2.1. Results for the threshold-one contact process

Denote by $\xi^{\lambda,\mathbb{T}_d}_t$ the threshold-one contact process on $\mathbb{T}_d$ . Observe that the system is increasing with respect to $\lambda$ , additive, and attractive with respect to the natural order of configurations, and that $\delta_{\textbf{0}}$ is the only absorbing state/trivial invariant distribution. Recalling the definitions of the different critical values, it is easy to see that $\lambda_2\left(d\right)\ge \max\left\{\lambda_1\left(d\right),\lambda_{\rm e}\left(d\right)\right\}$ . In Section 3 we will see that $\tilde \xi^{\lambda,\mathbb{T}_d}_t$ is the time reversal of $\xi^{\lambda,\mathbb{T}_d}_t$ , and thus $\tilde \xi^{\lambda,\mathbb{T}_d}_t$ survives from a finite population if and only if $\xi^{\lambda,\mathbb{T}_d}_t$ survives. Unlike the ordinary contact process which is self dual, it is clear that $\tilde \xi^{\lambda,\mathbb{T}_d}_t$ is not identically distributed as $\xi^{\lambda,\mathbb{T}_d}_t$ . So we first prove:

With Theorem 1, we will no longer discriminate between survival and survival from a finite population, and from now on we denote such a condition by survival. The following two results give sufficient conditions for $\xi^{\lambda,\mathbb{T}_d}_t$ to survive:

Note that the condition of survival in Theorem 2 is not necessarily sharp for each given d, although it has been shown in [Reference Xue23] to be asymptotically sharp as $d\to\infty$ . In fact, when $d=2$ , $\xi^{\lambda,\mathbb{T}_d}_t$ survives for substantially smaller $\lambda$ :

The following theorem gives an almost complete picture on the complete convergence theorem of the threshold-one contact process on $\mathbb{T}_d$ . In words, the complete convergence theorem holds if and only if $\xi^{\lambda,\mathbb{T}_d}_t$ dies out or survives strongly. When the process survives weakly, we can prove the convergence theorem for any translation-invariant $\mu$ or Dirac measure concentrating on the dense configuration. Recall the definition of dense configuration in [Reference Griffeath8] such that $\xi$ is dense if it satisfies

(2.1) \begin{align} &\text{there exists } N>0 \text{ such that, for any } x\in\mathbb{T}_d,\nonumber\\[2pt] &\quad \text{there exists } y\in B_x\left(N\right) \text{such that } \xi\left(y\right)=1 , \end{align}

where $B_x\left(N\right)$ is the ball of radius N centered at x. Note that the Dirac measure $\delta_\xi$ when $\xi$ is dense does not have to be translation invariant, and under the translation-invariant measure, the configuration $\xi$ does not have to be dense with probability one, so we need to give these two cases separately.

2.2. Results for the threshold-one voter model with positive spontaneous death

Denote by $\eta_t^{\varepsilon,\mathbb{T}_d}$ the threshold-one voter model with positive spontaneous death rate $\varepsilon$ on $\mathbb{T}_d$ , where when $\varepsilon=0$ it is the threshold-one voter model. Observe that $\eta_t^{\varepsilon,\mathbb{T}_d}$ is monotone with respect to $\varepsilon$ and is attractive with respect to the natural order of configurations, so that with the definition of the three types of survivals, we can define three critical values with respect to $\varepsilon$ :

\begin{equation*}\begin{aligned}&\varepsilon_1\left(d\right)=\sup\big\{\varepsilon\,{:}\,\text{$\eta^{\varepsilon,\mathbb{T}_d}_t$ survives}\big\}, \\[2pt] &\varepsilon_{\rm e}\left(d\right)=\sup\big\{\varepsilon\,{:}\,\text{$\eta^{\varepsilon,\mathbb{T}_d}_t$ survives from a finite population}\big\}, \\[2pt] &\varepsilon_2\left(d\right)=\sup\big\{\varepsilon\,{:}\,\text{$\eta^{\varepsilon,\mathbb{T}_d}_t$ survives strongly}\big\}.\end{aligned}\end{equation*}

Like the threshold-one contact process, we have $\varepsilon_2\left(d\right)\le \max\left\{\varepsilon_1\left(d\right),\varepsilon_{\rm e}\left(d\right)\right\}$ . Because the dual process is not attractive, there is no clear indication to our knowledge whether $\varepsilon_1\left(d\right)=\varepsilon_{\rm e}\left(d\right)$ . Since $\eta^{\varepsilon,\mathbb{T}_d}_t$ can dominate the threshold-one contact process $\xi^{1,\mathbb{T}_d}_t$ , by Theorem 2 and Theorem 3, we can get that for any $d\ge 2$ , $\varepsilon_{\rm e}\left(d\right)>0$ and $\varepsilon_1\left(d\right)>0$ , and by Remark 2, one can prove that $\varepsilon_2\left(d\right)>0$ when $d\ge 5$ . Next, for the the threshold-one voter model with positive spontaneous death, we also have the corresponding complete convergence theorem.

Next, we will consider the threshold-one voter model on $\mathbb{T}_d$ ; we denote the process by $\zeta^{\mathbb{T}_d}_{t}$ . Since $\zeta^{\mathbb{T}_d}_{t}$ can dominate the threshold-one contact process with parameter 1, by Theorems 2 and 3 and [Reference Liggett14, Proposition 2.11], we have:

Define $\beta\left(d\right)=\lim\limits_{n\to\infty}\left[\mathbb{P}^o\left(x_n\in \zeta_t^{\mathbb{T}_d}\text{ for some }t\right)\right]^{1/n}$ . Then $\beta\left(d\right)$ exists by [Reference Liggett14, Theorem B22]. Like the definition in (2.1), we define the configuration to be doubly dense if it satisfies

(2.3) \begin{align} &\text{there exists } N>0 \text{ such that, for any } x\in\mathbb{T}_d, \nonumber\\[3pt] & \quad \text{ there exist } y,z\in B_x\left(N\right) \text{ such that } \zeta\left(y\right)=1,\ \zeta\left(z\right)=0.\end{align}

3.1. Preliminaries

To precisely define the particle system, let $\mathbb{T}_d$ be the homogeneous tree where each vertex has $d+1$ neighbors. One can order $\mathbb{T}_d$ such that, for each vertex x, it has exactly one parent, denoted as $\overleftarrow{x}$ , and d children. At the same time, one can also label all the d children of each vertex by 1 to d according to this order. If a vertex x can be traced back to another vertex y through a path going up in the ‘family tree’, we call x the ancestor and y the descendant. For vertices $x,y\in \mathbb{T}_d$ , we write $x\sim y$ if y is a neighbor of x, and we use $|x-y|$ to denote the distance between x and y on $\mathbb{T}_d$ ; in particular, when $x=o$ , we denote the distance between y and o as $|y|$ .

According to [Reference Liggett14, Part 1, Section 4], one may embed $\mathbb{Z}$ into $\mathbb{T}_d$ by assigning consecutive integers to a doubly infinite connected lineage within $\mathbb{T}_d$ such that, for vertex x and y which are parent and child, we always have $l(y)=l(x)+1$ , where l(x) is the integer number assigned to vertex x. Thus, one can also see l(x) as the generation number of vertex x along this lineage, and denote by o the vertex with $l(x)=0$ . Also, for each $k\in \mathbb{Z}$ , denote by $x_k$ the vertex on this lineage with $l(x_k)=k$ . With generation numbers defined along one specific lineage, one can now discriminate all other vertices with the following finite sequence of integers: for each point $z\in \mathbb{T}_d$ , define $n_z\in \mathbb{Z}^+\cup \{0\}$ such that $x_{-n_z}$ is the most recent common ancestor of o and z. In particular, z is a descendent of o if and only if $n_z=0$ . If $z\not= x_{-n_z}$ , note that there is a unique simple path $\vec y$ from $x_{-n_z}$ to z denoted by $x_{-n_z}=y(0), y(1),\ldots, y(m_z)=z$ . Now define $\big(r(1),r(2),\ldots, r(m_z)\big)\in \{1,2,\ldots, d\}^{m_z}$ such that, for each i, y(i) is the r(i)th descendent of $y(i-1)$ . From the definitions above, one can see that each vertex $z\in \mathbb{T}_d$ now has a unique correspondence with $\big(n_z, r(1),r(2),\ldots, r(m_z)\big)$ , while $|z|=n_z+m_z$ . One may note that the discussion above also gives an explicit construction of the infinite homogeneous tree $\mathbb{T}_d$ .

For any $A\subseteq \mathbb{T}_d$ , denote by $\xi_A$ the configuration in $\{0,1\}^{T_d}$ such that $\xi_A(x)=1$ if and only if $x\in A$ .

3.2. Coalescing duality

The graphical representation for the threshold-one contact process is shown in [Reference Liggett11, Chapter I, Section 6]. Since the threshold-one contact process is additive, it has coalescing duality. Now we construct the threshold-one contact process and its dual process as follows.

For any $x\in \mathbb{T}_d$ , let $\big\{T_n^{x,\mathcal{N}_x}\big\}$ and $\{T_n^{x,\emptyset}\}$ be two independent Poisson flows with rate $\lambda$ and 1 respectively. Let $\mathcal{N}_x=\{y\,{:}\,|y- x|\le 1\}$ be the neighborhood of x. At each event time s of $T_n^{x,\mathcal{N}_x}$ we draw an arrow from y to x for every $y\in \mathcal{N}_x\backslash\{x\}$ , which can be thought of as the particle at y giving birth at x at time s. And at each event time s of $T_n^{x,\emptyset}$ put a $\delta$ at (x, s) where $\delta$ represents deaths. We say that there is an open path from (x, 0) to (y, t) if there is a pair of sequences $x_0=x,\dots,x_n=y$ and $s_0=0<s_1<\cdots< s_n<s_{n+1}=t$ such that:

1. • for $1\leq m\leq n$ there is an arrow from $x_{m-1}$ to $x_{m}$ at time $s_m$ ;

2. • for $0\leq m\leq n$ there are no $\delta$ s in $\{x_m\}\times(s_{m},s_{m+1})$ .

