First-passage competition with different speeds: positive density for both species is impossible

Consider two epidemics whose expansions on $\mathbb{Z}^d$ are governed by two families of passage times that are distinct and stochastically comparable. We prove that when the weak infection survives, the space occupied by the strong one is almost impossible to detect: for instance, it could not be observed by a medium resolution satellite. We also recover the same fluctuations with respect to the asymptotic shape as in the case where the weak infection evolves alone. In dimension two, we prove that one species finally occupies a set with full density, while the other one only occupies a set of null density. We also prove that the H\"{a}ggstr\"{o}m-Pemantle non-coexistence result"except perhaps for a denumerable set"can be extended to families of stochastically comparable passage times indexed by a continuous parameter.


Introduction
Consider two species both trying to colonize the graph Z d . The expansion of each species is governed by independent identically distributed random passage times attached to the bonds of the graph, as in first-passage percolation, and each vertex of the graph can only be infected once, by the first species that reaches it. Is it possible that both species simultaneously succeed in invading an infinite subset of the net, in other words that coexistence occurs? That is the kind of question which was asked in the middle of the 90's by Häggström and Pemantle in two seminal papers [9,10], where they gave the first results towards the following conjectures: • If the two species travel at the same speed, coexistence is possible.
• If one of them travels faster than the other one, coexistence is impossible. The passage times considered by Häggström and Pemantle follow exponential laws, which provides a Markov property and allows a description of the competition process in terms of particle system. However, the first-passage percolation setting naturally enables to consider competition with general passage times, even if the Markovianity is lost.
The problem of coexistence for two similar species has been solved by Häggström and Pemantle [9] in dimension two for exponential passage times, then by the authors of the present paper in any dimension for general passage times, under assumptions that are close to optimality [4]. Shortly later, Hoffman [11] gave a different proof involving tools that seem to allow an extension to a larger number of species -see Hoffman's manuscript [12].
On the contrary, the state of the art about the second conjecture -the noncoexistence problem -did not much change since its statement. More precisely, if one species travels according passage times following the exponential law with intensity 1, while the other one travels according passage times following the exponential law with intensity λ = 1, it is believed that coexistence is not possible. However, it is only known that coexistence is not possible "except perhaps for a denumerable set of values of λ", as it was proved by Häggström and Pemantle [10]. To sum up, if one denotes by Coex the set of intensities for the second particle that allow coexistence, we know that Coex ⊃ {1} and Coex is denumerable, but we would like to have Coex = {1}. It follows that we are currently in the following perplexing situation: we know that for almost every value of λ, coexistence does not happen, but we are unable to exhibit any value of λ such that coexistence does not occur.
Therefore, the aim of this paper is to prove a weakened version of non-coexistence for epidemics with distinct speeds. Let us first introduce our framework: we consider two epidemics whose expansions are governed by two families of independent and identically distributed passage times whose laws are distinct and stochastically comparable, which of course includes the case of exponential laws. We say that strong coexistence occurs when each species finally occupy a set with positive natural density.
In dimension two, we prove that, almost surely, strong coexistence does not occur. More precisely, we show that almost surely, at infinite time, one species fills a set with full natural density, whence the other one only fills a set with null natural density. In higher dimension, connectivity problems prevent us to obtain such a complete result. However, we show that, roughly speaking, a medium resolution satellite only sees one type of particles.
By the way, we also prove that the Häggström-Pemantle non-coexistence result "except perhaps for a denumerable set" can be extended to families of stochastically comparable passage times indexed by a continuous parameter. Note that the Häggström-Pemantle method [10] to prove denumerability of Coex has already been transposed to other models having familiarities with first-passage percolation: at first by Deijfen, Häggström, and Bagley for a model with spherical symmetry [2], then by the authors of the present paper for some percolating model [6].
Before giving more rigorous statements of our results, let us introduce general notations and give a formal description of the competition model.
General notations. We denote by Z the set of integers, by N the set of non negative integers.
We endow the set Z d with the set of edges E d between sites of Z d that are at distance 1 for the Euclidean distance: the obtained graph is denoted by L d . Two sites x and y that are linked by an edge are said to be neighbors and this relation is denoted: x ∼ y. If A is a subset of Z d , we define the border of A: ∂A = {z ∈ A : ∃y ∈ A c y ∼ z}.
A path in Z d is a sequence x 0 , x 1 , . . . , x l of points in Z d such that two successive points are neighbors. The integer l is called the length of the path.
The critical percolation for Bernoulli percolation (oriented percolation) on Z d is denoted by p c = p c (d) (respectively, − → p c = − → p c (d)).
Let us now recall the concept of stochastic domination: we say that a probability measure µ dominates a probability measure ν, which is denoted by ν ≺ µ, if f dν ≤ f dµ holds as soon as f in a non decreasing function.
The complementary event of A will mostly be denoted A c . But sometimes, to improve readability, we prefer to use ∁A.
