Multitype Contact Process on $\Z$: Extinction and Interface

We consider a two-type contact process on $\Z$ in which both types have equal finite range and supercritical infection rate. We show that a given type becomes extinct with probability 1 if and only if, in the initial configuration, it is confined to a finite interval $[-L,L]$ and the other type occupies infinitely many sites both in $(-\infty, L)$ and $(L, \infty)$. We also show that, starting from the configuration in which all sites in $(-\infty, 0]$ are occupied by type 1 particles and all sites in $(0, \infty)$ are occupied by type 2 particles, the process $\rho_t$ defined by the size of the interface area between the two types at time $t$ is tight.


Introduction
The contact process on is the spin system with generator for λ > 0 and p(·) a probability kernel. We take p to be symmetric and to have finite range R = max{x : p(x) > 0}.
The contact process is usually taken as a model for the spread of an infection; configuration ζ ∈ {0, 1} is the state in which an infection is present at x ∈ if and only if ζ(x) = 1. With this in mind, the dynamics may be interpreted as follows: each infected site waits an exponential time of parameter 1, after which it heals, and additionally each infected site waits an exponential time of parameter λ, after which it chooses, according to the kernel p, some other site to which the infection is transmitted if not already present.
We refer the reader to [13] for a complete account of the contact process. Here we mention only the most fundamental fact. Letζ and 0 be the configurations identically equal to 1 and 0, respectively, S(t) the semi-group associated to Ω, λ the probability measure under which the process has rate λ and ζ 0 t the configuration at time t, started from the configuration where only the origin is infected. There exists λ c , depending on p, such that • if λ ≤ λ c , then λ (ζ 0 t = 0 ∀t) = 0 and δζS(t) → δ 0 ; • if λ > λ c , then λ (ζ 0 t = 0 ∀t) > 0 and δζS(t) converges, as t → ∞, to some non-trivial invariant measure. Again, see [13] for the proof. Throughout this paper, we fix λ > λ c .
The multitype contact process was introduced in [15] as a modification of the above system. Here we consider a two-type contact process, defined as the particle system (ξ t ) t≥0 with state space {0, 1, 2} and generator (1 denotes indicator function). This is thought of as a model for competition of two biological species. Each site in corresponds to a region of space, which can be either empty or occupied by an individual of one of the two species. Occupied regions become empty at rate 1, meaning natural death of the occupant, and empty regions become occupied at a rate that depends on the number of individuals of each species living in neighboring sites, and this means a new birth. The important point is that occupancy is strong in the sense that, if a site has an individual of, say, type 1, the only way it will later contain an individual of type 2 is if the current individual dies and a new birth occurs originated from a type 2 individual.
Let us point out some properties of the above dynamics. First, it is symmetric for the two species: both die and give birth following the same rules and restrictions. Second, if only one of the two species is present in the initial configuration, then the process evolves exactly like in the one-type contact process. Third, if we only distinguish occupied sites from non-occupied ones, thus ignoring which of the two types is present at each site, again we see the evolution of the one-type contact process. We also point out that both the contact and the multitype contact processes can be constructed with families of Poisson processes which are interpreted as transmissions and healings. These families are called graphical constructions, or Harris constructions. Although we will provide formal definitions in the next section, we will implicitly adopt the terminology of Harris constructions in the rest of this Introduction.
The first question we address is: for which initial configurations does a given type (say, type 1) become extinct with probability one? By extinction we mean: for some time t 0 (and hence all t ≥ t 0 ), ξ t 0 (x) = 1 for all x. We prove (# denotes cardinality). This result is a generalization of Theorem 1.1. in [1], which is the exact same statement in the nearest neighbour context (i.e., p(1) = p(−1) = 1/2). Although there are some points in common between our proof and the one in that work, our general approach is completely different. Additionally, their methods do not readily apply in our setting, as we now briefly explain. Let = {ξ ∈ {0, 1, 2} : ξ(x) = 1, ξ( y) = 2 =⇒ x < y}. When the range R = 1, ξ 0 ∈ implies ξ t ∈ for all t ≥ 0. In [1], the proof of both directions of the equivalence of Theorem 1.1 rely on this fact (see for example Corollary 3.1 in that paper), which does not hold for R > 1.
If both types are present in finite number in the initial configuration, there is obviously positive probability that one of them is present for all times, because the process is supercritical. The following theorem says that there is positive probability that they are both present for all times. Theorem 1.2. Assume that 0 < #{x : ξ 0 (x) = 1}, #{x : ξ 0 (x) = 2} < ∞.
We have ρ 0 = −1, and at a given time t both events {ρ t > 0} and {ρ t < 0} have positive probability. If ρ t > 0, we call the interval [l t , r t ] the interface area. The question we want to ask is: if t is large, is it reasonable to expect a large interface? We answer this question negatively. Theorem 1.3. The law of (ρ t ) t≥0 is tight; that is, for any ε > 0, there exists L > 0 such that (|ρ t | > L) < ε for every t ≥ 0.
