2.1. Spaces and basic notation

Let $(\mathcal{X}, d)$ be a complete separable metric space. We write $\mathcal{B}$ for the Borel $\sigma$ -field on $\mathcal{X}$ and we assume that $(\mathcal{X}, \mathcal{B})$ is equipped with a locally finite reference measure $\nu$ (i.e. $\nu(B) < \infty$ for every bounded set $B \in \mathcal{B}$ ). We write $\mathcal{B}_{{b}}$ for the set of bounded Borel sets in $\mathcal{B}$ , and, for $\Lambda \in \mathcal{B}$ , we write $\mathcal{B}_{\Lambda}$ for the trace of $\Lambda$ in $\mathcal{B}$ .

We denote by $\mathcal{N}$ the space of locally finite counting measures on $(\mathcal{X}, \mathcal{B})$ , where we write $\mathbf{0} \in \mathcal{N}$ for the constant 0 measure. We equip $\mathcal{N}$ with the $\sigma$ -field $\mathcal{R}$ , generated by the functions

\begin{align*} N_\Lambda\,:\, \mathcal{N} \to \mathbb{N}_0 \cup \{\infty\}, \eta \mapsto \eta(\Lambda) \text{ for } \Lambda \in \mathcal{B}. \end{align*}

Further, we write $\mathcal{N}_{{f}}$ for the set of finite counting measures on $(\mathcal{X}, \mathcal{B})$ and equip it with $\mathcal{R}_{\text{f}}$ , the trace of $\mathcal{N}_{{f}}$ in $\mathcal{R}$ . For $\eta \in \mathcal{N}$ and $\Lambda \in \mathcal{B}$ , we write $\eta_{\Lambda}$ for the unique counting measure with $\eta_{\Lambda}(\Delta) = \eta(\Delta \cap \Lambda)$ for every $\Delta \in \mathcal{B}$ . We denote by $\mathcal{N}_{\Lambda} \,:\!=\, \{\eta \in \mathcal{N} \mid \eta(\mathcal{X} \setminus \Lambda) = 0\}$ the image of $\mathcal{N}$ under $\eta \mapsto \eta_{\Lambda}$ and write $\mathcal{R}_{\Lambda}$ for the trace of $\mathcal{N}_{\Lambda}$ in $\mathcal{R}$ .

2.2. Integration and factorial measures

In the above setting, it is well known that every $\eta \in \mathcal{N}$ can be written as a sum of Dirac measures $\eta = \sum_{i \in I} \delta_{x_i}$ , where $(x_i)_{i \in I}$ is a sequence of points in $\mathcal{X}$ , and the index set is either $I = \{1, \dots, \eta(\mathcal{X})\}$ if $\eta \in \mathcal{N}_{{f}}$ or $I = \mathbb{N}$ if $\eta \in \mathcal{N} \setminus \mathcal{N}_{{f}}$ (see [Reference Daley and Vere-Jones6, Proposition 9.1.III]). As a consequence, we have for every $\eta \in \mathcal{N}$ and every measurable $f\,:\, \mathcal{X} \to \overline{\mathbb{R}_{\ge 0}}$ that $\int f \mathrm{d} \eta = \sum_{i \in I} f(x_i)$ . This identity extends to measurable functions $f\,:\, \mathcal{X} \to \mathbb{R} \cup \{\infty\}$ provided that the negative or positive part of the integral is finite, which particularly holds if $\eta \in \mathcal{N}_{{f}}$ .

Further, given a locally finite counting measure $\eta = \sum_{i \in I} \delta_{x_i} \in \mathcal{N}$ and a positive integer $k \in \mathbb{N}$ , we write $\eta^{(k)}$ for the $k{\text{th}}$ factorial measure of $\eta$ , which is a counting measure on the product space $(\mathcal{X}, \mathcal{B})^{\otimes k}$ given by $\eta^{(k)} = \sum_{i_1, \dots, i_k \in I}^{\neq} \delta_{(x_{i_1}, \dots, x_{i_k})}$ , where the sum is taken over k tuples of distinct indices from I. Analogously to before, we have $\int_{\mathcal{X}^k} f \mathrm{d} \eta^{(k)} = \sum_{i_1, \dots, i_k \in I}^{\neq} f(x_{i_1}, \dots, x_{i_{k}})$ for every measurable $f\,:\, \mathcal{X}^k \to \mathbb{R} \cup \{\infty\}$ such that the negative or positive part of the integral is finite.

2.3. Support set and operations on counting measures

An alternative way of writing a locally finite counting measure $\eta \in \mathcal{N}$ on a complete separable metric space involves the notion of the support set

\begin{align*} S_{\eta} \,:\!=\, \{x \in \mathcal{X} \mid \eta(x) > 0\} , \end{align*}

where we write $\eta(x)$ for $\eta(\{x\})$ . It can be shown that $S_{\eta}$ is at most countable, and, for every $x \in S_{\eta}$ , the fact that $\eta$ is locally finite implies $\eta(x) < \infty$ (see again [Reference Daley and Vere-Jones6, Proposition 9.1.III]). We can now write $\eta$ as

\begin{align*} \eta = \sum_{x \in S_{\eta}} \eta(x) \cdot \delta_x . \end{align*}

Using this, we can define the following operations for pairs of locally finite counting measures $\eta, \xi \in \mathcal{N}$ :

\begin{align*} \eta \cap \xi &\,:\!=\, \sum_{x \in S_{\eta} \cap S_{\xi}} \min(\eta(x), \xi(x)) \cdot \delta_x, \\ \eta \cup \xi &\,:\!=\, \sum_{x \in S_{\eta} \cup S_{\xi}} \max(\eta(x), \xi(x)) \cdot \delta_x, \\ \eta \setminus \xi &\,:\!=\, \sum_{x \in S_{\eta}} \max(0, \eta(x) - \xi(x)) \cdot \delta_x, \\ \eta \ominus \xi &\,:\!=\, (\eta \setminus \xi) + (\xi \setminus \eta) . \end{align*}

Thinking of locally finite counting measures as point (multi-)sets, these operations correspond directly to intersection, union, set difference and symmetric difference. It is easily checked that $\eta \cup \xi = (\eta \ominus \xi) + (\eta \cap \xi)$ for all $\eta, \xi \in \mathcal{N}$ , which will turn out to be a particularly useful identity.

