Enrique D Andjel
email: enrique.andjel@univ-amu.fr

Nicolas Chabot
email: nicolas.chabot@cegetel.net

Ellen Saada
email: ellen.saada@mi.parisdescartes.fr

A shape

Keywords: 2000 Mathematics Subject Classification. 60K35, 82C22 Shape theorem, epidemic model, first passage locally dependent percolation, dynamic renormalization

come

1. Introduction [START_REF] Mollison | Spatial contact models for ecological and epidemic spread[END_REF][START_REF] Mollison | Markovian contact processes[END_REF] has introduced a stochastic spatial epidemic model on Z d called "general epidemic model", describing the evolution of individuals submitted to infection by contact contamination of infected neighbors. More precisely, on each site of Z d there is an individual who can be healthy, infected, or immune. At time 0, there is an infected individual at the origin, and all other sites are occupied by healthy individuals. Each infected individual emits germs according to a Poisson process, it stays infected for a random time, then it recovers and becomes immune to further infection. A germ emitted from x ∈ Z d goes to one of the neighbors y ∈ Z d of x chosen at random. If the individual at y is healthy then it becomes infected and begins to emit germs; if this individual is infected or immune, nothing happens. The germ emission processes and the durations of infections of different individuals are mutually independent. After Mollison's papers, this epidemic model has given rise to many studies, and other models that are variations of this "SIR" (Susceptible-Infected-Recovered) structure have been introduced. A first direction to study such models is whether the different states asymptotically survive or not, according to the values of the involved parameters (e.g. the infection and recovery rates). A second direction is the obtention of a shape theorem for the asymptotic behavior of infected individuals, when there is no extinction of the infection (throughout this paper, "extinction" is understood as "extinction of the infection"). [START_REF] Kelly | discussion of Mollison[END_REF] proved that for d = 1, extinction is almost sure for the general epidemic model. [START_REF] Kuulasmaa | The spatial general epidemic and locally dependent random graphs[END_REF] has studied the threshold behavior of this model in dimension d ≥ 2. He proved that the process has a critical infection rate below which extinction is almost certain, and above which there is survival, thus closing this question. His work (as well as the following ones on this model) is based on the analysis of an oriented percolation model, that he calls a "locally dependent random graph", in correspondence with the epidemic model. See also the related paper [START_REF] Kuulasmaa | On spatial general epidemics and bond percolation processes[END_REF].

In the general epidemic model on Z 2 , when there is no extinction, [START_REF] Cox | Limit theorems for the spread of epidemics and forest fires[END_REF] have derived a shape theorem for the set of infected or immune individuals when the contamination rule is nearest neighbor, and the durations of infection are positive with a positive probability. A second moment is required for those durations only to localize the infected but not immune individuals within the shape obtained. This result was extended to a finite range contamination rule by [START_REF] Zhang | A shape theorem for epidemics and forest fires with finite range interactions[END_REF]. The proofs in [START_REF] Cox | Limit theorems for the spread of epidemics and forest fires[END_REF]; [START_REF] Zhang | A shape theorem for epidemics and forest fires with finite range interactions[END_REF] are based on the correspondence with the locally dependent random graph; they refer to [START_REF] Chabot | Some limit theorems for percolation processes with necessary and sufficient conditions[END_REF], which deals with first passage percolation (see also [START_REF] Kesten | Aspects of first passage percolation[END_REF], including the possibility of infinite passage times. They rely on circuits to delimit and control open paths. This technique cannot be used for dimension greater than 2.

There was no investigation of the shape theorem for the general epidemic model in higher dimensions, until Chabot (1998, unpublished) proved it for a nearest neighbor contamination rule in dimension d ≥ 3, with the restriction to deterministic durations of infection: in that case the oriented percolation model is comparable to a non-oriented Bernoulli percolation model (as noticed in [START_REF] Kuulasmaa | The spatial general epidemic and locally dependent random graphs[END_REF], the case with constant durations of infection in the epidemic model is the only one where the edges are independent in the percolation model). Analyzing the epidemic model for d ≥ 3 required heavier techniques than before: [START_REF] Chabot | Some limit theorems for percolation processes with necessary and sufficient conditions[END_REF] used results from [START_REF] Antal | On the chemical distance for supercritical bernoulli percolation[END_REF] and [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF] for non-oriented Bernoulli percolation to derive, for the percolation model, exponential estimates in the subcritical case on the one hand, and estimates using percolation on slabs on the other hand. To apply those results to the epidemic model required to find an alternative, in the percolation model, to the neighborhoods (for points in Z 2 ) delimited by circuits of [START_REF] Cox | Limit theorems for the spread of epidemics and forest fires[END_REF]. [START_REF] Chabot | Some limit theorems for percolation processes with necessary and sufficient conditions[END_REF] introduced new types of random neighborhoods characterized by the properties of the percolation model in dimension d ≥ 3.

In the present work, we complete the derivation of the shape theorem for the set of infected or immune individuals in the general epidemic model with a nearest neighbor contamination rule in dimension d ≥ 3, by proving it for random durations of infection, which are positive with a positive probability. There, the comparison with non oriented percolation done in [START_REF] Chabot | Some limit theorems for percolation processes with necessary and sufficient conditions[END_REF] is no longer valid, and we have to deal with an oriented dependent percolation model, with possibly infinite passage times. Our approach consists in adapting the dynamic renormalization techniques of [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF] without calling on [START_REF] Antal | On the chemical distance for supercritical bernoulli percolation[END_REF]. This simplifies the paper, but we obtain sub-exponential estimates (which suffice for our purposes), instead of exponential estimates as in the paper [START_REF] Chabot | Some limit theorems for percolation processes with necessary and sufficient conditions[END_REF]. With this in hand, it is then possible to catch hold of the skeleton of the latter: We take advantage of the random neighborhoods introduced there (they turn out to be still valid in our setting) to derive the shape theorem. Similarly to [START_REF] Cox | Limit theorems for the spread of epidemics and forest fires[END_REF], we require a moment of order d of the durations of infection only to localize the infected but not immune individuals within the shape obtained.

Let us mention two recent works, [START_REF] Cerf | Weak shape theorem in first passage percolation with infinite passage times[END_REF] and [START_REF] Mourrat | Lyapunov exponents, shape theorems and large deviations for the random walk in random potential[END_REF], on shape theorems for (or related to) first passage (non dependent) percolation on Z d with various assumptions on the passage times, for which the approach in [START_REF] Kesten | Aspects of first passage percolation[END_REF] is extended.

Our paper is organized as follows. In Section 2 we define the general epidemic model, the locally dependent random graph, we explicit their link, and we state the shape theorem (Theorem 2.2). Section 3 is devoted to the necessary percolation estimates on the locally dependent random graph needed for Theorem 2.2. We prove the latter in Section 4, thanks to an analysis of the travel times for the epidemic. In Appendix A, we prove all the results of Section 3 requiring dynamical renormalisation techniques.

The set-up: definitions and results

Let d ≥ 3. The epidemic model on Z d is represented by a Markov process (η t ) t≥0 of state space Ω = {0, i, 1} Z d . The value η t (x) ∈ {0, i, 1} is the state of the individual located at site x at time t: state 1 if the individual is healthy (but not immune), state i if it is infected, or state 0 if it is immune. We will shorten this in "site x is healthy, infected or immune". We assume that at time 0, the origin o = (0, . . . , 0) is the only infected site while all other sites are healthy. That is, the initial configuration η 0 is given by

η 0 (o) = i, ∀ z = o, η 0 (z) = 1. (2.1)
We now describe how the epidemic propagates, then we introduce a related locally dependent oriented bond percolation model on Z d , and finally we link the two models. We assume that all the processes and random variables we deal with are defined on a common probability space, whose probability is denoted by P , and the corresponding expectation by E.

For x = (x 1 , . . . , x d ) ∈ Z d , y = (y 1 , . . . , y d ) ∈ Z d , x -y 1 = d i=1 |x i -y i | denotes the l 1 norm of x -
y, and we write x ∼ y if x, y are neighbors, that is x -y 1 = 1. Let (T x , e(x, y) : x, y ∈ Z d , x ∼ y) be independent random variables such that 1) the T x 's are nonnegative with a common distribution satisfying P (T x = 0) < 1; 2) the e(x, y)'s are exponentially distributed with a parameter λ > 0. We stress that the only assumption on the T x 's is that their distribution is not a Dirac mass on 0. They could be infinite, or without any finite moment. We define

X(x, y) = 1 if e(x, y) < T x ; 0 otherwise. (2.2)
In the epidemic model, for a given infected individual x, T x denotes the amount of time x stays infected; during this time of infection, x emits germs according to a Poisson process of parameter 2dλ; when T x is over, x recovers and its state becomes 0 forever. An emitted germ from x at some time t reaches y (say), one of the 2d neighbors of x, uniformly. If this neighbor y is in state 1 at time t -, it immediately changes to state i at time t, from t begins the duration of infection T y , and y begins to emit germs according to the same rule as x did; if this neighbor y is in state 0 or i at time t -, nothing happens.

In the percolation model, for x, y ∈ Z d , x ∼ y, the oriented bond (x, y) is said to be open with passage time e(x, y) (abbreviated λ-open, or open when the parameter is fixed) if X(x, y) = 1 and closed (with infinite passage time) if X(x, y) = 0. As in [START_REF] Kuulasmaa | The spatial general epidemic and locally dependent random graphs[END_REF], we call this oriented percolation model a locally dependent random graph. Indeed the fact that any of the bonds exiting from site x is open depends on the r.v. T x . For x, y ∈ Z d (not necessarily neighbors), "x → y" means that there exists (at least) an open path from x to y, that is a path of open oriented bonds, Γ x,y = (z 0 = x, z 1 , . . . , z n = y).

If x → y, x = y, we define the passage time on Γ x,y to be (see (2.2))

τ (Γ x,y ) = n-1 j=0 e(z j , z j+1 ) (2.3)
and, if x = y, we put τ (Γ x,x ) = 0. We then define the travel time from x to y to be

τ (x, y) =        inf {Γx,y} τ (Γ x,y ) if x = y, x → y, 0 if x = y, +∞ otherwise.
(2.4)

where the infimum is over all possible open paths from x to y.

Coming back to the epidemic model, note that when the initial configuration is η 0 defined in (2.1), for a given site z, τ (o, z) is the duration for the infection to propagate from o to z, changing successively the values on all the sites of the involved path Γ o,z from 1 to i.

To link the two models, we define, for t ≥ 0,

Ξ t = {x ∈ Z d : x is immune at time t} = {x ∈ Z d : η t (x) = 0}; (2.5) Υ t = {x ∈ Z d : x is infected at time t} = {x ∈ Z d : η t (x) = i}. (2.6) We have, for z ∈ Z d , t ≥ 0, z ∈ Υ t ∪ Ξ t if and only if τ (o, z) ≤ t.
(2.7) Indeed, τ (o, z) ≤ t means that the infection has reached site z before time t, so that site z is either still infected or already immune at time t, that is z

∈ Υ t ∪ Ξ t .
Conversely, if z is infected or immune at time t, it means that it has already been infected.

In the epidemic model, we denote by C out o the set of sites that will ever become infected, that is

C out o = {x ∈ Z d : ∃ t ≥ 0, η t (x) = i|η 0 (o) = i, ∀ z = o, η 0 (z) = 1}.
(2.8)

Then, by (2.7), C out o is the set of sites that can be reached from the origin following an open path in the percolation model. See also Cox and Durrett (1988, (1.2)), Mollison (1977, p. 322) and Kuulasmaa (1982, Lemma 3.1).

More generally, in the percolation model, for each x ∈ Z d we define the ingoing and outgoing clusters to and from x to be

C in x = {y ∈ Z d : y → x}, C out x = {y ∈ Z d : x → y}, (2.9)
and the corresponding critical values to be

λ in c = inf{λ : P (|C in x | = +∞) > 0}, λ out c = inf{λ : P (|C out x | = +∞) > 0},
(2.10) where |A| denotes the cardinality of a set A.

In Section 3, we will first prove the following proposition about these critical values.

Proposition 2.1. We have λ in c = λ out c . This common value will be denoted by

λ c = λ c (Z d ).
Assuming that λ > λ c , the most important part of our work in Section 3 will then be, thanks to dynamic renormalization techniques, to analyze for the percolation model percolation on slabs in Theorem 3.5, and, through a succession of lemmas, to establish in Proposition 3.11 subexponential estimates for the length of the shortest path between two points x and y given that x → y. This will imply (see Remark 4.1) uniqueness of the infinite cluster of sites connected to +∞. Proposition 3.11 contains the crucial properties we will need on the percolation model to derive our main result, the shape theorem, that we now state.