There are several works concerning interface tightness in one-dimensional particle systems, the first of which is [5], where interface tightness is established for the voter model. Others are [3], [4], [17] and [2].
In [2], it is shown that interface tightness also occurs on another variant of the contact process, namely the grass-bushes-trees model considered in [8], with both species having same infection rate and non-nearest neighbor interaction. The difference between the grass-bushes-trees model and the multitype contact process considered here is that, in the former, one of the two species, say the 1's, is privileged in the sense that it is allowed to invade sites occupied by the 2's. For this reason, from the point of view of the 1's, the presence of the 2's is irrelevant. It is thus possible to restrict attention to the evolution of the 1's, and it is shown that they form barriers that prevent entrance from outside; with this at hand, interface tightness is guaranteed regardless of the evolution of the 2's. Here, however, we do not have this advantage, since we cannot study the evolution of any of the species while ignoring the other.
Our results depend on a careful examination of the temporal dual process; that is, rather than moving forward in time and following the descendancy of individuals, we move backwards in time and trace ancestries. The dual of the multitype contact process was first studied by Neuhauser in [15] and may be briefly described as follows. Each site x ∈ at (primal) time s has a random (and possibly empty) ancestor sequence, which is a list of sites y ∈ such that the presence of an infection in ( y, 0) would imply the presence of an infection in (x, s) (in other words, such that there is an infection path connecting ( y, 0) and (x, s)). The ancestors on the list are ranked in decreasing order; the idea is that if the first ancestor is not occupied in ξ 0 , then we look at the second, and so on, until we find the first on the list that is occupied in ξ 0 , and take its type as the one passed to x. We denote this sequence (η x 1,s , η x 2,s , . . .). By moving in time in the opposite direction as that of the original process and using the graphical representation of the contact process for "negative" primal times, we can define the ancestry process of x, ((η x 1,t , η x 2,t , . . .)) t≥0 . The process given by the first element of the sequence, (η x 1,t ) t≥0 , is called the first ancestor process. We point out three key properties of the ancestry process: • First ancestor processes have embedded random walks. In [15] it is proven that, on the event that a site x has a nonempty ancestry at all times t ≥ 0, we can define an increasing sequence of random renewal times (τ x n ) n≥0 with the property that the space-time increments are independent and identically distributed. This fact enormously simplifies the study of the first ancestor process, which is not markovian and at first seems very complicated.
• Ancestries coalesce. If we are to use the dual process to obtain information about the joint distribution of the states at sites x and y at a given time, we must study the joint behavior of two ancestry processes, specially of two first ancestor processes. The intuitive picture is that this behavior resembles that of two random walks that are independent until they meet, at which time they coalesce. We give a new approach to formalizing this notion, one that we believe provides a clear understanding of the picture and allows for detailed results.
In order to follow two first ancestor processes simultaneously, we define joint renewals (τ x, y n ) n≥0 and argue that the law of the processes after a joint renewal only depends on their initial difference at the instant of the renewal. Thus, the discrete-time process defined by the difference between the two processes at the instants of renewals is a Markov chain on . For this chain, zero is an absorbing state and corresponds to coalescence of first ancestors. We also show that, far from the origin, the transition probabilities of the chain become close to a symmetric measure on , and from this fact we are able to show that the tail of the distribution of the hitting time of 0 for the chain looks like the one associated to a simple random walk on . From this construction and estimate we also bound the expected distance between ancestors at a given time.
• Ancestries become sparse with time. Consider the system of coalescing random walks in which each site of starts with one particle at time 0. The density of occupied sites at time t, which is equal to the probability of the origin being occupied, tends to 0 as t → ∞. We prove a similar result for our ancestry sequences. Fix a truncation level N and, at dual time t, mark the N first ancestors of each site at dual time 0 (this gives the set {η x n,t : 1 ≤ n ≤ N , x ∈ : the ancestry of x reaches time t}). We show that the density of this random set tends to 0 as t → ∞, and estimate the speed of this convergence depending on N .
From this last fact, we can immediately prove Theorem 1.1 under the stronger hypothesis that all sites outside [−L, L] are occupied by 2's in ξ 0 . To obtain the general case, we then use a structure called a descendancy barrier, whose existence was established in [2]. Theorem 1.2 is obtained quite easily from Theorem 1.1. The proof of Theorem 1.3 is more intricate, and follows the main steps of [5], which studies the voter model.
We believe that our results and general approach may prove useful in other questions concerning the multitype contact process, in particular those that relate to almost sure properties of the trajectories t → ξ t of the process, as opposed to properties of its limit measures.

Ancestry process
We will start describing the familiar construction of the one-type contact process from its graphical representation. We will then show how the same representation can be used to construct the multitype contact process, present the definition of the ancestry process together with some facts from [15], and finally prove a simple lemma.