2.4. Point processes

We call a probability measure $P \in \mathcal{P}(\mathcal{N}, \mathcal{R})$ a point process on $\Lambda \in \mathcal{B}$ if P is only supported on $\mathcal{N}_{\Lambda}$ (i.e. $P(\mathcal{N} \setminus \mathcal{N}_{\Lambda}) = 0$ ). Note that every point process P on $\Lambda$ can be uniquely identified with a probability measure on $(\mathcal{N}_{\Lambda}, \mathcal{R}_{\Lambda})$ . Moreover, for $\Lambda \in \mathcal{B}$ and a point process $P \in \mathcal{P}(\mathcal{N}, \mathcal{R})$ , we write $P[\Lambda]$ for the pushforward of P under $\eta \mapsto \eta_{\Lambda}$ , viewed as a map from $(\mathcal{N}, \mathcal{R})$ to itself. That is, for all $A \in \mathcal{R}$ , we have

\begin{align*} P[\Lambda](A) = \int_{\mathcal{N}} \unicode{x1D7D9}_{A}(\eta_{\Lambda}) P(\mathrm{d} \eta). \end{align*}

Note that $P[\Lambda]$ is a point process on $\Lambda$ . We call $P[\Lambda]$ the projection of P to $\Lambda$ .

In fact, point processes are fully characterized by their projections to finite-volume regions. This is implied by the following even stronger fact, which states that two point processes are equal if they induce the same joint distributions of point counts on families of bounded sets. To this end, recall that we defined a measurable map $N_{\Lambda}\,:\, \mathcal{N} \to \mathbb{N}_0 \cup \{\infty\}, \eta \mapsto \eta(\Lambda)$ for each $\Lambda \in \mathcal{B}$ .

Lemma 2.1 (Corollary 9.1.IX of [Reference Daley and Vere-Jones6].) Let $P, Q \in \mathcal{P}(\mathcal{N}, \mathcal{R})$ . Suppose, for all $\ell \in \mathbb{N}$ , $\Lambda_1, \dots, \Lambda_{\ell} \in \mathcal{B}_{{b}}$ and $m_1, \dots, m_{\ell} \in \mathbb{N}_0$ , it holds that

\begin{align*} P(N_{\Lambda_1} = m_1, \dots, N_{\Lambda_{\ell}} = m_{\ell}) = Q(N_{\Lambda_1} = m_1, \dots, N_{\Lambda_{\ell}} = m_{\ell}). \end{align*}

Then $P=Q$ .

2.5. Finite-volume Gibbs point processes

Let $\phi\,:\, \mathcal{X}^2 \to \mathbb{R} \cup \{\infty\}$ be symmetric and measurable. We call $\phi$ a (stable) pair potential if there is a constant $S \in \mathbb{R}_{\ge 0}$ such that, for all $\eta \in \mathcal{N}_{{f}}$ , it holds that

\begin{align*} H(\eta) \,:\!=\, \frac{1}{2} \int_{\mathcal{X}^2} \phi(x, y) \eta^{(2)} (\mathrm{d} x, \mathrm{d} y) \ge - S \cdot \eta(\mathcal{X}) . \end{align*}

We call every such S a stability constant for $\phi$ , and we call $H\,:\, \mathcal{N}_{{f}} \to \mathbb{R} \cup \{\infty\}$ the energy function induced by $\phi$ . Note that such an energy function satisfies the following important properties:

1. (i). It holds that $H(\mathbf{0}) = 0$ (nondegenerate).

2. (ii). Suppose $\eta, \xi \in \mathcal{N}_{{f}}$ are such that $\xi \le \eta$ and $H(\xi) = \infty$ ; then $H(\eta) = \infty$ (hereditary).

Given a pair potential $\phi$ and a bounded set $\Lambda \in \mathcal{B}$ , we define a function $H_{\Lambda}\,:\, \mathcal{N}_{\Lambda} \times \mathcal{N} \to \mathbb{R} \cup \{\infty\}$ as

\begin{align*} H_{\Lambda}(\eta \mid \xi) \,:\!=\, H(\eta) + \int_{\Lambda} \int_{\mathcal{X} \setminus \Lambda} \phi(x, y) \xi(\mathrm{d} y) \eta(\mathrm{d} x) , \end{align*}

where we set $H_{\Lambda}(\eta \mid \xi) = \infty$ whenever the latter integral fails to converge absolutely.

Next, we introduce the grand-canonical partition function, which serves as the normalizing constant of the Gibbs point process. For $\Lambda$ as above, we define the grand-canonical partition function on $\Lambda$ with boundary condition $\xi \in \mathcal{N}$ and activity $\lambda \in \mathbb{R}_{\ge 0}$ by

\begin{align*} \Xi_{\Lambda \mid \xi} (\lambda) \,:\!=\, 1 + \sum_{n = 1}^{\infty} \frac{\lambda^n}{n!} \int_{\Lambda^n} \exp \Big( - H_{\Lambda}\Big(\sum_{i=1}^{n} \delta_{x_i} \Big| \xi\Big) \Big) \nu^{n}(\mathrm{d} \pmb{x}). \end{align*}

Obviously, it holds that $\Xi_{\Lambda \mid \xi} (\lambda) \ge 1$ . Further, if $\xi$ is such that $\Xi_{\Lambda \mid \xi} (\lambda) < \infty$ , we define the finite-volume Gibbs point process (or finite-volume Gibbs measure) on $\Lambda$ with boundary condition $\xi$ at activity $\lambda$ as a probability distribution $\mu_{\Lambda \mid \xi} \in \mathcal{P}(\mathcal{N}, \mathcal{R})$ with

\begin{align*} \mu_{\Lambda \mid \xi}(A) \,:\!=\,\frac{{e}^{\nu(\Lambda)}}{\Xi_{\Lambda \mid \xi} (\lambda)} \int_{A} \lambda^{\eta(\Lambda)} {e}^{-H_{\Lambda}(\eta \mid \xi)} Q_{\Lambda}(\mathrm{d} \eta) \end{align*}

for $A \in \mathcal{R}$ , where $Q_{\Lambda}$ is the Poisson point process with intensity measure $\nu(\cdot \cap \Lambda)$ . Note that $\mu_{\Lambda \mid \xi}$ is a point process on $\Lambda$ since $Q_{\Lambda}$ is only supported on $\mathcal{N}_{\Lambda}$ .