Theorem 2.2. Assume λ > λ c , and the initial configuration of the epidemic model (η t ) t≥0 to be given by (2.1). Then there exists a convex subset D ⊂ R d such that, for all ε > 0 we have, for t large enough

(1 -ε)tD ∩ C out o ⊂ Ξ t ∪ Υ t ⊂ (1 + ε)tD ∩ C out o a.s.
(2.11)

and if E(T d o ) < ∞ we also have Υ t ⊂ (1 + ε)tD \ (1 -ε)tD a.s. for t large enough.
(2.12)

In other words, the epidemic's progression follows linearly the boundary of a convex set. Note that a moment assumption on T o is only required to localize the infected individuals, and not for the first part of the theorem, for which there is no assumption on the distribution of T o . It is also remarkable that the fact that T o could be either very small or very large with respect to the exponential variables e(x, y) does not play any role.

We prove Theorem 2.2 in Section 4. For this, we follow some of the fundamental steps of [START_REF] Cox | Limit theorems for the spread of epidemics and forest fires[END_REF], but since in dimensions three or higher, circuits are not useful as in dimension 2, we had to find other methods of proofs. By (2.7), we have to analyze travel times to prove Theorem 2.2. On the percolation model, we first construct, in Section 4.1, for each site z ∈ Z d a random neighborhood V(z) in such a way that two neighborhoods are always connected by open paths (these neighborhoods have to be different from those delimited by circuits of [START_REF] Cox | Limit theorems for the spread of epidemics and forest fires[END_REF]. For z, y ∈ Z d , we show that the travel time τ (z, y) is 'comparable' (in a sense precised in Lemma 4.8) to the travel time τ (z, y) to go from V(z) to V(y). Then we approximate the travel time between sites by a subadditive process, and we derive (in Theorem 4.12 and Section 4.3) a radial limit µ(x) (for all x), which is asymptotically the linear growth speed of the epidemic in direction x. In Theorem 4.17 we control how τ (o, •) grows. Finally we prove in Theorem 4.18 an asymptotic shape theorem for τ (o, •), from which we deduce Theorem 2.2.

Percolation estimates

In this section we collect some results concerning the locally dependent random graph, given by the random variables (X(x, y), x, y ∈ Z d ) introduced in (2.2). Our goal is to derive subexponential estimates in Proposition 3.11.

Remark 3.1. Although the r.v.'s (X(x, y), x, y ∈ Z d ) are not independent, if we denote by (e 1 , . . . , e d ) the canonical basis of Z d , then the random vectors {X(x, x + e 1 ), . . . , X(x, x + e d ), X(x, x -e 1 ), . . . , X(x, x -e d ) : x ∈ Z d } (in which each component depends on T x ) are i.i.d., since two different vectors for z, y ∈ Z d depend respectively on T z and T y which are independent. This small dependence forces us to explain why and how some results known for independent percolation remain valid in this context.

Remark 3.2. The function X(x, y) is increasing in the independent random variables T x and -e(x, y). It then follows that the r.v.'s (X(x, y) : x, y ∈ Z d , y ∼ x) satisfy the following property:

(FKG) Let U and V be bounded measurable increasing functions of the random variables (X(x j , y j ) :

x j , y j ∈ Z d , y j ∼ x j , j ∈ N), then E(U V ) ≥ E(U )E(V ).
For the proof of this property, we refer to Cox and Durrett (1988, Lemma (2.1)) with the help of Harris (1960, Lemma 4.1 and its Corollary) if U and V depend on a finite number of variables X(x j , y j ), and to Grimmett (1999, Chapter 2) to take the limit for an infinite number of variables.

We will use this property in the proofs of Theorem 3.5, Lemma 3.10 and Lemma 4.6 below for U, V indicator functions involving open paths without loops, thus we will speak of increasing events rather than increasing functions.

For n ∈ N \ {0}, let B(n) = [-n, n] d , let ∂B(n) denote the inner vertex boundary of B(n), that is ∂B(n) = {x ∈ Z d : x ∈ B(n), x ∼ y for some y / ∈ B(n)}; (3.1) and, for x ∈ R d , B x (n) = x + B(n). For A, R ⊂ Z d , "A → R" means that
there exists an open path Γ x,y from some x ∈ A to some y ∈ R.

Theorem 3.3. (i) Suppose λ < λ out c , then there exists β out > 0 such that for all n > 0, P (o → ∂B(n)) ≤ exp(-β out n).

(ii) Suppose λ < λ in c , then there exists β in > 0 such that for all n > 0, Van den Berg et al. (1998, Theorem (3.1)), whose proof can be adapted to obtain Theorem 3.3(ii). It is worth noting that in the context of our paper, by Remark 3.2, Van den Berg et al. (1998, Theorem (3.1)) can be proved using the BK inequality instead of the Reimer inequality (see Grimmett, 1999, Theorems (2.12), (2.19)). Theorem 3.3 yields Proposition 2.1:

P (∂B(n) → o) ≤ exp(-β in n). Theorem 3.3(i) is a special case of
Proof of Proposition 2.1. Suppose λ < λ in c .
Then by translation invariance and Theorem 3.3(ii) we have that for any x ∈ ∂B(n), P (o → x) ≤ exp(-β in n). Adding over all points of ∂B(n) we get

P (o → ∂B(n)) ≤ K n d-1 exp(-β in n) for some constant K , which implies that lim n→+∞ P (o → ∂B(n)) = 0. Therefore λ ≤ λ out c and λ in c ≤ λ out c .
The other inequality is obtained similarly.

From now on, we assume λ > λ c (Z d ) and define the following events: For x, y ∈ Z d , A ⊂ Z d , (i) The event {x → y within A} consists of all points in our probability space for which there exists an open path Γ x,y = (x 0 = x, x 1 , . . . , x n = y) from x to y such that x j ∈ A for all j ∈ {0, . . . , n -1}. Note that the end point y may not belong to A.

(ii) The event {x → y outside A} consists of all points in our probability space for which there exists an open path Γ x,y = (x 0 = x, x 1 , . . . , x n = y) from x to y such that none of the x j 's (j ∈ {0, . . . , n}) belongs to A.

Definition 3.4. For x ∈ Z d , A ⊂ Z d let C in x (A) = {y ∈ A : y → x within A} and C out x (A) = {y ∈ A : x → y within A}. Note that by this definition C in x (A) ⊂ A and C out x (A) ⊂ A.
The rest of this section relies heavily on the techniques of [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF] or Grimmett (1999, Chapter 7). We assume the reader familiar with them. We postpone to Appendix A the proofs of Theorem 3.5 and Lemma 3.7 below, which require a thoughtful adaptation of Grimmett (1999, Chapter 7) for our context of dependent percolation. Nonetheless, it is possible to go directly to Section 4, where these techniques are no longer used, assuming that Proposition 3.11 holds.

Next theorem is crucial, it states that there is percolation on slabs.

Theorem 3.5. Assume λ > λ c . For any k ∈ N \ {0}, let S k = {0, 1, . . . , k} × Z d-1 denote the slab of thickness k containing o. Then for k large enough we have inf

x∈S k P (|C in x (S k )| = +∞) > 0,
and inf

x∈S k P (|C out x (S k )| = +∞) > 0.
We introduce now some notation about the shortest path between two points x and y such that x → y. Notation 3.6. (a) For A ⊂ Z d we define the exterior vertex boundary of A as: The end of this section provides some upper bounds for the tail of the conditional distribution of D(x, y) given the event {x → y}. We derive Proposition 3.11, required in Section 4, thanks to Lemmas 3.7, 3.9, 3.10. These estimates are not optimal and better results could be obtained by a thoughtful adaptation of the methods of [START_REF] Antal | On the chemical distance for supercritical bernoulli percolation[END_REF]. Instead of getting exponential decays in x -y 1 (or in n) we get exponential decays in x -y 1/d 1 (or in n 1/d ). We have adopted this approach because those weaker results suffice for our purposes and are simpler to obtain, thus making our proof much easier to follow: it is possible to read our work knowing only [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF] and not [START_REF] Antal | On the chemical distance for supercritical bernoulli percolation[END_REF]. Next lemma is inspired by Grimmett and Marstrand (1990, Section 5

∆ v A = {x ∈ Z d : x / ∈ A,
(f ) p. 454). Lemma 3.7. Assume λ > λ c . There exist δ > 0, k ∈ N \ {0} and C 1 = C 1 (k) > 0 such that (i) ∀n > 0, x ∈ B(n + k) \ B(n), y ∈ (B(n + k) \ B(n)) ∪ ∆ v (B(n + k) \ B(n)) we have : P (x → y within B(n + k) \ B(n)) > δ. (ii) Let for (n, m) ∈ Z 2 with n < m, and for ≥ 0, A(n, m, ) = {z : -k + n ≤ z 1 < n, -∞ < z 2 ≤ + k}∪ {z : -k + n ≤ z 1 ≤ m + k, < z 2 ≤ + k}∪ {z : m < z 1 ≤ m + k, -∞ < z 2 ≤ + k}. (3.3) ∀n < m, ∀x ∈ A(n, m, 0), ∀y ∈ A(n, m, 0) ∪ ∆ v A(n, m ,
0), we have:

P (D(x, y) < C 1 ( x -y 1 + (-x 2 ) + + (-y 2 ) + ) within A(n, m, 0)) > δ.
We again introduce some notation, to decompose in Lemma 3.9 a path from the center of a box to its boundary through hyperplanes.

Notation 3.8. Let k be given by Lemma 3.7 and let x and y be points in Z d . For ∈ Z let H = {z ∈ Z d : z 1 = } and define the events, for n ∈ N,

J n = {x → H x1-1-jk within B x (nk), j = 0, . . . , n/2 }∩ {H y1+1+jk → y within B y (nk), j = 0, . . . , n/2 }, G n = {x → ∂B x (nk), ∂B y (nk) → y},
where, for any a ∈ R, a denotes the greatest integer not greater than a.

Lemma 3.9. Assume λ > λ c . Let k be given by Lemma 3.7 and let x, y be points in Z d . Then, for n ∈ N \ {0} there exists β > 0 such that

P (J n |G n ) ≥ 1 -exp(-βn).
Proof of Lemma 3.9. By translation invariance we may assume that x is the origin. We start showing that for some constant β > 0 and all n

P (o → H -1-jk within B(nk), j = 0, . . . , n/2 |o → ∂B(nk)) ≥ 1 -exp(-β n).
(3.4)

For this we first observe that

{o → H -1-jk within B(nk) for some n/2 ≤ j ≤ n} ⊂ {o → H -1-jk within B(nk), j = 0, . . . , n/2 }.
Hence (3.4) follows from

P (o → H -1-jk within B(nk) for some n/2 ≤ j ≤ n|o → ∂B(nk)) ≥ 1 -exp(-β n),
which is a consequence of Lemma 3.7(i). Since P (∂B y (kn) → y) is bounded below as n goes to infinity, (3.4) implies that

P (o → H -1-jk within B(nk), j = 0, . . . , n/2 |G n )
converges to 1 exponentially fast. Similarly one proves that

P (H y1+1+jk → y within B y (nk), j = 0, . . . , n/2 |G n )
converges to 1 exponentially fast, and the lemma follows.

In Lemma 3.10 below we prove a chemical distance bound that will be used later on to derive in Remark 4.1, through Proposition 3.11, the uniqueness of the infinite cluster of sites connected to +∞. The main technique is to construct an open path in a ring after independent attempts thanks on the one hand to Lemma 3.9 whose J n 's enable to get disjoint slabs, and on the other hand to Lemma 3.7(ii) once we find the appropriate ring.

Lemma 3.10. Assume λ > λ c . Let k be given by Lemma 3.7, and let G n be as in Lemma 3.9. Then, there exist constants C 2 , C 3 and α 2 > 0 such that, for all x, y ∈ Z d , n ∈ N \ {0}, we have

P (D(x, y) > C 2 x -y 1 + C 3 (nk) d | G n ) ≤ exp(-α 2 n).
Proof of Lemma 3.10. Again, by translation invariance we may assume that x is the origin and without loss of generality, we also assume that y 1 > 0 and y 2 ≥ 0. By Lemma 3.9 it suffices to show that

P (D(o, y) > C 2 y 1 + C 3 (nk) d | J n ) decays exponentially in n. For 0 ≤ j ≤ n/2 , let (see (3.3)) A j = A(-jk, y 1 + jk, y 2 + jk).
Note that the sets A 0 , . . . , A n/2 are disjoint. Figure 1 should help the reader to visualize them. Our aim is to find paths from o to y through independent attempts, which will enable to use Lemma 3.7(ii) in each set A j . This is why we have first replaced G n by J n to condition with.