A Harris construction H is a realization of all such processes. H can thus be understood as a point measure on ( ∪ 2 ) × [0, ∞). Sometimes we abuse notation and denote the collection of processes itself by H. Given (x, t) ∈ × [0, ∞), let θ (x, t)(H) be the Harris construction obtained by shifting H so that (x, t) becomes the space-time origin. By translation invariance of the space-time construction, θ (x, t)(H) and H have the same distribution. We will also write H [0,t] to denote the restriction of H to × [0, t], and refer to such restrictions as finite-time Harris constructions.

Given a Harris construction H and
One such function γ is called a path determined by H. The points in the processes {D x } are usually called death marks, and the points in {N (x, y) } are called arrows. Thus, a path can be thought of as a line going up from (x, s) to ( y, t) following the arrows and not crossing any death marks.
Under the law of H, (ζ A t ) has the distribution of the contact process with parameter λ, kernel p and initial state 1 A ; see [6] for details. From now on, we omit dependency on the Harris construction and write (for instance) ζ t instead of ζ t (H).
Before going into the multitype contact process, we list some properties of the one-type contact process that will be very useful. Fix (x, s) ∈ × [0, ∞) and t > s. Define the time of death and maximal distance traveled until time t for an infection that starts at (x, s), (these only depend on H and are thus well-defined regardless of ξ s (x)). When s = 0, we omit it and write T x , M x t . If A ⊂ , we also define T A = inf{t ≥ 0 : ∄ x ∈ A, y ∈ : (x, 0) ↔ ( y, t)}. We start by observing that M (x,s) t is stochastically dominated by a multiple of a Poisson random variable, so there exist κ, c, C > 0 such that Remark 2.1. Throughout this paper, letters that denote universal constants whose particular values are irrelevant, such as c, C, κ in the above, will have different values in different contexts and will sometimes change from line to line in equations.
Next, since we are taking λ > λ c , we have (T x = ∞) = (T 0 = ∞) > 0 for all x, and This follows from the self-duality of the contact process and the fact that its upper invariant measure has positive correlations; see [12]. Our last property is that there exist c, C > 0 such that, for any A ⊂ and t > 0, For the case R = 1, this is Theorem 2.30 in [13]. The proof uses a comparison with oriented percolation and can be easily adapted to the case R > 1.
We now turn to our treatment of the multitype contact process and its dual, the ancestry process, through graphical constructions. We will proceed in three steps.
Step 1: Graphical construction of the multitype contact process. The construction is done as for the one-type contact process, with the difference that we must ignore the arrows whose endpoints are already occupied. This was first done in [15]; there, an algorithmic procedure is provided to find the state of each site at a given time. Here we provide an approach that is formally different but amounts to the same. Fix (x, t) ∈ × [0, ∞), a Harris construction H and ξ 0 ∈ {0, 1, 2} . Let Γ x 1 be the set of paths γ that connect points of × {0} to (x, t) in H. Assume that #Γ x 1 < ∞; this happens with probability one if H is sampled from the processes described above. For the moment, also assume that Γ x 1 = . Given γ, γ ′ ∈ Γ x 1 , let us write γ < γ ′ if there existss ∈ (0, t) such that γ(s) = γ ′ (s) ∀s ∈ [s, t] and γ(s) = γ(s−), γ ′ (s) = γ ′ (s−). From the fact that these paths are all piecewise constant, have finitely many jumps and the same endpoint, we deduce that < is a total order on Γ x 1 . We can then find γ * 1 , the maximal path in Γ x , and for n ≥ N putξ x n,t = △. We claim that (Here ζ · continues to denote the one-type contact process defined from H). In words, if γ * n makes a jump that lands on a space-time point (x, s), then for some positive ε the set {x} × [s − ε, s) cannot be reached by paths coming fromξ x n+1,t , . . . ,ξ x N −1,t , and so the jump is not obstructed. If this were not the case, we could obtain m < n, s ∈ [0, t] and γ with γ(0) =ξ x n,t and γ * m (s−) = γ * m (s) = γ(s) = γ(s−). But we could then construct a path γ ′ coinciding with γ on [0, s] and with γ * m on (s, t], and γ ′ would contradict the maximality that defined γ * m . In this second case, using (2.4), we see that there is a path connecting (ξ x k,t , 0) to (x, t) which is not obstructed by any of the paths connecting { y =ξ x k,t : ξ 0 ( y) = 0} × {0} to (x, t). Finally, if Γ x 1 = , putξ x n,t = △ for every n and set ξ t (x) = 0. We can proceed similarly for every x ∈ and get a configuration (ξ t (x)) x∈ . It now follows that (ξ t (x)) x∈ has the distribution of the multitype contact process at time t with initial state ξ 0 . Additionally, by applying this construction to every t > 0, we get a trajectory (ξ t ) t≥0 of the process that is right continuous with left limits. We will sometimes write ξ t (H) to make the dependence on the Harris construction explicit.
