2. Proof of Theorem 2

Throughout the paper, we think of the process as being generated from a percolation substructure, also called Harris’ graphical representation [Reference Harris8]. The percolation substructure consists of independent Poisson processes with appropriate rates attached to each vertex and oriented edge of the d-dimensional integer lattice. The process is then constructed by assuming that, at the times of these Poisson processes, either a birth, or an infection, or a death, or a recovery occurs whenever the configuration of the system at that time is compatible with the event. Table 1 shows, for instance, how the stacked contact process can be constructed when $\lambda_{10} \geq \lambda_{20}$ , while Table 2 shows another example of contruction when $\lambda_{20} \geq \lambda_{21}$ .

Table 1.

Graphical representation of the process when $\lambda_{10} \geq \lambda_{20}$ (pathogen). The rates in the left column correspond to the different parameters of the independent Poisson processes, attached to either each oriented-edge-connected two neighbors (first three rows) or each vertex (last two rows).

Table 2.

Graphical representation of the process when $\lambda_{20} \geq \lambda_{21}$ . The rates in the left column correspond to the different parameters of the independent Poisson processes, attached to either each oriented-edge-connected two neighbors (first three rows) or each vertex (last two rows).

Proof of (4)–(7). In the limiting case when the recovery rate $\delta = \infty$ , all the symbionts die instantaneously so the host dynamics reduces to the basic contact process

\begin{equation*} \begin{array}{r@{\quad}c@{\quad}l@{\quad}c@{\quad}r@{\quad}c@{\quad}l} 0 \ \to \ 1 & \hbox{at rate} & \lambda_{10} \,f_1 (x, \xi), & \quad & 1 \ \to \ 0 & \hbox{at rate} & 1. \end{array} \end{equation*}

In this case, there is a critical value $\lambda_{\rm c} \in (0, \infty)$ such that above $\lambda_{\rm c}$ the host population survives whereas at and below $\lambda_{\rm c}$ the population goes extinct [Reference Bezuidenhout and Grimmett1]. This is in qualitative agreement with the mean-field equations.

Assume from now on that the recovery rate is finite and let

\begin{equation*} \xi_t^1 (x) = {\mathbf{1}} \{\xi_t (x) \neq 0 \} \quad \hbox{and} \quad \xi_t^2 (x) = {\mathbf{1}} \{\xi_t (x) = 2 \} \qquad \hbox{for all } x \in {\mathbb{Z}}^d \end{equation*}

be the process that keeps track of the hosts and the process that keeps track of the hosts associated to a symbiont, respectively. The transitions of the process $\xi_t^1$ satisfy

\begin{equation*} \begin{array}{r@{\quad}c@{\quad}r} 0 \ \to \ 1 & \hbox{at rate at least} & \min \,(\lambda_{10}, \lambda_{20})\quad f_1 (x, \xi^1) , \\[5pt] 0 \ \to \ 1 & \hbox{at rate at most} & \max \,(\lambda_{10}, \lambda_{20})\quad f_1 (x, \xi^1) , \end{array} \end{equation*}

while $1 \to 0$ at rate one. In particular, this process can be compared to the basic contact process to deduce that, for all $x \in {\mathbb{Z}}^d$ and starting from a translation-invariant distribution with infinitely many 1s and 2s,

\begin{equation*} \begin{array}{r@{\quad}c@{\quad}l} \displaystyle \liminf_{t \to \infty} \,{\rm P} (\xi_t (x) \neq 0) \gt 0 & \hbox{when} & \min \,(\lambda_{10}, \lambda_{20}) \gt \lambda_{\rm c} , \\ \\[-5pt] \displaystyle \lim_{t \to \infty} \,{\rm P} (\xi_t (x) \neq 0) = 0 & \hbox{when} & \max \,(\lambda_{10}, \lambda_{20}) \leq \lambda_{\rm c}. \end{array} \end{equation*}

This follows from Theorem III.1.5 in [Reference Liggett11], which applies to general two-state spin systems, together with obvious inequalities relating the transition rates of our process and their counterpart for the basic contact process. This proves (4) and (5). Alternatively, one can prove these results by using a coupling argument. For instance, when $\lambda_{10} \geq \lambda_{20}$ , the contact process $\zeta_t^2$ with parameter $\lambda_{20}$ can be constructed from the graphical representation in Table 1 by assuming that births can only occur through type-2 arrows, while the contact process $\zeta_t^1$ with parameter $\lambda_{10}$ can be constructed by assuming that births occur through both type-1 and type-2 arrows. In both processes, particles are killed at the death marks $\times$ . Constructing the stacked contact process and these two contact processes from this common graphical representation results in a coupling such that

\begin{equation*} \{x \in {\mathbb{Z}}^d \,:\, \zeta_t^2 (x) = 1 \} \subset \{x \in {\mathbb{Z}}^d \,:\, \xi_t^1 (x) = 1 \} \subset \{x \in {\mathbb{Z}}^d \,:\, \zeta_t^1 (x) = 1 \} \end{equation*}

at all times t, provided this holds at time zero. This shows (4) and (5) when $\lambda_{10} \geq \lambda_{20}$ .

The analogous property when the inequality is reversed can be proved similarly by using another graphical representation which is designed based on the ordering of the parameters. Looking now at the second process $\xi_t^2$ , the transitions satisfy

\begin{equation*} \begin{array}{r@{\quad}c@{\quad}r} 0 \ \to \ 1 & \hbox{at rate at least} & \min \,(\lambda_{20}, \lambda_{21})\quad f_1 (x, \xi^2) , \\ \\[-5pt] 0 \ \to \ 1 & \hbox{at rate at most} & \max \,(\lambda_{20}, \lambda_{21})\quad f_1 (x, \xi^2) , \end{array} \end{equation*}

while $1 \to 0$ at rate $1 + \delta$ , from which it follows that, for all $x \in {\mathbb{Z}}^d$ and starting from a translation-invariant distribution with infinitely many 1s and 2s,

\begin{equation*} \begin{array}{r@{\quad}c@{\quad}l} \displaystyle \liminf_{t \to \infty} \,{\rm P} (\xi_t (x) = 2) \gt 0 & \hbox{when} & \min \,(\lambda_{20}, \lambda_{21}) \gt (1 + \delta) \,\lambda_{\rm c} , \\ \\[-5pt] \displaystyle \lim_{t \to \infty} \,{\rm P} (\xi_t (x) = 2) = 0 & \hbox{when} & \max \,(\lambda_{20}, \lambda_{21}) \leq (1 + \delta) \lambda_{\rm c}, \end{array} \end{equation*}

which shows (6) and (7). This can again be proved by using Theorem III.1.5 in [Reference Liggett11], or by coupling the process with the two contact processes with respective birth rates $\lambda_{20}$ and $\lambda_{21}$ and common death rate $1 + \delta$ using the graphical representation in Table 2 when $\lambda_{20} \geq \lambda_{21}$ , while the analogous property when the inequality is reversed can be proved by using another graphical representation which is designed based on the ordering of the parameters.

The four parameter regions in (4)–(7) are illustrated in the diagrams of Figures 1 and 2.