We proceed by introducing some additional terminology related to pair potentials. As sketched in the introduction, we say a pair potential $\phi$ is weakly tempered if

\begin{align*} \hat{C}_{\phi} \,:\!=\, \sup_{x \in \mathcal{X}} \int_{\mathcal{X}} 1 - {e}^{- \left\lvert \phi(x, y) \right\rvert} \nu(\mathrm{d} y) < \infty, \end{align*}

and we call $\hat{C}_{\phi}$ the weak temperedness constant of $\phi$ . An alternative condition that is frequently considered is temperedness, which requires that $C_{\phi} \,:\!=\, \sup_{x \in \mathcal{X}} \int_{\mathcal{X}} |1 - {e}^{- \phi(x, y)}| \nu(\mathrm{d} y) < \infty$ . We remark that the term ‘weak’ does not refer to weak temperedness being a weaker condition (as $\hat{C}_{\phi} < \infty$ if and only if $C_{\phi} < \infty$ ), but to the fact that $\hat{C}_{\phi} \le C_{\phi}$ , where the inequality is strict if $\phi$ is tempered and has a nontrivial negative part. Moreover, we say that $\phi$ has bounded range $R \in \mathbb{R}_{\ge 0}$ if, for all $x, y \in \mathcal{X}$ with $d(x, y) \ge R$ , it holds that $\phi(x, y) = 0$ . Finally, we call $\phi$ locally stable with constant $L \in \mathbb{R}_{\ge 0}$ if, for all $\eta \in \mathcal{N}_{{f}}$ with $H(\eta) < \infty$ and all $x \in \mathcal{X}$ , it holds that

\begin{align*} H(\eta + \delta_x) - H(\eta) \ge - L . \end{align*}

We note that local stability of $\phi$ for some constant L immediately implies that it is also stable with constant L.

2.6. Infinite-volume Gibbs point process

Given an activity $\lambda \in \mathbb{R}_{\ge 0}$ and a pair potential $\phi$ , we denote by $\mathcal{G}(\lambda, \phi) \subseteq \mathcal{P}(\mathcal{N}, \mathcal{R})$ the set of infinite-volume Gibbs point processes that are compatible with $\lambda$ and $\phi$ , which we characterize via the DLR formalism.

As discussed earlier, we will mostly focus on proving conditions for which $\left\lvert \mathcal{G}(\lambda, \phi) \right\rvert \le 1$ (i.e. uniqueness of infinite-volume Gibbs point processes). However, at this point, we would like to point out that, in the setting studied in this paper, it also holds that $\mathcal{G}(\lambda, \phi) \neq \emptyset$ (i.e. there also exists an infinite-volume Gibbs point process). This is formally given by the following standard result.

We remark that this is not the most general setting in which existence of infinite-volume Gibbs measures can be proven, but it suffices for our purposes. The interested reader should refer to [Reference Dereudre7, Section 2] and [Reference Jansen21, Section 5] for a better overview.

2.7. Useful properties

We proceed by proving a list of properties of Gibbs point processes that will be useful throughout the rest of the paper. We start by constructing the space of feasible counting measures

(1) \begin{align} \mathcal{N}_{H} \,:\!=\, \{\eta \in \mathcal{N} \mid \text{ for all $\xi \in \mathcal{N}_{{f}}$ with $\xi \le \eta$, it holds that $H(\xi) < \infty$}\} . \end{align}

Feasible counting measures will play an important role in this work as boundary conditions. In particular, in light of the DLR formalism in Definition 2.1, the following lemma allows us to restrict to boundary conditions from $\mathcal{N}_{H}$ when characterizing infinite-volume Gibbs measures.

As we will see shortly, configurations from $\mathcal{N}_{H}$ are feasible in the sense that, for every $\xi \in \mathcal{N}_{H}$ , the finite-volume Gibbs measure $\mu_{\Lambda \mid \xi}$ is always well defined (i.e. the partition function converges). Before we prove this, it will be convenient to introduce the following notion of influence of a configuration $\eta \in \mathcal{N}$ on a single point $x \in \mathcal{X}$ by setting

\begin{align*} W(x, \eta) \,:\!=\, \int_{\mathcal{X}} \phi(x, y) \eta(\mathrm{d} y) \end{align*}

whenever the integral converges absolutely, and $W(x, \eta) \,:\!=\, \infty$ otherwise. This definition of influence is particularly useful due to the following decompositions of $H_{\Lambda}$ .

Lemma 2.3. For all $\Lambda \in \mathcal{B}_{{b}}$ , $\eta \in \mathcal{N}_{\Lambda}$ and $\xi \in \mathcal{N}$ , it holds that

(2) \begin{align} H_{\Lambda}(\eta \mid \xi) = \frac{1}{2} \int_{\Lambda} W(x, \eta \setminus \delta_x) \eta(\mathrm{d} x) + \int_{\Lambda} W(x, \xi_{\Lambda^c}) \eta(\mathrm{d} x). \end{align}

Moreover, for all $x \in \Lambda$ , it holds that

(3) \begin{align} H_{\Lambda}(\eta + \delta_x \mid \xi) = H_{\Lambda}(\eta \mid \xi) + W(x, \eta + \xi_{\Lambda^c}). \end{align}

Proof. To prove (2), we start by observing that $\eta \in \mathcal{N}_{\Lambda} \subseteq \mathcal{N}_{{f}}$ . First, note that

\begin{align*} H(\eta) = \frac{1}{2} \int_{\Lambda} W(x, \eta \setminus \delta_x) \eta(\mathrm{d} x) , \end{align*}

which can be seen by writing both sides as finite sum of terms in $(\infty, \infty]$ . Moreover, note that $\int_{\Lambda} \int_{\Lambda^c} \left\lvert \phi(x, y) \right\rvert \xi(\mathrm{d} y) \eta(\mathrm{d} x) < \infty$ if and only if $\int_{\Lambda^c} \left\lvert \phi(x, y) \right\rvert \xi(\mathrm{d} y) < \infty$ for every $x \in S_{\eta}$ . If this is the case, then

\begin{align*} \int_{\Lambda} \int_{\Lambda^c} \phi(x, y) \xi(\mathrm{d} y) \eta(\mathrm{d} x) = \int_{\Lambda} W(x, \xi_{\Lambda^c}) \eta(\mathrm{d} x) . \end{align*}

Otherwise, we have $\int_{\Lambda} W(x, \xi_{\Lambda^c}) \eta(\mathrm{d} x) = \infty$ and $H_{\Lambda}(\eta \mid \xi) = \infty$ by definition, proving identity (2).