Given x ∈ and 0 ≤ s ≤ t, we define η (x,s) n,t = η x n,t−s (θ (0, s)(H)) (that is, the nth ancestor in the graph that grows from (x, s) up to time t). Also, when n = 1, we omit it, writing η x t , η n,t ∈ : n ≥ 1} (notice that we are excluding the △ state), and similarly for η x * ,t . The set η x * ,t has the same distribution as ζ {x} t , the set of infected sites of a one-type contact process started from the configuration where only x is infected. For this reason, if we ignore the △ state, the process t → (η x 1,t , η x 2,t , . . .) can be seen as a one-type contact process started from 1 {x} and such that at each time, infected sites have numerical ranks.
Step 3: Joint construction of the multitype contact process and the ancestry process. We now explain the duality relation that connects the two processes described above. Fix t > 0 and let H = ((D x ), (N (x, y) )) be a Harris construction on ×[0, t] (this means that the Poisson processes that constitute H are only defined on [0, t]). Let t (H) be the Harris construction on [0, t] obtained from H by inverting the direction of time and of the arrows; formally, Given ξ 0 ∈ {0, 1, 2} , we now take the pair ( ξ s (H) ) 0≤s≤t , ( η x n,s ( t (H)) ) n∈ , 0≤s≤t, x∈ . We thus have a coupling of the contact process started at ξ 0 and the ancestry process of every site, both up to time t. We should think of time for the multitype contact process as running on the opposite direction as time for the ancestry processes. This is illustrated on Figure 1.
We now write our fundamental duality relation: for each x ∈ , where n * (x) = inf{n : ξ 0 (η x n,t ) = 0}. This can be seen at work in Figure 1. Its formal justification is straightforward and relies on the following. Given x ∈ , define a bijection t : ; t γ is thus γ ran backwards and repaired so that we get a right continuous path. We then have γ < γ ′ ⇔ t γ´ t γ ′ , so all maximality properties can be translated from one space of paths to the other.
The obvious utility of (2.5) is that it allows us to relate properties of the distributions of the multitype contact process and of the ancestry processes at a fixed time. Let us state two such relations. First, where n * (x) was defined after (2.5). This will be useful for our proof of Theorem 1.1. Second, taking ξ 0 = ξ h (the heaviside configuration defined in the Introduction) and ρ t as in the Introduction, This will be useful in our proof of Theorem 1.3. We have now concluded Step 3.
Even though the relations of Step 3 will be extremely useful, our standard point of view will be the one of Step 2. This means that, from now on, unless explicitly stated otherwise, we will have an infinite-time Harris construction H used to jointly define, for every x ∈ , the ancestry processes ((η x n,t ) n∈ ) t≥0 . Whenever we mention a function of the Harris construction, such as T (x,s) or M (x,s) t , we mean to apply it to the Harris construction used to define the ancestry process.
The following is an easy consequence of the definition of the ancestry process with the ordering of paths´defined above.
(that is, the first n − 1 ancestors of x at time s do not reach time t, but the n-th one does, with ancestors .
the first time after 1 at which the first ancestor of x lives forever. It is useful to think of τ x 1 as the result of a sequence of attempts, as we now explain and illustrate on Figure 2. Define σ x 1 ≡ 1 and, for n ≥ 1, , σ x n otherwise.
(2.8) σ x n is thought of as the time of the n-th attempt to find a first ancestor of x that lives forever. If and say that the first attempt succeeds. Otherwise, we must wait until σ x 2 = T (η x 1 ,1) to start the second attempt. This is , σ x 2 ) lives forever, then the second attempt succeeds and we have τ , σ x 2 ) to die, and so on.
On {T x = ∞}, also define τ x 0 ≡ 0 and, for n ≥ 1, For the sake of readability, we will sometimes write˜ x (·) and˜ x (·) instead of (·|T x = ∞) and (·|T x = ∞). In Proposition 1 in [15], it is shown that under˜ x , the times τ x n work as renewal times for the process η x t , that is, the (Time length, Trajectory) pairs are independent and identically distributed. This follows from an idea of Kuczek ([9]) which has become an important tool in the particle systems literature. In our current setting, it can be explained as follows. The probability˜ x is the original probability for the process conditioned on the event , τ x 1 ) living forever imply that (x, 0) lives forever, the event of the former conditioning. This and the fact that, under , restrictions of H to disjoint time intervals are independent yield that, under˜ x , the shifted Harris construction , τ x 1 )(H) has same law as H. The argument is then repeated for all τ x n , n ≥ 1.
We now list the properties of the renewal times that we will need.

Given an event A on finite-time Harris constructions and an event B on Harris constructions, we havẽ
(iii.) Under˜ x , the -valued process η x τ x n n≥0 is a symmetric random walk starting at x and with transitions (iv.) There exist c, C > 0 such that˜ Except for part (ii.), the above proposition is contained in Proposition (1), page 474, of [15] ((i.) and (iv.) are explicitly on the statement of the proposition and (iii.) is a direct consequence of (i.)). Part (ii.) is an adaption of Lemma 7 in [14] to our context; since its proof also uses ideas similar to the ones of Proposition (1) in [15], we omit it.