Proof of (8). Setting $\lambda_{21} = \delta = 0$ , the process reduces to the multitype contact process completely analyzed when the death rates are equal in [Reference Neuhauser13]. The transition rates become

\begin{equation*} \begin{array}{r@{\quad}c@{\quad}l@{\quad}c@{\quad}r@{\quad}c@{\quad}l} 0 \ \to \ 1 & \hbox{at rate} & \lambda_{10} \,f_1 (x, \xi) , & \quad & 1 \ \to \ 0 & \hbox{at rate} & 1 , \\ \\[-5pt] 0 \ \to \ 2 & \hbox{at rate} & \lambda_{20} \,f_2 (x, \xi) , & \quad & 2 \ \to \ 0 & \hbox{at rate} & 1. \end{array} \end{equation*}

In this case, the type with the larger birth rate outcompetes the other type provided its birth rate is also strictly larger than the critical value of the single-type contact process. This result was first proved in [Reference Neuhauser13] using duality techniques and again in [Reference Durrett and Neuhauser5] also using a block construction in two dimensions to prove that the long-term behavior of the process is not altered by small perturbations of the parameters. In particular, using a similar perturbation argument, it can be deduced from Propositions 3.1 and 3.2 in [Reference Durrett and Neuhauser5] that, for all $x \in {\mathbb{Z}}^2$ and regardless of the initial configuration,

\begin{equation*} \lim_{t \to \infty} \,{\rm P} (\xi_t (x) = 2) = 0 \quad \hbox{when} \quad \lambda_{10} \gt \lambda_{20} \ \hbox{and} \ \lambda_{21}, \delta \ \hbox{are small}. \end{equation*}

This proves property (8).

Proof of (9). The forest fire model, also referred to as epidemics with recovery, is the three-state spin system with a cyclic dynamics described by the following three transitions:

\begin{equation*} \begin{array}{r@{\quad}c@{\quad}l@{\quad}c@{\quad}r@{\quad}c@{\quad}l} 0 \ \to \ 2 & \hbox{at rate} & \alpha \,f_2 (x, \eta) , & \quad & 2 \ \to \ 1 & \hbox{at rate} & 1 , \\ \\[-5pt] 1 \ \to \ 0 & \hbox{at rate} & \beta. \end{array} \end{equation*}

The three states are interpreted as 0 = alive, 2 = on fire, and 1 = burnt, but can also be thought of respectively as healthy, infected, and immune in the context of epidemics. This process has been studied in [Reference Durrett and Neuhauser4], but note that we have interchanged the roles of the two states 1 and 2 to facilitate comparison with our model.

Denote the forest fire process by $\eta_t$ . The main result in [Reference Durrett and Neuhauser4] shows the existence of a critical value $\alpha_{\rm c} \in (0, \infty)$ such that, regardless of the value of $\beta \gt 0$ and starting from a translation-invariant distribution with infinitely many 1s and 2s in two dimensions,

\begin{equation*} \liminf_{t \to \infty} \,{\rm P} (\eta_t (x) = 2) \gt 0 \quad \hbox{when} \quad \alpha \gt \alpha_{\rm c}. \end{equation*}

Because the dynamics is cyclic, basic couplings between the forest fire model and our process do not lead to any useful stochastic ordering between the two systems. However, the proof in [Reference Durrett and Neuhauser4] easily extends to our process in a certain parameter region. Indeed, in addition to general geometrical properties and percolation results which are not related to the specific dynamics of the forest fire model, the key estimates in [Reference Durrett and Neuhauser4] rely on the following two ingredients:

1. (a) The set of burning trees dominates its counterpart in the process with no regrowth ( $\beta = 0$ ) provided both processes start from the same configuration.

2. (b) In regions that have not been on fire for at least S units of time, the set of trees which are alive dominates a product measure with density $1 - \e^{- \beta S}$ .

Now, fix $\delta \geq 0$ , let $\beta = 1 / (\delta + 1)$ , and consider the spin system on the two-dimensional integer lattice whose dynamics is described by the five transitions

\begin{equation*} \begin{array}{r@{\quad}c@{\quad}l@{\quad}c@{\quad}r@{\quad}c@{\quad}l} 0 \ \to \ 2 & \hbox{at rate} & \beta \lambda_{20} \,f_2 (x, \eta) , & \quad & 1 \ \to \ 0 & \hbox{at rate} & \beta , \\ \\[-6pt] 1 \ \to \ 2 & \hbox{at rate} & \beta \lambda_{21} \,f_2 (x, \eta) , & \quad & 2 \ \to \ 0 & \hbox{at rate} & \beta , \\ \\[-5pt] &&&& 2 \ \to \ 1 & \hbox{at rate} & \beta \delta. \end{array} \end{equation*}

Note that this is the process (1) with $\lambda_{10} = 0$ slowed down by the factor $\beta$ . Alternatively, one can see this process as the forest fire model modified so that burnt trees can catch fire ( $1 \to 2$ ) and trees on fire can spontaneously change to living trees ( $2 \to 0$ ), skipping the burnt phase. In this process, trees burn for an exponential amount of time with rate $\beta + \beta \delta = 1$ , as in the original forest fire model. It follows that the domination property (a) remains true: the set of burning trees in this new process dominates its counterpart in the forest fire model with no regrowth and in which the fire spreads by contact at rate $\alpha = \beta \lambda_{20}$ . Since the transition $1 \to 0$ again occurs spontaneously at rate $\beta$ , the domination of the product measure (b) remains true as well. In particular, starting from a translation-invariant distribution with infinitely many 1s and 2s,

\begin{equation*} \liminf_{t \to \infty} \,{\rm P} (\xi_t (x) = 2) \gt 0 \quad \hbox{when} \quad \lambda_{20} \gt (\delta + 1) \,\alpha_{\rm c}. \end{equation*}

This holds for all $\lambda_{21} \geq 0$ . Since the proof in [Reference Durrett and Neuhauser4] is based on a block construction, which supports small perturbations of the system, we also obtain coexistence in the process (1) under the same assumptions and provided $\lambda_{10}$ is sufficiently small. In conclusion, in $d = 2$ and starting from a translation-invariant distribution with infinitely many 1s and 2s,

\begin{equation*} \liminf_{t \to \infty} \,{\rm P} (\xi_t (x) = 2) \gt 0 \quad \hbox{when} \quad \lambda_{20} \gt (\delta + 1) \,\alpha_{\rm c} \ \hbox{and} \ \lambda_{10} \ \hbox{is small}. \end{equation*}

This proves (9), and completes the proof of Theorem 2.

3. Proof of Theorem 3

This section is devoted to the proof of (10) and (11), which together imply Theorem 3.

The first step is to obtain exponential bounds, in space-time, on the set of descendants (defined below in the natural way) of a type-2 individual, which is done in Proposition 3. To accomplish this we need two crucial observations, described in a moment, together with an iterative or ‘restart’ argument, and several estimates that build upon one another. We then show in Proposition 4 that, from any location, eventually the nearest type-2 individual will be at a distance which is exponentially far away as a function of time. This is then used to show that, from an initial configuration with infinitely many type-1 sites, at least one of them will produce a set of individuals of type 1 growing linearly in time, none of which ever interact with a type 2, completing the proof.

To obtain Proposition 3, the first crucial observation is the following asymmetry between sub- and supercritical contact processes. It is known that for a (single-type) contact process on ${\mathbb{Z}}$ with half-line initial condition $\xi_0(x) = {\mathbf{1}}(x \le 0)$ , and defining the right edge $r_t\,:\!= \sup\{x\,{:}\,\xi_t(x) \ne 0\}$ ,

(12) \begin{equation}\begin{array}{r@{\quad}c@{\quad}l@{\quad}l}\text{if} & \lambda>\lambda_c, & \text{eventually} & r_t \le Ct, \quad \text{while} \\ \\[-5pt] \text{if} & \lambda<\lambda_c, & \text{eventually} & r_t \le -e^{ct}, \end{array}\end{equation}

where $c,C>0$ depend only on the value of $\lambda$ in each case.