For (3), we proceed similarly. By rewriting $H(\eta + \delta_x)$ and $W(x, \eta)$ as a sum of finitely many terms in $(\infty, \infty]$ , we can observe that

\begin{align*} H(\eta + \delta_x) = H(\eta) + W(x, \eta) . \end{align*}

Further, if $\int_{\Lambda} \int_{\Lambda^c} \left\lvert \phi(y, z) \right\rvert \xi(\mathrm{d} z) (\eta + \delta_x)(\mathrm{d} y) < \infty$ , then

\begin{align*} \int_{\Lambda} \int_{\Lambda^c} \phi(y, z) \xi(\mathrm{d} z) (\eta + \delta_x)(\mathrm{d} y) &= \int_{\Lambda} \int_{\Lambda^c} \phi(y, z) \xi(\mathrm{d} z) \eta(\mathrm{d} y) + \int_{\Lambda^c} \phi(x, z) \xi(\mathrm{d} z)\\ &= \int_{\Lambda} \int_{\Lambda^c} \phi(y, z) \xi(\mathrm{d} z) \eta(\mathrm{d} y) + W(x, \xi_{\Lambda^c}) \end{align*}

with absolutely converging integrals. Hence, in this case we get

\begin{align*} H_{\Lambda}(\eta + \delta_x \mid \xi) &= H(\eta) + W(x, \eta) + \int_{\Lambda} \int_{\Lambda^c} \phi(y, z) \xi(\mathrm{d} z) \eta(\mathrm{d} y) + W(x, \xi_{\Lambda^c})\\ &= H_{\Lambda}(\eta \mid \xi) + W(x, \eta + \xi_{\Lambda^c}) . \end{align*}

Otherwise, if $\int_{\Lambda} \int_{\Lambda^c} \left\lvert \phi(y, z) \right\rvert \xi(\mathrm{d} z) (\eta + \delta_x)(\mathrm{d} y) = \infty$ then, by definition, $H_{\Lambda}(\eta + \delta_x \mid \xi) = \infty$ . Moreover, it must hold that $\int_{\Lambda} \int_{\Lambda^c} \left\lvert \phi(y, z) \right\rvert \xi(\mathrm{d} z) \eta(\mathrm{d} y) = \infty$ , which implies $H_{\Lambda}(\eta \mid \xi)$ , or $\int_{\Lambda^c} \left\lvert \phi(x, z) \right\rvert \xi(\mathrm{d} z) = \infty$ , which implies $W(x, \eta + \xi_{\Lambda^c}) = \infty$ . In both cases, identity (3) holds.

We use (2) in conjunction with the following lower bound on $W(x, \eta)$ to upper-bound the partition function under feasible boundary conditions.

Proof. First, we consider the case that $\eta \in \mathcal{N}_{H} \cap \mathcal{N}_{{f}}$ and observe that, by local stability,

\begin{align*} W(x, \eta) = H(\eta + \delta_x) - H(\eta) \ge - L. \end{align*}

Next, suppose that $\eta \in \mathcal{N}_{H} \setminus\mathcal{N}_{{f}}$ , and let $f\,:\, \mathcal{X} \to \overline{\mathbb{R}_{\ge 0}}, y \mapsto \max(0, -\phi(x, y))$ . Note that to prove our claim, it suffices to show that $\int f \mathrm{d} \eta \le L$ . Let $\eta = \sum_{i \in \mathbb{N}} \delta_{x_i}$ for some sequence $x_i \in \mathcal{X}$ . For $n \in \mathbb{N}$ , define $\eta_n = \sum_{i = 1}^{n} \delta_{x_i}$ and note that $\eta_n \to \eta$ setwise monotonically. Therefore, we obtain

\begin{align*} \int_{\mathcal{X}} f(y) \eta(\mathrm{d} y) = \lim_{n \to \infty} \int_{\mathcal{X}} f(y) \eta_n(\mathrm{d} y). \end{align*}

Next, for $n\in\mathbb{N}$ , let $S_n$ be the set of indices $i \in \{1, \dots, n\}$ such that $\phi(x, x_i) \le 0$ , and let $\eta_{S_n} = \sum_{i \in S_n} \delta_{x_i}$ . Observe that $\eta_{S_n} \in \mathcal{N}_{{f}}$ with $\eta_{s_n} \le \eta$ , and, since $\eta \in \mathcal{N}_{H}$ , $H(\eta_{S_n}) < \infty$ . Consequently, by local stability, we have

\begin{align*} \int_{\mathcal{X}} f(y) \eta_n(\mathrm{d} y) = \sum_{i=1}^{n} f(x_i) = - \sum_{i \in S_n} \phi(x, x_i) = -(H(\eta_{S_n} + \delta_x) - H(\eta_{S_n})) \le L . \end{align*}

Taking the limit $n \to \infty$ concludes the proof.

We proceed to show that $\mathcal{N}_{H}$ provides boundary conditions for which finite-volume Gibbs measures are well defined.

Proof. The main idea is to show that, for all $\eta \in \mathcal{N}_{\Lambda}$ , it holds that $H_{\Lambda}(\eta \mid \xi) \ge - C L \eta(\Lambda)$ for some absolute constant C. If $H(\eta) = \infty$ then $H_{\Lambda}(\eta \mid \xi) = \infty$ . Otherwise, we use identity (2) and Lemma 2.4 to obtain

\begin{align*} H_{\Lambda}(\eta \mid \xi) = \frac{1}{2} \int_{\Lambda} W(x, \eta \setminus \delta_x) \eta(\mathrm{d} x) + \int_{\Lambda} W(x, \xi_{\Lambda^c}) \eta(\mathrm{d} x) \ge - \frac{3}{2} L \eta(\Lambda) . \end{align*}

Thus, we get

\begin{align*} \Xi_{\Lambda \mid \xi} (\lambda) \le 1 + \sum_{n = 1}^{\infty} \frac{\lambda^n {e}^{\frac{3}{2} L n} \nu(\Lambda)^n}{n!}, \end{align*}

which proves the claim.

Knowing that $\mu_{\Lambda \mid \xi}$ is well defined for every $\xi \in \mathcal{N}_{\Lambda}$ , (3) can be used to derive the following identity, which was independently proven by Georgii [Reference Georgii15] and Nguyen and Zessin [Reference Xanh and Zessin41], and which is hence known as the GNZ equation. As some of our definitions differ slightly from those of the original papers, we briefly reprove it.

