contains a subset A ′ of N ′ points such that ∀x, y ∈ A ′ x -y ∞ ≥ 2n + 1.

Choose an integer M ′ such that

1 -P 2n η {0},Z d \{0} 1,2n ⊃ [0, 2n] × [-n, n] d-1 M ′ ≤ δ,
and then M such that every finite subset A of Z d contains a subset A ′ of M ′ points such that ∀x, y ∈ A ′ x -y ∞ ≥ 2n + 1.

As 1-2δ < 1-2δ 2 < P ∀t ∈ N η With lemmas 5.7 and 5.9, we obtain

P | LK η [-n,n] d ,Z d \[-n,n] d 1,TK ∩ [0, L) d | > 2 d N ≥ 1 -(2δ) 2 -d , P N [-n,n] d ,Z d \[-n,n] d + (L K , T K + 1) > d2 d M ≥ 1 -δ 2 -d /d .
Using the fact that edges and sites in disjoint areas are independent, this leads to

P ∃x ∈ [0, L) d LK+2n η [-n,n] d ,Z d \[-n,n] d 1,TK +1+2n ⊃ x + [-n, n] d ≥ (1 -(2δ) 2 -d )(1 -δ), P ∃x ∈ {L K + n} × [0, L) d-1 , t ∈ [0, T K + 1 + 2n) LK +2n η [-n,n] d ,Z d \[-n,n] d 1,t ⊃ x + [-n, n] d ≥ (1 -δ 2 -d /d )(1 -δ),
which ends the proof.

Using successively [START_REF] Remenik | The contact process in a dynamic random environment[END_REF] and [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF] with the help of an appropriate stopping time, we then get: Lemma 5.11 (Proposition 2.20 in Liggett [START_REF] Thomas | Stochastic interacting systems: contact, voter and exclusion processes[END_REF], Lemma 4.8 in Steif-Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF]). For every ε > 0, there exist positive integers n, L, T , with n ≤ N such that

P ∃x ∈ [L + n, 2L + n] × [0, 2L) d-1 , t ∈ [T, 2T ) L+3n η [-n,n] d ,Z d \[-n,n] d 1,t ⊃ x + [-n, n] d ≥ 1 -ε.
Next, Lemma 5.12 (Proposition 2.20 in Liggett [START_REF] Thomas | Stochastic interacting systems: contact, voter and exclusion processes[END_REF], Lemma 4.9 in Steif-Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF]). • θ t on the event {Y (u, x, t) = y, S(u, x, t) = s};

• P p,q,α (S(u, x, t) < +∞) ≥ 1 -ε;

Let us denote by χ n0 the law of the field ( n0 W n ( k,u) ) n≥0, k∈Z d , u 1 ≤1 under P p,q,α .

Lemma 5.13. We can choose the construction parameters a, b, n of Proposition 5.3 such that χ n0 belongs to C d (M, q 0 ).

Proof. Let ȳ ∈ Z d . For t = nT + r with 0 ≤ r < T , define F ′ȳ t as the σ-field generated by the background variables related to the space-times area (Z d × [0, nT ]) ∪ ([-7a + 2kaȳ, 7a + 2kaȳ] × [nT, nT + r])

Note that nT + t n+1 (ȳ) is a (F ′ȳ k ) k≥0 -stopping time. We define

G n = ∨ ȳ∈Z d F ′ȳ nT +tn+1(ȳ) .
It is not difficult to see that n0 W n+1 (ȳ,u) is G n+1 -measurable. It is more subtle to see that the identity holds for each n ≥ 0 and each e ∈ -→ E d . As in the proof of Theorem 1.2 for small α, space-time areas do not overlap too much, only generating local dependence. This is a classical argument. But space and time do not play the same role. While local spatial dependence is not a big deal, time dependence is strictly forbidden. This condition was for free in the proof of Theorem 1.2 for small α, because the temporal height of boxes was deterministic. This is no longer the case, then we can not use straight boxes and must have a finer control of the travel map of the infection, apply Lemma 5.11 several times, not juste once. Then, we ensure that the variables that define the state of a bond e at time n do not have information about what will happen to another bond at time n + 1.

From macroscopic to microscopic scale.