In Assumption (H 3 ), inf supp ν denotes the infimum of the support of the measure ν. Note that Assumptions (H2), (H3), and (H4) are clearly fulfilled when ν p1 and ν p2 are continuous with respect to Lebesgue's measure. Assumption (H2) exactly says that a sum of k independent random variables with common distribution ν p1 and a sum of l independent random variables with common distribution ν p2 , independent of the first family, have probability 0 to be equal: this will ensure that, during the competition process, no vertex of Z d can be reached exactly at the same time by the two epidemics. Assumptions (H 1 ), (H3) and (H 4 ) are the ones used by van den Berg and Kesten in [14] to prove that, in first-passage percolation, the time constant for ν p2 is strictly smaller than the one for ν p1 . Finally, assumption (H5) gives access to large deviations and moderate deviations related to the asymptotic shape in first-passage percolation.
Construction of the competition model. The first infection (second infection) will use independent identically distributed passage times with common law ν p1 (respectively, ν p2 ) and start from the source s 1 (respectively, s 2 , distinct from s 1 ). As ν p1 ≻ ν p2 , species 1 will be slower (or weaker) than species 2.
First, we couple the two measures ν p1 and ν p2 in agreement with the stochastic comparison relation (H 1 ): there exists a probability measure m on [0, +∞)×[0, +∞) such that m({(x, y) ∈ [0, +∞); x ≥ y}) = 1 and the marginals of m are ν p1 and ν p2 . Now, we consider, on Ω = ([0, +∞) × [0, +∞)) E d , the probability measure P = m ⊗E d . For ω = (ω 1 e , ω 2 e ) e∈E d , the number ω i e represents the time needed by species i to cross edge e. Note that under P, for each i ∈ {1, 2}, the variables (ω i e ) e∈E d are independent identically distributed with common law ν pi . Moreover, we almost surely have ∀e ∈ E d ω 1 e ≥ ω 2 e . It remains to construct the two infections in a realization ω ∈ Ω. Let E = ([0, +∞] × [0, +∞]) Z d . We recursively define a E-valued sequence (X n ) n≥0 and a non-negative sequence (T n ) n≥0 . The sequence (T n ) n≥0 contains the successive times of infections, while a point ε = (ε 1 (z), ε 2 (z)) z∈Z d ∈ E codes, for each site z, its times of infection ε 1 (z) (its time of infection ε 2 (z)) by the first infection (respectively, second infection). We start the process with two distinct sources s 1 and s 2 in Z d , and set T 0 = 0 and This means that at time 0, no point of Z d has yet been infected but the two initial sources s 1 and s 2 . Then, for n ≥ 0, define the next time of infection: Note that Assumption (H2) ensures that if this infimum is reached for several triplets (i 1 , y 1 , z 1 ), . . . , (i l , y l , z l ), then i 1 = · · · = i l = i. In this case, the next infections are of type i from each y j to each z j . The set of infected points of type 3 − i has not changed: , while the points z j has been infected by species i at time X i n (y j ) + ω i {yj,zj } : Note that X i n (y) and X 3−i n (y) can not be simultaneously finite, which corresponds to the fact that each site is infected by at most one type of infection. Moreover, once min(X 1 n (x), X 2 n (x)) is finite, its value -the time of infection of x -does not change any more with n.
Note however that this coupling is nothing else than a useful tool for our proofs: this does note constrain the evolution of the process. Particularly, the very definition of the evolution process tells us that the (ω e i ) e∈E d ,i∈{1,2} could be independent as well without the law of the evolution process being changed.
We can now define the sets η(t), η 1 (t), η 2 (t) that are respectively the sets of infected points, infected points of type 1, infected points of type 2 at time t, by setting We also introduce the the sets of points that are finally infected by each epidemic: The set G i = |η i (∞)| = +∞ for i = 1, 2, corresponds to the unbounded growth of species i, and coexistence is thus the event Coex = G 1 ∩ G 2 .
Coupling with first-passage percolation. The evolution of the two infections can be compared with the evolution of classical first-passage percolation.
On Ω = [0, +∞) E d , consider the probability measure P = ν ⊗E d , which makes the coordinates (ω e ) e∈E d independent identically distributed random variables with common law ν. Then for each x ∈ Z d and t ≥ 0, we define the set of points reached from x in a time less than t: The classical shape theorem gives the existence of a norm . ν on R d such that B 0 (t)/t almost surely converges to the unit ball B for . ν .
Note that the competition model contains two simple first-passage percolation models: for each i ∈ {1, 2}, x ∈ Z d and t ≥ 0, the set is the random ball of radius t of first-passage percolation starting from x with passage time law ν pi . For simplicity, the related norm will be denoted by . pi , and its associated discrete balls B x pi (t) = {y ∈ Z d : y − x pi ≤ t}. Here are the coupling relations between the competition model and first-passage percolation: We postpone the proof of these (not so) obvious properties to the next section.