Lemma 2.6. (Finite-volume GNZ equation [Reference Georgii15, Reference Xanh and Zessin41].) Suppose $\phi$ is locally stable with constant L, and let $\Lambda \in \mathcal{B}_{{b}}$ and $\xi \in \mathcal{N}_{H}$ . For all nonnegative measurable $F\,:\, \mathcal{X} \times \mathcal{N} \to \overline{\mathbb{R}_{\ge 0}}$ , it holds that

\begin{align*} \int_{\mathcal{N}_{\Lambda}} \int_{\Lambda} F(x, \eta) \eta(\mathrm{d} x) \mu_{\Lambda \mid \xi}(\mathrm{d} \eta) = \int_{\Lambda} \int_{\mathcal{N}_{\Lambda}} \lambda {e}^{- W(x, \eta + \xi_{\Lambda^c})} F(x, \eta + \delta_x) \mu_{\Lambda \mid \xi}(\mathrm{d} \eta) \nu(\mathrm{d} x). \end{align*}

Proof. Denoting by $Q_{\Lambda}$ a Poisson point process with intensity measure $\nu(\cdot \cap \Lambda)$ , we get

\begin{align*} \int_{\mathcal{N}_{\Lambda}} \int_{\Lambda} F(x, \eta) \eta(\mathrm{d} x) \mu_{\Lambda \mid \xi}(\mathrm{d} \eta) &= \frac{{e}^{\nu(\Lambda)}}{\Xi_{\Lambda \mid \xi} (\lambda)}\int_{\mathcal{N}_{\Lambda}} \int_{\Lambda} F(x, \eta) \eta(\mathrm{d} x) \lambda^{\eta(\Lambda)} {e}^{- H_{\Lambda}(\eta \mid \xi)} Q_{\Lambda}(\mathrm{d} \eta) \\ &= \int_{\mathcal{N}_{\Lambda}} \int_{\Lambda} G(x, \eta) \eta(\mathrm{d} x) Q_{\Lambda}(\mathrm{d} \eta) \end{align*}

for $G(x, \eta) \,:\!=\, \frac{{e}^{\nu(\Lambda)}}{\Xi_{\Lambda \mid \xi} (\lambda)} F(x, \eta) \lambda^{\eta(\Lambda)} {e}^{- H_{\Lambda}(\eta \mid \xi)}$ . Applying the Mecke equation (see [Reference Last and Penrose28, Proposition 4.1]) and the decomposition in (3) then yields

\begin{align*} \int_{\mathcal{N}_{\Lambda}} \int_{\Lambda} &G(x, \eta) \eta(\mathrm{d} x) Q_{\Lambda}(\mathrm{d} \eta)\\ &= \int_{\Lambda} \int_{\mathcal{N}_{\Lambda}} G(x, \eta + \delta_x) Q_{\Lambda}(\mathrm{d} \eta) \nu(\mathrm{d} x) \\ &= \int_{\Lambda} \frac{{e}^{\nu(\Lambda)}}{\Xi_{\Lambda \mid \xi} (\lambda)}\int_{\mathcal{N}_{\Lambda}} \lambda {e}^{- W(x, \eta + \xi_{\Lambda^c})} F(x, \eta + \delta_x) \lambda^{\eta(\Lambda)} {e}^{- H_{\Lambda}(\eta \mid \xi)} Q_{\Lambda}(\mathrm{d} \eta) \nu(\mathrm{d} x) \\ &= \int_{\Lambda} \int_{\mathcal{N}_{\Lambda}} \lambda {e}^{- W(x, \eta + \xi_{\Lambda^c})} F(x, \eta + \delta_x) \mu_{\Lambda \mid \xi}(\mathrm{d} \eta) \nu(\mathrm{d} x), \end{align*}

which concludes the proof.

Next, we introduce a restriction of $\mathcal{N}_{\Lambda}$ to configurations that are consistent with a given boundary condition $\xi \in \mathcal{N}$ by

(4) \begin{align} \mathcal{N}_{\Lambda \mid \xi} \,:\!=\, \{\eta \in \mathcal{N}_{\Lambda} \mid \eta + \xi_{\Lambda^c} \in \mathcal{N}_{H} \} . \end{align}

Using similar arguments as for $\mathcal{N}_{H}$ , it can be proven that $\mathcal{N}_{\Lambda \mid \xi} \in \mathcal{R}$ . We denote the trace of $\mathcal{N}_{\Lambda \mid \xi}$ in $\mathcal{R}$ by $\mathcal{R}_{\Lambda \mid \xi}$ . Rather than studying the finite-volume Gibbs point process $\mu_{\Lambda \mid \xi}$ on $(\mathcal{N}_{\Lambda}, \mathcal{R}_{\Lambda})$ , we will restrict it to $(\mathcal{N}_{\Lambda \mid \xi}, \mathcal{R}_{\Lambda \mid \xi})$ , which is justified for every $\xi \in \mathcal{N}_{H}$ due to the following lemma.

Proof. We show that $H_{\Lambda}(\eta \mid \xi) = \infty$ for all $\eta \in \mathcal{N}_{\Lambda} \setminus \mathcal{N}_{\Lambda \mid \xi}$ . If $H(\eta)=\infty$ this obviously holds. Hence, assume $H(\eta) < \infty$ . By definition of $\mathcal{N}_{\Lambda \mid \xi}$ and the fact that H is hereditary, we can find some $\zeta \in \mathcal{N}_{{f}}$ , $\zeta \le \xi_{\Lambda^c}$ such that $H(\eta + \zeta) = \infty$ . Since $\xi \in \mathcal{N}_{H}$ , it further holds that $H(\zeta) < \infty$ . Next, note that

\begin{align*} H(\eta + \zeta) = H(\eta) + H(\zeta) + \int_{\Lambda} \int_{\Lambda^c} \phi(x, y) \zeta(\mathrm{d} y) \eta(\mathrm{d} x). \end{align*}

Hence, there must be some $x \in S_{\eta}, y \in S_{\zeta}$ such that $\phi(x, y) = \infty$ . But then, since $\zeta \le \xi_{\Lambda^c}$ , we know that $\int_{\Lambda^c} \left\lvert \phi(x, y) \right\rvert \xi(\mathrm{d} y) \eta(\mathrm{d} x) = \infty$ and hence $H_{\Lambda}(\eta \mid \xi) = \infty$ . Thus, we have

\begin{align*} \mu_{\Lambda \mid \xi}(\mathcal{N}_{\Lambda \mid \xi}^c) = \int_{\mathcal{N}_{\Lambda} \setminus\mathcal{N}_{\Lambda \mid \xi}} \lambda^{\eta(\Lambda)} {e}^{-H_{\Lambda}(\eta \mid \xi)} Q_{\Lambda}(\mathrm{d} \eta) = 0, \end{align*}

which concludes the proof.