On the event J n , we can reach from the origin each of the sets A i by means of an open path contained in B(nk) and from each of these sets we can reach y by means of an open path contained in B y (nk). Hence, on J n for each j ∈ {0, . . . , n/2 } there exist a random point U j ∈ B(nk) ∩ A j and an open path from o to U j such

y 0 A A A A A A A 3 3 A 1 1 A 0 0 2 2 Figure 3.1. the event W 3 that all its sites except U j are in B(nk) ∩ (∩ n/2 =j A c
). If there are many possible values of U j we choose the first one in some arbitrary deterministic order. Similarly, there is a random point V j ∈ B y (nk) ∩ ∆ v A j and an open path from V j to y with all its sites in B y (nk) ∩ (∩ n/2 =j A c ). Let u j and v j be possible values of U j and V j respectively. Then let C 1 be as in Lemma 3.7 and define

F j (u j , v j ) = {U j = u j , V j = v j }, E j (u j , v j ) = {D(u j , v j ) < C 1 ( u j -v j 1 + |u j 2 | + |v j 2 |) within A j } and W j = ∪ u j ,v j F j (u j , v j ) ∩ E j (u j , v j ) ,
where the union is over all possible values of U j and V j . Now we define a subset of

Z d R j = B(nk) ∪ B y (nk) ∪ (A 0 ∪ • • • ∪ A j-1 ) ∩ A c j ∩ • • • ∩ A c n-1 , (3.5)
and we denote by σ j the σ-algebra generated by {T x , e(x, y) :

x ∈ R j , x ∼ y}. Then, noting that 1 Fj (u j ,v j ) Π j-1 =0 1 W c is σ j -measurable, write for j = 1, . . . , n/2 : P W j ∩ J n ∩ (∩ j-1 =0 W c ) = u j ,v j E 1 Fj (u j ,v j ) 1 Ej (u j ,v j ) 1 Jn (Π j-1 =0 1 W c ) = u j ,v j E 1 Fj (u j ,v j ) (Π j-1 =0 1 W c )E(1 Jn 1 Ej (u j ,v j ) |σ j ) ≥ u j ,v j P (E j (u j , v j ))E 1 Fj (u j ,v j ) (Π j-1 =0 1 W c )E(1 Jn |σ j ) = u j ,v j P (E j (u j , v j ))E 1 Fj (u j ,v j ) (Π j-1 =0 1 W c )1 Jn ≥ δ u j ,v j P F j (u j , v j ) ∩ J n ∩ (∩ j-1 =0 W c ) = δP J n ∩ (∩ j-1 =0 W c ) , (3.6)
where the sums are over all possible values of U j and V j , the first inequality follows from Remark 3.2 since both J n and E j (u j , v j ) are increasing events, and from the fact that E j (u j , v j ) is independent of σ j ; the second inequality follows from Lemma 3.7(ii) and the last equality from the fact that J n is contained in the union of the F j (u j , v j )'s which are disjoint. We rewrite (3.6) as

P J n ∩ (∩ j =0 W c ) ≤ (1 -δ)P W j ∩ J n ∩ (∩ j-1 =0 W c ) ≤ (1 -δ)P J n ∩ (∩ j-1 =0 W c
) Now, proceeding by induction on j one gets:

P J n ∩ (∩ n/2 -1 =0 W c ) ≤ (1 -δ) n/2 P (J n ).
Since we can choose C 2 and C 3 in such a way that the event {D(o, y) > C 2 y 1 + C 3 (nk) d } does not occur if any of the W i 's occurs, the lemma follows.

Next proposition concludes this section.

Proposition 3.11. Assume λ > λ c .

(i) Let C 2 be as in Lemma 3.10. Then, there exists α 3 > 0 such that for all x, y ∈ Z d , n ∈ N, we have

P (D(x, y) ≥ C 2 x -y 1 + n d |x → y) ≤ exp(-α 3 n); (ii) P (x → y| |C out x | = +∞, |C in y | = +∞) = 1.

The shape theorem

In the percolation model, let C ∞ be the cluster of sites connected to ∞:

C ∞ = {x ∈ Z d : x → ∞ and ∞ → x}. (4.1) Remark 4.1. As a consequence of Proposition 3.11(ii), C ∞ is a connected set: if two sites x, y of Z d belong to C ∞ , then x → y and y → x. 4.1. Neighborhoods in C ∞ . In this subsection, we construct neighborhoods V(•) of sites in Z d .
We first deal separately with finite clusters, which will have no influence on the asymptotic shape of the epidemic. We will include them in the neighborhoods V(•) of sites we construct.

Definition 4.2. For x ∈ Z d , let R out x = {y ∈ Z d : x → y outside C ∞ } (outgoing root from x); R in x = {y ∈ Z d : y → x outside C ∞ } (incoming root to x).
In particular x belongs to R out Lemma 4.3. There exists σ 1 = σ 1 (λ, d) > 0 such that, for all n ∈ N,

P (R out o ∪ R in o ) ∩ ∂B(n) = ∅ ≤ exp(-σ 1 n). Proof of Lemma 4.3. For n ∈ N \ {0}, R out o ∩ ∂B(2n) = ∅ means that there exists an open path o → ∂B(2n) outside C ∞ . This implies that there exists x ∈ ∂B(n) satisfying o → x → ∂B(2n) outside C ∞ . Similarly, R in o ∩ ∂B(2n) = ∅ implies that there exists x ∈ ∂B(n) satisfying ∂B(2n) → x → o outside C ∞ .
Then for such a point, either the cluster C out x or the cluster C in

x is finite, and has a radius larger than or equal to n. Relying on Proposition A.11,b) in Appendix A, we can follow the proof of Grimmett (1999, Theorems (8.18), (8.21)) to get the existence of σ 0 = σ 0 (λ, d) > 0 such that:

P (C out x ∩ ∂B x (n) = ∅, |C out x | < +∞) ≤ exp(-σ 0 n); P (C in x ∩ ∂B x (n) = ∅, |C in x | < +∞) ≤ exp(-σ 0 n). (4.2) Hence P (R out o ∪ R in o ) ∩ ∂B(2n) = ∅ ≤ P R out o ∩ ∂B(2n) = ∅ +P R in o ∩ ∂B(2n) = ∅ ≤ 2 x∈∂B(n) P (|C out x | < +∞, x → ∂B x (n)) +2 x∈∂B(n) P (|C in x | < +∞, ∂B x (n) → x) ≤ 4|∂B(n)| exp(-σ 0 n)
which induces the result.

To define the neighborhood V(x) on C ∞ of a site x, we introduce the smallest box whose interior contains R out x and R in x , which contains elements of C ∞ , and is such that two elements of C ∞ in this box are connected by an open path which does not exit from a little larger box. For this last condition, which will enable to bound the travel time through V(x), we use the parameter C 2 obtained in Lemma 3.10.

Definition 4.4. Let C = C 2 d + 2. Let κ(x) be the smallest l ∈ N \ {0} such that      (i) ∂B x (l) ∩ R out x ∪ R in x = ∅; (ii) B x (l) ∩ C ∞ = ∅; (iii) ∀ (y, z) ∈ (B x (l) ∩ C ∞ ) 2 , y → z within B x (C l). Remark 4.5. By (i) above, R out x ∪ R in x ⊂ B x (κ(x)
). In the next lemma, we bound the probability a box of size n does not admit properties (i)-(iii) above, that is, we prove that the random variable κ(x) has a sub-exponential tail.

Lemma 4.6. There exists a constant σ = σ(λ, d) > 0 such that, for any n ∈ N,

P (κ(x) ≥ n) ≤ exp(-σn 1/d ).
Proof of Lemma 4.6. We show that the probability that any of the 3 conditions in Definition 4.4 is not achieved for n decreases exponentially in n 1/d : (i) By translation invariance, we have by Lemma 4.3,

P ∂B x (n) ∩ R out x ∪ R in x = ∅ ≤ exp(-σ 1 n). (4.3) (ii) There exist k ∈ N, σ 2 = σ 2 (λ, d) > 0 such that for any n ∈ N, P (B x (n) ∩ C ∞ = ∅) ≤ exp(-σ 2 n/(k + 1) ). (4.4)
Indeed, let k = k(λ, d) be large enough for the conclusions of Theorem 3.5 to hold on the slab S k . Then we have

P (B x (n) ∩ C ∞ = ∅) ≤ P (∀ z ∈ {x + je 1 , 0 ≤ j ≤ n}, z / ∈ C ∞ ) = P (∀ z ∈ {x + je 1 , 0 ≤ j ≤ n}, C in z or C out z is finite) We denote by S k (l) = {l(k + 1), • • • , (l + 1)(k + 1) -1} × Z d-1 for l ≥ 0 the slab of thickness k to which z belongs. If C in z (or C out z ) is finite, so is C in z (S k (l)) (or C out z (S k (l))). Because {|C in z (S k (l))| = +∞} and {|C out z (S k (l))| =
+∞} are increasing events it follows from Theorem 3.5 and the FKG inequality (see Remark

3.2) that inf u∈S k (l) P (|C in u (S k (l))| = |C out u (S k (l))| = +∞) ≥ inf u∈S k (l) P (|C in u (S k (l))| = +∞)P (|C out u (S k (l))| = +∞) > 0. (4.5)
Since events occurring in two different slabs are independent, we have

P (∀ z ∈ {x + je 1 , 0 ≤ j ≤ n}, z / ∈ C ∞ ) ≤ P (∀ l ≥ 0, ∀ z ∈ {x + je 1 , 0 ≤ j ≤ n} ∩ S k (l), C in z (S k (l)) or C out z (S k (l)) is finite) ≤ (P (∀ z ∈ {je 1 , 0 ≤ j ≤ k}, C in z (S k (0)) or C out z (S k (0)) is finite n/(k+1) ≤ exp(-σ 2 n/(k + 1) )
with σ 2 = σ 2 (λ, d) > 0, independent of n, because, for z 0 = k/2 e 1 , using (4.5) we have

P (∃ z ∈ {x + je 1 , 0 ≤ j ≤ k}, |C in z (S k (0))| = |C out z (S k (0))| = +∞) ≥ P (|C in z0 (S k (0))| = |C out z0 (S k (0))| = +∞) > 0. (iii) There exists σ 3 = σ 3 (λ, d) > 0 such that P ∃ (y, z) ∈ (B x (n) ∩ C ∞ ) 2 , y → z within (B x (C n) ≤ exp(-σ 3 n 1/d ). (4.6) Indeed, if no open path from y to z (both in B x (n) ∩ C ∞ ) is contained in B x (C n), then D(y, z) ≥ 2(C -1)n.
Given our choice of C this implies that D(y, z) ≥ C 2 y -z 1 + n. Therefore (4.6) follows from Proposition 3.11(i).

We define the (site) neighborhood in C ∞ of x by We finally define an "edge" neighborhood Γ(x) of x:

V(x) = B x (κ(x)) ∩ C ∞ . ( 4
Γ(x) = {(y , z ) ⊂ B x (κ(x)), (y , z ) open}∪ {(y , z ) ∈ Γ * y,z , y, z ∈ V(x)}. (4.8)
Those neighborhoods satisfy

V(x) ⊂ B x (κ(x)); Γ(x) ⊂ B x (C κ(x)).
(4.9) 4.2. Travel times and radial limits. We now come back to the spatial epidemic model. In this subsection, we estimate the time needed by the epidemic to cover C ∞ , taking advantage of the analysis of paths in the percolation model done in Section 3. We first define an approximation for the passage time of the epidemic, then we prove the existence of radial limits for this approximation and for the epidemic. We will follow for this the spirit of the construction in [START_REF] Cox | Limit theorems for the spread of epidemics and forest fires[END_REF].

By analogy with Cox andDurrett (1981, 1988) (although neighborhoods in our context are defined differently), we define, for x, y ∈ Z d , the travel time from V(x) to V(y) and the time spent around x to be (remember (2.4))

τ (x, y) = inf

x ∈V(x),y ∈V(y)

τ (x , y ); (4.10) We now show that if y ∈ C out x \ R out x , τ (x, y) approximates τ (x, y). Lemma 4.8.

u(x) =      (y ,z )∈Γ(x) τ (y , z ) if Γ(x) = ∅, 0 otherwise.
For x ∈ Z d , if y ∈ C out x \ R out x , we have τ (x, y) ≤ τ (x, y) ≤ u(x) + τ (x, y) + u(y).
(4.12)

Proof of Lemma 4.8. Let Γ x,y be an open path from x to y such that τ (x, y) = τ (Γ x,y ). Since y / ∈ R out x this path must intersect C ∞ . Let c 1 and c 2 be the first and last points we encounter in C ∞ when moving from x to y along Γ x,y . By Definition 4.4(i), c 1 ∈ V(x) and c 2 ∈ V(y): indeed (for instance for c 1 ), either x ∈ C ∞ and c 1 = x, or the point a ∈ ∂B x (κ(x)) ∩ Γ x,y does not belong to R out x and c 1 is the first point on Γ x,y between x and a; we might have

c 1 = c 2 , if V(x) ∩ V(y) = ∅. We have, denoting by ∨ the concatenation of paths, Γ x,y = Γ x,c1 ∨ Γ c1,c2 ∨ Γ c2,y
where Γ x,c1 (resp. Γ c2,y ) is an open path from x to c 1 contained in B x (κ(x)) (resp. from c 2 to y contained in B y (κ(y))) and Γ c1,c2 is an open path from c 1 to c 2 . We then obtain the first inequality of (4.12) since:

τ (x, y) ≤ τ (Γ c1,c2 ) ≤ τ (Γ x,y ) = τ (x, y).
To prove the second inequality of (4.12), let Γ d1,d2 be an open path from

d 1 ∈ V(x) to d 2 ∈ V(y) such that τ (Γ d1,d2 ) = τ (x,
x,d1 = Γ x,c1 ∨ Γ * c1,d1 from x to d 1 satisfies τ (Γ x,d1 ) ≤ u(x).
Similarly, there is an open path Γ d2,y from d 2 to y such that τ (Γ d2,y ) ≤ u(y). We conclude with

τ (x, y) ≤ τ (Γ x,d1 ) + τ (Γ d1,d2 ) + τ (Γ d2,y ) ≤ u(x) + τ (x, y) + u(y).
We now prove that τ (., .) is almost subadditive, which will enable us later on in Theorem 4.12 to appeal to Kingman's Theorem.