The proof of Theorem 1.2 falls into two parts

• First prove that if the epidemy survives, then points not far from x will often be occupied at a reasonable time. • Then deduce that x itself will be hit at a reasonable time.

The first part can be formalized as follows.

  L η A,Z d \A 1,t only depends on the background random variables related to the spacetime zone [-L, L] d × [0, t]. Lemma 5.6 (Proposition 2.2 in Liggett [14], Lemma 4.3 in Steif-Warfheimer [19]). For every finite A ⊂ Z d , for every positive integer N , lim t→+∞ lim L→+∞

[- 2δ 2 1

 2 -n,n] d ,Z d \[-n,n] d 1,t= ∅ , there exist with Lemma 5.6 two increasing sequences (T k ) and (L k ) such that[START_REF] Stroock | Probability theory, an analytic view[END_REF] ∀kP | L k η [-n,n] d ,Z d \[-n,n] d 1,T k | > 2 d N ≥ 1 -2δ. Since lim t→+∞ P | L k η [-n,n] d ,Z d \[-n,n] d 1,t | > 2 d N = 0, by increasing T k if necessary, wecan assume that the following extra inequality is fullfilled:∀k P | L k η [-n,n] d ,Z d \[-n,n] d 1,T k +1 | > 2 d N < 1 -2δ.With Lemma 5.8 and (20), there exist K such thatP N [-n,n] d ,Z d \[-n,n] d (L K , T K + 1) ≤ d2 d M P | L η [-n,n] d ,Z d \[-n,n] d 1,TK +1 | ≤ 2 d N ≤ 2δ 2 ,which impliesP N [-n,n] d ,Z d \[-n,n] d (L K , T K + 1) > d2 d M ≥ 1-P | LK η [-n,n] d ,Z d \[-n,n] d 1,TK +1 | > 2 d N ≥ 1-δ.

Let ε > 0 .

 0 There exist n, a, b ∈ N with n < a such that for every(x, t) ∈ [-a, a] d × [0, b], each u ∈ V , we can define random variables Y (u, x, t) ∈ Z d and S(u, x, t) ∈ N such that • Y (u, x, t) ∈ 2au + [-a, a] d ; • S(u, x, t) ∈ [5b, 6b] ∪ {+∞}; • y + 2au + [-n, n] d ⊂ η x+[-n,n] d ,Z d \(x+[-n,n] d ) 1,s-t

P

  [ n0 W n+1 e = 1|G n ∨ σ( n0 W n+1 f , d(e, f ) ≥ 2)] = P[ n0 W n+1 e = 1|G n ] ≥ q 0

Lemma 5 . 14 . 1 =

 5141 Let E ⊂ Z d \{0}. There exists a ∈ N and positive constantsC 1 , C 2 , A, B such that if one defines R a n (x) with n ∈ N and x ∈ Z d by R a 0 (x) = 0 and R a i (x) = inf{t ≥ R a i-1 ; ∃y ∈ x + [-a, a] d ; y ∈ η 0,E 1,t }, then we have (22) ∀x ∈ Z d ∀n ≥ 0 P p,q,α (τ 0,E +∞, R a n (x) ≥ C 1 x + C 2 n) ≤ Ae -Bn .Thanks to our tools for dependent oriented percolation, Lemma 5.14 will appear as a consequence of Proposition 5.3. But first show how Lemma 5.14 implies the Theorem:Proof of Theorem 1.2. Let E ⊂ Z d \{0}, fix x ∈ Z d and define T ′ 0 = 0 and for i ≥ 1 T i = T i (x) = inf{t ≥ T ′ i-1 ; ∃y ∈ x + [-a, a] d ; y ∈ η 0,E 1,t } and T ′ i = T i + a + 1.

• The event {Y (u, x, t) = y, S(u, x, t) = s} belongs to the σ-algebra generated by the background random variables related to the space-time area

Proof. The idea is to use the previous lemma (or a reflected version of it) between 4 and 10 times. Note that we use the strong Markov property (to use independence of background random variables associated to disjoint time intervals) and the monotonicity to rule out the spatial dependencies.

Proof of Proposition 5.3. Using appropriate stopping times and monotonicity in the background process, we can use the previous lemma k-times repeatedly.