Finally, we introduce the main lemma that we use to show uniqueness of infinite-volume Gibbs point processes. It is a simple criterion for equality of infinite-volume Gibbs measures based on vanishing total variation distance of local projections of finite-volume Gibbs measures uniformly in the boundary condition. To this end, given a measurable space $(E, \mathcal{E})$ , the total variation distance between two probability measures $\pi_1, \pi_2 \in \mathcal{P}(E, \mathcal{E})$ is defined as

\begin{align*} \left\lvert \pi_1 - \pi_2 \right\rvert_{\mathrm{tv}} = \sup\nolimits_{A \in \mathcal{E}} \left\lvert \pi_1(A) - \pi_2(A) \right\rvert .\end{align*}

Moreover, for two point processes $P, Q \in \mathcal{P}(\mathcal{N}, \mathcal{R})$ and a region $\Lambda \in \mathcal{B}$ , we define the projected total variation distance as

\begin{align*} \left\lvert P - Q \right\rvert_{\Lambda} \,:\!=\, \left\lvert P[\Lambda] - Q[\Lambda] \right\rvert_{\mathrm{tv}} . \end{align*}

We now get the following criterion.

Proof. By Lemma 2.1, it suffices to show that $\mu_1(A) = \mu_2(A)$ for all $A \in \mathcal{R}$ of the form

\begin{align*} A = \{N_{\Delta_1} = m_1, \dots, N_{\Delta_{\ell}} = m_{\ell}\} \end{align*}

for $\ell \in \mathbb{N}$ , $\Delta_1, \dots, \Delta_{\ell} \in \mathcal{B}_{{b}}$ and $m_1, \dots, m_{\ell} \in \mathbb{N}_0$ . By assumption, we can find some $k \in \mathbb{N}$ such that $\Delta_j \subseteq \Lambda_k$ for all $1 \le j \le \ell$ . This implies $\eta(\Delta_j) = \eta_{\Lambda_{k}}(\Delta_j)$ for all $\eta \in \mathcal{N}$ and all $1 \le j \le \ell$ . In particular, for all $n \in \mathbb{N}_0$ , all $\eta \in \mathcal{N}_{\Lambda_{k + n}}$ and all $\xi \in \mathcal{N}$ , we have that $\eta + \xi_{\mathcal{X} \setminus \Lambda_{k + n}} \in A$ if and only if $\eta_{\Lambda_k} \in A$ . By the DLR formalism, we obtain that, for all $n \in \mathbb{N}_{0}$ ,

\begin{align*} \mu_1(A) = \int_{\mathcal{N}} \int_{\Lambda_{k + n}} \unicode{x1D7D9}_{A}(\eta_{\Lambda_k}) \mu_{\Lambda_{k + n} \mid \xi}(\mathrm{d} \eta) \mu_1(\mathrm{d} \xi) = \int_{\mathcal{N}} \mu_{\Lambda_{k + n} \mid \xi}[\Lambda_k](A) \mu_1(\mathrm{d} \xi) . \end{align*}

Applying the same argument to $\mu_2$ yields

\begin{align*} \left\lvert \mu_1(A) - \mu_2(A) \right\rvert &\le \int_{\mathcal{N}} \int_{\mathcal{N}} \left\lvert \mu_{\Lambda_{k + n} \mid \xi_1}[\Lambda_k](A) - \mu_{\Lambda_{k + n} \mid \xi_2}[\Lambda_k](A) \right\rvert \mu_1(\mathrm{d} \xi_1) \mu_2(\mathrm{d} \xi_2) \\ &\le \int_{\mathcal{N}} \int_{\mathcal{N}} \left\lvert \mu_{\Lambda_{k + n} \mid \xi_1} - \mu_{\Lambda_{k + n} \mid \xi_2} \right\rvert_{\Lambda_k} \mu_1(\mathrm{d} \xi_1) \mu_2(\mathrm{d} \xi_2) \\ &\le \mathop{\mathrm{ess} \sup}\limits_{\xi_1 \sim \mu_1, \xi_2 \sim \mu_2} \left\lvert \mu_{\Lambda_{k + n} \mid \xi_1} - \mu_{\Lambda_{k + n} \mid \xi_2} \right\rvert_{\Lambda_k} . \end{align*}

Since this holds for all n, taking the limit $n \to \infty$ shows $\mu_1(A) = \mu_2(A)$ , concluding the proof.

3.1. Terminology and notation

We recall basic terminology related to kernel and transition groups. Given a measurable space $(E, \mathcal{E})$ , a kernel on $(E, \mathcal{E})$ is a map $K\,:\, E \times \mathcal{E} \to \overline{\mathbb{R}_{\ge 0}}$ such that:

1. (i). for all $x \in E$ , $K(x, \cdot)$ is a measure on $(E, \mathcal{E})$ ;

2. (ii). for all $A \in \mathcal{E}$ , $K(\cdot, A)$ is measurable.

Further, we say that a kernel K is:

• • finite if $K(x, \cdot)$ is a finite measure for all $x \in E$ ;

• • sub-stochastic if $K(x, E) \le 1$ for all $x \in E$ ; and

• • stochastic if $K(x, E) = 1$ for all $x \in E$ .

In particular, if K is stochastic then $K(x, \cdot)$ is a probability measure for all $x \in E$ .

Occasionally, it will be useful to associate a kernel with an operator on the space of nonnegative measurable functions $f\,:\, E \to \overline{\mathbb{R}_{\ge 0}}$ . More precisely, we define the function $Kf\,:\, E \to \overline{\mathbb{R}_{\ge 0}}$ by setting

\begin{align*} (Kf)(x) \,:\!=\, \int_E f(y) K(x, \mathrm{d} y) \end{align*}

for $x \in E$ . Note that Kf is again measurable (this can be seen by taking monotone limits of simple functions).

Based on these definitions, a sub-Markov transition family on $(E, \mathcal{E})$ is a collection $(P_t)_{t \in \mathbb{R}_{\ge 0}}$ such that:

1. (i). for every $t \in \mathbb{R}_{\ge 0}$ , $P_t$ is a sub-stochastic kernel;

2. (ii). for all $x \in E$ and $A \in \mathcal{E}$ , we have $P_0(x, A) = \delta_x (A)$ ;

3. (iii). $(P_t)_{t \in \mathbb{R}_{\ge 0}}$ satisfies the Chapman–Kolmogorov equation, i.e. for all $t, s \in \mathbb{R}_{\ge 0}$ and all $x \in E$ , $A \in \mathcal{E}$ , it holds that

   \begin{align*} P_{t+s}(x, A) = \int_E P_s (y, A) P_t(x, \mathrm{d} y) . \end{align*}

When talking about a sub-Markov transition family, we will usually drop the index set and simply write $(P_t)$ instead of $(P_t)_{t \in \mathbb{R}_{\ge 0}}$ . Further, we say that $(P_t)$ is a Markov transition family if $P_t$ is stochastic for every $t \in \mathbb{R}_{\ge 0}$ .