Statement of results. We denote . 2 the euclidean norm on R d , by ., . the corresponding scalar product and by S the corresponding unit sphere: S = {x ∈ R 2 : x 2 = 1}. Let y, z ∈ R d , − → x ∈ S, and R, h > 0. We define: We can then define the following events: The event Shadow( − → x , t, R) means that each infinite path starting from η 1 (t) and contained in the cylinder Cyl + ( − → x , R) necessarily meets ∂η(t)∩η 2 (t). Loosely speaking, on the event Shadow( − → x , t, R), the strong infections casts a shadow of radius R on the weak infection.
Our main result says that if the strong infection occupies a too large portion of the frontier, i.e. if Shade(t, Rt 1/2+η ) occurs, then the survival probability of the slow species 1 is very small: Theorem 1.2. Consider M > 0 and η ∈ (0, 1/2). There exist two strictly positive constants A, B such that Corollary 1.3. Define R t as the supremum of the r for which Shade(t, r) occurs. Let η > 0. Then, on the event G 1 , we almost surely have lim t→+∞ R t t 1/2+η = 0. Remember that, by Lemma 1.1 and the asymptotic shape result for first-passage percolation, the diameter of η(t) is of order t.
To obtain the absence of strong coexistence, we also need a control on the number of such stains of species 2 on the surface of η(t): in dimension larger or equal to three, the set ∂η(t) ∩ η 2 (t) is not necessarily connected. On the contrary, in dimension two, the set ∂η(t) ∩ η 2 (t) is connected, which enables us to prove the absence of strong coexistence. Consider any norm . on R 2 and its discrete balls B(t) = {y ∈ Z 2 : y ≤ t}: Theorem 1.4. For the two dimensional lattice, we have 1. For every β ∈ (0, 1/2), there exists a constant C > 0 such that, almost surely on the event G 1

Strong coexistence almost surely does not happen.
The next corollary of Theorem 3.6 precises Lemma 5.2 in Häggström-Pemantle [10]: when coexistence occurs, the two species globally grow with the speed of the slow species, as if the slow species were alone. It also corresponds to a weak version of moderate deviations for first-passage percolation (see the results by Kesten and Alexander, recalled as Proposition 2.2 in the next section). Theorem 1.5. Let β > 0 and η ∈ (0, 1/2). There exist two strictly positive constants A, B such that for every t ≥ 0: The estimates we obtained in this paper finally allows us to recover the "except perhaps for a denumerable set" non-coexistence result by Häggström and Pemantle, and to extend it to more general families of passage times indexed by a continuous parameter: Theorem 1.6. Let (ν p ) p∈I be a family of probability measures indexed by a subset of R. We assume that for each p, q ∈ I, p < q ⇒ ν p ≻ ν q and ν p = ν q , (5) for each p, q ∈ I, ∀k, l ∈ N, ν * k p ⊗ ν * l q ({(x, x) : x ∈ [0, +∞)}) = 0. Denote by P p,q the law of the competition process where species 1 ( resp. 2) uses passage times with law ν p ( resp. ν q ). Then for each fixed q ∈ I, the set {p ∈ I : p ≤ q and P p,q (Coex) > 0} is a subset of the points of discontinuity of the non-decreasing map p → P p,q (G 1 ), and is therefore at most denumerable.
Organization of the paper. The paper is organized as follows: in Section 2, we give a series of useful results in first-passage percolation. Most of them are classical and are recalled without proof. We also give there the proof of Lemma 1.1. Section 3 is mainly devoted to the proof of Theorem 1.3, which is the technical core of the paper. Section 4 establishes Theorem 1.5). In Section 5, we improve for the two dimensional lattice the results of Section 3 into the more friendly Theorem 1.4. The last section extends the Häggström-Pemantle Theorem to the present context as announced in Theorem 1.6.

2.1.
First-passage percolation results. Let us recall some classical results about simple first-passage percolation. We assume here that the passage times are independent identically distributed with common law ν satisfying Denote by . ν the norm given by the shape theorem, and by B x (t) the discrete ball relatively to . ν with center x and radius t. The first two results give large deviations and moderate deviations for fluctuations with respect to the asymptotic shape, and the third one gives the strict monotonicity result for the asymptotic shape with respect to the distribution of the passage times: Proposition 2.1 (Grimmett-Kesten [8]). For any ε > 0, there exist two strictly positive constants A, B such that Proposition 2.2 (Kesten [13],Alexander [1]). For any β > 0, for any η ∈ (0, 1/2), there exist two strictly positive constants A, B such that There exists a constant C p1,p2 ∈ (0, 1) such that Note that in [14], the proof of this result is only written for the time constants. Nevertheless, it applies in any direction and computations can be followed in order to preserve a uniform control, whatever direction one considers. See for instance Garet and Marchand [6] for a detailed proof in an analogous situation. In the same way, the large deviation result Proposition 2.1 is only stated in [8] for the time constant, but the result can be extended uniformly in any direction, as it is done in Garet and Marchand [5] for chemical distance in supercritical Bernoulli percolation. As far as Proposition 2.2 is concerned, it is a by-product of the proof of Theorem 3.1 in Alexander [1]. We include here a short proof for convenience.