Dependent macroscopic Percolation

We will first, from the events defined in the preceding subsection, build a field

The idea is to construct an 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 at time t 0 = 0 from an area centered at 0 in the box with coordinate n0 ; for each u such that u 1 ≤ 1, say that the bond between ( n0 , 0) and ( n0 + u, 1) is open if A( n0 , u, 0, 0) holds and obtain an exit area centered at Y ( n0 , n0 + u, 0, 0); all bonds in this floor that are issued from another point than n0 are open. 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 1 (ȳ) and open the bond between (ȳ, 1) and (ȳ + u, 2) if A(ȳ, u, d 1 (ȳ), 0) • 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 0 (ȳ) = 0 for each ȳ ∈ Z d , t 0 ( n0 ) = 0, and also t 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:

To specify what "min" means, choose the smallest t in the natural order, and then the smallest s in the lexical order. If the set is empty, we consider that the min is (+∞, 0) Then, (t n+1 (ȳ), d n+1 (ȳ)) represents the relative position of the entrance area for the n0 W n+1 (ȳ,u) 's, with

It is know time to put the pieces together: now take M = 2 and choose q 0 < 1 such that g M (q 0 ) > -→ p c alt and q 0 satisfies the conclusion of corollary 3.3 with M = 2. Using Proposition 5.3 with 1 -ε = q 0 and k > 7, one can build an oriented percolation process ( n0 W n ( k,u) ) n≥0, k∈Z d , u 1 ≤1 . Among open bonds, only those corresponding to the realization of good events are relevant for the propagation of type 1 particles. Let us note however that the percolation cluster starting at n0 only contains bonds that are effectively used by the process.

Consider the event B

} and also, for n ≥ 1

Note that by construction, θ -Ti (B) is F Ti+1 -measurable, so P p,q,α (A n |F Tn ) = 1 1 An-1∩{Tn<+∞} P p,q,α (B c ).

It is easy to see that P(B) ≥ c for some c that does not depend on x. It follows that P p,q,α (A N ) ≤ (1 -c) n for each n ≥ 1. Note that the sequence (T k (x)) k≥1 does not consider all infections around x, but it is not difficult to see that T n (x) ≤ R a (a+1)n (x). So, Lemma 5.14 gives

It remains to prove Lemma 5.14.

Proof of Lemma 5.14. Note that the events in Equation ( 22) and in Corollary 3.3 control the density of times where a point (or a neighborood of a point) is occupied.

Using the events that are described in Proposition 5.3, we are going to exhibit (after a restart procedure), a macroscopic percolation that satisfies the assumptions of Corollary 3.3. This will prove Equation ( 22), hence the lemma.

Assume that τ 0,E 1 = +∞. Take M = 2 and choose q 0 < 1, θ, β such that g M (q 0 ) > -→ p c alt and q 0 ,θ,β satisfy the conclusion of corollary 3.3 with M = 2. By Lemma 5.13, we can choose the parameters a, b, n in Proposition 5.3 to ensure that the distribution of the macroscopic oriented percolation is in C d (M, q 0 ).

Then, using the events of Proposition 5.3, the construction of subsection 5.4.2 and Theorem 2.5, a restart argument gives the existence of some (Y,

Remember that χ n0 is the law of the field ( n0 W n ( k,u) ) n≥0, k∈Z d , u 1≤1 under P p,q,α , which has been defined in Subsection 5.4.2.

Then, Lemma 3.3 says that γ(θ, Y , x) ≤ β x -Y + k with probability at least 1 -Ae -Bk , where x and Y respectively stand for the coordinates of macroscopic blocks containing x and Y .

By the very definition of γ(), we have

This leads to

which concludes the proof.


Maxime Amblard

Paola Monachesi

Gerald Penn

Giorgio Satta

Treating clitics with Minimalist Grammars

Keywords: Minimalist Grammars, syntax-semantic interface, λ-calulus, clitics

We propose an extension of Stabler's version of clitics treatment for a wider coverage of the french language. For this, we present the lexical entries needed in the lexicon. Then, we show the recognition of complex syntactic phenomenae as (left and right) dislocation, clitic climbing over modal and extraction from determiner phrase. The aim of this presentation is the syntaxsemantic interface for clitics analyses in which we will stress on clitic climbing over verb and raising verb.

the next rules.