To get the threshold-one contact process, for any $A\subset \mathbb{T}_d$ , let

\begin{equation*}\xi_t^{\lambda,A,\mathbb{T}_d}(x)=\textrm{1}_{\{\text{there is an open path from (\textit{z}, 0) to (\textit{x}, \textit{t}) for some $z\in A$}\}}.\end{equation*}

To get the dual process up to time t, reverse the directions of the arrows and time in the graph representation before time t: for any $s\leq t$ , for any $B\subset \mathbb{T}_d$ , let

\begin{equation*}\tilde{\xi}^{\lambda,B,\mathbb{T}_d}_{t,s}(x)=\textrm{1}_{\{\text{there is an open path from (\textit{z}, \textit{t}) to $(x,t-s)$ for some $z\in B$}\}}.\end{equation*}

Since for any $s\leq t_1\leq t_2$ , $\tilde{\xi}^{\lambda,B,\mathbb{T}_d}_{t_1,s}$ and $\tilde{\xi}^{\lambda,B,\mathbb{T}_d}_{t_2,s}$ have the same distribution, we can define the dual process $\tilde{\xi}^{\lambda,B,\mathbb{T}_d}_{t}\triangleq\tilde{\xi}^{\lambda,B,\mathbb{T}_d}_{t,0}$ . Note that $\xi_t^{\lambda,A,\mathbb{T}_d}$ has the infinitesimal generator $\tilde\Omega^\lambda$ as follows: for any local function f,

\begin{equation*}\Omega^\lambda f(\xi)=\sum_{x\in \mathbb{T}_d} \Bigg(\textrm{1}_{\{\xi\left(x\right)=1\}}[\,f\left(\xi^x\right)-f\left(\xi\right)]+ \lambda\textrm{1}_{\Big\{\xi\left(x\right)=0,\sum\limits_{y\sim x}\xi\left(y\right)\ge 1\Big\}} [\,f\left(\xi^x\right)-f\left(\xi\right)]\Bigg) ,\end{equation*}

where $\xi^x$ has the same state as $\xi$ at any vertex except x. Also, $\tilde{\xi}^{\lambda,B,\mathbb{T}_d}_{t}(x)$ has the infinitesimal generator $\tilde\Omega^\lambda$ as follows: for any local function f,

\begin{equation*}\tilde\Omega^\lambda f(\xi)=\sum_{x\in \mathbb{T}_d}\left( \textrm{1}_{\{\xi(x)=1\}}[\,f\left(\xi^ x\right)-f\left(\xi\right)]+ \lambda [\,f\left(\xi\cup \xi_{\mathcal{N}_x}\right)-f\left(\xi\right)]\right) ,\end{equation*}

where $\xi\cup\eta=\xi\vee\eta$ .

Let $\mathbb{P}^{\xi_A}(\cdot)$ and $\tilde{\mathbb{P}}^{\xi_B}(\cdot)$ denote the distribution of ${\xi}^{\lambda,A,\mathbb{T}_d}$ and $\tilde{\xi}^{\lambda,B,\mathbb{T}_d}$ respectively, and $\mathbb{E}^{\xi_A}(\cdot)$ and $\tilde{ \mathbb{E}}^{\xi_B}(\cdot)$ denote the corresponding expectations. When $A=\{x\}$ is a singleton, we abbreviate $\mathbb{P}^{\xi_{\{x\}}}$ , $\mathbb{E}^{\xi_{\{x\}}}$ , and ${\xi}^{\lambda,A,\mathbb{T}_d}$ to $\mathbb{P}^x$ , $\mathbb{E}^x$ , and ${\xi}^{\lambda,x,\mathbb{T}_d}$ for convenience, and we sometimes omit A in ${\xi}^{\lambda,A,\mathbb{T}_d}$ . From the construction, we can get the following important dual relation:

(3.1) \begin{equation}\begin{aligned} \mathbb{P}^{\xi_A} \big(\xi^{\lambda,\mathbb{T}_d}_t\cap\xi_B\ne \textbf{0}\big)=\tilde{\mathbb{P}}^{\xi_B} \big( \tilde{\xi}^{\lambda,\mathbb{T}_d}_t\cap \xi_A\ne\textbf{0} \big) \qquad \text{for all}\ A, B \subset \mathbb{T}_d , \end{aligned}\end{equation}

where $\xi\cap\eta=\xi\land\eta$ .

The graph representation is a useful tool to get the monotonicity of $\xi_t^{\lambda,\mathbb{T}_d}$ in $\lambda$ and the initial distribution such that:

1. • For any $\lambda_{(1)}\leq\lambda_{(2)}$ , regarding the Poisson arrows with parameter $\lambda_{(2)}$ as superpositions of independent Poisson arrows with parameter $\lambda_{(1)}$ and $\lambda_{(2)}-\lambda_{(1)}$ , we can couple $\xi_t^{\lambda_{(1)},\mathbb{T}_d}$ and $\xi_t^{\lambda_{(2)},\mathbb{T}_d}$ on the common graph representation, so that

   \begin{equation*}\begin{aligned} \xi_t^{\lambda_{(1)},\mathbb{T}_d}\subseteq \xi_t^{\lambda_{(2)},\mathbb{T}_d}\end{aligned}\end{equation*}
   for any t when both starting from A.

2. • For any finite subsets $A\subseteq B$ , we can couple ${\xi}^{\lambda,A,\mathbb{T}_d}$ and ${\xi}^{\lambda,B,\mathbb{T}_d}$ with the same $\lambda$ such that

   \begin{equation*}\begin{aligned} \xi_t^{\lambda,A,\mathbb{T}_d}\subseteq \xi_t^{\lambda,B,\mathbb{T}_d}\end{aligned}\end{equation*}
   for any t by using the same Poisson flows on a common graph representation.

The same results also hold for the dual process. Moreover, $\xi^{\lambda,\mathbb{T}_d}_t$ and $\tilde \xi^{\lambda,\mathbb{T}_d}_t$ are additive in the sense that

\begin{equation*}\xi^{\lambda,A,\mathbb{T}_d}_t=\cup_{x\in A}\xi^{\lambda,x,\mathbb{T}_d}_t, \qquad \tilde \xi^{\lambda,A,\mathbb{T}_d}_t=\cup_{x\in A}\tilde \xi^{\lambda,x,\mathbb{T}_d}_t.\end{equation*}

Let $A=\mathbb{T}_d$ in (3.1); we obtain

\begin{equation*}\tilde{\mathbb{P}}^{\xi_B}\big(\tilde{\xi}^{\lambda,\mathbb{T}_d}_t\ne\textbf{0}\big)=\delta_\textbf{1}S^\lambda(t)(\xi\,{:}\,\xi\cap \xi_B\ne\textbf{0}).\end{equation*}

Thus, the limit of $\delta_\textbf{1}S^\lambda(t)$ exists; denote it as $\bar{\nu}^\lambda$ , such that

(3.2) \begin{equation}\begin{aligned}\tilde{\mathbb{P}}^{\xi_B}\big(\tilde{\xi}^{\lambda,\mathbb{T}_d}_t\ne\textbf{0}\text{ for all}\ t\big)=\bar{\nu}^\lambda(\xi\,{:}\,\xi\cap \xi_B\ne\textbf{0})\end{aligned}\end{equation}

for any B.

3.3. Annihilating duality

Since the threshold-one voter model is not additive, it has no coalescing duality. However, it has another important duality, the annihilating duality (see [Reference Liggett11, Chapter 3.4]), which was introduced in [Reference Cox and Durrett4].

For any $x\in \mathbb{T}_d$ , let $\{T_n^{x,S_x}\}_{S_x\subseteq \mathcal{N}_x}$ and $\{T_n^{x,\emptyset}\}$ be independent Poisson processes with rate $\beta(S_x)$ and $\varepsilon$ respectively, where

\begin{equation*} \beta(S_x)= \left \{ \begin{array}{l@{\quad}l} \frac{1}{2^{d}}, \quad &S_x\subset\mathcal{N}_x \text{ and $|S_x|\ge 2$ is even,}\\[3pt] 0,\quad & \text{otherwise}. \end{array}\right.\\\end{equation*}

At any event time $t\in\{T_n^{x,S_x}\}$ draw an arrow from y to x for every $y\in S_x$ , and if $x\in S_x$ put a $\delta$ at (x, t), while if $t\in\{T_n^{x,\emptyset}\}$ , put a $\delta$ at (x, t).

Let $\eta_t^{\varepsilon,A,\mathbb{T}_d}(x)=\textrm{1}_{\{\text{there are an odd number of paths from}\ (z,\ 0)\ to\ (x,\ t)\ \text{for some }\ z\ \in\ A \}}$ . Reverse the direction of the arrows, let

\begin{equation*}\begin{aligned} \tilde{\eta}_{t,s}^{\varepsilon,B,\mathbb{T}_d}(y)=\textrm{1}_{\{\text{there are an odd number of paths from (\textit{z}, \textit{t}) to (\textit{y}, \textit{s}) for some $z\in B$}\}}.\end{aligned}\end{equation*}

Then the dual process starting from $\eta_B$ can be defined as $\tilde{\eta}_{t,0}^{\varepsilon,B,\mathbb{T}_d}$ . From the above argument we can define the infinitesimal generator of $\eta_t^{\varepsilon,A,\mathbb{T}_d}$ as: for any local function f,

\begin{equation*}\Omega^\varepsilon f(\eta)=\sum_{x\in \mathbb{T}_d}\Bigg(\sum_{S_x} \beta\left(S_x\right)[\,f\left(\eta\triangle \eta_{S_x}\right)-f\left(\eta\right)]+\varepsilon\textrm{1}_{\{\eta(x)=1\}}\big[\,f\left(\eta^x\right)-f\left(\eta\right)\big]\Bigg) ,\end{equation*}

where $\eta\triangle \eta_{S_x}(y)=1$ if and only if $\eta(y)[1-\eta_{S_x}(y)]=1$ or $[1-\eta(y)]\eta_{S_x}(y)=1$ $\eta$ being odd means that there are an odd number of 1s in $\eta$ .