To conclude this section, we prove some simples properties of the first ancestor process.
Using Proposition 2.3(ii.) and (iv.), Observe that the above expectation is less than 1, because there is at most one renewal in each unit interval. (2.9) is thus less than since this does not depend on t, we get˜ for some c, C > 0 and all t ≥ 0. Let us now prove the two statements of the lemma.
(ii.) The definition of ψ t and (2.10) imply By the reflection principle (see [7], page 285), the expectation on the right-hand side is less than With (2.11), (2.12) and (2.13) at hand, we are ready to estimatẽ Let us treat each of the three terms separately.
since, by the independence of increments between different pairs of renewals and symmetry, 0 (η 0 By (2.13), this is less than Putting this, (2.16) and (2.17) back in (2.14) completes the proof.

The first term is less than
where κ is as in (2.1). Now use (2.1) and (2.3) to get that this last sum is less than C e −cl . Next, we have˜ because there are at most ⌊t⌋ renewals until time t. By (2.10),˜ 0 (ψ t > l/2) ≤ C e −cl . By Proposition This completes the proof.

Pairs and sets of ancestries
In this section, we study the joint behavior of ancestral paths. For pairs of ancestries, we define joint renewal points that have properties similar to the ones just discussed for single renewals, and then use these properties to study the speed of coalescence of first ancestors. For sets of ancestries, we show that, given N > 0, the overall density of sites of occupied by ancestors of rank smaller than or equal to N at time t tends to 0 as t → ∞.
Let us define our sequence of joint renewal times. Fix x, y ∈ and on {T x = T y = ∞} define τ x, y the first time after 1 at which both the first ancestor of x and the one of y live forever. In parallel with (2.8), define σ x, y 1 ≡ 1 and, for n ≥ 1, The sequence of attempts in this case works as follows. We start asking if both (η Otherwise, we wait until one of them dies out; this happens at time σ 2 ), and so on. Also define τ x, y For x, y ∈ , we write˜ x, y (·) = (·|T x = T y = ∞) and˜ x, y (·) = (·|T x = T y = ∞). Note that x,x =˜ x ,˜ x,x =˜ x and τ x,x n = τ x n for any x and n. We have the following analog of Lemma 2.3: We omit the proof since it is an almost exact repetition of the one of Lemma 2.3; the only difference is that, when looking for renewals, we must inspect two points instead of one.

Given an event A on finite-time Harris constructions, an event B on Harris constructions and z, w
We now study the behavior of the discrete time Markov chain mentioned in part (iii.) of the above proposition. Our first objective is to show that the time it takes for two ancestries to coalesce has a tail that is similar to that of the time it takes for two independent simple random walks on to meet. This fact will be extended to continuous time in Lemma 3.3; in Section 5, we will establish other similarities between pairs of ancestries and pairs of coalescing random walks. Lemma 3.2. (i.) For z ∈ , let π z denote the probability on given by There exist a symmetric probability π on and c, C > 0 such that where || · || T V denotes total variation distance. (ii.) There exists C > 0 such that, for all x, y ∈ and n ∈ , Proof. (i.) Fix z ∈ . For simplicity of notation, we will go through the proof in the case z > 0; however, it will be clear how to treat the case z < 0. Let us take two random Harris constructions H 1 and H 2 defined on a common space with probability measure , under which H 1 and H 2 are independent and both have the original, unconditioned distribution obtained from the construction with Poisson processes. Define H 3 as a superposition of H 1 and H 2 , as follows. We include in H 3 : • from H 1 , all death marks in sites that belong to (−∞, ⌊z/2⌋] and all arrows whose starting points belong to (−∞, ⌊z/2⌋]; • from H 2 , all death marks in sites that belong to (⌊z/2⌋, ∞) and all arrows whose starting points belong to (⌊z/2⌋, ∞). Then, H 3 has same law as H 1 and H 2 . We will write all processes and times defined so far as functions of these Harris constructions: for i ∈ {1, 2, 3}, we may take as defined before and nothing new is involved. On the event Our definition ofτ 0,z is similar to the one of first joint renewal time of two first ancestor processes. However, forτ 0,z , we follow a different Harris construction for each ancestor process. We can also think ofτ 0,z as the result of a "sequence of attempts", and define corresponding stopping times similar to the ones illustrated on Figure 2. The same proof that establishes Proposition 3.1(iv.) can be repeated here to show that there exist c, C > 0 such that Now define Note that π z (·) = (X 0,z = z + · | X 0,z = △), where π z is defined in the statement of the lemma. Also define By the definition of Y 0,z from independent Harris constructions, π is symmetric and does not depend on z. To conclude the proof, we have two tasks. First, to show that X 0,z = Y 0,z with high probability when z is large. Second, to show that this implies that, when z is large, ||π z − π|| T V is small.