The case $\lambda>\lambda_c$ follows from Theorem 4 in [Reference Griffeath7], while the case $\lambda<\lambda_c$ is the content of the proof of Theorem 8 in [Reference Griffeath7]. If $\lambda>\lambda_c$ we actually have $r_t/t \to \alpha(\lambda)$ , but the above is the more pertinent fact here. In words, the invasion front of a supercritical contact process advances at most linearly, while the front for a subcritical contact process falls back exponentially fast. Thus, if, in our process, we begin with $\xi_0(x) = {\mathbf{1}}(x<0) + 2{\mathbf{1}}(x>0)$ , i.e. type 1 to the left of the origin and type 2 to the right, then if the right-hand boundary of type 1 and the left-hand boundary of type 2 do not meet within a short time, with high probability the two types will never interact, with the 2s vanishing rapidly while the 1s gradually advance. Naturally, this argument is also applicable if we start with a small patch of type 2s surrounded by type 1s.

The second observation is a comparison property that lets us reduce the study of the descendants of a type-2 site to the setting where there is a collection of 2s surrounded by 1s on either side. Namely, if in the initial configuration we replace all 2s with 1s, then the resulting process has at least as many occupied sites as it did before. Given $\xi_t$ with $\xi_0=\xi$ , if we define an auxiliary copy $\xi'$ on the same graphical representation, with initial configuration

\begin{equation*}\xi_0^{\prime} = {\mathbf{1}} \{\xi_0(y) \neq 0\},\end{equation*}

then since ${\lambda}_{20} \le {\lambda}_{10}$ it follows that

(13) \begin{equation} \hbox{for all} \ t \ge 0, \ \{x\,{:}\,\xi_t(x)\ne 0\} \subseteq \{x\,{:}\,\xi_t^{\prime}(x)=1\}. \end{equation}

Notice that the same is not true if we replace some but not all 2s with 1s, as can be seen by simple counterexamples. Next, we define descendant and ancestor. Suppose $\xi_s (x) = \xi_t (y) = i \ne 0$ for some x, y and $s \leq t$ . Then (y, t) is a descendant of (x, s), and (x, s) is an ancestor of (y, t), if either $(y,t)=(x,s)$ or if there are times and sites

\begin{equation*} s = t_0 \lt t_1 \lt \cdots < t_{k - 1} \leq t_k = t \quad \hbox{and} \quad x = x_1, x_2, \ldots, x_k = y \end{equation*}

such that the following two conditions hold:

1. • For $j = 1, 2, \ldots, k$ , we have $\xi_r (x_j) = i$ for all times $r \in [t_{j - 1}, t_j]$ .

2. • For $j = 1, 2, \ldots, k - 1$ , we have $\xi_{t_j^-} (x_{j + 1}) \neq \xi_{t_j} (x_{j + 1}) = i$ as a result of a birth or infection event along the edge $(x_j, x_{j + 1})$ at time $t_j$ .

For a set $S\subset {\mathbb{Z}}$ and $s\le t$ , let

\begin{equation*}A(s,t;\,S) = \{y\,:\,(y,t) \ \hbox{is a descendant of} \ (x,s) \ \hbox{for some} \ x \in S\}.\end{equation*}

Use the shorthand $A(s,t;\,x)$ for $A(s,t;\,\{x\})$ and $A_t(S)$ for $A(0,t;\,S)$ , and for $i=1,2$ and a configuration $\xi$ let $S_i(\xi) =\{x\,{:}\,\xi(x)=i\}$ . It follows from the definition of descendant that for $t\ge 0$ and $i=1,2$ ,

\begin{equation*}\{A(s,t;\,x) \colon x \in S_i(\xi_s)\} \quad \hbox{is a partition of} \quad S_i(\xi_t).\end{equation*}

Thus, to control $S_2(\xi_t)$ , which is our goal, it is enough to get good bounds on $A_t(x)$ for $x \in S_2(\xi_0)$ .

Figure 3.

Picture related to the proof of Proposition 3.

Given an interval S disjoint from $S_1(\xi_0)$ , let $A_t=A_t(S)$ and let $\ell_t = \inf A_t$ , $r_t = \sup A_t$ ,

\begin{equation*} a_t = \sup \,\{x < \ell_t \,:\, \xi_t (x) \neq 0 \} , \quad \hbox{and} \quad b_t = \inf\{x \gt r_t \,:\, \xi_t (x) \neq 0 \}. \end{equation*}

Also, let $d_t = \ell_t - a_t$ , $h_t = b_t - r_t$ , and $m_t = \min(d_t,h_t)$ . Figure 3 gives an illustration of these quantities. Notice that, since interactions are with nearest neighbors and $S\cap S_1(\xi_0)=\varnothing$ by assumption, it follows that $[\ell_t,r_t]\cap S_1(\xi_t) = \varnothing$ for $t\ge 0$ , and a fortiori that

(14) \begin{equation}A(s,t;\,[\ell_s,r_s]) = A_t \ \hbox{for} \ s\le t.\end{equation}

Time intervals when $m_t=1$ we call invasion, and when $m_t\ge 2$ we call struggle. The basic recipe for controlling $A_t$ is to control the duration and extent of each invasion, and to show that each struggle results, with positive probability, in the rapid and total collapse of $A_t$ .

We begin with struggle, which is the toughest to address; in fact, invasion will be surreptitiously taken care of in Proposition 1. In the next result we show that, starting from a fixed initial configuration $\xi_0$ , with positive probability, collapse of $A_t$ occurs before invasion, uniformly over finite intervals S and $\xi_0$ such that $S_1(\xi_0)$ is disjoint from S. In addition, we show that if collapse occurs, then it is exponentially fast. This makes use of the comparison property (13) as well as the asymmetry (12), and the fact that in the absence of invasion (that is, when $m_t>1$ ) the particles in $A_t$ do not interact with the particles outside $A_t$ . Using these estimates and an iterative restarting argument, we can then prove Proposition 1, which then easily leads to Propositions 2 and 3, at which point we have enough to tackle the proof of (10) and (11). In the following proofs, c and C are strictly positive constants such that the given statements hold for c and all smaller values than c, or C and all larger values than C, which will mean that $c, 1/C$ are allowed to decrease from step to step.

Lemma 1. There are $p,c,C \gt 0$ such that, if $S\subset {\mathbb{Z}}$ is a finite interval, $S \cap S_1(\xi_0) = \varnothing$ , and $m_0\ge 2$ , then letting $\tau=\inf\{t\colon m_t=1\}$ , ${\rm P}(\tau = \infty) \ge p$ and, for $t>0$ ,

\begin{equation*}\max({\rm P}(r_s \gt r_0 - \e^{cs} \ \hbox{for some} \ s \gt t \ \hbox{and} \ \tau = \infty), \ {\rm P}(t \lt \tau < \infty) )\leq C \,\e^{-ct}.\end{equation*}

Proof. Using a pair of independent substructures that we denote $\Omega_1,\Omega_2$ , define generalized stacked contact processes $\zeta_t^{1-}, \zeta_t^{1+}$ and $\zeta_t^2$ with the same parameters as $\xi_t$ and with initial configurations

\begin{align*} \zeta_0^{1-} (x) &= {\mathbf{1}}(\xi_0(x) \ne 0, x < \min S), \quad \zeta_0^{1+}(x) = {\mathbf{1}}(\xi_0(x) \ne 0, x \gt \max S) \quad \text{and} \ \\ \\[-5pt] \zeta_0^2 (x) &= 2 \,{\mathbf{1}} (\xi_0(x)=2, \ x \in S ). \end{align*}