Also, the infinitesimal generator of $\tilde\eta_t^{\varepsilon,A,\mathbb{T}_d}$ can be defined as: for any local function f,

\begin{equation*}\tilde\Omega^\varepsilon f(\eta)=\sum_{x\in \mathbb{T}_d}\Bigg(\sum_{S_x} \beta\left(S_x\right)[\,f\left(\eta\triangle \eta_{S_x}\right)-f\left(\eta\right)]+\varepsilon\textrm{1}_{\{\eta\left(x\right)=1\}}\big[\,f\left(\eta^x\right)-f\left(\eta\right)\big]\Bigg).\end{equation*}

The threshold-one voter model is the special case when $\varepsilon=0$ .

From the construction above, ${\eta}^{\varepsilon,\mathbb{T}_d}_t$ and $\tilde{\eta}^{\varepsilon,\mathbb{T}_d}_t$ can be seen as annihilating branching processes in the sense that when two particles meet, they annihilate. Thus, we get the duality relation:

(3.3) \begin{equation}\begin{aligned}\mathbb{P}^{\eta_A}\big( \eta_t^{\varepsilon,\mathbb{T}_d}\cap \eta_B\text{ is odd} \big)=\tilde{\mathbb{P}}^{\eta_B}\big( \tilde{\eta}_t^{\varepsilon,\mathbb{T}_d}\cap \eta_A\text{ is odd} \big) \qquad \text{for all}\ A, B.\end{aligned}\end{equation}

4.1. Equivalence of survivals

In this subsection we extend the arguments in [Reference Liggett14, Part I, Section 4] to $\xi^{\lambda,\mathbb{T}_d}_t$ and $\tilde \xi^{\lambda,\mathbb{T}_d}_t$ with $\rho=1$ and prove Theorem 1. The proofs of Lemmas 1 and 2 are parallel, with no significant difference from those in [Reference Liggett14] since all three processes $\xi^{\lambda,\mathbb{T}_d}_t$ , $\tilde \xi^{\lambda,\mathbb{T}_d}_t$ , and the ordinary contact process are attractive and additive spin systems, so we omit the proofs here.

Lemma 1. (Adaption of Proposition 4.27(b), [Reference Liggett14])

(4.1) \begin{equation}\begin{aligned}\Phi\left(\lambda\right)=\lim\limits_{t\to\infty}\mathbb{E}^o|\xi^{\lambda,\mathbb{T}_d}_t|^{1/t}\end{aligned}\end{equation}

exists, and there exists $ C<\infty$ such that

\begin{equation*}\begin{aligned} \left[\Phi\left(\lambda\right)\right]^t\leq \mathbb{E}^o|\xi^{\lambda,\mathbb{T}_d}_t|\leq C\left[\Phi\left(\lambda\right)\right]^t.\end{aligned}\end{equation*}

Lemma 2. (Adaption of Proposition 4.39, [Reference Liggett14])

(4.2) \begin{equation}\begin{aligned}&\lambda_{\rm e}\left(d\right)=\sup\{\lambda\,{:}\,\Phi\left(\lambda\right)\leq1\},\\[2pt] &\tilde{\lambda}_{\rm e}\left(d\right)=\sup\{\lambda\,{:}\,\tilde{\Phi}\left(\lambda\right)\leq1\}.\end{aligned}\end{equation}

Moreover, $\xi^{\lambda,\mathbb{T}_d}_t$ and $\tilde\xi^{\lambda,\mathbb{T}_d}_t$ die out at the first critical value.

Proof of Theorem 1. By duality, it suffices to prove $\lambda_{\rm e}\left(d\right)=\tilde{\lambda}_{\rm e}\left(d\right)$ . When starting from a single vertex $o$ , it is easy to calculate that

(4.3) \begin{equation}\begin{aligned}\mathbb{E}^o|\xi^{\lambda,\mathbb{T}_d}_t|=\sum\limits_{n=0}^{\infty}\left(d+1\right)d^n\mathbb{P}^o\big(x_n\in \xi^{\lambda,\mathbb{T}_d}_t\big)=\sum\limits_{n=0}^{\infty}\left(d+1\right)d^n\mathbb{P}^o\big(x_n\in \tilde{\xi}^{\lambda,\mathbb{T}_d}_t\big)=\mathbb{E}^o|\tilde{\xi}^{\lambda,\mathbb{T}_d}_t|.\end{aligned}\end{equation}

By the definition of $\Phi$ in (4.1), $\Phi\left(\lambda\right)=\tilde{\Phi}\left(\lambda\right)$ . Thus, it follows from (4.2) that $\lambda_{\rm e}\left(d\right)=\tilde{\lambda}_{\rm e}\left(d\right)$ . Moreover, by Lemma 2, both $\xi^{\lambda,\mathbb{T}_d}_t$ and $\tilde{\xi}^{\lambda,\mathbb{T}_d}_t$ die out at the first critical value, so $\xi^{\lambda,\mathbb{T}_d}_t$ neither survives nor survives from a finite population at $\lambda_{\rm e}\left(d\right)$ by duality.

Remark 3. Define another two critical values:

\begin{align*}&\lambda_{\rm T}=\sup\left\{\lambda\,{:}\left(\int_0^\infty\mathbb{E}^o|\xi^{\lambda,\mathbb{T}_d}_t|\,{\rm d} t\right)<\infty\right\},\\[3pt] &\tilde\lambda_{\rm T}=\sup\left\{\lambda\,{:}\left(\int_0^\infty\mathbb{E}^o|\tilde\xi^{\lambda,\mathbb{T}_d}_t|\,{\rm d} t\right)<\infty\right\}.\end{align*}

Then, by (4.3), we get $\lambda_{\rm T}=\tilde\lambda_{\rm T}$ . Once we have

(4.4) \begin{equation}\begin{aligned}\lambda_{\rm e}=\lambda_{\rm T}, \qquad \tilde\lambda_{\rm e}=\tilde\lambda_{\rm T},\end{aligned}\end{equation}

Theorem 1 holds immediately. For any transitive graph with bounded degree, [Reference Aizenman and Jung1] showed that for the ordinary contact process $\hat\lambda_{\rm e}=\hat\lambda_{\rm T}$ , where $\hat\lambda_{\rm e}, \hat\lambda_{\rm T} $ represent the corresponding critical values for the ordinary contact process. Applying their proof to the threshold-one contact process and its dual process, we should also be able to obtain (4.4). Thus, Theorem 1 holds for any transitive graph with bounded degree, such as $\mathbb{Z}^d$ .

4.2. Conditions for survival

In this subsection we will give conditions for the dual process $\tilde\xi^{\lambda_d,\mathbb{T}_d}_t$ to survive; as a result, we can get an upper bound for the critical value $\lambda_1\left(d\right)$ .

Proof of Theorem 2. For any non-empty $A\in Y$ , define $g\left(A\right)=|A|$ . Then, for $\lambda_d=\left(d-1\right)^{-1}$ and our process $\xi^{\lambda_d,\mathbb{T}_d}_t$ on $\mathbb{T}_d$ , we can similarly define

\begin{equation*}\begin{aligned}h^{\lambda_d}\left(A\right)&=\frac{{\rm d}}{{\rm d} t}\mathbb{E}^{\xi_A} \left[g\left(\xi^{\lambda_d,\mathbb{T}_d}_t\right)\right]\Big|_{t=0}\\[2pt] &=\lambda_d |\partial A|-|A|\\[2pt] &\ge \frac{1}{d-1}\left[\left(d-1\right)|A|+2\right]-|A|\\[2pt] &=\frac{2}{d-1}.\end{aligned}\end{equation*}

The third inequality comes from the fact that $|\partial A|\ge\left(d-1\right)|A|+2$ , which can be proved by induction. First, when $|A|=1$ , $|\partial A|=d+1=\left(d-1\right)|A|+2$ . Then we assume that the inequality is true for all $|A|\le n-1$ . We come to the case when $|A|=n$ . Assume $x\in A$ has the maximum generation, i.e. $m_{x}-n_{x}=\max\left\{m_y-n_y\,{:}\,y\in A\right\}$ . If there is more than one such point, we choose the ‘smallest’ one according to the lexicographical order of $\left(n_z, r\left(1\right),r\left(2\right),\ldots, r\left(m_z\right)\right)$ . It follows that

\begin{equation*} |\partial A|\ge |\partial \left(A\cup\left\{x\right\}\right)|-1+d ,\end{equation*}

where the inequality holds if and only if $\overleftarrow x \in A$ . Hence, by the inductive hypothesis we have

\begin{align*} |\partial A|&\ge |\partial \left(A\cup\left\{x\right\}\right)|-1+d \\[2pt] &\ge\left(d-1\right)\left(n-1\right)+2+d-1 \\[2pt] &=n\left(d-1\right)+2 \\[2pt] &=\left(d-1\right)|A|+2. \end{align*}