Let κ be as in (2.1) and define t * = z/3κ. Consider the events We now claim that, if the event := ( 1 ∩ 2 ) ∪ ( 1 ∩ 3 ) occurs, then X 0,z = Y 0,z . To see this, assume first that 1 ∩ 2 occurs. Then, by the definition of 2 , we either have T 0 (H 1 ) < t * or We now use (3.5), (3.6) and the definition of a 2 , b 2 in Lemma 2.2(i.) to conclude that By the definition of t 2 , we also have T (a 2 ,t 2 ) (H 3 ) = T (b 2 ,t 2 ) (H 3 ) = ∞; together with (3.7) this yields we see from (3.5) and (3.8) that t 2 ≤ t 1 , so t 2 = t 1 . A similar set of arguments show that t 2 ≤ t 1 implies t 1 = t 2 . This completes the proof of the claim. Now note that the event c is contained in the union of: We thus have (3.9) Then, Using Proposition 3.1(iii.) and translation invariance, we see that under˜ x, y , X x, y n is a Markov chain that starts at y − x and has transitions x, y (X In particular, 0 is an absorbing state. (ii.) follows from Theorem 6.1 in Section 6. Here, let us ensure that the four conditions in the beginning of that section are satisfied by π z and π. Conditions (6.1) and (6.4) are already established. Condition (6.2) is straightforward to check and (6.3) follows from (3.1) and Proposition 3.1(iv.).
We now want to define a random time J x, y that will work as a "first renewal after coalescence" for the first ancestors of x and y, a time after which the two processes evolve together with the law of a single first ancestor process. Some care should be taken, however, to treat the cases in which the ancestries of x or of y die out. With this in mind, we put This definition is symmetric: J x, y = J y,x .
Iterating, we get˜ x, y (τ Note that if T x = ∞, T y < t/2 and there exists some n such that τ x n ∈ [t/2, t], then J x, y ≤ t, and similarly exchanging the roles of x and y. Using (2.3), Lemma 2.5(i.) and (3.10), we thus have (ii.) Let A be a borelian of [0, ∞) and B be an event in the σ-field of Harris constructions. Using Proposition 2.3 (ii.), Using Proposition 3.1 (ii) and the fact that˜ z,z =˜ z for any z, Putting things together we get The claim is a direct consequence of this equality.
Let x, y ∈ ; assume that T x = T y = ∞ and J x, y + σ N • θ (η x J x, y , J x, y ) ≤ t. This means that first, (η x ) and (η y ) have the first joint renewal at some space-time point (η x J x, y , J x, y ) = (η y J x, y , J x, y ) with J x, y ≤ t, and second, that the ancestry process of (η x J x, y , J x, y ) never has less than N elements after time t. We must then have z 1 , . . . , z N ∈ such that Lemma 2.2 then implies that η x n,t = η y n,t = z n , 1 ≤ n ≤ N , and we have thus shown that Then, where in the last inequality we used Lemma 3.3(i.) in the first term and Lemma 3.3(ii.) and (3.12) in the second.
Finally, we have Proposition 3.5. There exist C, γ > 0 such that, for any N ≥ 1 and t ≥ 0, Proof. Fix a real t ≥ 0 and a positive integer l with l > N . Define Γ = {0, . . . , l − 1} and that is, for all sites in Γ that have non-empty ancestry at time t, the first N terms of the ancestor sequence at time t must coincide. We can use Lemma 3.4 to bound the probability of Λ c : since there are less than l 2 choices for {x, y} and for any of them, |x − y| ≤ l.
A consequence of the above inequality is that there exists r * ∈ {0, . . . , l − 1} such that (3.14) Finally, for z ∈ let Γ z = −r * + lz + Γ. The idea is that 0 seen from Γ 0 is the same as r * seen from Γ. Let Λ z , η Γ z n,t and η Γ z N −,t be defined from Γ z as Λ, η Γ n,t and η Γ N −,t are defined from Γ. We can now proceed to our upper bound: Noting that We now put l = t 1 9 ; we have thus obtained The first two terms are already in the form we want, and it is straightforward to show that, for some

Extinction, Survival and Coexistence
In this section we prove Theorems 1.1 and 1.2. Our three key ingredients will be a result about extinction under a stronger hypothesis (Lemma 4.1), an estimate for the edge speed of one of the types when obstructed by the other (Lemma 4.2) and the formation of "descendancy barriers" for the contact process on (Lemma 4.3).
We recall our notation from the Introduction: the letters ξ and η will be used for the multitype contact process and the ancestry process, respectively. Throughout this section, in contrast with the rest of the paper, Harris constructions and statements related to them, such as "(x, s) ↔ ( y, t)", refer to the construction for ξ rather than the one for η. Proof. Using (2.6), for any t 0 > 0 we have By (3.5) each of the probabilities in the last sum converges to 0 as t 0 → ∞.