Define $\zeta_t^1$ by $\zeta_t^1(x) = \max(\zeta_t^{1-}(x),\zeta_t^{1+}(x))$ , noting $\zeta_t^1$ is a stacked contact process with initial configuration ${\mathbf{1}}(\xi_0(x) \ne 0, x \notin S)$ , independent of $\zeta_t^2$ . For all t, $\zeta_t^1$ has no sites of type 2, and since $\delta=0$ , $\zeta_t^2$ has no sites of type 1. Let

\begin{equation*}\begin{array}{r@{\,}c@{\,}l@{\quad}r@{\,}c@{\,}l} \ell_t^{\prime} &=& \inf\{y\,:\,\zeta_t^2(y) \ne 0\}, & r_t^{\prime} &=& \sup\{y\,:\,\zeta_t^2(y) \ne 0\} , \quad\hbox{and}\\ \\[-5pt] a_t^{\prime} &=& \sup\{y\,:\,\zeta_t^{1-}(y) \ne 0\}, & b_t^{\prime} &=& \inf\{y\,{:}\,\zeta_t^{1+}(y) \ne 0\}.\end{array}\end{equation*}

Let $d_t^{\prime} = \ell_t^{\prime}-a_t^{\prime}$ , $h_t^{\prime} = b_t^{\prime}-r_t^{\prime}$ , and $m_t^{\prime}=\min(d_t^{\prime},h_t^{\prime})$ .

Since $A_t(S)$ and $A_t({\mathbb{Z}} \setminus S)$ are separated (have no adjacent sites) for $t<\tau$ , we can define $\xi_t$ as follows. For $t<\tau$ , use $\Omega_2$ and the initial configuration $\xi_0\, {\mathbf{1}}(S)$ to determine $\xi_t(x)$ for $x\in A_t(S)$ and use $\Omega_1$ and the initial configuration $\xi_0\,{\mathbf{1}}({\mathbb{Z}}\setminus S)$ to determine $\xi_t(x)$ for $x\notin A_t(S)$ . Then use $\xi_\tau$ and $\Omega_1$ to determine $\xi_t$ for $t\ge \tau$ . This furnishes a coupling of $\xi_t$ with $\zeta_t^{1-},\zeta_t^{1+},\zeta_t^2$ , and this coupling has the property that, for all $t \le \tau$ ,

\begin{equation*}A_t(S) = \{x \,:\, \zeta_t^2 (x) = 1 \}\quad \hbox{and} \quad A_t({\mathbb{Z}} \setminus S) \subseteq \{x \,:\, \zeta_t^1 (x) = 1 \} .\end{equation*}

The first statement is clear, while the second statement follows from (13). It follows in particular that $a_t \le a_t^{\prime}$ , $\ell_t \ge \ell_t^{\prime}$ , $r_t \le r_t^{\prime}$ , and $b_t \ge b_t^{\prime}$ , and thus $d_t \ge d_t^{\prime}$ , $h_t \ge h_t^{\prime}$ , and $m_t \ge m_t^{\prime}$ for $t\le \tau$ . To simplify matters we note that $\tau = \tau_{\ell} \wedge \tau_r$ , where

\begin{equation*}\tau_\ell = \inf\{t\,:\,d_t=1\} \quad \hbox{and} \quad \tau_r = \inf\{t\,:\,h_t=1\}.\end{equation*}

In the estimates that follow, c, C, and D are positive constants and $c, 1/C$ may get smaller from step to step. By the monotonicity of the contact process, the set of occupied sites $\{x \,:\, \zeta_t^{1-} (x) = 1 \}$ of $\zeta_t^{1-}$ is dominated by the pure birth process in which particles do not die and give birth onto neighboring sites at rate $\lambda_{10}$ , similarly for $\zeta_t^{1+}$ , and for $\zeta_t^2$ with $\lambda_{20}$ instead of $\lambda_{10}$ . In particular,

\begin{equation*} {\rm P} (b_t^{\prime} - r_t^{\prime} \lt n) \leq {\rm P} (h_0 - \mathrm{Poisson} \, ((\lambda_{10} + \lambda_{20}) \,t) < n) \qquad \hbox{for all } n \gt 0, \end{equation*}

and, applying a standard large deviations estimate, we get

(15) \begin{equation} {\rm P} (h_s^{\prime} < 2 \ \hbox{for some} \ s \leq h_0 / (2 \,(\lambda_{10} + \lambda_{20}))) \leq C \,\e^{-ch_0}.\end{equation}

Also, for each $t \gt 0$ ,

(16) \begin{equation} {\rm P} (b_s^{\prime} < b_0 - 2 \lambda_{10} \,t \ \hbox{for some} \ s \leq t) \leq \e^{-ct}.\end{equation}

To control $r_s^{\prime}$ , we use a known estimate at integer times, then a Poisson estimate at in-between times.

From (12) and the assumption $\lambda_{20} \lt \lambda_c$ , for $t>0$ ,

(17) \begin{equation} {\rm P} (r_t^{\prime} \gt r_0 - e^{ct}) \leq C \,\e^{-ct}.\end{equation}

In addition, since the displacement in one unit of time is dominated by a Poisson random variable with parameter $\lambda_{20}$ , for any integer $k \geq 0$ we have

(18) \begin{equation} {\rm P} (r_s^{\prime} - r_n^{\prime} \gt k \ \hbox{for some} \ s \in [n, n + 1]) \leq C \,\e^{-ck}.\end{equation}

Combining (17) with $t=n$ and (18), we deduce that

(19) \begin{equation} {\rm P} (r_s^{\prime} \gt r_0 - \e^{cn} + n \ \hbox{for some} \ s \in [n, n + 1]) \leq C \,\e^{-cn} .\end{equation}

Then, combining with (16) evaluated at $t = n + 1$ , for each integer $n \geq 1$ ,

(20) \begin{equation} {\rm P} (h_s^{\prime} < h_0 + \e^{cn} - (1 + 2 \,\lambda_{10})(n + 1) \ \hbox{for some} \ s \in [n, n + 1]) \leq C \,\e^{-cn}. \end{equation}

To deduce the first estimate we distinguish two cases, where $D>0$ is a large enough constant.

Case 1: $m_0 \gt D$ For one side of the argument, $h_0 \gt D$ suffices; an analogous argument applies to the other side assuming $d_0>D$ . The bound on $h_s^{\prime}$ in (20) is at least two for all n. Recalling that $h_t \ge h_t^{\prime}$ for $t\le\tau$ , then combining (15) with (20) summed over $n \geq \lfloor h_0 / (2 \,(\lambda_{10} + \lambda_{20})) \rfloor$ ,

(21) \begin{equation} \begin{array}{r@{\quad}c@{\quad}l} {\rm P}(\tau_r < \infty) &=& {\rm P}(\inf_{t \ge 0}h_t \le 1) \le {\rm P} (\inf_{t \ge 0}h_t^{\prime} \le 1) \\ \\[-5pt] & \leq & C\e^{-ch_0} + \sum_{n \geq ch_0} \,C \,\e^{-cn} \leq C \,\e^{-ch_0} \qquad \hbox{when } h_0 \gt D, \end{array}\end{equation}

where c, C do not depend on D. By reflection invariance, the same holds for $\tau_\ell$ . If D is large enough that $C\e^{-cD}<1/2$ we find that ${\rm P}(\tau = \infty) \ge 1-2C\e^{-cD} = {\varepsilon}$ for some ${\varepsilon}>0$ .