Proof of Proposition 2.2. The outer bound for B 0 (t) follows from Kesten [13], Equation (1.19): there exist positive constants A 1 , B 1 such that for all t > 0, we have Turning to the inner bound, we follow the lines of Alexander's proof: for x, y ∈ Z d , let us define the travel time between x and y by T (x, y) = inf{t ≥ 0 : y ∈ B x (t)}. Then we have: where A and B are determined by Proposition 2.1 with ε = 1/2. By Alexander [1], Theorem 3.2, there exist positive constants C ′ 4 , M such that where C ′ M is a positive constant, and then ). By Kesten's result [13], Equation (2.49) (see also Equation (3.7) in Alexander [1]), there exist positive constants C 5 , C 6 such that where C ′ 5 , C ′ 6 are positive constants.
The next lemma ensures that the minimal time needed to cross the cylinder Cyl z ( − → x , h, r) from bottom to top, using only edges in the cylinder, can not be much larger than the expected value h − → x ν .
as the minimal time needed to cross it from s 0 to s f , using only edges in the cylinder.
Then for any ε > 0, and any function f : R + → R + with lim +∞ f = +∞, there exist two strictly positive constants A and B such that Note that this gives the existence of a nearly optimal path from z to z + h − → x that remains at a distance less than f (h) of the straight line. This result can be interesting on its own as we often miss information on the position of the real optimal paths.
Proof. For x, y ∈ Z d , denote by I x,y the length of the shortest path from x to y which is inside B x (1, 25 x − y ν ) ∩ B y (1, 25 x − y ν ). Of course I x,y as the same law that I 0,x−y , and we simply write I x = I 0,x . We begin with an intermediary lemma.
Lemma 2.5. Let ε, a in (0, 1) and . be any norm on R d . There exists M 0 such that for each M ≥ M 0 , there exist ρ ∈ (0, 1) and t > 0 such that Proof of Lemma 2.5. Note that by norm equivalence, we can restrict ourselves to . 1 . Let Y be a random variable with law ν and let γ > 0 be such that E e 2γY < +∞. First, the large deviations result, Proposition 2.1, easily implies the following almost sure convergence: By considering a deterministic path from 0 to x with length x 1 , we see that I x is dominated by a sum of x 1 independent copies of Y denoted by Y 1 , . . . , Y x 1 , and thus I x / x 1 is dominated by This family is equi-integrable by the law of large numbers. So (T x / x 1 ) x∈Z d \{0} and then (I x / x ν ) x∈Z d \{0} are also equi-integrable families, which implies that Note now that for every y ∈ R and t ∈ (0, γ], the previous inequality implies that As Therefore, we take t = min γ, γ 2 e 1 ν ε 3 R −M/a > 0 and ρ = 1− 1 3 ε e 1 ν t < 1. Let us come back now to the proof of Lemma 2.4. Let ε ∈ (0, 1) and consider the integer M 0 ∈ N given by Lemma 2.5. As . ν is a norm, there exist two strictly positive constants c and C such that Let M 1 ≥ M 0 be an integer large enough to have Consider h > M 1 and set N = 1 + Int(h/M 1 ) -Int(x) denotes the integer part of x -and, for each i ∈ {0, . . . , N } denote by x i the integer point in the cylinder which is the closest to and that for each i, j ∈ {0, . . . , N − 1}, 1. Applying (5), (2) and (4), we obtain that for each So we can find a > 0 such that, by increasing M 1 if necessary, If we take now h larger than h 1 , and if y ∈ B xi (1, 25 (2) and (5), On the other hand, We choose then i 0 ∈ N such that: Then, if h 2 is such that 1 + Int(h 2 /M 1 ) ≥ 3i 0 , we obtain: 3. There exists a deterministic path inside the cylinder from N + 2 edges: we denote by L start (respectively, L end ) the random length of this path. By Equation (4), we have Chernoff's theorem gives the existence of two strictly positive constants A 1 , B 1 such that 4. So, provided that h ≥ h 2 , we have by (8), inside the cylinder, a path from x 0 to x N with length as soon as |j − i| ≥ 2. We thus introduce, for j ∈ {0, 1}, the sums: Note that, with (5), (2) and (3) for each j ∈ {0, 1}, Then, by independence of the terms in each S j , Together with (8), this proves the estimate of the lemma.

2.2.
Comparisons with first-passage percolation. We now prove Lemma 1.1, using an algorithmic building analogous to the one used to define the competition process in the introduction.
Proof of Lemma 1.1. The inclusion η 1 (t) ⊂ B s1 p1 (t) is obvious: by construction of the process, if x ∈ η 1 (t), there exists a path between s 1 and x included in η 1 (t), and whose travel time is thus less than t. The second inclusion η 2 (t) ⊂ B s2 p1 (t) is proved in the same way.