τ (x, z) ≤ τ (Γ a,b ) + τ (Γ * b,c ) + τ (Γ c,d ) ≤ τ (x, y) + u(y) + τ (y, z).
We introduce a new notation, for the length of the shortest path between two neighborhoods. For x, y ∈ Z d , let D(x, y) = inf

x ∈V(x),y ∈V(y) D(x , y ). Proposition 4.10. There exist constants C 4 and α 4 > 0 such that

P (D(x, y) ≥ C 4 x -y 1 + n) ≤ exp(-α 4 n 1/d ), ∀ x, y ∈ Z d , n ∈ N.
Proof of Proposition 4.10. Let C 2 be as in Lemma 3.10 and Proposition 3.11. Then

P (D(x, y) ≥ C 2 x -y 1 + (2d + 1)C 2 n) ≤ P (κ(x) > n) + P (κ(y) > n) + P (D(x, y) ≥ C 2 x -y 1 + (2d + 1)C 2 n, κ(x) ≤ n, κ(y) ≤ n) ≤ P (κ(x) > n) + P (κ(y) > n) + x ∈Bx(n),y ∈By(n) P (D(x , y ) ≥ C 2 x -y 1 + (2d + 1)C 2 n, x → y ) ≤ P (κ(x) > n) + P (κ(y) > n) + x ∈Bx(n),y ∈By(n) P (D(x , y ) ≥ C 2 x -y 1 + C 2 n, x → y ).
The result follows from Proposition 3.11 and Lemma 4.6.

Of course, the random variables u(x) and τ (x, y) are almost surely finite. But we will need later on repeatedly a better control of their size, provided by our next lemma.

Lemma 4.11. For all x, y ∈ Z d , r ∈ N \ {0}, u(x) and τ (x, y) have a finite r-th moment.

Proof of Lemma 4.11. By Lemma 4.6, u(x) is bounded above by a sum of passage times e(y, z) with y and z in the box B x (Y ), where Y is a random variable whose moments are all finite. By Lemmas 4.6 and 4.10 the same happens to τ (x, y) (if x ∈ V(x), y ∈ V(y) are the sites that achieve D(x, y), then τ (x, y) ≤ τ (x , y )). Therefore it suffices to show that if (X i , i ∈ N) is a sequence of i.i.d. random variables and N is a random variable taking values in N, then the moments of N i=1 X i are all finite if it is the case for both the X i 's and N . To prove this write:

E(| N i=1 X i | r ) = ∞ n=1 E(|X 1 + • • • + X n | r 1 {N =n} ) ≤ ∞ n=1 [E(|X 1 + • • • + X n | 2r )P (N = n)] 1/2 ≤ ∞ n=1 [E(|X 1 | + • • • + |X n |) 2r P (N = n)] 1/2 ≤ ∞ n=1 [n 2r C 2r P (N = n)] 1/2
where the second line comes from Cauchy-Schwartz' inequality, the factor n 2r counts the number of terms in the development of (|X 1 |+• • •+|X n |) 2r and the constant C 2r depends on the distribution of the X i 's. As N has all its moments finite P (N = n) decreases faster than n -2r-4 and the sum is finite.

We now construct a process (ϑ • ) which will be subadditive in every direction, and will have a.s., by Kingman's Theorem, a radial limit denoted by µ. We will then check that τ (o, •) also has, in every direction, the same radial limit, and we will extend this conclusion to τ (o, •) on the set C out o of sites that have ever been infected. Hence we first prove Theorem 4.12. For all z ∈ Z d , there exists µ(z) ∈ R + such that almost surely

lim n→+∞ τ (o, nz) n = µ(z) and (4.15) lim n→+∞ τ (o, nz) n -µ(z) 1 {nz∈C out o } = 0. (4.16) Proof of Theorem 4.12. (i) For all z ∈ Z d , (m, n) ∈ N 2 , let ϑ z (m, n) = τ (mz, nz) + u(nz).
(4.17)

The process (ϑ z (m, n)) (m,n)∈N 2 satisfies the hypotheses of Kingman' subadditive ergodic theorem (see Liggett, 2005, Theorem VI.2.6) by (4.13). Hence (noticing also that ϑ z (0, n) = ϑ nz (0, 1)) there exists µ(z

) ∈ R + such that lim n→+∞ 1 n ϑ z (0, n) = lim n→+∞ E ϑ z (0, n) n = lim n→+∞ E ϑ nz (0, 1) n = inf n∈N E ϑ z (0, n) n = inf n∈N E ϑ nz (0, 1) n = µ(z) (4.

Extending µ.

We have proved the existence of a linear propagation speed in every direction of Z d . Now, to derive an asymptotic shape result, in particular for the approximating travel times ( τ (x, y), x, y ∈ Z d ), we need to extend µ from Z d to a Lipschitz, convex and homogeneous function on R d . The asymptotic shape of the epidemic will be given by the convex set D defined in (4.20) below. As a first step, we prove properties of µ on Z d .

Lemma 4.13. The function µ satisfies the following properties for all x, y ∈ Z d , k ∈ N:

(i) µ(x) = lim n→+∞ E τ (o, nx) n , (ii) µ(x + y) ≤ µ(x) + µ(y), (iii) µ(x) = µ(-x), (iv) µ(e i ) = µ(e ), ∀i, ∈ {1, . . . , d}, (v) µ(kx) = kµ(x), (vi) µ(x) ≤ µ(e 1 ) x 1 .
Proof of Lemma 4.13. Since ϑ x (0, n) = τ (o, nx)+u(nx), part (i) follows from (4.18) and (4.19). To prove part (ii) write:

µ(x + y) = lim n→+∞ E τ (o, n(x + y)) n ≤ lim n→+∞ E τ (o, nx) n + lim n→+∞ E τ (nx, n(x + y)) n + lim n→+∞ E u(nx) n = lim n→+∞ E τ (o, nx) n + lim n→+∞ E τ (nx, n(x + y)) n = µ(x) + µ(y),
where the first equality follows from part (i), the inequality from (4.13), the second equality from (4.19) and the third one from part (i) and translation invariance of τ . Parts (iii)-(iv) follow immediately from part (i) and the corresponding properties of τ (o, x). To prove part (v) write:

µ(kx) = lim n→+∞ E ϑ nkx (0, 1) n = k lim n→+∞ E ϑ nkx (0, 1) nk = kµ(x),
where the first and third equalities follow from (4.18). Finally, part (vi) follows from parts (ii)-(iv).

Next corollary extends Lemma 4.13(iii)-(iv).

Corollary 4.14. For any permutation σ of {1, )) for any choice of the signs and any permutation σ, hence the corollary follows from Lemma 4.13(i).

Lemma 4.15. Let γ * = µ(e 1 ). Then γ * is a Lipschitz constant for µ. For all u, v ∈ Z d we have

|µ(u) -µ(v)| ≤ γ * u -v 1 .
Proof of Lemma 4.15. Let y = u -v, x = v. We have

µ(u) -µ(v) = µ(x + y) -µ(x) ≤ µ(y) = µ(u -v) ≤ µ(e 1 ) u -v 1 ,
where the inequalities follow from Lemma 4.13(ii) and (vi). Similarly, taking x = u, y = v -x gives µ(v) -µ(u) ≤ µ(e 1 ) v -u 1 , and the lemma follows.

In a second step, we extend µ to R d and we introduce the set D. Proposition 4.16. There exists an extension of µ to R d , which is Lipschitz with Lipschitz constant γ * given by Lemma 4.15, convex and homogeneous on R d . Moreover, µ(x) = 0 if and only if x = o and the set

D = {x ∈ R d : µ(x) ≤ 1} (4.20)
is convex, bounded and contains an open ball centered at o.

Proof of Proposition 4.16. We start by extending µ to

Q d . For x ∈ Q d \ {o} let N x = min{k ≥ 1, k ∈ N : kx ∈ Z d } and (4.21) µ(x) = µ(N x x) N x . (4.22)
We now prove that this extension is homogeneous: let α ∈ Q be positive and let

x ∈ Q d , x = o. Then, there exist k 1 , k 2 ∈ N multiples of N x and N αx respectively, such that k 1 x, k 2 αx ∈ Z d and k 1 x = k 2 αx. Write µ(αx) = µ(N αx αx) N αx = µ(k 2 αx) k 2 = µ(k 1 x) k 2 = k 1 k 2 µ(k 1 x) k 1 = α µ(N x x) N x = αµ(x),
using (4.22) for the first equality, Lemma 4.13(v) for the second and fifth ones.

To prove that µ is Lipschitz on

Q d , let x, y ∈ Q d \ {o}. Then, |µ(x) -µ(y)| = µ(N x x) N x - µ(N y y) N y = µ(N y N x x) N y N x - µ(N x N y y) N x N y = |µ(N x N y x) -µ(N x N y y)| N x N y ≤ γ * N x N y x -N x N y y 1 N x N y = γ * x -y 1 ,
using Lemma 4.13(v) for the second equality and Lemma 4.15 for the inequality.

To prove that µ is convex on

Q d , take x, y ∈ Q d and α ∈ Q ∩ (0, 1). Then let k 1 , k 2 be elements in N such that k 1 α ∈ N, k 2 x ∈ Z d , k 2 y ∈ Z d and write: µ(αx + (1 -α)y) = lim n→+∞ E τ (o, nαx + n(1 -α)y) n = lim n→+∞ E τ (o, nk 1 αk 2 x + nk 1 (1 -α)k 2 y) nk 1 k 2 ≤ lim n→+∞ E τ (o, nk 1 αk 2 x) + τ (o, nk 1 (1 -α)k 2 y) + u(nk 1 αk 2 x) nk 1 k 2 = lim n→+∞ E τ (o, nk 1 αk 2 x) + τ (o, nk 1 (1 -α)k 2 y) nk 1 k 2 = µ(k 1 k 2 αx) + µ(k 1 k 2 (1 -α)y) k 1 k 2 = αµ(x) + (1 -α)µ(y),
where the first equality follows from Lemma 4.13(i), the inequality from Lemma 4.9, the third equality from (4.19), the fourth from Lemma 4.13(i) and the last one from the homogeneity of µ on Q d . Because µ is homogeneous, Lipschitz and convex on Q d , we can extend µ by continuity to R d .

To prove that µ(x) > 0 if x = o we argue by contradiction: assume µ(x) = 0 and without loss of generality that x = (x 1 , . . . , x d ) with x 1 = 0. First note that since µ is Lipschitz and homogeneous, the conclusion of Corollary 4.14 also holds for any (x 1 , . . . , x d ) ∈ R d , then write

µ(2x 1 , 0, • • • , 0) = µ(2x 1 , 0, • • • , 0) -µ(x 1 , x 2 , • • • , x d ) ≤ µ(x 1 , -x 2 , • • • , -x d ) = 0,
using Lemma 4.13(ii) for the inequality, and Corollary 4.14 with the assumption µ(x) = 0 for the last equality. Then since µ is homogeneous we get µ(e 1 ) = 0. However, considering a standard first passage percolation model with passage times e(z, y) and adding a 'tilde' to quantities associated to this model, we have τ (o, z) ≤ τ (o, z) a.s. for all z ∈ Z d . Since by Kesten (1986, Theorem (2.18)), lim n→+∞ τ (o, ne 1 ) = µ(e 1 ), it follows from (4.16) that µ(e 1 ) ≤ µ(e 1 ) = 0. But from Kesten (1986, Theorems (1.7) and (1.15)) we get µ(e 1 ) > 0, thus reaching a contradiction.