Case 2: $m_0 \le D$ Given S, let E denote the event where, in one unit of time, there is a death at every site in $[\inf S-D,\inf S-1]\cup [\sup S+1,\sup S+D]$ and no birth onto any vertex in the same set. Since the birth rate onto any vertex is at most $2{\lambda}_{10}$ and the death rate at any site is 1,

\begin{equation*}{\rm P}(E) \ge (1-\e^{-1})^{4D}(\e^{-2{\lambda}_{10}})^{4D} = \delta>0,\end{equation*}

with $\delta$ depending on D but not on S. Note that on E, $\tau>1$ and $m_1>D$ . Using the Markov property and the previous result,

\begin{equation*}{\rm P}(\tau=\infty) \ge {\rm P}(\tau>1 \ \hbox{and} \ m_1 \gt D){\rm P}(\tau=\infty \mid \tau>1 \ \hbox{and} \ m_1>D) \ge \delta {\varepsilon},\end{equation*}

which gives the first statement with $p=\delta{\varepsilon}>0$ .

Now, in (19) above, for $n \geq n_0$ for some $n_0$ , absorb n into $-\e^{cn}$ by decreasing c, then increase C to account for $n \lt n_0$ . Then, for any $t \gt 0$ , summing (19) over $n \geq \lfloor t \rfloor$ ,

\begin{equation*} {\rm P} (r_s^{\prime} \gt r_0^{\prime} - \e^{-cs} \ \hbox{for some} \ s \gt t) \leq C \,\e^{-ct}.\end{equation*}

On the event $\{\tau = \infty \}$ , we have $r_t \le r_t^{\prime}$ for all $t \geq 0$ , and the second statement follows. Using the two bounds (16) and (17) above, and noting that $m_0\ge 2$ ,

\begin{equation*} {\rm P} (h_t^{\prime} < 2 + \e^{ct} - 2 \,\lambda_{10} \,t) \leq C \,\e^{-ct} \end{equation*}

and for t large enough, $2 + \e^{ct} - 2 \, \lambda_{10} \, t \ge t$ . Since $h_t \ge h_t^{\prime}$ for $t\le \tau$ , it follows that

\begin{equation*} {\rm P}( h_t \lt t, \ t<\tau) \le C \e^{-ct},\end{equation*}

and an analogous estimate applies to $d_t$ . For t large enough, using the above and (21),

\begin{equation*} \begin{array}{r@{\,}c@{\,}l} {\rm P} (t \lt \tau \lt \infty) & = & {\rm P}(m_t \lt t, \ t<\tau<\infty) + {\rm P}(m_t \ge t, \ t<\tau<\infty) \\ \\[-5pt] &\le& {\rm P}(m_t \lt t, \ t<\tau) + {\rm P}(t<\tau<\infty \mid m_t \ge t)\\ \\[-5pt] &\le& {\rm P}(h_t \lt t, \ t<\tau) + {\rm P}(d_t<t, \ t<\tau) \\ \\[-5pt] &+& {\rm P}(t<\tau_r<\infty \mid m_t \ge t) + {\rm P}(t<\tau_\ell<\infty \mid m_t \ge t)\\ \\[-5pt] &\le& 4C \e^{-ct}. \end{array} \end{equation*}

This completes the proof.

Lemma 2. Let $T=\inf\{t\,:\,m_t \ge 2\}$ . There are positive constants c,c such that, if $S\subset {\mathbb{Z}}$ is a finite interval and $S\cap S_1(\xi_0)=\varnothing$ , then

\begin{equation*}{\rm P}( T >t) \le C\e^{-ct}.\end{equation*}

Proof. Let $s_0=0$ and $s_1,s_2,\dots$ denote the times when either death occurs at $\ell_t-1$ or $r_t+1$ , or infection occurs across either the edge $(\ell_t,\ell_t-1)$ or $(r_t,r_t+1)$ . Then $T \le s_{2K}$ where

\begin{equation*}K = \inf\{k\,:\, \hbox{death occurs at } (\ell_t-1,s_{2k-1}) \hbox{ and } (r_t+1,s_{2k})\}.\end{equation*}

Clearly, $K \preceq \mathrm{Geometric \,}((1/2{\lambda}_{21})^2)$ and $\{s_{k+1}-s_k \colon k\ge 0\} \preceq \{\sigma_k \colon k\ge 0\}$ , an independent and identically distributed sequence of $\exp(2)$ random variables. A routine estimate gives the result.

Next, we show the position of the rightmost pathogen in any $A_t$ goes to $- \infty$ exponentially fast. This is the analog of the second estimate in Lemma 1 but dropping the condition $\tau=\infty$ . This result is then used in the subsequent lemma to show that the probability that the rightmost pathogen moves n steps to the right of its initial position decays exponentially with n.

Proposition 1. There are positive constants c,C such that, if $S\subset {\mathbb{Z}}$ is a finite interval and $S\cap S_1(\xi_0)=\varnothing$ , then

\begin{equation*} {\rm P}(r_t \gt r_0 - \e^{ct}) \leq C \,\e^{-ct}. \end{equation*}

Proof. Define recursively the two sequences of stopping times $(\tau_i)_{i\ge 0}$ and $(T_i)_{i\ge 0}$ by $\tau_0=0$ , $T_0 = \inf\{t \ge 0\,{:}\,m_t=2\}$ , and, recursively for $i\ge 1$ ,

\begin{equation*}\begin{array}{r@{\,\,}c@{\,\,}l}\tau_i &=& \inf \,\{t \gt T_{i-1} \,:\, m_t = 1 \}, \\ \\[-5pt] T_i &=& \inf \,\{t \gt \tau_i \,:\, m_t = 2 \},\end{array}\end{equation*}

with the convention $\inf \varnothing = \infty$ . Let $N = \inf\{i\,{:}\,\tau_i=\infty\}$ . Recursively at each time $\tau_i$ , applying the strong Markov property and noting (14), then applying the first result of Lemma 1 with $S = [\ell_{\tau_i},r_{\tau_i}]$ we find that

\begin{equation*} N \preceq - 1 + \mathrm{Geometric \,} (p) , \end{equation*}

where $\preceq$ means stochastically smaller. Doing the same, but applying the third result of Lemma 1,

\begin{equation*}\{(\tau_i - T_{i-1}) \,{\mathbf{1}}(N \geq i) \colon i>0\} \preceq \{\sigma_i \,{\mathbf{1}} (N \geq i ) \colon i>0\},\end{equation*}

where $\sigma_1,\sigma_2,\ldots$ is a sequence of independent, identically distributed random variables independent of the random variable N and such that

\begin{equation*} {\rm P} (\sigma_i \gt t) = \min \,(C \e^{-ct}, 1) \qquad \hbox{for all } t \gt 0. \end{equation*}

Using Lemma 2, the same holds for $\{(T_i-T_{i-1}) {\mathbf{1}} (N \ge i) \colon i>0\}$ . Letting $n = \lceil t /4 {\rm E} [\sigma_i]\rceil$ , and applying a large deviations bound,

(22) \begin{equation}\begin{array}{r@{\,}c@{\,}l} {\rm P} (\tau_N \gt t) &\le& {\rm P}((\tau_N-T_{N-1}) + \cdots + (\tau_1 - T_0) \gt t/2) \\ \\[-5pt] &&+\ {\rm P}((T_{N-1}-\tau_{N-1}) + \cdots + (T_0 - \tau_0) \gt t/2)\\ \\[-5pt] & \leq & 2{\rm P} (\sigma_1 + \cdots + \sigma_N \gt t/2) \\ \\[-5pt] & \leq & 2{\rm P} (N \gt n) + 2{\rm P} (\sigma_1 + \cdots + \sigma_n \gt 2 n \,{\rm E} [\sigma_i]) \leq 2C \,\e^{-cn} \leq 2C \,\e^{-ct}. \end{array} \end{equation}