Let us now prove the third inclusion B s1 p1 (t) ⊂ η(t). We are going to build the first-passage process by Dijkstra's algorithm, in a formalism analogous to the one used to define the competition process.
Let E ′ = [0, +∞] Z d . We recursively define a E ′ -valued sequence (X ′ n ) n≥0 and a non-negative sequence (T ′ n ) n≥0 . The sequence (T ′ n )n ≥ 0 contains the successive times of infections, while a point ε = (ε(z)) z∈Z d ∈ E codes, for each site z, its times of infection ε(z). We start the process with the single source s 1 , and set T ′ 0 = 0 and Then, for n ≥ 0, define the next time of infection: The infimum is reached for some couples (y i , z i ), meaning that the z i are being infected from the y i : We also note that η ′ (t), the set of infected points at time t by By Dijkstra's algorithm, η ′ (t) is exactly the set B s1 p1 (t). We now proceed by induction to prove that for every n ∈ N ). Clearly, (H 0 ) is true. Assume that (H n ) holds. We have the following alternative: ) is also finite, and thus min( n+1 through the edge e from the point y, which is consequently such that

Coexistence can not be observed by a medium resolution satellite
The proof of Theorem 1.2 follows, at least in its main lines, the strategy initiated by Häggström and Pemantle: the aim is to prove that an event, suspected to be incompatible with the survival of the weak, allows the strong to grant themselves, with high probability, a family of shells that surround the weak, preventing thus coexistence. In the Häggström-Pemantle paper [10], the objective is to show that when coexistence occurs, it is unlikely that the strong can advance significantly beyond the weak. The event considered here is of a different nature: we must prove that the strong can not occupy a too large region on the frontier of the infected zone. Obviously, this requires finer controls. Moreover, the use of the shape theorem is not sufficient: moderate deviations for the fluctuations with respect to the asymptotic shape provide sharper estimates. Some more technical difficulties also follow from the loss of some nice properties of exponential laws. However, this last kind of difficulties has already been overcome by the authors of this paper in the previous article [6]. We refer the reader to this paper for some more comments.
To prove Theorem 1.2, we first prove an analogous result, Lemma 3.6, in a fixed given direction. Theorem 1.3 follows then from a Borel-Cantelli type of argument.
Definitions. Let S p2 be the unit sphere for the norm . p2 . We define the shells: for each A ⊂ S p2 , and every 0 < r < r ′ , we set So roughly speaking, A is to think about as the set of possible directions for the points in the shell, while [r, r ′ ] is the set of radii.
For A ⊂ S p2 and ϕ > 0, define the following enlargement of A: Let us state first three geometric lemmas: Lemma 3.2. For any norm |.| on R d , there exist a constant C > 0 such that the unit sphere for |.| can be covered with C(1 + 1 ε ) d−1 balls of radius ε having their centers on the unit sphere.
Proof. When |.| = . ∞ , it is easy to see that the sphere can be covered with 2d(1 + 2 ε ) d−1 balls of radius ε. Now let A, B be two strictly positive constants such that Suppose, by the previous step, that the unit sphere {x ∈ R d : So, the unit sphere for |.| can be covered with 2d(1 + 4B A 1 ε ) d−1 balls of radius ε having their centers on the unit sphere.
which ends the proof.
By condition (12), δ + β < δ ′ . Thus, using the moderate deviation result (Proposition 2.2), there exist two strictly positive constants A 1 , B 1 such that ∀t > 0 We can thus assume in the following that E 1 occurs.
Step 2. Control of the competition process. Denote: By Proposition 2.2 and Lemma 1.1, there exist two strictly positive constants A 2 , B 2 such that ∀t > 0 . We can thus assume in the following that F 1 occurs.
Define an integer approximation of the line R − → x as follows: Step 3. The weak infection can not fill the hole. Define As δ ′ > δ, by the large deviation result (Proposition 2.1), there exist two strictly positive constants A 3 , B 3 such that ∀t > 0 ). We can thus assume in the following that F 2 occurs. Note that as B x p1 (δ ′ ) ⊂ B x 2 (K 1 δ ′ ) by the very definition of K 1 , the event Shadow( − → x , t, Rt 1/2+η ) ∩ F 1 ∩ F 2 prevents the weak infection to bother the strong one in its progression after time t inside Cyl( − → x , rt 1/2+η ).
The next lemma describes the typical progression of the strong infection from one shell to the next one.