Say that x ∈ forms a ρ-descendancy barrier if the origin forms a ρ-descendancy barrier according to θ (x, 0)(H).

Lemma 4.3.
For any ε > 0, there exists β, K > 0 such that The proof is in [2]; see Proposition 2.7 and the definition of the event 2 in page 10 of that paper.
Before the proof of Theorem 1.1, we state two more lemmas. Their proofs are straightforward and we omit them. For the following, as is usual, we abbreviate {x : ξ t (x) = i} as {ξ t = i}.
(ii.) Assume condition (A) of Theorem 1.1 is satisfied but condition (B) is not (say, with finitely many 2's in [0, ∞)) and, for a given a ∈ , we have ξ 0 (a) = 1. Then, with positive probability, Proof of Theorem 1.1. We first prove that, if conditions (A) and (B) in the statement of the theorem are satisfied, then the 1's almost surely become extinct. Fix ε > 0. As in Lemma 4.3, choose β, K 1 corresponding to ε, then as in Lemma 4.2, choose K 2 corresponding to ε and β. Let K = K 1 +K 2 +2R. Using Lemma 4.5(i.) with this value of K and relabeling time so as to start looking at the process at time 1, we may assume that there exist a 1 , a 2 , L ′ such that (A ′ ), (B ′ ) and (C ′ ) are satisfied by ξ 0 (rather than by ξ 1 ).
Let (ξ 1 t ), (ξ 2 t ), (ξ 12 t ) and (ξ 21 t ) be realizations of the multitype contact process all built using the same Harris construction as the original process (ξ t ) and having initial configurations By a series of comparisons and uses of the previous lemmas, we will show that in ξ 1 , the 1's become extinct with high probability. An application of Lemma 4.4 to the pair ξ 1 , ξ then implies that in ξ, the 1's become extinct with high probability.
• y − b 2 , x − b 2 have opposite signs. We get ξ 1 t (x) = 0 as in the previous case.
We now claim that, on show that, if condition (A) of the theorem is satisfied but condition (B) is not, then the 1's have positive probability of surviving. We will first treat the case of ξ 0 = ξ 21K 0 := 2 · 1 (−∞,0) + 1 [0,K] ; let us show that lim Fix ε > 0 and choose β, K 1 and K 2 as before. We will need another constant K 3 whose choice will depend on the following. Let α > 0 be the edge speed for our contact process (i.e., the almost sure limit as t → ∞ of 1 t sup{ y : ∃x ∈ (−∞, 0] : (x, 0) ↔ ( y, t)}. See Theorem 2.19 in [12] for the existence of the limit; the proof is easily seen to apply to our non-nearest neighbor context). Given α ′ ∈ (0, α), we have This is a consequence of the definition of α and the fact that lim K ′ →∞ (∀t, ∃x ∈ [0, K ′ ], y ∈ : (x, 0) ↔ ( y, t)) = 1; we omit the details. We may assume that the β we have chosen is strictly smaller than α: it is readily verified that if x ∈ forms a β-descendancy barrier, then it forms a β ′descendancy barrier for any β ′ < β; hence, we may decrease β if required. We choose K 3 such that, putting K ′ = K 3 and α ′ = β, the probability in (4.4) is larger than 1 − ε.

Interface tightness
We now carry out the proof outlined at the end of the Introduction. It is instructive to restate Theorem 1.3 in its dualized form, which follows from (2.7): Theorem 1.3, dual version For any ε > 0, there exists L > 0 such that We start with two Lemmas concerning the expectation of the distance between two first ancestors. Lemma 5.1 shows a resemblance to the case of two random walks that evolve independently until they meet, at which time they coalesce. Lemma 5.2 is a generalization that allows us to integrate over the event of death of a preassigned set of sites.
Lemma 5.1. There exists C > 0 such that, for all x < y ∈ and t ≥ 0, Proof. By translation invariance, it suffices to treat x = 0 < y. It also suffices to prove (i.) and (ii.) for t sufficiently large (not depending on x, y), because , and these expectations grow polynomially in t, by comparisons with Poisson random variables. Finally, by symmetry. By Cauchy-Schwarz, this last expectation is less than Let us estimate the expectation.
Using (5.4) and (5.5) and ignoring the term (η 0 J 0, y ) 2 , the expression in (5.3) is less thañ by Lemma 2.5(ii.) and Lemma 3.3(i.). Now we can continue as in Lemma 1 in [5]: the above is less than when t ≥ 1. This and another application of Lemma 3.3(i.) show that (5.2) is less than C y t · C y t ≤ C y; going back to (5.1), we get (ii.) To treat the expectation on the event {T 0 = T y = ∞}, we will separately consider two cases, depending on whether or not the ancestor processes of 0 and y had a joint renewal in inverted order before time t. To this end, define ) and (2.2). Then, (5.6) is less than which is bounded by Lemma 6.6.