3.2. Jump kernel and associated transition families

We proceed by introducing the basic terminology related to Markov jump processes with values in a general measurable space $(E, \mathcal{E})$ . We will always assume that $\mathcal{E}$ contains all singletons (i.e. $\{x\} \in \mathcal{E}$ for all $x \in E$ ).

Throughout the paper, we mostly characterize Markov jump processes by their jump kernel. A function $K\,:\, E \times \mathcal{E} \to \mathbb{R}_{\ge 0}$ is a jump kernel on $(E, \mathcal{E})$ if:

1. (i). K is a finite kernel;

2. (ii). for all $x \in E$ , we have $K(x, \{x\}) = 0$ .

Given a jump kernel K, we call $\kappa: E \to \mathbb{R}_{\ge 0}, x \mapsto K(x, E)$ the associated rate function. Further, note that $\Pi\,:\, E \times \mathcal{E} \to [0, 1]$ with

(5) \begin{align} \Pi(x, A) \,:\!=\, \begin{cases} \frac{1}{\kappa(x)} \cdot K(x, A) &\text{ if $\kappa(x) > 0$,} \\ \delta_x(A) &\text{ otherwise}, \end{cases}\end{align}

defines a Markov kernel on $(E, \mathcal{E})$ , which we call the associated transition kernel.

Given a jump kernel K with jump rate function $\kappa$ , we recursively define a sub-Markov transition family $(P^{(n)}_{t})$ for every $n \in \mathbb{N}_{0}$ as follows. For every $x \in E$ , $A \in \mathcal{E}$ and $t \in \mathbb{R}_{\ge 0}$ we set

(6) \begin{gather} P_{t}^{(0)}(x, A) = {e}^{- \kappa(x) t} \cdot \delta_{x}(A). \end{gather}

Further, for $n \ge 1$ , we set

(7) \begin{gather} P_{t}^{(n)} (x, A) = {e}^{-\kappa(x) t} \cdot \delta_{x}(A) + \int_{0}^{t} {e}^{- \kappa(x) s} \int_{E} P_{t-s}^{(n-1)}(y, A) K(x, \mathrm{d} y) \mathrm{d} s . \end{gather}

The following properties can easily be checked via induction over n (see, for example, [Reference Feller11]):

1. (i). For every $n \in \mathbb{N}_{0}$ , $(P_{t}^{(n)})$ is a well-defined sub-Markov transition family, and for every $A \in \mathcal{E}$ the map $(t, x) \mapsto P_{t}^{(n)}(x, A)$ is measurable.

2. (ii). For every $x \in E$ , $A \in \mathcal{E}$ and $t \in \mathbb{R}_{\ge 0}$ , it holds that $P_{t}^{(n)}(x, A)$ is nondecreasing in n.

It immediately follows by monotone convergence that $\lim_{n \to \infty} P_{t}^{(n)}(x, A)=P_{t}(x, A)$ exists and defines a sub-Markov transition family $(P_t)$ . We call $(P_t)$ the sub-Markov transition family associated to K. Note that $(P_t)$ is in fact Markov whenever $P_t(x, E) = 1$ for all $x \in E$ and $t \in \mathbb{R}_{\ge 0}$ . In that case, we say that the jump kernel K is nonexplosive and simply call $(P_t)$ the transition family associated to K.

To check if a given jump kernel is nonexplosive, we are going to use the following simple criterion.

3.3. Stationary distribution and reversibility

Given a Markov transition family $(P_t)$ on $(E, \mathcal{E})$ , we call a probability distribution $\pi$ on $(E, \mathcal{E})$ a stationary distribution (or invariant probability measure) for $(P_t)$ if, for all $t \in \mathbb{R}_{\ge 0}$ and $A \in \mathcal{E}$ , it holds that

\begin{align*} \int_{E} P_t(x, A) \pi(\mathrm{d} x) = \pi(A) . \end{align*}

A sufficient criterion for $\pi$ to be a stationary distribution is that, for all $t \in \mathbb{R}_{\ge 0}$ and $A, B \in \mathcal{E}$ , it holds that

\begin{align*} \int_{A} P_t(x, B) \pi(\mathrm{d} x) = \int_{B} P_t(x, A) \pi(\mathrm{d} x), \end{align*}

in which case we say that $(P_t)$ is reversible with respect to $\pi$ . Note that the definition of reversibility extends to sub-Markov transition families. However, it only implies that $\pi$ is an invariant measure if $(P_t)$ is stochastic (hence Markov).

If $(P_t)$ is generated by a jump kernel K, reversible measures can be identified using K. This is given by the following lemma, which we believe was first proven by Chen [Reference Chen4].

Lemma 3.2. (Proposition 2 of [Reference Serfozo40].) Let K be a jump kernel on $(E, \mathcal{E})$ with associated (sub-)Markov transition family $(P_t)$ . If a probability measure $\pi$ on $(E, \mathcal{E})$ satisfies

\begin{align*} \int_{A} K(x, B) \pi(\mathrm{d} x) = \int_{B} K(x, A) \pi(\mathrm{d} x) \end{align*}

for all $A, B \in \mathcal{E}$ , then $(P_t)$ is reversible with respect to $\pi$ . Further, if K is nonexplosive, then $\pi$ is invariant under $(P_t)$ .

3.4. Coupling and convergence

Given probability measures $\pi_1, \pi_2$ on $(E_1, \mathcal{E}_1)$ and $(E_2, \mathcal{E}_2)$ , a coupling of $\pi_1, \pi_2$ is a probability measure $\pi$ on the product space $(E_1 \times E_2, \mathcal{E}_1 \otimes \mathcal{E}_2)$ with $\pi(A_1 \times E_2) = \pi_1(A_1)$ for all $A_1 \in \mathcal{E}_1$ and $\pi(E_1 \times A_2) = \pi_2(A_2)$ for all $A_2 \in \mathcal{E}_2$ . A useful aspect of couplings is that they provide a way of bounding the total variation distance between probability distributions.

Note that the definition of total variation distance in [Reference den Hollander19] differs by a factor of 2 from the definition that is used in this paper. The notion of a coupling extends directly to Markov transition families as follows.

We wish to construct couplings between (jump) Markov transition families based on their jump kernels. Following the ideas in [Reference Chen5], we introduce the following definition.