An advantage of this system is that the structure of the calculus is constant. The coverage of the grammar is extended by adding new elements to the lexicon, never by adding new structural rules. The structural system of these grammars contains only two kinds of rules: move and merge (but extensions exist for both). We refer the reader to Stabler's articles and others for presentation of the use of MG, [START_REF] Stabler | Derivational minimalism. Logical Aspect of Computational Linguistic[END_REF], [START_REF] Vermaat | Controlling movement: Minimalism in a deductive perspective[END_REF].

Clitics are the normal form for pronoun in romance language. The syntactic and semantic behavior of clitics in these languages are complex. For French, clitics often climb over auxiliary verb. Ed Stabler proposes in [START_REF] Stabler | Logical Aspects of Computational Linguistics[END_REF] a partial lexicon for french clitics recognition and analysis.

We propose here to extend this lexicon to several well-known linguistic problems. These problems interfere at different levels of analysis. Subject raising is typically a semantic question whereas the clitic climbing over modals is a syntactic question. We propose a new lexicon for its syntactic analysis and then we will show how our semantic interface solves semantic questions.

We use the description of clitics proposed by Perlmutter in Perlmutter (1971). He proposes a filter to recognize the right order of clitics for romance languages, from where we extract the subfilter:

[{je/tu/ • • • }|ne|{me/te/se/ • • • }|{le/la/les/ • • • }|{lui/leur}|y|en]. [nominative | negative | reflexive | accusative | dative | locative | genitive].
In the first part, we propose an extension of Stabler's version of clitics treatment for a wider coverage of the french language. For this, we will present the lexical entries needed in the lexicon. Then, we will show the recognition of complex syntactic phenomena as (left and right) dislocation, clitic climbing and extraction from determiner phrase. The aim of this presentation is the last part: the syntax-semantic interface for clitics analyses in which we will stress on clitic climbing over verb and raising verb.

Lexicon for french clitics 1.Stabler analysis

Stabler's works on clitics are inspired by [START_REF] Sportiche | Clitic constructions. Phrase Structure and the Lexicon[END_REF], who proposes the following treatment:

Clitics are not elements moved from position XP * , but are coreferent with this position. The clitics appearing in the structure bear all the features their co-refering XP * would bear. Furthermore, clitics do not form an autonomous syntactic object, but they are built into a unit with some host. In this work, two parts in the cliticization are distinguished. The first one is an empty element which takes an argumental position from the verb. The second is the phonological treatment of the unit -the clitic in the surface structure.

We introduce lexical entries which are phonologically empty but carry special features which need to be unified with features of the phonological part of the clitic. The two different parts are connected by a move operation. If just one of these items occurs in the sentence, derivation fails.

We sum up this treatment in the derivation as follow -the annotation recall the main feature of the word and the annotation on the ǫ recall the word which ǫisthetrace: In more details, the derivation is the following:

(1) donne ǫ -F Jean -k la +F donne ǫ -F ⇒ Jean -k la
Derivation 1 Derivation of the simple french sentence : Jean la donne. In his presentation, Stabler proposes a lexicon for accusative, dative and reflexive clitics recognition. He ensures the right order with several verbal types. The analysis is driven by the head and the next clitics to introduce will have to be assigned verbal type as they occur in the Perlmutter filter's order. Stabler uses the SMC -shortest move condition -to exclude the use of a reflexive and an accusative clitics in the same sentence.

Lexicon: Jean D -k ǫ =T C ǫ D -k -G donne V ǫ =>V =D +k =D v ǫ =Acc3 +k T la =v +G Acc3

Extension: genitive, oblique and nominative clitics

We can extend this first approach of french clitics treatment to other cases, in particular genitive, oblique and nominative. This section will present the lexical entries and the process of acceptance of derivations.

We call "state of a verb" the basic type of the head currently handled. For example, if a verb has a accusative clitic its type will be "Acc".

For genitive and oblique clitics, we just add in the lexicon two new empty argumental positions and a list of possible types for each clitic.