We may assume that $\lambda_{20} \leq \lambda_{21}$ , since otherwise $A_t$ is dominated by a contact process with infection rate $\lambda_{20}<\lambda_c$ and initial sites S, and the result then follows directly from (12). Comparing the set of sites in state 2 to a pure birth process with no deaths and with birth to adjacent sites at rate $\lambda_{21}$ , a large deviations estimate gives $c, C \gt 0$ such that

(23) \begin{equation} {\rm P} (r_t \gt r_0 + 2 \,\lambda_{21} \,t) \leq C \,\e^{-ct} \qquad \hbox{for all } t \gt 0.\end{equation}

For any $c \gt 0$ , there is $t_0$ such that $\lambda_{21} \,t \lt \e^{ct} - \e^{ct/2}$ for all $t \gt t_0$ , in which case

(24) \begin{equation} \begin{array}{r@{\,}c@{\,}l} {\rm P} (r_t \gt r_0 - \e^{ct/2}) & \leq & {\rm P} (r_{t/2} \gt r_0 + \lambda_{21} \,t) \\ \\[-5pt] & & \hspace*{10pt} + \ {\rm P} (r_t \gt r_0 - \e^{ct/2} \ \hbox{and} \ r_{t/2} \leq r_0 + \lambda_{21} \,t) \\ \\[-5pt] & \leq & {\rm P} (r_{t/2} \gt r_0 + \lambda_{21} \,t) + {\rm P} (r_t \gt r_{t/2} - \lambda_{21} \,t - \e^{ct/2}) \\ \\[-5pt] & \leq & {\rm P} (r_{t/2} \gt r_0 + \lambda_{21} \,t) + {\rm P} (r_t \gt r_{t/2} - \e^{ct}) \end{array}\end{equation}

for all $t \gt t_0$ . On the other hand,

(25) \begin{equation} {\rm P} (r_t \gt r_{t/2} - \e^{ct}) \le {\rm P} (\tau_N \gt t/2) + {\rm P}(r_t \gt r_{t/2}- \e^{ct}, \ \tau_N \le t/2) .\end{equation}

Letting $\tau(t) = \inf\{s>t\,:\,m_s=1\}$ , the event $\tau_N \le t/2$ is equivalent to $\tau(t/2)=\infty$ . Applying the second result of Lemma 1 with $S=[\ell_{t/2},r_{t/2}]$ we find that

\begin{equation*}{\rm P}(r_t \gt r_{t/2} - \e^{ct} \ \hbox{and} \ \tau(t/2)=\infty) \le C\e^{-ct}.\end{equation*}

Combining with (22)–(25) gives the desired estimate when $t \gt t_0$ . If $t\leq t_0$ then, after increasing C if necessary, the estimate holds for all values of t.

Proposition 2. There is $C \gt 0$ such that, if $S\subset {\mathbb{Z}}$ is a finite interval and $S \cap S_1(\xi_0) = \varnothing$ , then

\begin{equation*} {\rm P} (r_t \gt r_0 + n \ \hbox{for some} \ t \geq 0) \leq C \,\e^{-cn}.\end{equation*}

Proof. Comparing to a pure birth process as above,

\begin{equation*} {\rm P} (r_s \gt r_0 + n \ \hbox{for some} \ s \leq m_0) \leq C \,\e^{-cn} \quad \hbox{for} \quad m_0 = \lfloor n / 2 \lambda_{21} \rfloor. \end{equation*}

If $n \ge n_0 = \sup_{m\ge 0}- \e^{cm} + m$ , then using Proposition 3 and large deviations for the Poisson distribution with parameter $\lambda_{21}$ ,

\begin{equation*} \begin{array}{l} {\rm P} (r_s \gt r_0 + n \ \hbox{for some} \ s \in [m, m + 1]) \\ \\[-5pt] \hspace*{18pt} \leq \ {\rm P} (r_m \gt r_0 - \e^{-cm}) + {\rm P} (r_s \gt r_m + m \ \hbox{for some} \ s \in [m, m + 1]) \leq C \,\e^{-cm}. \end{array} \end{equation*}

Summing over $m \geq m_0$ gives the desired estimate for $n\ge n_0$ ; for $n<n_0$ increase C if necessary.

We can now show that the set of descendants is exponentially bounded in both space and time.

Proposition 3. There are $c, C \gt 0$ such that, if $x \in {\mathbb{Z}}$ and $\xi_0(x)=2$ ,

\begin{equation*} {\rm P}(A_t(x) \neq \varnothing) \leq C \,\e^{-ct} \quad \hbox{and} \quad {{\rm P}(A_t(x) \nsubseteq [x - n, x + n] \ \hbox{for some} \ t \geq 0) \leq C \,\e^{-cn}.} \end{equation*}

Proof. Defining $a_t$ , $\ell_t$ , etc. with $S=\{x\}$ , $\ell_0=r_0=x$ , and $A_t(x)\ne\varnothing$ is equivalent to $\ell_t \le r_t$ . Using Proposition 1, reflection invariance, and a union bound,

\begin{equation*}\begin{array}{r@{\,}c@{\,}l}{\rm P}(\ell_t \gt r_t) &\ge& {\rm P}(\ell_t \ge \ell_0 + \e^{ct} \ \hbox{and} \ r_t \le r_0-\e^{ct}) \\ \\[-5pt] &\ge & 1 - ({\rm P}(\ell_t < \ell_0+\e^{ct}) + {\rm P}(r_t \gt r_0-\e^{ct}) \ge 1-2C\e^{-ct},\end{array}\end{equation*}

and the first result follows by taking the complement. The second result follows in the same way, except using Proposition 2.

Next, we use Proposition 3 to prove (10), which says that in an exponentially growing neighborhood of any site, eventually there are no sites in state 2. We also record an exponential estimate. Note the change in the definition of $\ell_t^x,r_t^x$ .

Proposition 4. For a site $x \in {\mathbb{Z}}$ , let

\begin{equation*} \ell_t^x = \sup \,\{y \leq x \,:\, \xi_t(y) = 2 \} \quad \hbox{and} \quad r_t^x = \inf \,\{y \geq x \,:\, \xi_t (y) = 2 \}. \end{equation*}

Then, there exist $c, C \gt 0$ such that, for any $\xi_0$ and $t_0$ ,

\begin{equation*} {\rm P}(\ell_t^x \gt x - \e^{ct} \ \hbox{or} \ r_t^x < x + \e^{ct} \ \hbox{for some} \ t \gt t_0) \leq C \,\e^{-ct_0}. \end{equation*}

Also, there exists $c \gt 0$ such that, for any $\xi_0$ and any site x,

\begin{equation*} {\rm P}(\sup \,\{t \,:\, \xi_t (y) = 2 \ \hbox{for some} \ y \ \hbox{such that} \ |y - x| \leq \e^{ct} \} < \infty) = 1. \end{equation*}

Proof. Throughout this proof, y refers to a site which is initially in state 2. Let c, C be two constants as in Proposition 3, so that for all y, $\xi$ , and n,

\begin{equation*} \max({\rm P}(A_n(y) \neq \varnothing),{\rm P}(A_t(y) \nsubseteq [x-n,x+n] \ \text{for some} \ t\ge 0)) \leq C \,\e^{-cn}. \end{equation*}

Using a union bound over $y \in [x - \e^{cn/2}, x + \e^{cn/2}]$ and that $A_t(y) = \varnothing$ is an absorbing property,

(26) \begin{equation}\begin{array}{l} {\rm P} (A_t(y) \neq \varnothing \ \hbox{for some} \ y \ \hbox{such that} \ \xi_0 (y) = 2 \\ \\[-5pt] \hspace*{80pt} \hbox{and} \ |y - x| \leq \e^{cn/2} \ \hbox{and some} \ t \geq n) \leq C \,\e^{-cn/2}. \end{array}\end{equation}