Now the fact is true for all A satisfying $|A|=n$ and thus for all finite subsets A. From the transition rates above, one can see that the stochastic process $|\xi^{\lambda_d,\mathbb{T}_d}_t|$ dominates a Markov process $\hat A^d_t$ on $\mathbb{Z}^+\cup\left\{0\right\}$ with 0 as absorbing state and transition rates as follows: for each $n\in \mathbb{Z}^+$ ,

\begin{equation*}\left\{\begin{aligned}&n\rightarrow n+1 \text{ at rate } n+\tfrac{2}{d-1},\\&n\rightarrow n-1 \text{ at rate } n.\\\end{aligned}\right.\end{equation*}

Note that in Lemma 2 we have proved that survival of $A^{\lambda_{d},\mathbb{T}_d}_t$ is equivalent to $\Phi\left(\lambda\right)>1$ . Define $\mathbb{E}^n$ as the expectation of a random walk starting from n. By Lemmas 1 and 2 it suffices to prove that $\lim\limits_{t\to\infty}\mathbb{E}^o\left|A^{\lambda_d,\mathbb{T}_d}_t\right|= \infty$ , which, by the discussion above, is immediate if one can show that

(4.5) \begin{equation}\begin{aligned}\lim_{t\to\infty}\mathbb{E}^1\left|\hat A^{d}_t\right|= \infty.\end{aligned}\end{equation}

Again, note that for $\hat A^d_t$ , its infinitesimal mean is given by

\begin{equation*}\mu^d\left(n\right)=\frac{{\rm d}}{{\rm d} t} \mathbb{E}^n \hat A^d_t\Big|_{t=0}=\frac{2}{d-1}\textrm{1}_{\left\{n\not=0\right\}}.\end{equation*}

Recalling that 0 is absorbing for $\hat A^d_t$ , for stopping time $\hat \tau_0=\inf\big\{t\,{:}\,\hat A^d_t=0\big\}$ and any $t\in \left(0,\infty\right)$ , by Dynkin’s formula,

\begin{equation*}\begin{aligned}\mathbb{E}^1 \hat A^d_t= 1+\mathbb{E}^1\left[\int_0^t \frac{2}{d-1}\textrm{1}_{\left\{\hat \tau_0>s\right\}}\,{\rm d} s \right]=1+ \frac{2}{d-1}\int_0^t \mathbb{P}^1\big(\hat \tau_0>s\big)\,{\rm d} s.\end{aligned}\end{equation*}

Note that $\int_0^\infty \mathbb{P}^1\big(\hat \tau_0>s\big)\,{\rm d} s=\mathbb{E}^1\left[\hat \tau_0\right]$ , thus in order to prove (4.5), it is sufficient to prove that $\mathbb{E}^1\left[\hat \tau_0\right]=\infty$ .

Moreover, note that $\hat A^d_t$ further dominates $S_t$ , a (inhomogeneously) rescaled continuous-time simple random walk with 0 as absorbing state and transition rates as follows: for each $n\ge 1$ ,

\begin{equation*}\frac{{\rm d}}{{\rm d} t} \mathbb{P}^n\left(S_t=n+1\right)\Big|_{t=0}=\frac{{\rm d}}{{\rm d} t} \mathbb{P}^n\left(S_t=n-1\right)\Big|_{t=0}=n.\end{equation*}

Let $\tau_0=\inf\left\{t\,{:}\,S_t=0\right\}$ . Thus, we only need to prove that $\mathbb{E}^1\left[\tau_0\right]=\infty$ . Now define the following sequence of stopping times for $S_t$ :

\begin{equation*}\tau_n=\inf\left\{t\ge 0\,{:}\,S_t=n\right\} \qquad \text{for all}\ n\ge 1,\end{equation*}

with the convention $\inf\emptyset=\infty$ , and disjoint events $A_n=\left\{\tau_n<\infty, \ \tau_{n+1}=\infty\right\}$ . Then

\begin{equation*}\begin{aligned}\mathbb{E}^1\left[\tau_0\right]=\sum_{n=1}^\infty \mathbb{E}^1\left[\tau_0\textrm{1}_{A_n}\right].\end{aligned}\end{equation*}

Now, for each sufficiently large n, note that on the event $A_n$ , $\tau_n<\tau_0<\tau_{n+1}=\infty$ , which implies that

(4.6) \begin{equation}\begin{aligned}\mathbb{E}^1\left[\tau_0\textrm{1}_{A_n}\right]\ge \mathbb{E}^1\left[\tau_n\textrm{1}_{A_n}\right].\end{aligned}\end{equation}

Also, by the strong Markov property,

(4.7) \begin{equation}\begin{aligned}\mathbb{E}^1\left[\tau_n\textrm{1}_{A_n}\right]=\mathbb{E}^1\left[\tau_n\textrm{1}_{\left\{\tau_n<\infty\right\}}\right]\mathbb{P}^n\left(\tau_0<\tau_{n+1}\right)=\frac{1}{n+1} \mathbb{E}^1\left[\tau_n\textrm{1}_{\left\{\tau_n<\infty\right\}}\right],\end{aligned}\end{equation}

where the last equality is a result that the embedded chain of $S_t$ is a simple random walk with 0 as absorbing state.

Now define $Y_n$ as the number of transitions within $\left[0,\tau_n\wedge \tau_0\right]$ , and let $X_n$ , $n\ge 0$ , be a discrete simple random walk on $\mathbb{Z}$ with stopping time

\begin{equation*}T_n=\min\left\{k\,{:}\,X_k=n\right\} \qquad \text{for all}\ n\ge 0.\end{equation*}

Then, again by the fact that the embedded chain of $S_t$ is a simple random walk with 0 as absorbing state, we have, for all $m\ge 1$ ,

(4.8) \begin{equation}\begin{aligned}\mathbb{P}^1(\tau_n<\infty, Y_n\ge m)=\mathbb{P}^1(T_n<T_0, T_n\ge m).\end{aligned}\end{equation}

Note that $\mathbb{P}^1(T_n<T_0)=n^{-1}$ , while

\begin{equation*}\mathbb{P}^1(T_n<m)= \mathbb{P}^0(T_{n-1}<m)\le \mathbb{P}^0\bigg(\max_{i\le m}X_i\ge n-1 \bigg).\end{equation*}

By the reflection principle,

\begin{equation*}\mathbb{P}^0\bigg(\max_{i\le m} X_i\ge n-1 \bigg)=2 \mathbb{P}^0(X_m\ge n-1).\end{equation*}

Now let $m_n=\left[n^2/\left(3\log n\right)\right]$ ; by Chernoff’s inequality,

\begin{equation*}\begin{aligned} \mathbb{P}^0\big(X_{m_n}\ge n-1\big)\le \exp\bigg(-\frac{\left(n-1\right)^2}{m_n}\bigg)\le \exp\left(-2\log n\right)=\frac{1}{n^2},\end{aligned}\end{equation*}

which implies that

\begin{equation*}\mathbb{P}^1(T_n<T_0, T_n\ge m_n)\ge \mathbb{P}^1(T_n<T_0)-\mathbb{P}^1(T_n<m_n)\ge \frac{1}{n}-\frac{2}{n^2}\ge \frac{1}{2n}\end{equation*}

when n is large enough. At the same time, note that, given any nonzero and nearest-neighbor trajectory $1=x_{\left(0\right)}, x_{\left(1\right)}, \ldots, x_{\left(m\right)}=n$ with $x_{\left(0\right)}, x_{\left(1\right)}, \ldots, x_{\left(m-1\right)}\in \left[1,n-1\right]$ as the embedded chain of $S_t$ within $\left[0,\tau_n\right]$ , the conditioned expectation of $\tau_n$ is given by

\begin{equation*}\frac{1}{x_{\left(0\right)}}+\frac{1}{x_{\left(1\right)}}+\cdots+\frac{1}{x_{\left(m-1\right)}}\ge \frac{m}{n}.\end{equation*}

Moreover, in the event $\left\{\tau_n<\infty, Y_n\ge m_n\right\}$ , each trajectory of the embedded chain within $\left[0,\tau_n\right]$ has to satisfy the condition above with $m=m_n$ . Thus,

(4.9) \begin{equation}\begin{aligned}\mathbb{E}\left[\tau_n \mid \tau_n<\infty, Y_n\ge m_n\right]\ge \frac{m_n}{n}\ge \frac{n}{4\log n}.\end{aligned}\end{equation}

By (4.6), (4.7), (4.8), and (4.9) we have

\begin{align*}\mathbb{E}^1\left[\tau_0\textrm{1}_{A_n}\right] & \ge \frac{1}{n+1} \mathbb{E}^1\left[\tau_n\textrm{1}_{\left\{\tau_n<\infty\right\}}\right] \\ & \ge \frac{1}{n+1} \mathbb{P}^1(\tau_n<\infty, Y_n\ge m_n)\, \mathbb{E}\left[\tau_n \mid \tau_n<\infty, Y_n\ge m_n\right]\\& \ge \frac{1}{8\left(n+1\right)\log n}\end{align*}

when n is large enough, which is unsummable. Thus, $\mathbb{E}^1[\tau_0]=\infty$ and the proof of Theorem 2 is complete.

Idea of the proof of Theorem 3 Recall that in Theorem 1 we have proved that $\xi^{\lambda,\mathbb{T}_d}_t$ survives if and only if $\tilde{\xi}^{\lambda,\mathbb{T}_d}_t$ survives, and in Lemma 2 we proved that ${\xi}^{\lambda_1\left(2\right),\mathbb{T}_d}_t$ dies out. Thus, it suffices to prove that when $d=2$ and $\lambda\geq 0.637$ , the dual process $\tilde{\xi}^{\lambda,\mathbb{T}_d}_t$ survives. The constant $0.637$ is the same as the weaker upper bound for survival of the ordinary contact process on $\mathbb{T}_2$ found in [Reference Liggett13]. The key ingredient in this proof is to show that in the finite system, although $\tilde \xi^{\lambda,\mathbb{T}_d}_t$ and the ordinary contact process have different transition dynamics, their infinitesimal generators will behave exactly the same when applied to the special limiting test function f defined in [Reference Liggett13]. Specifically,

\begin{equation*} h\left(A\right)=\frac{{\rm d}}{{\rm d} t}\mathbb{E}^{\xi_A}f\big(\tilde{A}^\lambda_t\big)\Big|_{t=0} \end{equation*}

is the same as [Reference Liggett13, (4.2)], which corresponds to the ordinary contact process.