In a first time, we introduce a new verbal type for beginning the cliticization and another where the cliticization is finished. We call them "clitic" and "endclitic".

Following the Perlmutter filter [START_REF] Perlmutter | Deep and surface structure constraint in syntax[END_REF], the first clitic we have to treat for keeping the right order is the genitive one. We add a genitive state which is connected to the "clitic" state. The verbal state passes to the genitive state by means of a lexical entry the phonological form of which is "en" and carries a licensee feature"en":

[en] :: [clitic <=, +EN, genitif ].
From this state we pass to all the other states of the cliticization, for example :

[le] :: [genitif <=, +G, acc]. and if there is only a genitive clitic, we use phonologically empty entry to pass to the end of the cliticization.

[] :: [genitif <=, endclitic].

The "oblique" clitics are treated in the same way, except that from "oblique" it is impossible to go back to "genitive". All lexical entries of this type have a "y" phonological form.

[y] :: [clitic <=, +Y, oblique].

[y] :: [genitive <=, +Y, oblique]. In the same way, from oblique we can pass to other possible clitic states, as for example :

[le] :: [oblique <=, +G, acc].

[leur] :: [oblique <=, +F, dat].

[] :: [oblique <=, endclitic]. The nominative case is treated the same way. But the use of this procedure to add new clitic treatment is quadratic in the number of lex-ical entries. For the nominative pronoun, a discussion could be opened around its clitic state. We consider here that they are clitics.

Another discussion about negative form could rise around the status of the negation marker whose position is after the pronoun.

For the moment, we do not treat the negative form in a right way so we will not include it in this presentation, but we assume that the treatment of nominative clitics is outside the clitic cluster. All the phonological pronoun entries take a verbal form in "endclitic" state and give a new verbal form in "Nom"(inative) state.

We add an empty verb argument which must be included in the derivation before the clitic treatment:

[] :: [d, -Subj, -case].
The sketch of the analysis is:

. la donne ǫ -N om ǫ . Je +Subj la donne ǫ -N om ǫ . Je t Je la donne ǫ

I t I it give ǫ I give it
We add in the lexicon a basic feature "Nom" and the lexical entries of the nominative pronouns, for example:

[je] :: [= endclitic, +Subj, N om].

[nous] :: [= endclitic, +Subj, N om]. The derivation continues with a phonologically empty entry at the end of the derivation.

[] :: [= nom, +case, t].

Recognition of complex phenomena

This treatment of french clitics is simple and can be integrated easily into a larger analysis.

climbing over modal

We treat the clitic climbing over the whole verbal cluster in particular over modal.

The modal is combine with the verb in the inflexion step. The inflexion is treated with head movement and all clitics take their own place after this treatment.

If there are words which must be inserted between the verb and the modal -for sentences with adverbs -we first build the verbal constituent after which we treat the clitics. In this situation, the clitics could climb over the verb constituent or stand after.

For example, in french we can analyse a sentence as:

(2) Je l'ai vu.

I him have seen.

I have see him.

by building the constituent ai vu. We can extend to sentences with inserted word: "Je l'ai souvent vu" / "I have often see him" with a derivation as :

(3) ai souvent vu ǫ -N om ǫ -F l' +F ai souvent vu ǫ -N om ǫ -F → l'ai souvent vu ǫ -N om ǫ Je +N om l'ai souvent vu ǫ -N om ǫ -F → Je l'ai souvent vu ǫ ǫ I it often sawn I often sawn him. dislocation Clitic can be a direct recovery of a not-"empty verbal argument", for example in case of nominal dislocation.

There is a non empty verbal argument which must be extracted from the main sentence and become an indirect argument of the verb.

We build a verb with an "argument which must be extracted" -a determiner phrase -DP -must be outside the main sentence. This state is introduced by a pause or comma. It modifies the determiner phrase in two different ways which depend on the side of the extraction:

. it adds a licensee for the left dislocation and cliticization.

. it adds a licensee for cliticization (and nothing for right dislocation).

The main problem is to include in the sentence the right part which will be replaced by the clitic.

Left dislocation: the DP is extracted from the sentence, placed in first position and recovered by a clitic.

(4) Marie le i voit trop ce type i , → Ce type i , Marie le i voit trop.