Let $n_0$ be such that $n \geq n_0$ implies $\e^{cn/2} - \e^{c (n + 1)/4} \gt n$ . If $|y - x| = \lceil \e^{cn/2} \rceil + m$ for integer $m \geq 0$ and any $n \geq n_0$ , then, using the second half of Proposition 3,

\begin{equation*} {\rm P} (\inf_t \min \{|z-x| \colon z \in A_t(y)\} \leq \e^{c (n + 1)/4}) \leq C \,\e^{-c (n + m)}. \end{equation*}

Taking C larger if necessary makes the previous inequality also true for all $n \lt n_0$ . Then, taking a union bound and increasing C at the last step gives

(27) \begin{equation} \begin{array}{l} {\rm P} (\inf_t \min \{|z-x|\colon z \in A_t(y)\} \leq \e^{c (n + 1)/4} \ \hbox{for some} \ y \\ \\[-5pt] \hspace*{50pt} \hbox{such that} \ |y - x| \gt \e^{cn/2} \ \hbox{and} \ \xi_0 (y) = 2) \leq {C (1 - \e^{-c}) \,\e^{-cn} \leq C \,\e^{-cn}}. \end{array} \end{equation}

Combining (26) and (27) gives

\begin{equation*} {\rm P} (\xi_t (y) = 2 \ \hbox{for some} \ y \in [x - \e^{ct/4}, x + \e^{ct/4}] \ \hbox{and} \ t \in [n, n + 1]) \leq C \,\e^{-cn/2}. \end{equation*}

Given $t_0$ , summing over $n \geq \lfloor t_0 \rfloor$ gives the first statement. Summing over all n and using the Borel–Cantelli lemma finishes the proof.

In what follows, a mark refers to a Poisson point in the graphical representation. Marks are named by their effect on the target site, so, for example, a $0 \to 1$ mark is an edge mark at some time t along a directed edge (x, y) such that if $\xi_{t^-} (x) = 1$ and $\xi_{t^-} (y) = 0$ then $\xi_t (y) = 1$ . Also, a death mark refers to a $\star \to 0$ event, while a birth mark is a $0 \to \star$ event, where $\star \neq 0$ . Note that since the graphical representation consists of at most a countably infinite number of Poisson point processes, with probability one, no two marks occur at the same time.

Next, we use Proposition 4 to produce an interval that grows linearly in time and is devoid of pathogens.

Lemma 3. For any $\mu \gt 0$ , there are $c, C \gt 0$ such that, for all integer $n>0$ ,

\begin{equation*} {\rm P} (\xi_t (x) = 2 \ \hbox{for some} \ |x| \leq n / 2 + \mu t) \leq C \,\e^{-cn} \end{equation*}

for all $\xi_0$ such that $\xi_0(x) \neq 2$ for all $|x| \leq n$ .

Proof. Let $y = n + m$ with $m \gt 0$ ; a similar argument applies to $-y$ . Using Proposition 3 gives the existence of constants $c, C \gt 0$ such that, for all ${\varepsilon} \gt 0$ ,

\begin{equation*} \begin{array}{l} {\rm P} (A_{{\varepsilon} y}^y \neq \varnothing \ \hbox{or} \ \inf_t A_t^y < 2y/3) \leq C \,\e^{-c {\varepsilon} y}. \end{array} \end{equation*}

On the complement of the above event, the descendants of (y, 0) are contained in

\begin{equation*} \Lambda \,:\!= \{(x, t) \,:\, x \geq 2y/3 \ \hbox{and} \ 0 \leq t \leq {\varepsilon} y \}. \end{equation*}

A quick sketch (see Figure 4) shows that the rectangle $\Lambda$ is disjoint from the set

\begin{equation*} \{(x, t) \,:\, t \geq 0 \ \hbox{and} \ |x| \leq n / 2 + \mu t \} \end{equation*}

provided the top left corner of $\Lambda$ lies to the right of the line $x = n / 2 + \mu t$ , which is the condition

\begin{equation*} n / 2 + \mu {\varepsilon} y < 2y / 3. \end{equation*}

Since $n \leq y$ , this condition is satisfied if ${\varepsilon} \lt 1/ 6 \mu$ . Summing over $m \gt 0$ for both $y = n + m$ and $y = - n - m$ then gives the desired result.

Figure 4.

Picture related to the proof of Lemma 3.

A related notion to the descendants is the cluster, that we need only define for type 1, as follows. Suppose $\xi_s (x) = \xi_t (y) = 1$ for some x, y and $s \leq t$ . Then, we say that (y, t) belongs to the cluster of (x, s) if there are times and sites

\begin{equation*} s = t_0 \lt t_1 \lt \cdots < t_{k - 1} \leq t_k = t \quad \hbox{and} \quad x = x_1, x_2, \ldots, x_k = y \end{equation*}

such that the following two conditions hold:

1. • For $j = 1, 2, \ldots, k$ , we have $\xi_r (x_j) = 1$ for all times $r \in [t_{j - 1}, t_j]$ .

2. • For $j = 1, 2, \ldots, k - 1$ , there is a $0 \to 1$ birth mark along the edge $(x_j, x_{j + 1})$ at time $t_j$ .

In contrast to the definition of descendants, it is permitted to have $\xi_{t_j^-} (x_{j + 1}) = 1$ .

If $\xi_s (x) = 1$ then, for $t \geq s$ , let $B_t (x, s)$ denote the cluster of (x, s) at time t, that is,

\begin{equation*} B_t (x, s) \,:\!= \{y \in {\mathbb{Z}} \,:\, (y, t) \ \hbox{is in the cluster of} \ (x, s) \}, \end{equation*}

and denote it $B_t (x)$ for $s = 0$ . Again, since interactions are nearest-neighbor,

\begin{equation*} \xi_t(y) \neq 2 \quad \hbox{for all} \quad y \in [\inf B_t (x, s), \sup B_t (x, s)] \quad \hbox{and} \quad t \geq s. \end{equation*}

As a warm-up to (11), we prove the following.

Lemma 4. Suppose $\xi_0(0)=1$ . Let $\ell_t = \inf B_t (0)$ and $r_t = \sup B_t (0)$ , and let

\begin{equation*} \tau = \inf \,\{t \gt 0 \,:\, \xi_t (\ell_t - 1) = 2, \ \xi_t (r_t + 1) = 2 , \ \text{or} \ B_t = \varnothing\} .\end{equation*}

Then, there are $p, c, C \gt 0$ such that

\begin{equation*} {\rm P}(\tau = \infty) \geq p \quad \hbox{and} \quad {\rm P} (t \lt \tau < \infty) \leq C \,\e^{-ct} \end{equation*}

uniformly over $\xi_0$ such that $\xi_0(0) = 1$ .