Let $\tau_a=\inf\left\{t\,{:}\,f\left(\xi_t^A\right)\le a\right\}$ where a is large enough. As a result, we can show that, for any finite set A, $f\Big(\tilde{\xi}^{\lambda,A,\mathbb{T}_d}_{t\land\tau_a}\Big)^{-1}$ is a nonnegative supermartingale whose transition rate has a uniform lower bound when $\lambda\geq 0.637$ . Applying the optional stopping theorem, we obtain that $\mathbb{P}^{\xi_A}\left(\tau_{a}=\infty\right)>0$ when $f\left(A\right)>a$ . Since $f\left(\textbf{0}\right)=0$ , we get $\mathbb{P}^{\xi_A}\left(\tau_{\textbf{0}}=\infty\right)>0$ . So we have proved that $\tilde{\xi}^{\lambda,\mathbb{T}_d}_t$ survives when $\lambda\geq 0.637$ . Since the approach follows that in [Reference Liggett13, Theorem 1.2(a)], for completeness we put the detailed proof in Appendix A.

4.3. Results for strong survival

It is easy to generalize some results for the ordinary contact process on $\mathbb{T}_d$ , which are proved in [Reference Liggett14], to all attractive spin systems we consider in this paper (see Remark 4). For convenience, we just prove some for the threshold-one contact process.

Let $\mathbb{T}^d_x$ be the subtree containing x together with all its descendants, and $\hat{\mathbb{T}}^d_x=\left(\mathbb{T}_d\backslash\mathbb{T}^d_x\right)\cup\left\{x\right\}$ . Denote by $\xi^{\lambda,\hat{\mathbb{T}}^d_x}_t$ the process restricted on $\hat{\mathbb{T}}^d_x$ which is always 0 outside. Define

\begin{equation*}\beta\left(\lambda,d\right)=\lim\limits_{n\to\infty}\left[\mathbb{P}^o\big(x_n\in\xi^{\lambda,\mathbb{T}_d}_t \text{ for some }t\big)\right]^{1/n} ; \end{equation*}

this exists by [Reference Liggett14, Theorem B22].

Among the key statements proven below is

Lemma 3. (Adaption of Proposition 4.57, [Reference Liggett14]) If $\beta\left(\lambda,d\right)>\frac{1}{\sqrt{d}\,}$ , then

\begin{equation*}\inf\limits_t \mathbb{P}^o\Big(o\in \xi^{\lambda,\mathbb{T}^d_o}_t\Big)>0.\end{equation*}

Since the proof follows the approach used in [Reference Liggett14, Proposition 4.57], for completeness we put it in Appendix B.

Once we have Lemma 3, we immediately have the next three lemmas.

Proof. When $\inf_t\mathbb{P}^o\big(o\in \xi^{\lambda,\mathbb{T}^d_o}_t\big)>0$ , $\mathbb{P}^o\big(o\in \xi^{\lambda,\mathbb{T}_o^d}_t\text{ i.o.}\big)>0$ is immediate simply by definition. To prove the other direction, when $\mathbb{P}^o\big(o\in \xi^{\lambda,\mathbb{T}_d}_t\text{ i.o.}\big)>0$ , by a simple argument, we have

\begin{equation*}\mathbb{P}^o\big(x_n\in \xi^{\lambda,\mathbb{T}_d}_t \text{ i.o.}\big)=\mathbb{P}^o\big(o\in \xi^{\lambda,\mathbb{T}_d}_t\text{ i.o.}\big)\qquad \text{for all}\ n\ge 1.\end{equation*}

Recalling the definition of $\beta\left(\lambda,d\right)$ ,

\begin{equation*}\beta\left(\lambda,d\right)\ge \lim_{n\to\infty}\left[\mathbb{P}^o\big(x_n\in \xi^{\lambda,\mathbb{T}_d}_t \text{ i.o.}\big)\right]^{1/n}=1>\frac{1}{\sqrt{d}\,}.\end{equation*}

Thus, from Lemma 3 we have proved the other direction.

Lemma 5. When $\mathbb{P}^o\big(o\in\xi^{\lambda,\mathbb{T}^d}_t\text{ i.o.}\big)>0$ ,

\begin{equation*}\lim\limits_{n\to\infty}\sup_t\mathbb{P}^{\xi_{B\left(n\right)}}\left(|\xi_t^{\lambda,\mathbb{T}_d}\cap B\left(n\right)|\le k\right)= 0 \qquad \text{for any } k,\end{equation*}

where $B\left(n\right)$ is the ball of radius n centered at o.

Proof. By Lemma 4,

\begin{equation*}\inf_{t\ge 0}\mathbb{P}^o\left(o\in\xi_t^{\lambda,\mathbb{T}^d_o}\right)\ge p>0.\end{equation*}

Since $\mathbb{T}^d_x\cap\mathbb{T}^d_y=\emptyset$ for any $ x\ne y$ but $|x|=|y|$ , $\xi_t^{\lambda,\mathbb{T}^d_x}$ and $\xi_t^{\lambda,\mathbb{T}^d_y}$ are independent, so that, for any k,

\begin{align*}\sup_t\mathbb{P}^{\xi_{B\left(n\right)}}\left(\big|\xi_t^{\lambda,\mathbb{T}_d}\cap B_o\left(n\right)\big|\le k\right)&\le \sup_t\mathbb{P}^{\xi_{B\left(n\right)}}\left(\left|\left(\cup_{|x|=n}\xi_t^{\lambda,x,\mathbb{T}_x^d}\right)\cap B\left(n\right)\right|\le k\right)\\&\le \mathbb{P}\left(B\left(\left(d+1\right)^n,p\right)\ge k\right)\!,\end{align*}

where $B\left(\left(d+1\right)^n,p\right)$ is the binomial distribution with parameters $\left(d+1\right)^n$ and p, which converges to 0 as $n\to\infty$ .

Lemma 6. When $\xi_t^{\lambda,\mathbb{T}_d}$ survives strongly,

\begin{equation*}\lim\limits_{n}\sup\limits_{|A|=n}\mathbb{P}^{\xi_A}\left(\xi_t^{\lambda,\mathbb{T}_d}\ne\textbf{0}\text{ for any }t\right)=1.\end{equation*}

Proof. Assume $A\subset\mathbb{T}_o^d$ and $|A|=n$ . Define

\begin{align*} H\left(A\right) & = \left\{x\in A\,{:}\,\text{all children of \textit{x} are in $\bar{A}$}\right\}, \\F\left(A\right) & = A\backslash H\left(A\right), \\\hat A & = \left\{x\in A,\mathbb{T}_x^d\cap\bar A=\left\{x\right\}\right\}, \\p_1 & = \mathbb{P}^o\Big(\xi^{\lambda,\mathbb{T}^d_{x_1}\cup\left\{o\right\}}_t\text{ dies out}\Big), \\p_2 & =\mathbb{P}^o\Big( \xi^{\lambda,\mathbb{T}^d_o}_t\text{ dies out}\Big) ,\end{align*}

where $x_1$ is a child of o and $\bar A$ is the minimum connected component containing A.

From Lemma 4, $ \xi^{\lambda,\mathbb{T}_d}_t$ surviving strongly implies $ \xi^{\lambda,\mathbb{T}^d_o}_t$ survives strongly, so that $p_2<1$ and thus $p_1<1$ .

If $ |F\left(A\right)|\geq \frac{n}{2}$ , for any $x\in F\left(A\right)$ we can choose a child $x^*\in\bar{A}^{\rm c}$ , and thus the processes $\xi^{\lambda,x,\mathbb{T}^d_{x^*}\cup\left\{x\right\}}_t,{x\in F\left(A\right)}$ are identically independent. Then, by monotonicity,

(4.10) \begin{equation}\begin{aligned}\mathbb{P}^{\xi_A}\big(\xi^{\lambda,\mathbb{T}_d}_t \text{ dies out}\big)&\leq \mathbb{P}\bigg(\text{for all}\ x\in F\left(A\right),\ \xi^{\lambda,x,\mathbb{T}^d_{x^*}\cup\left\{x\right\}}_t\text{ dies out}\bigg)\\ &=\prod_{x\in F\left(A\right)}\mathbb{P}\bigg( \xi^{\lambda,x,\mathbb{T}^d_{x^*}\cup\left\{x\right\}}_t\text{ dies out}\bigg)\\ &\leq p_1^{\frac{n}{2}}.\end{aligned}\end{equation}

Otherwise, when $|H(A)|\geq \frac{n}{2}$ , we claim that $|\hat{A}|\geq |H(A)|$ . We prove the claim by induction. First, when $|A|=1$ , we have $|H(A)|=0$ , so that $|\hat{A}|\geq |H(A)|$ holds immediately. Next, we assume that the claim is true for any A satisfying $|A|=n$ . Now we come to the case when $|A|=n+1$ . Assume $x\in A$ has the maximum generation, i.e. $m_{x}-n_{x}=\max\left\{m_y-n_y\,{:}\,y\in A\right\}$ . If there is more than one such point, we choose the ‘smallest’ one according to the lexicographical order of $\left(n_z, r\left(1\right),r\left(2\right),\ldots, r\left(m_z\right)\right)$ . Then,

(4.11) \begin{equation}|H(A\backslash\left\{x\right\})|=\begin{cases}|H(A)|-1,&\text{ if } \overleftarrow x\in H(A),\\[3pt] |H(A)|,&\text{ otherwise} , \end{cases} \end{equation}

where $\overleftarrow x$ represents the parent of x. Let $\tilde{x}$ be (the unique) point in $\overline{A\backslash\left\{x\right\}}$ such that $d\left(x,\tilde x\right)=\min\left\{d\left(x,y\right), \ y\in \overline{A\backslash\left\{x\right\}}\right\}$ . That is,

1. • if x is a descendant of the common ancestor in $\overline{A\backslash\left\{x\right\}}$ , then $\tilde{x}$ has to be the nearest ancestor of x in $\overline{A\backslash\left\{x\right\}}$ ;

2. • otherwise, $\tilde{x}$ has to be the common ancestor itself.