Lemma 3.5. Let ϕ ∈ (0, 2], h ∈ (0, 1/2) and α ∈ (1, 2) be fixed parameters such that For any S subset of S and for any r, s > 0, we define the following event E = E(S, r, s): "Any point in the big Shell(S ⊕ ϕr 2(s+r) ⊕ ϕ(1+h)r 2(s+(1+h)r) , s + (1 + h)r, s + (1 + h) 2 r) is linked to a point in the small Shell(S ⊕ ϕr 2(s+r) , s + r, s + (1 + h)r) by an open path whose length is less than αhr." Then there exist two strictly positive constants A and B, only depending on ϕ, h, α, such that for any r, s > 0 and any S of S, we have

The triangle ensures the first inclusion
by Equation (25) and definition of v z ≤ ϕr 2(s + r) thanks to Equation (21) and thus u u p 2 ∈ S ⊕ ϕr 2(s+r) . For the norm of u, by definition of v z and Equation (25), we have: where the union is for z ∈ Shell(T, s + (1 + h)r, s + (1 + h) 2 r). By Proposition 2.1, there exist two strictly positive constants A 2 and B 2 such that for every S, for every s, r > 0, thanks to (26). Then, for every z ∈ Shell(T, (1 + h)r, (1 + h) 2 r), thanks to (25) and (23), one has (1 + ε)(1 + ρ) z − v z p2 ≤ αhr, which proves the exponential estimate of the lemma.
Control of the infection paths: It remains to estimate the minimal room needed to perform this infection, or in other words to control z∈Shell(T,s+(1+h)r,s+(1+h) 2 r) This last term tends to s + r(1 + h − ϕ − (1 + h) 3 ϕ) when ε and ρ tend to 0. By decreasing if necessary ε and ρ, we obtain, as h < 1/2: Finally, by applying Lemma 3.1 and then Inequality (25), we have Thus u ∈ Shell(T ⊕ 2αhr s+r , s + (1 − 3ϕ)(1 + h)r, ∞), which ends the proof of the lemma.
Idea of the proof: The idea is quite natural: start the progression by the initialization Lemma 3.4, and apply recursively the progression Lemma 3.5 until the stronger infection surrounds the weaker one. The point is to ensure that this progression is not disturbed by the spread of the weaker infection.
Notations: We still need to introduce a certain number of notations, inspired by Lemma 3.5. k = 1 Define also the following events, for k ≥ 2 and x ∈ Z d \{0}: . The aim is the following: we want to apply Lemma 3.5 to prove that if E 2 k (x, t) is realized, then with high probability E 2 k+1 (x, t) is also realized. But we need first to control the spread of the slow p 1 -infection, and to see that it will not disturb the spread of the fast p 2 -infection from S k (x, t) to S k+1 (x, t).
Step 2. Rough control of the slow p 1 -infection: Here, for convenience, the complementary event of A is denoted by ∁(A). Let ε > 0. Denote, for every k ≥ 2 ) . Let us prove that there exist two strictly positive constants A 2 and B 2 such that (34) and thus that In our context, this gives The large deviation result, Proposition 2.1, gives then two strictly positive constants A, B such that Step 3. Estimates for angles: Let us see that for anyx ∈ S p2 , for any ϕ, ψ ≥ 0, Let z ∈ (x ⊕ ϕ) ⊕ ψ: there exist y ∈x ⊕ ϕ and v ∈ B p2 (ψ) such that z = y + v. As y ∈x ⊕ ϕ, there exists w ∈ B p2 (ϕ) such that y =x + w. Thus

This implies
ϕr j x p1 2(t + r j x p1 ) Then, for all k ≥ 2, for t large enough, Step 4. The weak can not bother the strong: Let us see that for all k ≥ 2 where the last inequality follows from (35). Then We want to prove that this quantity is negative for every k ≥ 2. Conditions (31) and (32) ensures that the coefficient in (1 + h) k−1 in (37) is negative. Thus, asymptotically in k, the right-hand side in (37) is negative. To ensure it is negative for every k ≥ 2, we only need to see that it is true for k = 2, which is ensured by Conditions (31) and (32). This proves (36). Note that this also implies (38) ∀k ≥ 2 F 1 k ⊂ E 1 k . Equations (38) and (34) together give: Step 5. Control of the fast p 2 -infection: Let A 3 and B 3 be the two strictly positive constants given by Lemma 3.5 for the choice for α, h, ϕ we made in (28), (30) and (31). Note that for every k ≥ 2, we have where the event E(., ., .) was defined in Lemma 3.5. Thus, the application of Lemma 3.5 implies that for anyx ∈ S p2 , any t > 0, for every k ≥ 2, where A 4 , A 5 and B 4 , B 5 are strictly positive constants.
Conclusion: For k large enough, the set S k disconnects 0 from infinity, and thus the event k≥1 E k implies that the slow p 1 -infection is surrounded by the fast p 2 -infection and thus dies out. So, using (33), (39) and (40), we obtain: which completes the proof.
Proof of Theorem 1.2. Proposition 2.1 and Lemma 1.1 give the existence of strictly positive constants α, β, A 1 , B 1 such that the event F t = {∀y ∈ ∂η(t) y 2 ∈ (αt, βt)} has a probability larger than 1 − A 1 exp(−aB 1 t), so we only have to control the probability of the event Assume that t is large enough to have αt > M t 1/2+η and set By Lemma 3.2, there exists a subset T of the unit sphere S with |T | ≤ C(1 2 t 1/2+η ) happens. Let γ be an infinite path in Cyl + ( − → x , M 2 t 1/2+η ) starting at some point y ∈ η(t). We must prove that γ meets ∂η(t)∩η 2 (t). We can suppose without loss of generality that y is the last point of γ in η(t) and thus y ∈ ∂η(t). Note that the points in γ after y are in B 2 (αt) c ⊂ B 2 (M t 1/2+η ) c .