Proof. Define a pair of independent copies $\xi_t^1$ and $\xi_t^2$ of the generalized stacked contact process with initial configurations

\begin{equation*} \xi_0^1 (y) = \xi_0 (y) \,{\mathbf{1}} \{y = 0 \} \quad \hbox{and} \quad \xi_0^2 (y) = \xi_0 (y) \,{\mathbf{1}} \{y \neq 0 \}. \end{equation*}

Also define

\begin{equation*} \begin{array}{r@{\quad}c@{\quad}l} \ell_t^1 = \inf \,\{x \,:\, \xi_t^1 (x) = 1 \} & \hbox{and} & r_t^1 = \sup \,\{x \,:\, \xi_t^1 (x) = 1 \} , \\ \\[-5pt] a_t^2 = \sup \,\{x < \ell_t^1 \,:\, \xi_t^2 (x) = 2 \} & \hbox{and} & b_t^2 = \inf \,\{x \gt r_t^1 \,:\, \xi_t^2 (x) = 2 \} , \end{array} \end{equation*}

so that $\tau$ can also be expressed as

\begin{equation*} \tau = \inf \,\{t \gt 0 \,:\, \ell_t^1 - a_t^2 \leq 1, \ b_t^2 - r_t^1 \leq 1 , \ \text{or} \ \xi_t^1 \equiv 0\}. \end{equation*}

To show that $\tau = \infty$ with positive probability, first we fix n and consider the case $\min (|a_0^2|, |b_0^2|) \geq n$ . Using large deviations estimates for the Poisson distribution, we can show that

\begin{equation*} {\rm P} (\max (|\ell_t^1|, |r_t^1|) \geq 2 \lambda_{10} \,t + n / 2 - 1 \ \hbox{for some} \ t \gt 0) \leq C \,\e^{-cn}. \end{equation*}

To do so, it suffices to first make an estimate for $t \leq m_0 \,:\!= \lfloor n / 4 \lambda_{10} \rfloor$ , then for $t \in [m, m + 1]$ for each $m \geq m_0$ , then to take a union bound. Then, taking $\mu = 2 \lambda_{10}$ and using Lemma 3, for integer $n>0$ we have

\begin{equation*} {\rm P} (\min (|a_t^2|, |b_t^2|) \leq 2 \lambda_{10} \,t + n/2 \ \hbox{for some} \ t \gt 0) \leq C \,\e^{-cn}. \end{equation*}

Since $\lambda_{10}>\lambda_c$ by assumption, $q\,:\!={\rm P}(\xi_t^1 \not\equiv 0 \ \text{for all } t)>0$ . Moreover, if $\xi_t^1 \not\equiv 0 \ \text{for all } t$ and $\min \,(|a_t^2|, |b_t^2|) \gt \max (|\ell_t^1|, |r_t^1|) \ \hbox{for all} \ t \gt 0$ then $\tau=\infty$ , so taking n large enough that $2C\e^{-cn}<q/2$ , we find that if $\xi (x) \neq 2$ for $|x| \leq n$ then

\begin{equation*} {\rm P} (\tau = \infty) \geq q-q/2 = q/2. \end{equation*}

For $\xi$ such that $\xi (0) = 1$ , the probability

\begin{equation*} {\rm P} (\xi_1 (0) = 1 \ \hbox{and} \ \xi_1 (x) = 0 \ \hbox{for all} \ 0 < |x| \leq n) \end{equation*}

is at least the probability that, on the time interval [0, 1], there are no birth marks along edges touching $[\!-n, n]$ , there is no death mark at 0, and there is a death mark at every x with $0 \lt |x| \leq n$ , and this probability is at least $2p/q$ for some $p \gt 0$ . Using the Markov property and the estimate on $\tau = \infty$ in the previous case then gives ${\rm P} (\tau = \infty) \geq (2p/q)(q/2)=p \gt 0$ as desired.

To deduce the estimate on ${\rm P} (t \lt \tau \lt \infty)$ , we note that

\begin{equation*} {\rm P} (\max (|\ell_s^1|, |r_s^1|) \geq 2 \lambda_{10} \,s \ \hbox{for some} \ s \geq t) \leq C \,\e^{-ct}, \end{equation*}

which can be proved by applying an estimate at each integer time $n \gt t$ and summing over n. Then, combining with the first statement in Proposition 3 and noting that

\begin{equation*} t \lt \tau < \infty \quad \hbox{implies that} \quad \max (|\ell_s^1|, |r_s^1|) \geq \min (|a_s^2|, |b_s^2|) - 1 \ \hbox{for some} \ s \gt t, \end{equation*}

we deduce the estimate on ${\rm P} (t \lt \tau \lt \infty)$ .

We are now ready to establish (11), which states the existence of a linearly growing region starting from a random space-time point in which the process agrees with the contact process with parameter $\lambda_{10}$ . This will also complete the proof of Theorem 2.

Proof of (11). Given (x, s), for $t \geq s$ recall that $B_t (x, s)$ denotes the cluster of (x, s) at time t. Let $\tau_0 = 0$ , and $x_0$ be any site with $\xi (x_0) = 1$ . Without loss of generality, suppose that the set $\{x \gt 0 \,:\, \xi (x) = 1 \}$ is infinite, and define $x_i$ and $\tau_i$ recursively for $\lambda_{21} \lt \infty$ by letting

\begin{equation*} \begin{array}{r@{\,}c@{\,}l} \ell_t^i & = & \inf \,B_t (x_i, \tau_i) , \\ \\[-5pt] r_t^i & = & \sup \,B_t (x_i, \tau_i) , \\ \\[-5pt] \tau_{i + 1} & = & \inf \,\{t \gt \tau_i \,:\, \xi_t (\ell_t^i - 1) = 2 \ \hbox{or} \ \xi_t (r_t^i + 1) = 2 \} , \\ \\[-5pt] x_{i + 1} & = & \inf \,\{x \gt x_i \,:\, \xi_{\tau_{i + 1}} (x) = 1 \}, \end{array} \end{equation*}

with the value of $x_i$ being unimportant if $\tau_i = \infty$ . Note that if time $\tau_i \lt \infty$ then site $x_{i + 1}$ is well defined due to the fact that

\begin{equation*} \{\xi \,:\, \xi (x) = 1 \ \hbox{for infinitely many} \ x \gt 0 \} \end{equation*}

is an invariant set for the dynamics. Let $N = \sup \,\{i \,:\, \tau_i \lt \infty \}$ . Applying the strong Markov property and using Lemma 4, we obtain that N is at most geometric with parameter p. In addition, by the second part of Lemma 4, for $i = 0, 1, 2, \ldots$ ,

\begin{equation*} \tau_{i + 1} - \tau_i \preceq T_i \quad \hbox{where} \quad {\rm P} (T_i \gt t) \leq \max \,(1, C \e^{-ct}) \end{equation*}

and the random variables $T_i$ are independent. In particular, $\tau_N$ is almost surely finite. Let $T = \tau_N$ , $X = x_N$ , $\ell_t = \ell_t^N$ , and $r_t = r_t^N$ .

Recall that $\zeta_t$ denotes the process with initial configuration $\zeta_0 (x) = 1$ for all x. Since $\lambda_{10} \gt \lambda_{20}$ by assumption, a straightforward coupling argument shows that, for any configuration $\xi_0$ ,

\begin{equation*} \{x \in {\mathbb{Z}} \,:\, \xi_t (x) \neq 0 \} \subseteq \{x \in {\mathbb{Z}} \,:\, \zeta (x) = 1 \}. \end{equation*}

Therefore, $\zeta_{\tau_i} (x_i) = 1$ whenever $\tau_i \lt \infty$ . By definition of time $\tau_{i + 1}$ , the set $B_t(x_i,\tau_i)$ is the set of infected sites in a (single-type) contact process started from the single infected site $x_i$ at time $\tau_i$ , so a coupling found in [Reference Durrett3] shows that if $\tau_i \lt \infty$ then

\begin{equation*} \xi_t (x) = \zeta_t (x) \ \hbox{for all} \ x \in [\ell_t^i, r_t^i] \ \hbox{and all} \ \tau_i \lt t < \tau_{i + 1}. \end{equation*}

In [Reference Durrett3] it is shown that, for the contact process, on the event of single-site survival, $-\ell_t^i/ t$ and $r_t^i / t \to \alpha \gt 0$ , so the same is true here provided $\tau_{i + 1} = \infty$ , which is the case for $i = N$ . The proof is now complete.