The convexity of µ implies that D is convex. We prove by contradiction that D contains an open ball centered at o: otherwise, there exists a sequence (x n ) n∈N such that x n / ∈ D, lim n→+∞ x n = 0; therefore on the one hand µ(x n ) > 1, and on the other hand lim n→+∞ µ(x n ) = 0 because µ(o) = 0 and µ is continuous, hence a contradiction.

Finally we argue again by contradiction to prove that the set D is bounded: otherwise there would exist a sequence (y n ) n∈N with y n ∈ D and y n 1 > n. Then x n = y n / y n 1 satisfies x n 1 = 1, and, since µ is homogeneous, lim n→+∞ µ(x n ) = 0. By compactness (x n ) n∈N has a converging subsequence to some x such that µ(x) = 0 with x 1 = 1; since we have already proved there is no such x we get a contradiction.

4.4. Behavior of τ . Our next result establishes how τ (o, z) grows for z ∈ Z d . Theorem 4.17. There exist K = K(λ, d) > 0 and α > 0 such that

P ( τ (o, z) > K z ∞ ) ≤ exp(-α( z 1/d ∞ ), ∀ z ∈ Z d , P ( τ (o, z) > K( z ∞ + n)) ≤ exp(-αn 1/d ), ∀ z ∈ Z d , n ∈ N, z∈Z d P ( τ (o, z) > K z ∞ ) < +∞. Proof of Theorem 4.17. Let K ≥ 0, z ∈ Z d and let B = B(o, ( z ∞ + n)/4) × B(z, ( z ∞ + n)/4
). Then write:

P ( τ (o, z) > K( z ∞ + n)) ≤ P (4κ(z) > z ∞ + n) + P (4κ(o) > z ∞ + n) + P (A) (4.23)
where

A = { τ (o, z) > K( z ∞ + n), 4κ(z) ≤ z ∞ + n, 4κ(o) ≤ z ∞ + n} ⊂ ∪ (x,y)∈B {x → y, τ (x, y) > K( z ∞ + n)}. (4.24) Note that if (x, y) ∈ B we have z ∞ -n ≤ 2 x -y ∞ ≤ 3 z ∞ + n and 3( z ∞ + n) = 3 z ∞ + n + 2n ≥ 2( x -y ∞ + n).
(4.25)

From (4.24), (4.25), for C 2 given in Proposition 3.11, we get:

P (A) ≤ x∈B(o,( z ∞ +n)/4) y∈B(z,( z ∞ +n)/4) P (3τ (x, y) > 2K( x -y ∞ + n), D(x, y) < (C 2 + 1)( x -y 1 + n)) +P (x → y, D(x, y) ≥ (C 2 + 1)( x -y 1 + n)) . (4.26)
It now follows from Proposition 3.11(i) that we have, for some α 5 > 0,

P (x → y, D(x, y) ≥ (C 2 + 1)( x -y 1 + n)) ≤ exp(-α 5 ( x -y 1 + n) 1/d ) ≤ exp(-α 5 ( x -y ∞ + n) 1/d ).
(4.27)

Then, taking K large enough, by large deviation results for exponential variables, we also have, for some α 6 > 0,

P (3τ (x, y) > 2K( x -y ∞ + n), D(x, y) < (C 2 + 1)( x -y 1 + n)) ≤ P (3τ (x, y) > 2K( x -y ∞ + n), D(x, y) < (C 2 + 1)d( x -y ∞ + n)) ≤ exp(-α 6 ( x -y ∞ + n)). (4.28)
Hence, from (4.25)-(4.28), for some constants R and α 7 > 0 we have:

P (A) ≤ R( z ∞ + n) 2d exp(-α 7 ( z ∞ + n) 1/d ),
which gives, by modifying the constants,

P (A) ≤ R exp(-α 8 ( z ∞ + n) 1/d ). (4.29)
The theorem's statements now follow from (4.29), (4.23) and Lemma 4.6.

4.5. Asymptotic shape for τ . Next theorem is the last necessary step to prove the shape theorem.

Theorem 4.18. Let ε > 0, and

A t = {z ∈ Z d : τ (o, z) ≤ t}.
Then, a.s. for t large enough, for D defined in (4.20),

(1 -ε)tD ∩ Z d ⊂ A t ⊂ (1 + ε)tD ∩ Z d . (4.30)
In the sequel K and α are fixed constants satisfying the conclusions of Theorem 4.17, γ * is the Lipschitz constant of µ (see Lemma 4.15) and N x was defined in (4.21) for any x ∈ Q d \ {o}. To prove Theorem 4.18 we need the two following lemmas.

Lemma 4.19. Let ρ > 0 and let δ ≤ ρ/(2K). Then, for all x ∈ Q d \ {o}, k>0 P ( sup

z∈B kNx x (δkNx)∩Z d τ (kN x x, z) ≥ kN x ρ) < ∞, (4.31) k>0 P ( sup z∈B kNx x (δkNx)∩Z d τ (z, kN x x) ≥ kN x ρ) < ∞. (4.32)
Proof of Lemma 4.19. We derive only (4.31), since the proof of (4.32) is analogous. By translation invariance P ( sup

z∈B kNx x (δkNx)∩Z d τ (kN x x, z) ≥ kN x ρ) = P ( sup z∈B(δkNx)∩Z d τ (o, z) ≥ kN x ρ).
Hence it suffices to show that k>0 P ( sup

z∈B(δkNx)∩Z d τ (o, z) ≥ kN x ρ) < ∞. Let k > 0, z ∈ B(δkN x ) ∩ Z d
. By Theorem 4.17 we have:

P ( τ (o, z) ≥ kN x ρ) ≤ P ( τ (o, z) ≥ K z ∞ + kN x ρ/2) ≤ exp -α kN x ρ 2K 1/d .
Therefore, for some constant C, k>0 P ( sup

z∈B(δkNx)∩Z d τ (o, z) ≥ kN x ρ) ≤ k>0 C(δkN x ) d exp -α kN x ρ 2K 1/d < ∞.
For x ∈ Q d \ {o}, δ > 0, we define the cone associated to x of amplitude δ as

C x (δ) = Z d ∩ ∪ t≥0 B tx (δt) . (4.33) Lemma 4.20. Let x ∈ Q d \ {o}. Then for any 0 < δ < δ the set C x (δ ) \ ∪ k≥0 B kNxx (δkN x ) is finite.
Proof of Lemma 4.20. Let

t 0 = N x x 1 δ -δ . Since Z d ∩ ∪ t0 t≥0 B tx (δ t) is finite, it suffices to show that ∪ ∞ t≥t0 B tx (δ t) ⊂ ∪ k≥0 B kNxx (δkN x ). To prove this, pick z ∈ B t1x (δ t 1 ) for some t 1 ≥ t 0 . Let k 0 = inf{i ∈ N : iN x ≥ t 1 }. Hence 0 ≤ k 0 N x -t 1 < N x , and z -k 0 N x x 1 ≤ z -t 1 x 1 + |t 1 -k 0 N x | x 1 < z -t 1 x 1 + N x x 1 ≤ δ t 1 + N x x 1 = δ t 1 + (δ -δ )t 0 ≤ δt 1 ≤ δk 0 N x .
Therefore z ∈ B k0Nxx (δk 0 N x ) and the lemma is proved.

In the next proof we use that since the Lipschitz constant of µ is γ * for the norm . 1 (by Proposition 4.16), it is γ = γ * d for the norm . ∞ .

Proof of Theorem 4.18. Fix ε ∈ (0, 1) and let ρ, δ and ι be three small positive parameters such that δ ≤ ρ/(2K), whose values will be determined later. The set Y = {x ∈ Q d : 1 -2ι < µ(x) < 1 -ι} is a ring between two balls with the same center but with a different radius, because by Proposition 4.16, µ is homogeneous and positive except that µ(o) = 0. Hence the (compact) closure of Y, which is recovered by balls of the same radius centered on the rational points of Y, is in fact covered by a finite number of such balls. Thus there exists a finite subset Y of Y such that Z d ⊂ ∪ x∈Y C x (δ/2) (if the balls recover the ring, the cones associated to them recover the whole space). Hence, to prove the first inclusion of (4.30) it suffices to show that for any x ∈ Y and any sequences (t n ) n>0 and (z n

) n>0 such that t n ↑ ∞ in R + , z n ∈ C x (δ/2) ∩ Z d with z n ∞ ≥ n and µ(z n ) ≤ (1 -ε)t n ,
∈ N be such that z n ∈ B knNxx (δk n N x ), hence k n ≥ Cn for some constant C. Since µ is Lipschitz, write k n N x (1 -2ι) ≤ µ(k n N x x) ≤ µ(z n ) + γδk n N x ≤ (1 -ε)t n + γδk n N x , so that k n N x ≤ 1 -ε 1 -2ι -γδ t n . (4.34)
It now follows from (4.34) and the subadditivity property (4.13) of τ that:

τ (o, z n ) t n ≤ 1 -ε 1 -2ι -γδ τ (o, k n N x x) k n N x + u(k n N x x) k n N x + τ (k n N x x, z n ) k n N x .
Therefore, by Theorem 4.12, Lemma 4.11 (the variables u(.) are identically distributed, and k n ≥ Cn), Lemmas 4.16 and 4.19 we obtain:

lim sup n→+∞ τ (o, z n ) t n ≤ 1 -ε 1 -2ι -γδ µ(x) + ρ a.s. Since x ∈ Y this implies: lim sup n→+∞ τ (o, z n ) t n ≤ 1 -ε 1 -2ι -γδ 1 -ι + ρ a.s.
Taking ι, ρ and δ small enough, the right hand side is strictly less than 1 which proves that τ (o, z n ) ≤ t n a.s. for n sufficiently large. Similarly, to prove the second inclusion of (4.30) it suffices to show that for any x ∈ Y and any sequences

t n ↑ ∞ in R + and z n in C x (δ/2) ∩ Z d such that µ(z n ) ≥ (1 + ε)t n we have τ (o, z n ) > t n a.s.
for n sufficiently large. As before, taking subsequences if necessary, we let (t n ) n>0 and (z n ) n>0 be such sequences, and k n ∈ N be such that z n ∈ B knNxx (δk n N x ). Proceeding then as for the first inclusion, we get:

k n N x (1 -ι) ≥ µ(k n N x x) ≥ µ(z n ) -γδk n N x ≥ (1 + ε)t n -γδk n N x , k n N x ≥ 1 + ε 1 -ι + γδ t n , τ (o, z n ) t n ≥ 1 + ε 1 -ι + γδ τ (o, k n N x x) k n N x - u(z n ) k n N x - τ (z n , k n N x x) k n N x , and lim inf n→+∞ τ (o, z n ) t n ≥ 1 + ε 1 -ι + γδ µ(x) -ρ a.s. ≥ 1 + ε 1 -ι + γδ 1 -2ι -ρ a.s.
Now, taking ι, ρ and δ small enough, the right hand side is strictly bigger than 1 and the second inclusion of (4.30) is proved.

4.6. Asymptotic shape for the epidemic. We can now prove our main result, the shape theorem.

Proof of Theorem 2.2. Let ε > 0 be given.

(i) We first show that the infection grows at least linearly as t goes to infinity, that is,

P Υ t ∪ Ξ t ⊃ (1 -ε)tD ∩ C out o for all t large enough = 1. (4.35) Since R out o
is finite a.s. this will follow from:

P Υ t ∪ Ξ t ⊃ (1 -ε)tD ∩ (C out o \ R out o ) for all t large enough = 1. (4.36) By Theorem 4.18, if 0 < a < b then atD ∩ Z d ⊂ A bt a.s. for t large enough. Hence, for t large enough z ∈ (1 -ε)tD ∩ (C out o \ R out o ) implies τ (o, z) ≤ (1 -ε/2)t a.s. (4.37)
and by Lemma 4.8, τ (o, z) ≤ (1 -ε/2)t + u(o) + u(z). Since u(o) < +∞ a.s. we have u(o) < (ε/4)t a.s. for t large enough. Hence, by (2.7), (4.36) will follow if we show that sup z∈tD u(z) ≤ (ε/4)t a.s. for t large enough, which is implied by sup z∈(n+1)D u(z) ≤ (ε/4)n a.s. for n = t . By Proposition 4.16, D is bounded, hence the number of points in (n+1)D with coordinates in Z is less than C 5 (n+1) d for some constant C 5 . Then write

P sup z∈(n+1)D u(z) ≥ εn 4 ≤ C 5 (n + 1) d P u(o) ≥ εn 4
We introduce parameters whose values will be settled in Lemma A.9 below. We fix λ > λ c and adopt the terminology of Grimmett (1999, Chapter 7). Nonetheless, we might change names of constants if this creates confusions with the rest of our paper. In the sequel n, m and N are positive integers such that 2m < n and

N = n + m + 1. (A.1)
We consider our percolation model on the slab Z 2 × [-3N, 3N ]. Recall that we denote by (e 1 , e 2 , e 3 ) the canonical basis of Z 3 . For

x = (x 1 , x 2 , x 3 ) ∈ Z 3 and k ∈ N such that -3N + k ≤ x 3 ≤ 3N -k we recall that B(k) = [-k, k] 3 and B x (k) = x + [-k, k] 3 . We divide the face F (n) = {x : x ∈ ∂B(n), x 1 = n} of ∂B(n)
in 4 quadrants:

T +,+ (n) = {x : x ∈ ∂B(n), x 1 = n, x 2 ≥ 0, x 3 ≥ 0}, T +,-(n) = {x : x ∈ ∂B(n), x 1 = n, x 2 ≥ 0, x 3 ≤ 0}, T -,+ (n) = {x : x ∈ ∂B(n), x 1 = n, x 2 ≤ 0, x 3 ≥ 0}, T -,-(n) = {x : x ∈ ∂B(n), x 1 = n, x 2 ≤ 0, x 3 ≤ 0}.
and, for any choice of (i, j) ∈ {+, -} 2 , we define a box of thickness 2m+1 composed of translates of the corresponding quadrant, by

T i,j (m, n) = ∪ 2m+1 =1 { e 1 + T i,j (n)}. (A.2)
For any of the above sets a subindex as y means we translate it by y.