• Or y ∈ η 1 (t). Let z be the point after y along γ where γ exits from B 2 (βt): between y and z, the path is in Cyl βt) thanks to Lemma 3.3. We build now a path γ ′ inside the Cyl + ( − → u , M t 1/2+η ) by concatenating the portion of γ between y and z and any infinite path starting from z and staying in Cyl + ( − → u , M t 1/2+η ) ∩ B 2 (βt) c : this path prevents the occurrence of Shadow( − → u , t, M t 1/2+η ), as it starts from a point in η 1 (t) and do not visit any other point in η(t). The assumption y ∈ η 1 (t) is thus contradicted.
So y ∈ η 2 (t), which means that Shadow( − → x , t, M 2 t 1/2+η ) happens, and implies Finally, Theorem 3.6 give two strictly positive constants A ′ , B ′ such that which ends the proof.
Note that the order t 1/2+η that appears in our results follows from moderate deviations for fluctuations with respect to the asymptotic shape given in Proposition 2.2. However, the conjectured order for these fluctuations is rather 1/3. The proofs would apply with an estimate of the type which would lead to replace t 1/2+η by t 1/3+η in our results.

Moderate deviations for the global growth of the epidemics
This section aims to prove Theorem 1.5: when the weak survives, we recover the same fluctuations with respect to the asymptotic shape as in the case where the weak infection evolves alone.
In fact, the only point is to see that if the strong infection at time t admits points outside B p1 t + βt 1/2+η , then this positional advantage enables to create an event of type Shade(T, M T 1/2+η ) at the slightly larger time T = t + αt 1/2+η , and the probability of such an event is controlled by Lemma 1.2.

Density of the strong in the two dimensional case
This section is devoted to the proof of Theorem 1.4: we prove that in dimension two, when coexistence occurs, the strong epidemic finally occupies a subset of Z 2 with null density. We first need some definitions: Definition 5.1. For any t ≥ 0, denote by C ext (t) the infinite connected component of η(t) c , and by ∂ ∞ η(t) the external boundary of η(t): What is specific to the two dimensional case is that the external boundary of the fast infection ∂ ext η(t) ∩ η 2 (∞) is * -connected -this will be proved in Lemmas 5.2 and 5.3. Loosely speaking, by Theorem 1.5, the external boundary ∂ ext η(t) of the infection at time t is included in a very thin annulus with radius t and width t 1/2+η . Then, Theorem 1.3, combined with the * -connectivity of ∂ ext η(t) ∩ η 2 (∞), ensures that the shadow cast by the fast infection on the slow infection has a diameter smaller than t 1/2+η . However, it remains to control the points in η 2 (∞) that are never in a position to create shadow, see Figure 5. The proof breaks down in higher dimension, as we can imagine a configuration where the fast infection occupies a tree whose branches simultaneously widen and ramify. For instance, we can assume that the radius at height t is of order t 1/2 and the number of branches at height t is of order t d−3/2 .
Let us now recall the graphical duality of the square lattice. Let Z 2 , which is isomorphic to L 2 . For each bond e = {a, b} of L 2 (resp. L 2 * ), let us denote by s(e) the only subset {i, j} of Z 2 * (resp. Z 2 ) such that the quadrangle aibj is a square in R 2 . The application s is clearly an involution.
For any finite set A ⊂ Z 2 , we denote by Peierls(A) the set of Peierls contours associated to A, that is Note also that if γ is a Jordan curve on L 2 * , the set Int(γ) (Ext(γ)) composed by the points in s(γ) that are in the bounded (respectively, unbounded) connected component of R 2 \γ is * -connected.
We begin with two lemmas to prove the * -connectivity of the set η 2 (t) ∩ ∂ ext η(t) in dimension 2. Assume that x 0 ∈ A and suppose by contradiction that E A A∪B is not connected: there exist p, q with 1 < p < q < f with Since A is connected, there exists a simple path in A from x 0 to x q−1 which corresponds to a path in Z 2 * from the start of e 0 to the end of e q−1 . The union of this path with the path (e 0 , e 1 , . . . e q−1 ) makes a Jordan curve γ.
Obviously, m q / ∈ γ. Since m q is on the outer boundary of the connected set (A ∪ B) + [−1/2, 1/2] 2 , there exists an (infinite) path in (A ∪ B) c joining m q to infinity. So, we can say that m q is in the infinite component of γ c . Since B is connected, there exists a path γ ′ in B from x q to x 1 . Let e be the first edge of γ ′ which crosses γ.