Note that $\bar{A}=\overline{A\backslash\left\{x\right\}}\cup\overline{\left\{x,\tilde{x}\right\}}$ , $\overline{A\backslash\left\{x\right\}}\cap\overline{\left\{x,\tilde{x}\right\}}=\left\{\tilde{x}\right\}$ , and $\hat A=\hat {\bar A}$ , so that

(4.12) \begin{equation}|\hat A|\begin{cases} =|\widehat{A\backslash\left\{x\right\}}|+1,&\text{ if } \overleftarrow x\in H( A),\\[3pt] \ge|\widehat{A\backslash\left\{x\right\}}|,&\text{ otherwise}. \end{cases} \end{equation}

Then, by the inductive hypothesis and Equations (4.11) and (4.12), the claim holds for the case when $|A|=n+1$ and thus holds for any n. For any $x\in\hat{A}$ , we consider the process $\xi_t^{\lambda,x,\mathbb{T}_x^d}$ . Obviously, $\xi_t^{\lambda,x,\mathbb{T}_x^d}$ , $x\in\hat A$ , are identically independent, so that

(4.13) \begin{equation}\begin{aligned}\mathbb{P}^{\xi_A}\big(\xi^{\lambda,\mathbb{T}_d}_t \text{ dies out}\big)&\leq \mathbb{P}\big(\text{for all}\ x\in\hat A,\ \xi^{\lambda,x,\mathbb{T}^d_{x}}_t\text{ dies out}\big)\\&=\prod_{x\in \hat A}\mathbb{P}\left( \xi^{\lambda,x,\mathbb{T}^d_{x}}_t\text{ dies out}\right)\\&\leq p_2^{\frac{n}{2}}.\end{aligned}\end{equation}

Thus, by (4.10), (4.13), and the translation invariance,

\begin{align*}\lim\limits_{n}\sup\limits_{|A|=n}\mathbb{P}\left( \xi^{\lambda,x,\mathbb{T}^d_{x}}_t\text{ dies out}\right)\leq\lim\limits_n\left[\max\left\{p_1, p_2\right\}\right]^{\frac{n}{2}}=0.\\[-40pt] \end{align*}

4.4. Complete convergence theorem

Inspired by [Reference Bramson, Ding and Durrett3, Theorem 1.2] and [Reference Liggett14, Theorem 1.12], we will prove the following statement and then prove the complete convergence theorem for the threshold-one contact process; both are also applicable to the threshold-one voter model with positive spontaneous death and only annihilating duality.

Proposition 1. For any $\mu$ satisfying $\mu\left(\xi=\textbf{0}\right)=0$ , $\mu$ is translation invariant, or $\mu=\delta_\xi$ where $\xi$ is dense, we have

\begin{equation*}\lim_{t\to\infty}{\mu} S^\lambda\left(t\right)=\bar \nu^\lambda,\end{equation*}

where $S^\lambda\left(t\right)$ is the Markov semigroup and $\bar \nu^\lambda$ is the upper invariant measure of ${\xi}^{\lambda,\mathbb{T}_d}_t$ .

Proof. Let $\Omega_{\infty}=\big\{\tilde{\xi}^{\lambda,\mathbb{T}_d}_t\ne\textbf{0}\text{ for all}\ t\big\} $ . By (3.2), for any $\xi_A$ ,

\begin{equation*}\bar{\nu}^\lambda\left(\xi\,{:}\,\xi\cap \xi_A\ne\textbf{0}\right)=\tilde{\mathbb{P}}^{\xi_A}\left(\Omega_{\infty}\right).\end{equation*}

Thus, it suffices to prove that, for any $A\subseteq\mathbb{T}_d$ ,

(4.14) \begin{equation}\begin{aligned}\mathbb{P}^{\mu,\xi_A}\left( \xi^{\lambda,\mathbb{T}_d}_{2}\cap\tilde{\xi}_t^{\lambda,\mathbb{T}_d}\ne\textbf{0}\right)\to\tilde{\mathbb{P}}^{\xi_A}\left(\Omega_{\infty}\right)\end{aligned}\end{equation}

when $\mathbb{\tilde{P}}^{\xi_A}\left(\Omega_{\infty}\right)>0$ , otherwise the convergence becomes obvious, where $\mathbb{P}^{\mu,\xi_A}$ is the coupling measure of $\xi^{\lambda,\mathbb{T}_d}_{2}$ and $\tilde{\xi}_t^{\lambda,\mathbb{T}_d}$ starting from $\mu$ and $\delta_{\xi_A}$ , which is constructed by the graph representation.

First, we claim that for any $\epsilon>0 , N$ , we can find K such that, for any $|A|\geq K$ ,

(4.15) \begin{equation}\begin{aligned}\mathbb{P}^\mu\left( |\xi^{\lambda,\mathbb{T}_d}_1\cap A|\leq N \right)<\epsilon.\end{aligned}\end{equation}

When $\mu=\delta_\xi$ and $\xi$ is dense, denote $\mathcal{I}_x=\left\{\xi, \xi\cap B_x\left(N\right)\ne\textbf{0}\right\}$ ; we have

(4.16) \begin{equation}\begin{aligned}\inf\limits_x\mathbb{P}^\xi\left(\xi_1^{\lambda,\mathbb{T}_d}\left(x\right)=1\right) &\geq\inf\limits_{x}\inf\limits_{\xi\in\mathcal{I}_x}\mathbb{P}^\xi\left(\xi_1^{\lambda,\mathbb{T}_d}\left(x\right)=1\right) \\&=\inf\limits_{\xi\in\mathcal{I}_o}\mathbb{P}^\xi\left(\xi_1^{\lambda,\mathbb{T}_d}\left(o\right)=1\right)\qquad\text{ by translation invariance} \\&=\inf\limits_{\xi\in\mathcal{I}_o}\mathbb{P}^o\left(\tilde{\xi}^{\lambda,\mathbb{T}_d}_1\cap\xi\text{ $\ne\textbf{0}$}\right)\qquad\text{by duality} \\&\geq \inf\limits_{\xi\in\mathcal{I}_o}\mathbb{P}^o\left(\tilde{\xi}^{\lambda,\mathbb{T}_d}_1\subset B_N\left(o\right)\text{ and }\tilde{\xi}^{\lambda,\mathbb{T}_d}_1 \cap\xi\text{ $\ne\textbf{0}$}\right)\\&=\inf\limits_{\xi\subseteq B_N\left(o\right),\xi\in\mathcal{I}_o}\mathbb{P}^o\left(\tilde{\xi}^{\lambda,\mathbb{T}_d}_1\subseteq B_N\left(o\right),\tilde{\xi}^{\lambda,\mathbb{T}_d}_1 \cap\xi\text{ $\ne\textbf{0}$}\right)\\&=p>0,\end{aligned}\end{equation}

where the last equality holds because there are finite configurations satisfying $\xi\subseteq B(N)$ .

By [Reference Liggett11, Chapter I, Theorem 4.6], for any $A\in Y$ , there exists $\lambda=\lambda\left({\rm diam}\left(A\right)\right)$ , where ${\rm diam}\left(A\right)=\max\left\{|x-y|\,{:}\,x,y\in A\right\}$ , with $\lambda\left(k\right)\to 0$ as $k\to\infty$ , such that

(4.17) \begin{equation}\begin{aligned}\mathbb{E}^{\xi}\prod\limits_{x\in A}\left[1-\xi^{\lambda,\mathbb{T}_d}_1\left(x\right)\right]\le \prod\limits_{x\in A}\mathbb{E}^{\xi}\Big[1-\xi^{\lambda,\mathbb{T}_d}_1\left(x\right)\Big]+\lambda|A|.\end{aligned}\end{equation}

It follows from (4.16) and (4.17) that

(4.18) \begin{equation}\begin{aligned}\mathbb{P}^\xi\left( \xi_1^{\lambda,\mathbb{T}_d}\cap \xi_A=\textbf{0} \right)&=\mathbb{E}^{\xi}\prod\limits_{x\in A}\Big[1-\xi^{\lambda,\mathbb{T}_d}_1\left(x\right)\Big]\\[2pt] &\le\prod\limits_{x\in A}\mathbb{P}^\xi\left(\xi_1^{\lambda,\mathbb{T}_d}\left(x\right)=0\right)+\lambda|A|\\[2pt] &\leq \left(1-p\right)^{|A|}+\lambda|A|.\end{aligned}\end{equation}

When $\mu$ is translation invariant, by (4.17) and the arithmetic–geometric mean inequality, we have

(4.19) \begin{equation}\begin{aligned}\mathbb{E}^{\mu}\prod\limits_{x\in A}\Big[1-\xi^{\lambda,\mathbb{T}_d}_1\left(x\right)\Big]& \le \int\prod\limits_{x\in A}\mathbb{E}^{\xi}\Big[1-\xi^{\lambda,\mathbb{T}_d}_1\left(x\right)\Big]\mu\left({\rm d}\xi\right)+\lambda|A|\\[3pt] &\le \mathbb{E}^{\mu}\big[1-\xi^{\lambda,\mathbb{T}_d}_1\left(o\right)\big]^n+\lambda|A|,\end{aligned}\end{equation}

where the first term converges to 0 when n converges to $\infty$ .