We then have As in the proof of Lemma 2.5, we can then show that (φ t ; T 0 = T y = ∞) is bounded uniformly in y and t. Putting together (5.7) and (5.8), we get the result.

Lemma 5.2.
There exist c, C > 0 such that, for all x < y ∈ , t ≥ 0 and finite A ⊂ , Proof. Since both estimates are treated similarly, we will only show part (ii.): the expectation is less than C k, and the above sum is less than where we have written y = x + z, used the facts that } are independent and that ([x, x +z]∩R = {x, x +z}∪(x +A)) = ([0, z] ∩R = {0, z} ∪ A) by translation invariance. Applying Lemma 5.3 to the inner sum, we get that the above is less than We thus have Step 2. Our next goal is to bound the probability of  (5.13). Hence, at least one of x, y is in R −R, so at least one of the events in (5.14) occurs.
The probability of the first event in (5.14) is less than z≥1 A⊂(0,z) a∈ ,m>N (a 1 ,...,a m−1  The inner sum is less than a∈ 0 ∈ η a * ,s = #{a ∈ : 0 ∈ η a * ,s }. By a routine comparison with a Poisson random variable, the latter is less than Cs for some C > 0. Hence the expression in (5.16) is less than By symmetry, the same bound applies to the second event in (5.14), so we have: Using this and Lemma 5.3, we see that the expression in (5.18) is less than Recall that the choice of t in (s, ∞) was arbitrary, so the above derivation shows that we can choose L large enough that L < ε 2 for all t > s, and thus Noticing that the trajectories of (B t ) are right continuous with left limits, we can increase L if necessary so that this inequality also holds for t ≤ s, completing the proof.
Proof of Theorem 1.3. We separately show that (ρ t ∧ 0) and (ρ t ∨ 0) are tight. We start with the first. Given L > 0, for the event {ρ t > L} to occur, there necessarily exist two sites x, y such that y − x > L and η y t ≤ 0 < η x t . If N < L and {B t < N } also occurs, then we cannot have more than N sites z ∈ (x, y) such that η z * ,t = , because every such site makes a [0, t]-inversion either with x or with y and thus increases B t by one. So we have, for all t ≥ 0, Using Lemma 5.3 on the innermost sum and counting the possible choices of A, the above is less than which tends to 0 as L → ∞. So, given ε > 0, choose N > 0 such that < ε ∀t, and we are done. Now we treat (ρ t ∨ 0). This is easier: given L > 0, for {ρ t < −L} to occur we must have x < y such that η x t < 0 ≤ η y t and η w * ,t = ∀w ∈ (x, y). Then, for any t, which tends to zero as L −→ ∞.
Theorem 6.1. There exists C > 0 such that, for x ∈ , The proof of Theorem 6.1 will be carried out in a series of results. Fix L > 0 such that Ge −g L < 1 and let I = [−L, L]. Put ε z = Ge −g|z| for z ∈ I c and ε z = 1 for z ∈ I. A consequence of (6.4) is that, for all z ∈ , there exist probabilities g z , b 1 z , b 2 z on such that π z = ε z b 1 z + (1 − ε z )g z ; (6.5) π = ε z b 2 z + (1 − ε z )g z . (6.6) (Of course, if z ∈ I we must have b 1 z = π z , b 2 z = π). We will construct the process (X n ) coupled with other processes of interest. Let (X n , Z n ) be a Markov chain on × {0, 1} with transitions We write x to represent any probability for this chain with X 0 = x, regardless of the law of Z 0 . This abuse of notation is justified by the fact that Z 0 has no influence on the distribution of the other variables of the chain, nor on the random variables to be defined below. Let ( n ) be the natural filtration of the chain, and T = inf{n ≥ 1 : Z n = 1}.
Let (Ψ z ) z∈ be random variables defined on the same probability space as the chain above, independent of the chain and with laws Ψ z d = b 2 z . Additionally, let (Φ n ) n≥0 be a random walk with increment law π, initial state 0, also defined on the same space as the previous variables and independent of them. For n ≥ 0, define Y n = X n , if n < T ; X T −1 + Ψ X T −1 + Φ n−T , if n ≥ T. (6.8) We can use (6.5) and (6.6) to check that under x , (X n ) is a Markov chain with transitions P(z, w) = π z (w − z) and initial state x, and (Y n ) is a random walk with increment distribution π and initial state x. They satisfy X n = Y n on {T > n}.
Lemma 6.5. There exists C > 0 such that, for all x ∈ , Proof. If x ∈ I, the left-hand side is zero. Assume that x / ∈ I.
We can now write and then, as in the preceeding proof, use (6.15), Lemma 6.5, (6.13), (6.16), and (6.18) to show that the above sum is less than C|x| N for some C > 0. To conclude, we mention the following result, for use in the proof of Lemma 5.1. We omit its proof since it is simply a repetition of the above arguments.