Definition A.1. A seed is a translate of B(m) such that all its edges are λ -open.

We will be looking for oriented open paths starting in a seed inside B(2N ), and: either (a) contained in the union of boxes B(3N ) ∪ B 6N e1 (3N ) and reaching a seed inside B 6N e1 (2N ) ∩ B 8N e1 (2N ); or (b) contained in the union of boxes B(3N ) ∪ B 6N e2 (3N ) and reaching a seed inside B 6N e2 (2N ) ∩ B 8N e2 (2N ). We will construct those paths in Lemma A.7 below.

An important tool in Grimmett (1999, Chapter 7) is the sprinkling technique, which enables some bonds, that would be closed otherwise, to be independently open with a probability larger than some ε > 0. We therefore need to find a way to proceed similarly, in spite of the fact that we work with dependent percolation. For this, it is convenient to define the processes for different values of the rate of propagation λ on our common probability space and compare them. Let (e 1 (x, y), x, y ∈ Z 3 ) be a collection of independent exponential r.v.'s with parameter 1. Then let e λ (x, y) = λ -1 e 1 (x, y), and

X λ (x, y) = 1 if e λ (x, y) < T x ; 0 otherwise. (A.3)
We recall that the event {T x > e λ (x, y)} occurs if and only if the oriented bond (x, y) is λ-open.

The following lemma implies that given λ > δ 1 > 0, there exists ι > 0 such that for any λ such that λ + δ 1 < λ the random field {X λ+δ1 (u, v) : u, v ∈ Z 3 } is stochastically above the random field {max{X λ (u, v), Y (u, v)} : u, v ∈ Z 3 } where the random variables Y (u, v) are i.i.d. Bernoulli with parameter ι and are independent of the random variables X λ (u, v). This lemma justifies the use of the sprinkling technique needed to prove Lemmas A.5 and A.7 below.

Lemma A.2. Assume λ > δ 1 > 0. There exists ι > 0 such that for any λ > 0 such that λ + δ 1 < λ, and any x, y ∈ Z 3 , with y ∼ x,

P (X λ+δ1 (x, y) = 1 | X λ+δ1 (u, v), u, v ∈ Z 3 , u ∼ v, (u, v) = (x, y); X λ (u, v), u, v ∈ Z 3 , u ∼ v) > ι a.s. Proof of Lemma A.2. Recall that X λ+δ1 (x, y) is independent of the random vari- ables X λ+δ1 (u, v), X λ (u, v), u = x, v ∼ u, hence it suffices to show that P (X λ+δ1 (x, y) = 1 | X λ+δ1 (x, z), z ∼ x, z = y; X λ (x, z), z ∼ x) > ι a.s. (A.4)
Since we are now conditioning on a finite number of random variables taking only values 0 and 1, (A.4) will follow from:

P (X λ+δ1 (x, y) = 1 | X λ+δ1 (x, z) = a z , z ∼ x, z = y; X λ (x, z) = b z , z ∼ x) > ι (A.5)
for all choices a z and b z in {0, 1} with b z ≤ a z .

To prove (A.5), we denote by N x the union of all partitions of the set {z ∈ Z 3 : z ∼ x} of neighbors of x into three disjoint sets called N 1 x , N 0 x , N 0,1

x such that y ∈ N 1

x ∪ N 0,1 x . Using the inequality P (A|B) ≥ P (A ∩ B) for two events A, B, and taking an arbitrary partition in N x gives

P (X λ+δ1 (x, y) = 1 | X λ+δ1 (x, u) = 0, ∀u ∈ N 0 x , X λ (x, v) = 1, ∀v ∈ N 1 x ; X λ (x, w) = 0, X λ+δ1 (x, w) = 1, ∀w ∈ N 0,1 x ) ≥ P (X λ+δ1 (x, y) = 1, X λ+δ1 (x, u) = 0, ∀u ∈ N 0 x , X λ (x, v) = 1, ∀v ∈ N 1
x ; X λ (x, w) = 0, X λ+δ1 (x, w) = 1, ∀w ∈ N 0,1 x ). Let a > 0 be such that P (T x ∈ [a, a + γ]) > 0 for all γ > 0, and let δ 2 ∈ (0, a) be such that b defined by

b := (a + δ 2 ) λ λ + δ 1 satisfies b < a. On the event {T x ∈ [a, a + δ 2 /2), e λ (x, w) ∈ [a + δ 2 /2, a + δ 2 ), ∀w ∈ N 0,1 x , e λ (x, v) ∈ [a -δ 2 /2, a), ∀v ∈ N 1 x , e λ+δ1 (x, u) ∈ [a + δ 2 /2, a + δ 2 ), ∀u ∈ N 0 x }, (A.6)
for all sites v such that v ∈ N 1

x we have e λ (x, v) < T x hence X λ (x, v) = 1; for all sites u such that u ∈ N 0

x we have T x < e λ +δ1 (x, u) hence X λ+δ1 (x, u) = 0; and for all sites w such that w ∈ N 0,1 x we have T x < e λ (x, w) and

e λ+δ (x, w) = λ λ + δ 1 e λ (x, w) ≤ λ λ + δ 1 (a + δ 2 ) = b < a ≤ T x
hence X λ (u, w) = 0 and X λ+δ1 (u, w) = 1. Since the probability p(N 0 x , N 1 x , N 0,1 x ) of the event (A.6) is strictly positive, we conclude the proof of (A.5) by taking

ι = inf Nx p(N 0 x , N 1 x , N 0,1 x ).
It is in view of this sprinkling procedure that we chose a propagation rate λ > λ c . As in Grimmett (1999, Section 7.2), we go on with two key geometrical lemmas.

The first one, Lemma A.3, corresponds to Grimmett (1999, Lemma 7.9), with a very similar proof that we omit consequently. The second one, Lemma A.5, corresponds to Grimmett (1999, Lemma 7.17), that it generalizes in view of its applications for Theorem 3.5 and Lemma 3.7.

Lemma A.3. If λ c < λ and η > 0, then there exist integers m = m(λ , η) and n = n(λ , η) satisfying (A.1) and such that

P there exists a λ -open path in B(n) ∪ T i,j (m, n) from B(m) to a seed contained in T i,j (m, n) > 1 -η,
for any choice of (i, j) ∈ {+, -} 2 .

Notation A.4. Given a subset V of Z 3 , δ > 0 and x ∈ V , we let σ(x, V, λ , δ) be the σ-algebra generated by the indicator functions of the following collection of events:

{T y > e λ (y, z) : y ∈ V, z ∼ y} ∪ {T y > e λ +δ (y, z) :

y ∈ V ∩ B x (n) c , z ∼ y}. (A.7)
Note that when V ∩ B x (n) c = ∅, σ(x, V, λ , δ) is simply the σ-algebra generated by the indicator functions of {T y > e λ (y, z) : y ∈ V, z ∼ y}, which we will denote by σ(V, λ ).

For A a subset of Z 3 , recall from (3.2) that ∆ v A denotes the exterior vertex boundary of A.

Lemma A.5. If λ c < λ and , δ > 0, there exists m = m(λ , , δ) and n = n(λ , , δ) satisfying (A.1) and with the following property: For any choice of (i, j) ∈ {+, -} 2 , any x ∈ Z 3 , any set L ⊂ Z 3 such that

B x (m) ⊂ L ⊂ Z 3 \ T i,j x (m, n) (A.8)
and for any σ(x, L, λ , δ)-measurable event H of strictly positive probability we have:

P (G i,j |H) ≥ 1 -, (A.9)
where G i,j = there exists a path contained in B x (n) ∪ T i,j x (m, n) going from L to a seed contained in T i,j

x (m, n) and such that its first edge (u, v) Since the model is invariant under translations and 90 degree rotations, it suffices to show the lemma when x is the origin and (i, j) = (+, +). We will hence drop x, i and j from the notation. Let Here it is understood that there always is a λ -open path from a point to itself. Therefore any

with u ∈ L, v ∈ ∆ v L is (λ + δ)-open
z ∈ ∆ v (L ∩ B(n)) contained in a seed in T (m, n) is in V (L). Note also that since the path is contained in B(n) ∪ T (m, n), V (L) is a subset of B(n) ∪ T (m, n). Now write G = ∪ K {L λ +δ ⇒ K} ∩ {V (L) = K} (A.12)
where the union is over all possible values of V (L). Our next step is to show that if m and n are properly chosen, the set V (L) is large with probability close to 1. Let k be a positive integer. Note that if V (L) has at most k elements, by the FKG inequality (see Remark 3.2), the probability that all the bonds entering V (L) are λ -closed is at least (1 -α) 6k (recall that α is the probability that any given bond is λ -open, and that we are working in Z 3 ). All the λ -open paths going from L to a seed contained in T (m, n) have to pass through a point in V (L). Hence if all the bonds entering V (L) are λ -closed, such a path does not exist. Thus we have

P there exists a λ -open path contained in B(n) ∪ T (m, n) \ L going from L to a seed contained in T (m, n) | |V (L)| ≤ k ≤ 1 -(1 -α) 6k .
(A.13) Now write:

P (G, V (L) = K | H) = P (L λ +δ ⇒ K, V (L) = K | H) = P (L λ +δ ⇒ K | V (L) = K, H)P (V (L) = K | H).
But the event {V (L) = K} is measurable with respect to the σ-algebra generated by the random variables (T x , e λ (x, y) : x / ∈ L) while both {L λ +δ

⇒ K} and H are measurable with respect to the σ-algebra generated by the random variables (T x , e λ (x, y), e λ +δ (x, y) : x ∈ L). Therefore {V (L) = K} is independent of the pair of events H, {L λ +δ ⇒ K}, so that

P (G, V (L) = K | H) = P (L λ +δ ⇒ K | H)P (V (L) = K)
Then summing up over all sets K such that |K| ≥ 6r, it follows from (A.12) and (A.19) that P (G|H) ≥ (1 -(1 -ι) r )P (|V (L)| ≥ 6r). To complete the proof of Lemma A.5 first pick r such that (1 -ι) r < /2 and then use (A.17) to pick m and n such that P (|V (L)| ≥ 6r) ≥ 1 -/2. Notation A.6. For a given x ∈ Z 3 and i = 1, 2, 3, H i

x will denote the hyperplane perpendicular to e i passing through x. Before stating the next lemma where x ∈ B(2N -m), we define T x (m, n) to be the thickened box built from the quadrant opposite to the one x belongs to in the face H

1 x ∩ B(2N -m), that is T +,+ x (m, n) if x 2 ≤ 0 and x 3 ≤ 0, T +,- x (m, n) if x 2 ≤ 0 and x 3 > 0, T -,+ x (m, n) if x 2 > 0 and x 3 ≤ 0, T -,- x (m, n) if x 2 > 0 and x 3 > 0.
Thanks to Lemma A.5, in the following lemma we construct open paths starting in a seed inside B(2N ), and reaching either a seed inside B 6N e1 (2N )∩B 8N e1 (2N ) or inside B 6N e2 (2N ) ∩ B 8N e2 (2N ). For i ∈ {1, 2}, the successive seeds in these open paths will have centers belonging to the hyperplanes H i x+N ei , H i x+2N ei , H i x+3N ei , H i x+4N ei , . . .; these successive seeds will be respectively contained in B N ei (2N ), B 2N ei (2N ), B 3N ei (2N ), B 4N ei (2N ), . . ., and we will stop as soon as we will get a seed in B 8N ei (2N ). This construction will use a steering procedure in which, at each stage, the choice of a seed in T . (m, n) compensates from an earlier deviation. Proof of Lemma A.7. We consider C 1

x . Since the model is invariant under 90 degree rotations, the proof will also be valid for C 2

x . Let V 1 be the set of vertices of all paths starting at B x (m) and contained in B x (n) ∪ T x (m, n) whose first edge is (λ + δ)-open and all the other edges are λ -open. Let