Then the claim (4.15) follows from (4.18) and (4.19). Since $|\tilde{\xi}_t^{\lambda,\mathbb{T}_d}|\rightarrow\infty$ in probability on $\Omega_{\infty}$ , by a simple calculation we have

(4.20) \begin{equation}\begin{aligned}\big|\xi_1^{\lambda,\mathbb{T}_d}\cap\tilde{\xi}_t^{\lambda,\mathbb{T}_d}\big|\rightarrow\infty \text{ in probability under $\mathbb{P}^{\mu,\xi_A}$ on $\Omega_{\infty}$}.\end{aligned}\end{equation}

Next, let $U_t=\xi_1^{\lambda,\mathbb{T}_d}\cap\tilde{\xi}_t^{\lambda,\mathbb{T}_d}$ and $V_t=\left\{x\,{:}\,x\in U_t, \text{ \textit{x} is isolated}\right\}$ , where x is isolated if and only if $T_n^{x,\mathcal{N}_x}\cap\left(\left\{x\right\}\times\left[1,2\right]\right)=\emptyset$ and $T_n^{y,\mathcal{N}_y}\cap\left(\left\{y\right\}\times\left[1,2\right]\right)=\emptyset$ for all $x\in \mathcal{N}_y$ . This ensures that

(4.21) \begin{equation}\begin{aligned}\big|\xi_{2}^{\lambda,\mathbb{T}_d}\cap\tilde{\xi}^{\lambda,\mathbb{T}_d}_t\cap V_t\big|=\sum\limits_{x\in V_t}\textrm{1}_{\left\{T_n^{x,\emptyset}\cap \left(\left\{x\right\}\times\left[1,2\right]\right)=\emptyset\right\}}.\end{aligned}\end{equation}

Since the events $\left\{x\in V_t\right\}_{x\in U_t}$ are 2-dependent (i.e. independent when the distance is larger than 2), by (4.20),

(4.22) \begin{equation}\begin{aligned}|V_t|\to\infty \text{ in probability under $\mathbb{P}^{\mu,\xi_A}$ on }\Omega_{\infty}.\end{aligned}\end{equation}

Let

(4.23) \begin{equation}\begin{aligned}g_x=\textrm{1}_{\left\{T_n^{x,\mathcal{N}_x}\cap \left(\left\{x\right\}\times\left[1,2\right]\right)=\emptyset\right\}}.\end{aligned}\end{equation}

Given $ \mathcal{G}_t=\sigma\left(V_t\right)$ , $\left\{g_x, x\in V_t\right\}$ are identically independent and independent of $V_t$ ; moreover, each has positive probability $p_0$ of being 1. By (4.21), (4.22), and (4.23),

(4.24) \begin{equation}\begin{aligned}1\ge \mathbb{P}^{\mu,\xi_A}\left( \xi_{2}^{\lambda,\mathbb{T}_d}\cap\tilde{\xi}^{\lambda,\mathbb{T}_d}_t\text{ $\ne\textbf{0}$ } \mid \mathcal{G}_t\right)&\ge \mathbb{P}^{\mu,\xi_A}\left(\sum\limits_{x\in V_t}g_x\ge 1 \mid \mathcal{G}_t\right)\\[3pt] &=\mathbb{P}\left(B\left(n,p_0\right)\ge1\right)|_{n=|V_t|}\to 1\end{aligned}\end{equation}

in probability on $\Omega_{\infty}$ . The interpretation is that when $|V_t|$ is large enough, we can find at least one vertex in $V_t$ whose state does not change in unit time.

Taking the expectation of (4.24), by the bounded convergence theorem,

\begin{align*}\mathbb{P}^{\mu,\xi_A}\left( \xi_{2}^{\lambda,\mathbb{T}_d}\cap\tilde{\xi}^{\lambda,\mathbb{T}_d}_t\text{ $\ne\textbf{0}$} \mid \Omega_{\infty}\right)\to1, \end{align*}

which implies (4.14) and hence the proof is complete.

Proof of Theorem 4. When $\lambda\le\lambda_1\left(d\right)$ we can get the result by Lemma 2 When $\lambda_1\left(d\right)<\lambda\le\lambda_2\left(d\right)$ we can get the result by Proposition 1. Therefore, we only need to prove the result when $\lambda>\lambda_2\left(d\right)$ . By Lemma 4, $\inf_t\mathbb{P}^o\big(\xi_{t}^{\lambda,\mathbb{T}_d}\left(o\right)=1\big)>0$ , so that $\tilde\xi_{t}^{\lambda,\mathbb{T}_d}$ survive strongly.

Define the stopping times

\begin{align*} \tau_{B\left(n\right)} & = \inf\left\{t\ge0\,{:}\,B\left(n\right)\subset \xi^{\lambda,\mathbb{T}_d}_t\right\}, \\[3pt] \tilde{\tau}_{B\left(n\right)} & = \inf\left\{t\ge0\,{:}\,B\left(n\right)\subset \tilde{\xi}^{\lambda,\mathbb{T}_d}_t\right\}.\end{align*}

By Levy’s 0–1 law, we can prove that

\begin{align*} \left\{\xi_{t}^{\lambda,\mathbb{T}_d}\text{ survives strongly}\right\} & \subseteq {\Omega}_A \text{ almost surely (a.s.)}, \\[3pt] \left\{\tilde\xi_{t}^{\lambda,\mathbb{T}_d}\text{ survives strongly}\right\} & \subseteq \tilde{\Omega}_A \text{ a.s.},\end{align*}

which implies that, for any finite B,

(4.25) \begin{equation}\begin{aligned}&\mathbb{P}^{\xi_A}\left(\xi_{t}^{\lambda,\mathbb{T}_d}\cap\xi_B\text{ $\ne\textbf{0}$},\Omega^c_{B\left(n\right)}\right)\to 0,\\[3pt] &\tilde{\mathbb{P}}^{\xi_{A}}\left(\tilde{\xi}^{\lambda,\mathbb{T}_d}_{t}\cap \xi_{B}\text{ $\ne\textbf{0}$},\tilde{\Omega}_{B\left(n\right)}^c\right)\to 0 ,\end{aligned}\end{equation}

where $\Omega_A=\left\{\tau_{A}<\infty\right\}$ and $\tilde{\Omega}_A=\left\{\tilde{\tau}_{A}<\infty\right\}$ .

For any A and finite B, by the strong Markov property,

\begin{equation*}\begin{aligned} &\lim\limits_{t\to\infty}\mathbb{P}^{ \xi_A}\left( \xi_{t+1}^{ \lambda,\mathbb{T}_d}\cap \xi_B\text{ $\ne\textbf{0}$}\right)\\[3pt] &=\lim\limits_{t\to\infty}\mathbb{P}^{ \xi_A}\left( \xi_{t+1}^{ \lambda,\mathbb{T}_d}\cap \xi_B\text{ $\ne\textbf{0}$},\Omega_{B\left(n\right)}\right)\qquad\text{by (\linkref{eqn32}{4.25})}\\[3pt] &=\lim\limits_{t\to\infty}\mathbb{E}^{ \xi_A}\Big[\mathbb{P}^{ \xi^{ \lambda,\mathbb{T}_d}_{\tau_{B\left(n\right)}}}\Big[ \xi^{ \lambda,\mathbb{T}_d}_{t+1-\tau_{B\left(n\right)}}\cap \xi_B\text{ $\ne\textbf{0}$}\Big],\Omega_{B\left(n\right)}\Big]\\[3pt] &=\lim\limits_{t\to\infty}\mathbb{E}^{ \xi_A}\Big[\tilde{\mathbb{P}}^{ \xi_B}\Big[\tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{t+1-\tau_{B\left(n\right)}}\cap \xi^{ \lambda,\mathbb{T}_d}_{\tau_{B\left(n\right)}}\text{ $\ne\textbf{0}$}\Big],\Omega_{B\left(n\right)}\Big]\qquad\text{by (\linkref{eqn5}{3.1})}\\[3pt] &=\lim\limits_{t\to\infty}\mathbb{E}^{ \xi_A}\Big[\tilde{\mathbb{P}}^{ \xi_B}\Big[\tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{t+1-\tau_{B\left(n\right)}}\cap \xi^{ \lambda,\mathbb{T}_d}_{\tau_{B\left(n\right)}}\text{ $\ne\textbf{0}$},\tilde{\Omega}_{B\left(n\right)}\Big],\Omega_{B\left(n\right)}\Big]\qquad\text{by (\linkref{eqn32}{4.25}) }\\[3pt] &=\lim\limits_{t\to\infty}\mathbb{E}^{ \xi_A}\bigg[\tilde{\mathbb{P}}^{ \xi_B}\bigg[\tilde{\mathbb{P}}^{\tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}}}\Big[\tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{t+1-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap \xi^{ \lambda,\mathbb{T}_d}_{\tau_{B\left(n\right)}}\text{ $\ne\textbf{0}$}\Big],\tilde{\Omega}_{B\left(n\right)}\bigg],\Omega_{B\left(n\right)}\bigg]\\[3pt] &=\lim\limits_{t\to\infty}\mathbb{E}^{ \xi_A}\bigg[\tilde{\mathbb{E}}^{ \xi_B}\bigg[\mathbb{P}^{ \xi^{ \lambda,\mathbb{T}_d}_{\tau_{B\left(n\right)}}}\Big[{ \xi}^{ \lambda,\mathbb{T}_d}_{t+1-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap\tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}} \text{ $\ne\textbf{0}$}\Big],\tilde{\Omega}_{B\left(n\right)}\bigg],\Omega_{B\left(n\right)}\bigg] \quad\text{by (\linkref{eqn5}{3.1})}.\end{aligned}\end{equation*}