That guy, Marie him sees too much.

Lexical entry of modifier of DP.

[, ] :: [=> d, d, -H, -disloc].

Remark that we use coma to caring this treatment, but it can be through an empty lexical entry. The analyse would be the same. The comma will be placed after the DP by a head movement. The first licensee will be cancelled with the licensor of the clitic and the second with another entry that we must add in the classical "comp" entry (this last entry is used to finish the derivation).

[] :: [= t, c, +DISLOC].

Right dislocation : In this case the determiner phrase is placed at the end of the sentence. For the homogeneity of the mechanism, we add a licencee of recovered by a clitic, and another for the extraction at the end of the sentence.

[, ] :: [d <=, d, -H, -disloc].

The "comp" phase uses a weak move which lets the phonological form of the constituent in its place -here, at the end of the sentence.

(5) Marie le i voit trop , ce type i . → Marie le i voit trop, ce type i .

Marie him sees too much, that guy.

This extraction seems to be very similar to questions: in questions, an argument of the verb is extracted to take another position in the surface level of the sentence.

Extraction from DP

With the same kind of mechanism, we can extract an argument of any constituent. The determiner phrase can be complex and we extract an argument of the DP. For example:

(6) Pierre en voit la fin -(Pierre voit la fin du film).

Peter of-it sees the end -Peter sees the end of the movie.

We build "la fin ǫ -en " and the cliticization allowed the extraction of the genitive. "Pierre en voit la fin."

Raising verb

Raising verbs are verbs where one of the arguments is a verb and one of the other arguments is shared by both verbs, like in the sentence:

(7) Il semble le lui donner.

He seems it him give. He seems give it to him.

where the pronoun "Il" is subject of the two verbs "semble" and "donner". The second verb must be in infinitive form.

In this case, the sentence has the following structures: [ subject raising verb clitic infinitive verb ].

A raising verb takes as an argument a verb in infinitive form -with a special inflexion "infinitive" -and without subject. The infinitive inflexion has the lexical entry:

[-inf]::[=>v, verbe].
"verbe" is the feature needed before starting the clitic treatment. A verbal form gets a "verbe" type after the verb receives its inflexion.

The raising verb selects such a "verb", then a DP subject and then becomes a VP of type "raisingv" which means a VP which has not yet received the inflexion feature and will be able to receive new clitics (in particular pronoun).

For example:

[semble]:: [=verbe, =d, raisingv].

This verb should receive its inflexion and its subject. It follows this mechanism until the end of the derivation:

. semble la répare-inf . semble -ǫ la répare-inf . Je semble -ǫ la répare-inf I seem -ǫ it repare-inf I seem repare it 1.3 Semantic interface 1.3.1 How to use the syntax/semantic interface From a sentence, we build a formula of higher order logic which represents its propositional structure. We associate to each lexical entry a λ-term and to each syntactic rule an equivalent semantic rule. We assume that the syntactic analysis drives the semantic calculus.

λ-terms application occurs only when an element has no features. We assume the following functions:

feat(x) =
1 if the number of feature of x = 0 0 else sem(x, y) = 1 if feat(x) = 1 or feat(y) = 1 0 else Syntactic and semantic synchronisation: after any operation in the syntactic calculus, the semantic counter part computes the sem function and if sem(x, y) = 1, we perform the functionnal application of the two λ-terms. To known which application to perform, we look at the type of the semantic terms.

A semantic tree represents the semantic counter part of the sentence. It is a tree where the leaves are the semantic part of the lexical entries and the inner nodes contain the λ-term built and the direction of the head (of the syntactic part). We use the following notation:

In the semantic tree of the part of the clitisization above, we do not represent the identity operator (except for the clitic one).

<⊢ t(la) @ Infl @ donne, je <⊢ Infl @ donne, je, t(la) ¨¨¨¨r r r r r la :: Id <⊢ Infl @ donne, je, t(la) The last part of the tree is built by a move which creates a link between the phonological part of the clitic and the argumental part.

Over raising verbs

For the semantic calculus, raising verbs are predicates which take a subject and an action as argument. They apply a variable at this action.

We present the analysis of the sentence:

(9) Je semble la réparer. I seem it repair. I seem repair it.