By construction, we know that each edge e ∈ L 2 which crosses γ ′ from the unbounded component to a point in B must be one of the s(e i )'s. But there is a contradiction because no s(e i ) can have both ends in A ∪ B.
We can now proceed to the proof of Theorem 1.4.
Let us now prove the second point: for every β > 0, It is clearly sufficient to consider β ∈ (0, 1/2). We can choose α ′ > α > 0 such that For t large enough, choose n such that n 2+α < t ≤ (n + 1) 2+α . Then, by (43), there exists y ∈ R 2 such that As n ∼ t 1 2+α , this ends the proof of the second point. Turning to the proof of the last assertion, consider the following alternative: • If the weak species (type 1) does not unboundedly grow, its natural density is zero, while the density of the strong is one. • If the weak species grows unboundedly, the first point of the present theorem ensures that the strong species has null density, and therefore that the weak have full density.
6. Non-coexistence except perhaps for a denumerable set In this section, we prove Theorem 1.6: Remember that Häggström and Pemantle proved non-coexistence for two epidemics progressing according exponential laws with parameter 1 and λ "except perhaps for a denumerable set" for λ, and Theorem 1.6 extends this result to families of laws depending on a continuous parameter.
The first step consists in coupling all possible competition models on the same probability space, respecting the stochastic order of the laws. This will give natural inclusions between sets of infected points for competition with distinct parameters, as stated in Lemma 6.1. Assume that p < r, that the slow infection (the strong one) uses the law with parameter p (respectively, r) and that both infections manage to grow unboundedly. Let also choose q ∈ (p, r). Then, we can expect than strengthening the slow infection by increasing its parameter from p to q makes it strong enough to win and surround the fast one. Proving this is the aim of Lemma 6.3, and the proof is based on Theorem 1.5. Finally, to prove Theorem 1.6, we show that for a fixed q ∈ I, the set of p < q such that P(G 1 p,q ∩ G 2 p,q ) > 0 is a subset of the discontinuity set of an increasing function, and is thus at most denumerable.
Coupling. We first couple all passage times for varying parameters thanks to the generalized inverse of the repartition function, which allows to build all the competition models on the same probability space. This generalizes the construction presented in the introduction with only two parameters.
This means that at time 0, no point of Z d has been infected yet but the two initial sources s 1 and s 2 . Then, for n ≥ 0, define the next time of infection: Note that the infimum in the definition of T n+1 is always taken on a finite set. Moreover, Assumption (5) ensures that if this infimum is reached by several triplets (i, y, z), all these triplets have the same first coordinate, which means that a point can be infected by the same species from distinct neighbors at the same time, but not by the two species simultaneously. For such a triplet, the next infection is of type i from (one of the) y to z. The set of infected points of type 3 − i has not changed: ∀x ∈ Z d X 3−i,p1,p2 n+1 (x) = X 3−i,p1,p2 n (x), while the point z has been infected by species i at time X i,p1,p2 n (y) + t pi {y,z} : ∀x ∈ Z d \{z} X i,p1,p2 n+1 (x) = X i,p1,p2 n (x) and X i,p1,p2 n+1 (z) = X i,p1,p2 n (y) + t pi {y,z} . Note that X i,p1,p2 n (y) and X 3−i,p1,p2 n (y) can not be simultaneously finite, which corresponds to the fact that each site is infected by at most one type of infection. Moreover, once min(X 1,p1,p2 n (x), X 2,p1,p2 n (x)) is finite, its value -the time of infection of x -does not change any more.
The set G i,p1,p2 corresponds to the survival of type i.

Concluding remarks
Since we are coming to the end of our study, it is worth questioning the relevance of the notion of strong coexistence and its relationship with the Häggström-Pemantle conjecture.
At first, let us say a word about the notion of strong coexistence. It is obviously stronger, but how far away is it from the notion of coexistence? In the case where the two species have the same passage times law, a partial answer is given by a recent work by Gouéré [7]: some of these results imply that -under classical assumptions implying coexistence -one can find initial configurations that give rise to strong coexistence with positive probability. The restriction on the initial conditions can however be dropped by a modification argument as in Garet-Marchand [4]. Actually, in the case where the two species have the same passage times law, simulations let think that when coexistence occurs, each species grants itself a cone, and thus strong coexistence occurs.
The above remarks seem to show the relevance of the notion of strong coexistence, but we must now wonder how far we are from the Häggström-Pemantle conjecture. First, it could be interesting to note that Theorem 1.6 allows a reformulation of this conjecture: it is sufficient to prove that the map p → P p,q (G 1 ) has a unique point of discontinuity. Note also that this reformulation is quite close to some recent result by Deijfen and Häggström concerning competition with exponential speeds on general graphs (Theorem 4.1 in [3]). However, it is not evident that this remark can be exploited to prove the conjecture. In this perspective, it it more natural to look at Theorem 1.2 and Corollary 1.3: it seems highly unlikely that the strong could survive being constrained to occupy only a negligible portion of the aerial surface, but we did not success to prove it at this time.