When $t>\tau_{B\left(n\right)}+\tilde{\tau}_{B\left(n\right)}$ , for any n, k,

\begin{align*} \bigg|\mathbb{P}^{ \xi^{ \lambda,\mathbb{T}_d}_{\tau_{B\left(n\right)}}}&\left( \xi^{ \lambda,\mathbb{T}_d}_{t+1-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap \tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}}\text{ $\ne\textbf{0}$}\right)-1\bigg| \\[3pt] & \le \mathbb{P}^{ \xi^{ \lambda,\mathbb{T}_d}_{\tau_{B\left(n\right)}}}\left(\left| \xi^{ \lambda,\mathbb{T}_d}_{t-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap \tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}}\right|\ge k\right) \\[3pt] & \quad \times \left|\mathbb{P}^{ \xi^{ \lambda,\mathbb{T}_d}_{\tau_{B\left(n\right)}}}\left( \xi^{ \lambda,\mathbb{T}_d}_{t+1-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap \tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}}\text{ $\ne\textbf{0}$} \,\big| \left| \xi^{ \lambda,\mathbb{T}_d}_{t-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap \tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}}\right|\ge k\right)-1\right| \\[3pt] & \quad + 2\mathbb{P}^{ \xi^{ \lambda,\mathbb{T}_d}_{\tau_{B\left(n\right)}}}\left(\left| \xi^{ \lambda,\mathbb{T}_d}_{t-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap \tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}}\right|\le k\right) \\[3pt] & \le \left|\mathbb{P}^{ \xi^{ \lambda,\mathbb{T}_d}_{\tau_{B\left(n\right)}}}\left( \xi^{ \lambda,\mathbb{T}_d}_{t+1-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap \tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}}\text{ $\ne\textbf{0}$} \,\big| \left| \xi^{ \lambda,\mathbb{T}_d}_{t-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap \tilde{ \xi}^{ \lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}}\right|\ge k\right)-1\right| \\[3pt] & \quad + 2\mathbb{P}^{ \xi^{ \lambda,\mathbb{T}_d}_{\tau_{B\left(n\right)}}}\left(\left| \xi^{ \lambda,\mathbb{T}_d}_{t-\tilde{\tau}_{B\left(n\right)}-\tau_{B\left(n\right)}}\cap B\left(n\right)\right|\le k\right) .\end{align*}

Let $U_t=\xi^{\lambda,\mathbb{T}_d}_{t-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap \tilde{\xi}^{\lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}}$ and $V_t$ be the corresponding variable; by (4.24), for any $\epsilon>0$ ,

\begin{equation*} \left|\mathbb{P}^{\xi_{\tau_{B\left(n\right)}}}\left(\xi^{\lambda,\mathbb{T}_d}_{t+1-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap \tilde{\xi}^{\lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}}\text{ $\ne\textbf{0}$} \,\big| \left|\xi^{\lambda,\mathbb{T}_d}_{t-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap \tilde{\xi}^{\lambda,\mathbb{T}_d}_{\tilde{\tau}_{B\left(n\right)}}\right|\ge k\right)-1\right|<\epsilon/2\end{equation*}

for any t, n when k is large enough.

By Lemma 5 and Remark 4, for any k, $\mathbb{P}^{\xi_{B\left(n\right)}}\left(|\xi_{t-\tau_{B\left(n\right)}-\tilde{\tau}_{B\left(n\right)}}\cap B\left(n\right)|\le k\right)<\epsilon/2$ for any $t>\tau_{B\left(n\right)}+\tilde{\tau}_{B\left(n\right)}$ when n is large enough. Thus, for any $\epsilon>0$ , we can find large enough $t_0,n_0$ such that, for any $t\ge t_0$ , $n\ge n_0$ ,

\begin{equation*}\left|\mathbb{P}^{\xi_A}\left(\xi_{t+1}^{\lambda,\mathbb{T}_d}\cap\xi_B\text{ $\ne\textbf{0}$}\right)-\mathbb{P}^{\xi_A}\left(\Omega_{B\left(n\right)}\right)\tilde{\mathbb{P}}^{\xi_B}\big(\tilde{\Omega}_{B\left(n\right)}\big)\right|<\epsilon.\end{equation*}

Letting $\epsilon\to 0$ , we can get

(4.26) \begin{equation}\begin{aligned}\lim\limits_{t\to\infty}\mathbb{P}^{\xi_A}\left(\xi_{t+1}^{\lambda,\mathbb{T}_d}\cap\xi_B\text{ $\ne\textbf{0}$}\right)&=\lim\limits_{n\to\infty}\mathbb{P}^{\xi_A}\left(\Omega_{B\left(n\right)}\right)\tilde{\mathbb{P}}^{\xi_B}\big(\tilde{\Omega}_{B\left(n\right)}\big).\end{aligned}\end{equation}

We claim that

(4.27) \begin{equation}\begin{aligned}\lim\limits_{n\to\infty}\mathbb{P}^{\xi_A}\left(\Omega_{B\left(n\right)}\right)=\mathbb{P}^{\xi_A}\big(o\in\xi_t^{\lambda,\mathbb{T}_d}\text{ i.o.}\big).\end{aligned}\end{equation}

Since, on one hand,

(4.28) \begin{equation}\begin{aligned}\mathbb{P}^{\xi_A}\big(\tau_{B\left(n\right)}<\infty,o\in\xi_t^{\lambda,\mathbb{T}_d}&\text{ finitely often (f.o.)}\big) \\[3pt] & =\mathbb{P}^{\xi_A}\Big[\mathbb{P}^{\xi_{\tau_{B\left(n\right)}}}\Big[o\in\xi_t^{\lambda,\mathbb{T}_d}\text{ f.o.}\Big],\tau_{B\left(n\right)}<\infty\Big]\\[3pt] &\le \mathbb{P}^{\xi_A}\Big[\mathbb{P}^{\xi_{\tau_{B\left(n\right)}}}\Big[o\in\xi_t^{\lambda,\mathbb{T}_d}\text{ f.o.}\Big]\Big]\\[3pt] &=\mathbb{P}^{\xi_A}\Big[\mathbb{P}^{\xi_{\tau_{B\left(n\right)}}}\Big[\tau_{\textbf{0}}<\infty\Big]\Big]\\[3pt] & \le \mathbb{P}^{\xi_A}\big[\mathbb{P}^{B\left(n\right)}\big[\tau_{\textbf{0}}<\infty\big]\big]\\[3pt] &\to 0 ,\end{aligned}\end{equation}

where the third equality is the equivalence of the survival and strong survival, and the last term is from Lemma 6. On the other hand, by Levy’s 0–1 law,

(4.29) \begin{equation}\begin{aligned}\mathbb{P}^{\xi_A}\big(\tau_{B\left(n\right)}<\infty \mid o\in\xi_t^{\lambda,\mathbb{T}_d}\text{ i.o.}\big)=1.\end{aligned}\end{equation}

Then, (4.27) comes from (4.28) and (4.29).

Similarly, we can get

(4.30) \begin{equation}\begin{aligned}\lim\limits_{n\to\infty}\tilde{\mathbb{P}}^{\xi_B}\big(\tilde \Omega_{B\left(n\right)}\big)=\tilde{\mathbb{P}}^{\xi_B}\big(o\in\tilde\xi_t^{\lambda,\mathbb{T}_d}\text{ i.o.}\big).\end{aligned}\end{equation}

By (4.26), (4.27), (4.30), and the equivalence of survival and strong survival when $\xi_t^{\lambda,\mathbb{T}_d}$ survives strongly,

\begin{align*} \lim\limits_{t\to\infty}\mathbb{P}^{\xi_A}\Big(\xi_{t+1}^{\lambda,\mathbb{T}_d}\cap\xi_B & \text{ $\ne\textbf{0}$}\Big) \\& = \mathbb{P}^{\xi_A}\Big(o\in\xi_t^{\lambda,\mathbb{T}_d}\text{ i.o.}\Big)\mathbb{P}^{\xi_B}\Big(o\in\tilde\xi_t^{\lambda,\mathbb{T}_d}\text{ i.o.}\Big) \\& = \mathbb{P}^{\xi_A}\Big(\xi_t^{\lambda,\mathbb{T}_d}\ne\textbf{0}, \text{ for all}\ t\Big)\tilde{\mathbb{P}}^{\xi_B}\Big(\tilde\xi_t^{\lambda,\mathbb{T}_d}\ne\textbf{0},\text{ for all}\ t\Big). \end{align*}

By (3.2),

\begin{align*} \lim\limits_{t\to\infty}\mathbb{P}^{\xi_A} & \left(\xi_{t+1}^{\lambda,\mathbb{T}_d}\cap\xi_B\text{ $\ne\textbf{0}$}\right)\\& = \mathbb{P}^{\xi_A}\left(\xi_t^{\lambda,\mathbb{T}_d}\ne\textbf{0},\text{ for all}\ t\right)\bar \nu^\lambda\left(\xi\,{:}\,\xi_B\cap\xi\ne\textbf{0}\right) \\& = \delta_{\textbf{0}}\left(\xi\,{:}\,\xi_B\cap\xi\ne\textbf{0}\right)+\mathbb{P}^{\xi_A}\left(\xi_t^{\lambda,\mathbb{T}_d}\ne\textbf{0},\text{ for all}\ t\right)\bar \nu^\lambda\left(\xi\,{:}\,\xi_B\cap\xi\ne\textbf{0}\right)\!,\end{align*}

which implies (2.2), and the proof is complete.