A 1 x = {V 1 contains a seed in T x (m, n)}. Note that since x ∈ B(2N -m) a path contained in B x (n) ∪ T x (m, n) is also contained in B(3N ). Note also that the center of a seed in T x (m, n) belongs to H 1 x+N e1
, and that by our definition of T x (m, n) (in Notation A.6) this seed is contained in B N e1 (2N ). Thanks to Lemma A.5 with L = B x (m) and H the whole probability space (that is without conditioning), there exist n, m such that

P (A 1 x ) > 1 -. (A.20) If A 1
x occurs, of all the seeds in T x (m, n) ∩ V 1 we choose one according to some arbitrary deterministic order. We now define a random variable Z 1 as follows: on A 1

x , Z 1 is the center of the chosen seed and on (A 1 x ) c , Z 1 = ∆ where ∆ is an extra point we add to Z 3 . Note that on A 1

x , Z 1 takes values in the hyperplane H 1 x+N e1 . The random variable Z 1 is a function of V 1 which we denote by F 1 . We now wish to give a lower bound to the conditional probability given {Z

1 = z 1 } with z 1 = ∆ that there is a path contained in B z1 (n) ∪ T z1 (m, n) from V 1 to a seed in T z1 (m, n)
and having the following properties:

(i) all its bonds are (λ + δ)-open;

(ii) all its bonds which are to the right of H 1 x+(N +m)e1 are λ -open. Therefore to obtain the lower bound we let L 1 be a value of V 1 containing a seed in T x (m, n) and consider the event: We also define the event

A 1 (x, L 1 ) = there exist v 1 ∈ L 1 ∩ B F1(L1) (n) and a path from v 1 to a seed in T F1(L1) (m, n), contained in B F1(L1) (n) ∪ T F1 (
A 2 x = {V 2 contains a seed centered in H 1 x+2N e1 ∩ B 2N e1 (2N -m)}. Noting that A 2 x contains A 1 (x, L 1 ) ∩ {V 1 = L 1 } for any L 1 containing a seed in T x (m, n),
summing over all such L 1 's we get by (A.20) and (A.21)

P (A 2 x ) ≥ L1 P (A 1 (x, L 1 )|V 1 = L 1 )P (V 1 = L 1 ) ≥ L1 (1 -)P (V 1 = L 1 ) = (1 -)P (A 1 x ) ≥ (1 -) 2 ≥ 1 -2 . (A.22)
Now we define a random variable Z 2 as follows: on the event A 2 x among the seeds contained in V 2 and centered in H 1 x+2N e1 ∩ B 2N e1 (2N -m) we choose one according to some arbitrary deterministic order and we let Z 2 be its center. On (A 2

x ) c we let Z 2 = ∆. Thus, Z 2 is a function F 2 of V 2 . As before we let L 2 be a possible value of V 2 containing a seed centered in H 1 x+2N e1 ∩ B 2N e1 (2N -m) and consider the event:

A 2 (x, L 2 ) = there exist v 2 ∈ L 2 ∩ B F2(L2) (n) and a path from v 2 to a seed in T F2(L2) (m, n), contained in B F2(L2) (n) ∪ T F2(L2) (m, n)
whose edges are all (λ + δ)-open and those to the right of

H 1 x+(2N +m)e1 are λ -open . The event {V 2 = L 2 } is σ(F 2 (L 2 ), L 2 , λ , δ)-measurable, hence it follows from Lemma A.5 that P (A 2 (x, L 2 )|V 2 = L 2 ) ≥ 1 -.
We now let V 3 be the set of vertices belonging to all the paths with the following properties:

(i) they start from B x (m); We also define the event:

A

3 x = {V 3 contains a seed centered in H 1 x+3N e1 ∩ B 3N e1 (2N -m)}. Since A 3 x contains A 2 (x, L 2 ) ∩ {V 2 = L 2 }
we can argue as before and get: P (A 3 x ) ≥ 1 -3 . The argument is then repeated until we reach a seed in B 8N e1 (2N ). The total number of steps needed is at most 8. Since at each step the probability is reduced by , the lemma is proved. .23) be looking for paths reaching a seed whose center is above z 1 . We then move to the second point in {(x, y) ∈ Z 2 + : x + y = n -1} which has been reached in the renormalized model from (0, 0) following open paths. Let (x 2 , n -1 -x 2 ) be that point and let z 2 be the center of the seed located inside B 8N x2e1+8N (n-1-x2)e2 (2N ) which was reached in the original model following open paths starting at B(m). Two different cases arise: either x 2 = x 1 -1 or x 2 < x 1 -1. In the first case the point (x 2 + 1, n -1 -x 2 ) = (x 1 , n -x 1 ) has already been declared open or closed and remains in that state. Then, we declare (x 2 , n -x 2 ) open (closed) if C 2 z2 occurs (does not occur) in the original model. In the second case (when x 2 < x 1 -1) we declare (x 2 + 1, n -1 -x 2 ) open (closed) if C 1 z2 occurs (does not occur) in the original model and we declare (x 2 , n -x 2 ) open (closed) if C 2 z2 occurs (does not occur) in the original model. Then we go on.

Then define

C x = C 1 x ∩ C 2 x . ( A
We now note that for all n each site examined in the set {(x, y) : x + y = n} has probability bigger than 1 -8 of being open and that such sites are dependent at most by pairs. This implies, as explained in the following lines, that the open cluster of the origin is stochastically above the open cluster of an independent oriented site percolation model of parameter 1 -16 .

For this, we again proceed by induction on n. We denote by a 1 , a 2 , . . . , a k the points in the open cluster of the origin that belong to {(x, y) ∈ Z 2 + : x + y = n}. We assume that they are ordered according to their second coordinates. Point a 1 has two neighbors b 1 , b 2 on {(x, y) ∈ Z 2 + : x + y = n + 1}. They are both open with probability at least 1-16 , which is stochastically larger than if they were both independently open with probability 1-16 . In other words, if a random vector (Y 1 , Y 2 ) with coordinates taking values in {0, 1} is such that P (Y 1 = Y 2 = 1) ≥ 1 -16 , then the vector (Y 1 , Y 2 ) is stochastically larger than the vector (X 1 , X 2 ) where X 1 and X 2 are independent Bernoulli r.v.'s of parameter 1-16 . Going on, if a 2 = a 1 +(-1, 1), then we just have to consider the point b 3 = a 2 + (0, 1), because a 2 + (1, 0) has already been examined. This point b 3 will be open with probability at least 1 -8 independently of what happened with b 1 and b 2 . Otherwise if a 2 is more distant from a 1 we have to examine b 3 = a 2 + (1, 0) and b 4 = a 2 + (1, 0): they will both be open with probability at least 1 -16 independently of what happened with b 1 and b 2 , and so on. In the end, to each examined point on {(x, y) ∈ Z 2 + : x + y = n + 1} is attached a r.v. with value 1 if it is open and 0 if it is closed. The r.v.'s thus obtained are stochastically larger than a sequence of independent Bernoulli r.v.'s of parameter 1 -16 . Thus, for our choice of the renormalized model is supercritical and there exists a constant ι > 0 such that P ((0, 0) → (k, k)) ≥ ι for all k ∈ N. Note also that the existence of an open oriented path from (0, 0) to (k, k) (which has length 2k) in the renormalized model implies the existence of a (λ + δ)-open path in the original model from B(m) to some seed in B 8N ke1+8N ke2 (2N ) whose number of edges is bounded above by 2Kk where K is some constant that depends on N but not on k. Indeed suppose that the point following (0, 0) in the path of the renormalized model is (1, 0). This means that there exists an open path in the original model from a seed in B(2N ) to a seed in B 8N e1 (2N ). This last path is not oriented, but being contained in B(3N ) ∪ B 6N e1 (3N ), it uses only edges in this set. The total number of edges in the latter is a function of N which does not depend on k, that we denote by K(N ). Hence the derived open path in the original model from a seed in B(2N ) to a seed in B 8N ke1+8N ke2 (2N ) has a number of edges bounded by 2K(N )k.

For our next result we define the boxes:

B i,j = B (3+8i)N e1+(3+8j)N e2 (2N )
where i and j are non-negative integers.

Corollary A.10. Assume λ > λ c . Let N be as in the conclusion of Lemma A.9. Then, there exist ι > 0 and K ∈ N such that: For any k ∈ N and any 0 ≤ i 1 , i 2 , j 1 , j 2 ≤ k we have Proof of Corollary A.10. We wish to join B i1,i2 to B j1,j2 . Lemma A.9 enables to go from a box to another one along a diagonal direction issued from that box. Hence applying Lemma A.9 we get P (there exists a λ-open path contained in [0, (6 + 8k)N ] × [0, (6 + 8k)N ] × [-3N, 3N ] from B i1,i2 to B i1+r,i2+r whose number of edges is at most 2Kr) ≥ ι for all r ∈ N such that i 1 + r, i 2 + r ≤ k. Since the percolation model is invariant under 90 degree rotations, the same inequality holds if instead of adding (r, r) to (i 1 , i 2 ) we add (r, -r),(-r, r) or (-r, -r). That is, instead of going in one direction of one diagonal issued from (i 1 , i 2 ), we may take this diagonal in the other direction, or one direction of the other diagonal issued from (i 1 , i 2 ). This depends on the relative positions of (i 1 , i 2 ) and (j 1 , j 2 ) within the square [0, k] × [0, k] to which they both belong. More precisely, from (i 1 , i 2 ) and from (j 1 , j 2 ) is issued a diagonal, and those two diagonals intersect within [0, k] × [0, k]. If this intersection point has integer coordinates, it can be written (i 1 + r 1 , i 2 + r 2 ) as well as (j 1 + 1 , j 2 + 2 ), with r 1 = r 2 or r 1 = -r 2 (depending on which diagonal issued from (i 1 , i 2 ) was used), and with 1 = 2 or 1 = -2 similarly. If this intersection point does not have integer coordinates, on each of the involved diagonals there is one point with integer coordinates, with those two points at distance 1, of the form (i 1 + r 1 , i 2 + r 2 ) and (j 1 + 1 , j 2 + 2 ), always with r 1 = r 2 or r 1 = -r 2 , and 1 = 2 or 1 = -2 . To summarize, there exist integers r 1 , r 2 , 1 , 2 with the following properties (1) r 2 = r 1 or r 2 = -r 1 and 2 = 1 or 2 = -1 ;

(2) 0 ≤ i 1 + r 1 , i 2 + r 2 , j 1 + 1 , j 2 + 2 ≤ k;

(3) either |i 1 + r 1 -

(j 1 + 1 )| + |i 2 + r 2 -(j 2 + 2 )| = 0 or |i 1 + r 1 -(j 1 + 1 )| + |i 2 + r 2 -(j 2 + 2 )| = 1; (4) |r 1 | + | 1 | ≤ |i 1 -j 1 | + |i 2 -j 2 |.
The corollary now follows from Lemma A.9, the FKG inequality (see Remark 3.2) and the fact that the distance from a point in B i1+r1,i2+r2 to a point in B j1+ 1,j2+ 2 is bounded above by 20N .

  Proof of Proposition 3.11. (i) Modifying the constant α 2 , the statement of Lemma 3.10 above holds for C 3 = 1/k d . (ii) We have that {x → ∞ and ∞ → y} = ∩ n G n . Hence for all k, P (D(x, y) = +∞, x → ∞ and ∞ → y) ≤ P (D(x, y) = +∞, G k ) ≤ P (D(x, y) = +∞|G k ), which converges to 0 when k goes to infinity by Lemma 3.10. We thus have P (D(x, y) = +∞|x → ∞ and ∞ → y) = 0.

x

  and R in x if and only if x / ∈ C ∞ . Otherwise R out x and R in x are empty. By next lemma, the distribution of the radius of R out o ∪ R in o decreases exponentially.

. 7 )

 7 Remark 4.7. (a) By Definition 4.4(ii), V(x) = ∅. (b) By Definition 4.4(iii), for all y, z in V(x), there exists at least one open path from y to z, denoted by Γ * y,z , contained in B x (C κ(x)). If there are several such paths we choose the first one according to some deterministic order.

  a), τ (x, y) is finite. If V(x) ∩ V(y) = ∅, then τ (x, y) = 0.

  y). Since the open paths Γ x,c1 from x to c 1 and Γ * c1,d1 (which exists by Remark 4.7(b)) from c 1 to d 1 have edges in Γ(x) (see (4.8)), the open path Γ

Lemma 4. 9 .