The λ-terms, semantic counter-part of lexical entries are: sembler λSλv.(seem v, S(v)) Je I ǫ la Y * réparer λ x λ y . repair (y, x) * this variable is bound in the context The semantic counter part of the pronoun is a constant refering to the speaker "I". The clitic subject climbs over the raising verb. It can be the subject of both verbs in the sentence due to the semantic structure of the raising verb. If the main verb of the sentence has a subject, the application will not introduce a new variable in the formula, else the main verb needs a variable which stands at the subject place. The raising verb involves this variable by duplication of its subject.

The syntactic analysis builds the following structure:

(I@(inf lexion@(seem@(la(inf initive@repare)))))

which allows the computation of the formula: "la reparer" λx.repair(x, Y ) and this term is applied to the raising verb: λSλv.(seem v, S(v))

λv.seem(v, repair(v, Y ))

At the end of the calculus, we construct the formula:

pres(seem(I, repair(I, Y )))

where Y is bound in the context. This is the formula we want to construct for representing the propositionnal semantics of the sentence. The subject clitic syntactically climbs over the main verb, and semantically climbs over the two verbs.

Conclusion and future work

In this paper, we presented an extension of Ed Stabler's propositions on french clitics in minimalist grammars. The new lexicon makes it possible to treat several other syntactic phenomena, the same way as clitic climbing, e.g. extraction from NP or right and left dislocation.

Then, we proposed a syntax-semantic interface for Minimalist Grammars. The aim of this calculus is to build a formula of higher order logic. The semantic calculus -λ-calculus -is driven by the syntactic one. We emphasize on the way to recognize clitics and semantic implication of climbing with raising verbs.

For future work, we want to integrate the negation into the grammar. We consider that the neg-marker "ne" is a clitic and must be incorporated in the treatment of french clitics. There is another complex phenomenon to consider concerning with clitics in the imperative mode (and negation).

Other cases of raising verbs exist which are more complex, allowing several syntactic clitic climbings as in:

(10) Je la laisse le lui donner.

I her let it (to) him give.

I let her give it to him.

where clitics take place in different orders. Moreover, we want to continue to modelize the semantic effect of clitics in sentences, in particular for interaction between quantifier scope and clitics, which can introduce ambiguities in sentences like:

(11) Je la laisse tous les lui donner.

I her let all them him give. I let her gives all to him.

  of lexical entry : [ǫ:: =T C] -empty "complement" position. 16. merge ; end of the derivation with feature 'c' : acceptance.

  donne ǫ la t ǫ Jean -k la donne ǫ la Jean t ǫ t Jean la donne ǫ la . John t ǫ t Jean it gives ǫ it . John give it.

. breaker between direction head and λ-term : ⊢. . application: @ Applications are carried out when syntax allows it, therefore when the function sem = 1 for one of the two terms. The following applications are possible:

If a move operation cancelled the last feature, we represent it by a unary branch in the tree.

Remark. There are two different possibilities for the semantic calculus: either waiting for elements completely discharged either immediately perform the application. But both fail in different cases: immediate application fails in case of "late adjunction" and the other possibility fails in questions treatment. The right solution seems to be intermediate: it consists in determining a subset of features which must be consumed before applications will be performed. For the moment, we choose the first possibility. Later on, we shall do differently but this only involve changes in the feat function.

Example of semantic treatment

Clitic semantics

We present a syntactic treatment of clitics in two different parts. One is phonologically empty and is the non empty argument of the verb, the other is syntactically empty but it is a phonological recovery of the first one. The semantical part of the clitic is in the argumental position and this is a free variable which must be bound in the context. The phonological recovery is an identity. lexical entries syntactic form semantic form la dat <= +G acc Id t(la) p -case -G x * * Free variable, bound in the context -we could use the Bonato algorithm to determinated how this variables are bounded, [START_REF] Bonato | An Integrated Computational Approach to Binding Theory[END_REF] We briefly present a semantic tree for a clitic treatment:

(8) Jean la répare. John it repairs. John repairs it.

Acknowledgement

The writer like to thank Christian Retoré and Alain Lecomte for crucial supports and one of the anonymous TALN 2006 referees for important comments and examples reuse in this paper.