 9 For all x, y, z ∈ Z d , we have the subadditivity property τ (x, z) ≤ τ (x, y) + u(y) + τ (y, z). (4.13) Proof of Lemma 4.9. Let Γ a,b be an open path from a ∈ V(x) to b ∈ V(y) such that τ (x, y) = τ (Γ a,b ). Similarly, let Γ c,d be an open path from c ∈ V(y) to d ∈ V(z) such that τ (y, z) = τ (Γ c,d ) (we might have a = b, c = d or b = c). Since both b and c are in V(y) there exists an open path Γ * b,c from b to c such that τ (Γ * b,c ) ≤ u(y) (see Remark 4.7(b) and (4.8)). The lemma then follows since the concatenation of these three paths is an open path from a point of V(x) to a point of V(z) and

  D(x, y), D(x, y) is always finite. Next proposition corresponds to Proposition 3.11(i) for D(x, y) instead of D(x, y). It will be used in Lemma 4.11 which follows.

  and all its other edges are λ -open . Note that since a λ -open edge is also (λ + δ)-open, all the involved edges in the path in G i,j are (λ + δ)-open, but we do not know if the first edge of this path is λ -open.

  Figure A.2. In the open path, the first edge going out of L is (λ + δ)-open, and all other edges are λ -open.

V

  (L) = z ∈ ∆ v (L ∩ B(n)) : there exists a λ -open path contained in B(n) ∪ T (m, n) \ L going from z to a seed contained in T (m, n) . (A.11)

Lemma A. 7 .

 7 Figure A.3. Event C 1 x

  L1) (m, n) whose edges are all (λ + δ)-open and those to the right of H 1 x+(N +m)e1 are λ -open . The event {V1 = L 1 } is σ(L 1 , λ )-measurable (recall Notation A.4), hence it follows from Lemma A.5 that P (A 1 (x, L 1 )|V 1 = L 1 ) ≥ 1 -. (A.21)Let V 2 be the set of vertices of all the paths with the following properties: (i) they start from B x (m); (ii) they are contained in B(3N ) ∪ B N e1 (3N ) and lie entirely to the left of H 1 x+(2N +m)e1 ; (iii) all their edges are (λ + δ)-open and those to the right of H 1 x+(N +m)e1 are λ -open.

  ii) they are contained in B(3N ) ∪ B N e1 (3N ) and lie entirely to the left of H 1 x+(3N +m)e1 ; (iii) all their edges are (λ + δ)-open and those to the right of H 1 x+(2N +m)e1 are λ -open.

P

  (there exists a λ-open path contained in[0, (6 + 8k)N ] × [0, (6 + 8k)N ] × [-3N, 3N ] from B i1,i2 to B j1,j2 whose number of edges is at most 2K (|i 1 -j 1 | + |i 2 -j 2 |)) ≥ ι .

  • • • , d}, any y = (y 1 , y 2 , • • • , y d ) ∈ Z d and any choice of the signs ±, µ(±y σ(1) , ±y σ(2) , • • • , ±y σ(d) ) = µ(y 1 , y 2 , • • • , y d ).

	Proof

of Corollary 4.14. Clearly τ (o, (y 1 , y 2 , . . . , y d )) has the same distribution as τ (o, (±y σ(1) , ±y σ(2) , • • • , ±y σ(d)

Acknowledgements. We thank Geoffrey Grimmett for useful discussions. We thank referees for helpful comments and suggestions. This work was initiated during the semester "Interacting Particle Systems, Statistical Mechanics and Probability Theory" at CEB, IHP (Paris), whose hospitality is acknowledged. Part of this paper was written while E.A. was visiting IMPA, Rio de Janeiro and thanks are given for the hospitality encountered there.

E. D. Andjel was partially supported by PICS no. 5470, by CNRS and by CAPES.. E. Saada was partially supported by PICS no. 5470.

≤ C 5 (n + 1) d 4 d+2

(εn) d+2 E(u(o) d+2 ).

Thus, by Lemma 4.11, n∈N P (sup z∈(n+1)D u(z) ≥ εn/4) < ∞, and (4.36) follows from Borel-Cantelli's Lemma.

(ii) Next we show that Let z ∈ (1 -ε)tD ∩ C out o , then, by (2.7), (4.35) and the same reasoning as for (4.37), we have τ (o, z) ≤ (1 -ε/2)t if t is large enough. Hence, (4.39) will follow if we show that T z ≥ (ε/2)τ (o, z) occurs only for a finite number of z's. Indeed otherwise T z ≤ (ε/2)(1 -ε/2)t so that τ (o, z) + T z < t if t is large enough: it means that the infection has reached site z and the time of infection from z is over before time t, hence z has recovered by time t, that is z

then by (2.7) and (4.38), for δ z ∞ larger than some t 0 , we have

Therefore, it suffices to show that for any δ > 0 the event {T z ≥ δ z ∞ } can only occur for a finite number of z's. This will follow from Borel-Cantelli's Lemma once we prove that z∈Z d P (T z ≥ δ z ∞ ) < ∞. To do so we write, since the T z 's are identically distributed:

for some constant c, and this last series converges because T o has a finite moment of order d. Putting together (2.11) and (4.39) yields (2.12).

Appendix A.

In this appendix we prove Theorem 3.5, Lemma 3.7 and (4.2) in the proof of Lemma 4.3. These proofs rely on dynamic renormalisation techniques introduced in [START_REF] Barsky | Percolation in half-spaces: equality of critical densities and continuity of the percolation probability[END_REF]. In applying these techniques we follow Grimmett (1999, Chapter 7) and [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF], but we introduce some modifications. In particular by considering some larger boxes we avoid using the sprinkling technique more than once on any given bond. Because of this we only need to consider two different values of the infection parameter and we do not need to introduce the updating functions of [START_REF] Grimmett | Percolation[END_REF]. To simplify the notation we write the proofs for d = 3, but their generalizations to higher dimensions presents no problems.

But according to Lemma A.3 there exist m and n such that 2m < n and

is as close to 1 as we wish. This implies that P there exists a λ -open path contained (except its initial point) in

is as close to 1 as we wish uniformly in L's such that

The probability (A.14) is equal to P there exists a λ -open path contained (except its initial point) in

where the inequality comes from (A.13). For this upper bound to be close to 1, P (|V (L)| ≤ k) has to be small. Hence, it follows from (A.13)-(A.16) that for any 0 > 0 and any k ∈ N, we can choose m and n with 2m < n in such a way that

for all sets L satisfying (A.15). Let K be a subset of ∆ v (L ∩ B(n)). We will now provide a lower bound to

which depends on the cardinality of K but is independent of H. Suppose K has at least 6r elements. Each point u ∈ K has a neighbor v ∈ L∩B(n), that we associate to u. But since each point of Z 3 has 6 nearest neighbors, v could be a neighbor of up to 6 points of K, to which it could have been associated. Then, there exist distinct x 1 , . . . , x r ∈ L ∩ B(n) and distinct y 1 , . . . , y r ∈ K such that x i ∼ y i for i = 1, . . . , r.

Since .18) for i = 1, . . . , r. It now follows from (A.18) and an inductive argument that

Hence for all K ⊂ ∆ v (L ∩ B(n)) such that |K| ≥ 6r we have:

From Lemma A.7 we get:

Corollary A.8. Given λ > λ c , for any , δ > 0 there exist n, m, N satisfying (A.1) and such that for any x ∈ B(2N -m) we have

Next lemma fixes the values of all the parameters introduced up to now.

Lemma A.9. Assume λ > λ c . Then, there exist constants m, N, K and ι > 0 such that for all k, P (there exists a λ-open path contained in

whose number of edges is at most 2Kk) ≥ ι.

Proof of Lemma A.9. We first fix > 0 small enough for the two dimensional oriented site percolation of parameter 1-16 to be supercritical. Then we take λ > λ c and δ > 0 such that λ +δ < λ. Finally for those values of , δ and λ we fix n, m and N = n+m+1 satisfying (A.1) and such that the conclusion of Corollary A.8 is valid.

We create a two dimensional oriented site percolation on (Z + ) 2 associated to the percolation model we already have. We will refer to this model as the "renormalized model", while the percolation model we already had on Z 3 will be referred to as the "original model". On the renormalized model all the paths are oriented upwards and towards the right; moreover, two subsequent sites of a path are at euclidean distance 1. We now explain the way in which these models are associated. In the renormalized model site (0, 0) is always considered open, site (0, 1) is open (closed) if C 1 0 occurs (does not occur) in the original model. Similarly, (1, 0) is open (closed) if C 2 0 occurs (does not occur) in the original model. Note that although the states of these last two sites (0, 1) and (1, 0) are dependent, by Corollary A.8 they are both open with probability at least 1 -16 . We then proceed recursively as follows.

At the n-th step we will look at the points in {(x, y) ∈ Z 2 + : x + y = n -1} which have been reached in the renormalized model from (0, 0) following open paths and order them according to their second coordinates. We start from the point having the lowest second coordinate. Assume it is (x 1 , n -1 -x 1 ). This point was reached from either (x 1 -1, n -1 -x 1 ) or (x 1 , n -2 -x 1 ). In the first case, in the original model a seed is reached in the left portion of B 8N x1e1+8N (n-1-x1)e2 (2N ) (remember the description given before the statement of Lemma A.7). Let z 1 be the center of this seed. If C 1 z1 occurs (does not occur) in the original model we say that site (x 1 + 1, n -1 -x 1 ) in the renormalized model is open (closed). And if C 2 z1 occurs (does not occur) in the original model we say that site (x 1 , n -x 1 ) is open (closed). Note that since z 1 is in the left portion of B 8N x1e1+8N (n-1-x1)e2 (2N ), when we attempt to move upwards, the first seed we are seeking is centered to the right of z 1 due to our steering procedure, thus avoiding regions where we have already used (λ + δ)-open edges. In the second case, the seed reached in the original model (we again denote its center by z 1 ) is in the lowest portion of B 8N x1e1+8N (n-1-x1)e2 (2N ) and when we want to establish if C 1 z1 occurs we will

. Two possible cases

Proposition A.11. Suppose λ > λ c . Then there exist constants C, N and

P there exists an open path from x to y contained in

Proof of Proposition A.11. It follows from Lemma A.9 that the probability of having an open path of length n starting in B(m) and contained in the slab Z × Z × [-3N, 3N ] does not converge to 0 as n goes to infinity. This proves part b). To prove part a) consider the boxes B (3N +8N i)e1+(3N +8N j)e2 with 0 ≤ i, j ≤ ( M N -6) 1 8 . Then, note that for any point in [0, M ] × [0, M ] × [-3N, 3N ] there is such a box at distance at most 12N . The result now follows from this, the FKG inequality (see Remark 3.2) and Corollary A.10.

We have now all the ingredients for the proofs of Theorem 3.5, Lemma 3.7, and (4.2) of Lemma 4.3.

Proof of Theorem 3.5. Let x and y be two points in a slab of thickness 6N . By Proposition A.11, the probability to have an open path from x to y in the slab is larger than δ 1 . Therefore the probability for the outgoing cluster from x in the slab to be infinite, as well as the probability for the incoming cluster to y in the slab to be infinite, is at least δ 1 .

Note that Proposition A.11 gives more precise information, since it restricts the involved open paths to a part of the slab, and gives an upper bound on the lengths of the paths.

Proof of Lemma 3.7. For two points x and y, the idea to build an open path from x to y is to combine paths in different slabs using in each one Proposition A.11,a).

(i) Let δ 1 , M and C be given by Proposition A.11, and let k ≥ M . For n > 0, .11,a) there exist with a probability larger than δ 1 an open path from x to u, as well as from u to v and from v to y. By FKG inequality (see Remark 3.2) there exists therefore with a probability larger than δ 3 1 an open path from x to y. Since this particular case gives the maximal distance between x and y, δ = δ 3 1 enables us to conclude.

(ii) Let n < m, x ∈ A(n, m, 0), y ∈ A(n, m, 0) ∪ ∆ v A(n, m, 0). We proceed similarly to (i). Assume for instance that x 1 < n, x 2 < 0 and m < y 1 , y 2 < 0. Let u, v ∈ A(n, m, 0) be such that u 1 < n, 0 < u 2 and m < v 1 , 0 < v 2 . By Proposition A.11,a) there exist with a probability larger than δ 1 an open path from x to u, as well as from u to v and from v to y. We conclude with δ = δ 3 1 and C 1 = C.

Note that we have to add (-x 2 ) + + (-y 2 ) + in part (ii) of the lemma because if x ∈ {z : -k + n ≤ z 1 < n, -∞ < z 2 ≤ 0} and y ∈ {z : m < z 1 ≤ m + k, -∞ < z 2 ≤ 0}, to move from x to y staying in A(n, m, 0) we need to reach first the set {z : -k + n ≤ z 1 ≤ m + k, 0 < z 2 ≤ k} (i.e. to increase the second coordinate until it is positive).

Proof of (4.2) of Lemma 4.3. Relying on Proposition A.11,b), we can follow the proof of Grimmett (1999, Theorems (8.18), (8.21)) to derive (4.2).