Definition 3.2. Let $K_1, K_2$ be jump kernels on $(E_1, \mathcal{E}_1)$ and $(E_2, \mathcal{E}_2)$ . We call a jump kernel K on $(E_1 \times E_2, \mathcal{E}_1 \otimes \mathcal{E}_2)$ a coupling of $K_1$ and $K_2$ if, for all nonnegative bounded measurable functions $f_1\,:\, E_1 \to \mathbb{R}_{\ge 0}$ , $f_2\,:\, E_2 \to \mathbb{R}_{\ge 0}$ and all $x_1 \in E, x_2 \in E_2$ , it holds that

\begin{align*} \int_{E_1 \times E_2} (f_1(y_1) - f_1(x_1)) K((x_1, x_2), \mathrm{d} y_1 \times \mathrm{d} y_2) = (K_1 f_1)(x_1) - \kappa_1(x_1) \cdot f_1(x_1),\\ \int_{E_1 \times E_2} (f_2(y_2) - f_2(x_2)) K((x_1, x_2), \mathrm{d} y_1 \times \mathrm{d} y_2) = (K_2 f_2)(x_2) - \kappa_2(x_2) \cdot f_2(x_2). \end{align*}

The following result relates the two notions of a coupling that are given in Definitions 3.1 and 3.2.

A central part of this paper will be to bound the total variation distance of certain jump Markov transition families from their stationary distribution (provided it exists) after a sufficiently large time span t. The following technical lemma will be useful for that.

Lemma 3.5. Let K be a jump kernel on $(E, \mathcal{E})$ with jump rate function $\kappa$ and (sub-)Markov transition family $(P_t)$ . Suppose $f\,:\, E \to \mathbb{R}_{\ge 0}$ is a nonnegative measurable function such that there exists some $\delta > 0$ such that, for all $x \in E$ , it holds that

\begin{align*} (Kf)(x) \le (\kappa(x) - \delta) \cdot f(x) . \end{align*}

Then, for all $x \in E$ and $t \in \mathbb{R}_{\ge 0}$ , we have

\begin{align*} (P_{t}f)(x) \le {e}^{- \delta t} f(x) . \end{align*}

Proof. We show by induction that, for all $n \in \mathbb{N}_{0}$ , it holds that

\begin{align*} (P_{t}^{(n)} f)(x) \le {e}^{- \delta t} f(x) , \end{align*}

where $(P^{(n)}_{t})$ is as defined in (6) and (7). The desired result then follows from the fact that, for all $t \in \mathbb{R}_{\ge 0}$ and $x \in E$ , $P_t^{(n)}(x, \cdot) \to P_t(x, \cdot)$ setwise and monotonically as $n \to \infty$ .

For the base case $n=0$ , we have by (6) that

\begin{align*} (P_{t}^{(0)} f)(x) = {e}^{- \kappa(x) t} f(x) . \end{align*}

Hence, we need to show ${e}^{- \kappa(x) t} f(x) \le {e}^{- \delta t} f(x)$ for all $x \in E$ . Since this holds trivially if $f(x) = 0$ , we focus on $x \in E$ such that $f(x) > 0$ . Because f is nonnegative, our assumptions imply $0 \le (Kf)(x) \le (\kappa(x) - \delta) \cdot f(x)$ . Thus, we have $\delta \le \kappa(x)$ for all $x \in E$ with $f(x) > 0$ and the desired inequality follows.

For the induction step, combining (7) and the induction hypothesis yields

\begin{align*} (P_{t}^{(n)} f)(x) &= {e}^{-\kappa(x) t} \cdot f(x) + \int_{0}^{t} {e}^{- \kappa(x) s} \int_{E} (P_{t-s}^{(n-1)} f)(y) K(x, \mathrm{d} y) \mathrm{d} s \\ &\le {e}^{-\kappa(x) t} \cdot f(x) + \int_{0}^{t} {e}^{- \kappa(x) s} \int_{E} {e}^{-\delta \cdot (t - s)} f(y) K(x, \mathrm{d} y) \mathrm{d} s \\ &= {e}^{-\kappa(x) t} \cdot f(x) + {e}^{-\delta t} \int_{0}^{t} {e}^{- (\kappa(x) - \delta) \cdot s} (Kf)(x) \mathrm{d} s . \end{align*}

Using $(Kf)(x) \le (\kappa(x) - \delta) \cdot f(x)$ , we obtain

\begin{align*} (P_{t}^{(n)} f)(x) &\le \left( {e}^{-\kappa(x) t} + {e}^{-\delta t} \int_{0}^{t} {e}^{- (\kappa(x) - \delta) \cdot s} (\kappa(x) - \delta) \mathrm{d} s\right) \cdot f(x) \\ &= \left( {e}^{-\kappa(x) t} + {e}^{-\delta t} \left(1 - {e}^{-(\kappa(x) - \delta) \cdot t} \right) \right) \cdot f(x) \\ &= {e}^{-\delta t} f(x) , \end{align*}

which concludes the induction step and proves the claim.

We can now combine Lemma 3.5 with a suitable coupling of a nonexplosive jump kernel to derive information about the speed of convergence of the associated Markov transition family to its stationary distribution.

Lemma 3.6. Let K be a nonexplosive jump kernel on $(E, \mathcal{E})$ , and assume that the associated Markov transition group $(P_t)$ has a stationary distribution $\pi$ . Suppose there is a coupling $\tilde{K}$ of K with itself (in the sense of Definition 3.2) and a nonnegative measurable function $f\,:\, E \times E \to \mathbb{R}_{\ge 0}$ such that the following hold.

1. (i). There exists some $c > 0$ such that $f(x, y) \ge c$ if and only if $x \neq y$ .

2. (ii). There exists some $\delta > 0$ such that, for all $x, y \in E$ ,

   \begin{align*} (\tilde{K} f)(x, y) \le (\tilde{\kappa}(x, y) - \delta) \cdot f(x, y) , \end{align*}
   where $\tilde{\kappa}$ is the rate function of $\tilde{K}$ .

Then, for all $x \in E$ and all $t \in \mathbb{R}_{\ge 0}$ , it holds that

\begin{align*} \left\lvert P_t (x, \cdot) - \pi \right\rvert_{\mathrm{tv}} \le \frac{1}{c} \cdot {e}^{- \delta t} \int_{E} f(x, y) \pi(\mathrm{d} y). \end{align*}

Proof. By Lemma 3.4 we know that $\tilde{K}$ is nonexplosive and that the associated Markov transition family $(Q_t)$ is a coupling of $(P_t)$ with itself in the sense of Definition 3.1. Further, by assumption 3.6 and Lemma 3.5, we have, for all $x, y \in E$ and all $t \in \mathbb{R}_{\ge 0}$ , that $(Q_t f)(x, y) \le {e}^{- \delta t} f(x, y)$ .

Next, for $x \in E$ , $t \in \mathbb{R}_{\ge 0}$ and $A \in \mathcal{E} \otimes \mathcal{E}$ define

\begin{align*} Q_{x, t}(A) \,:\!=\, \int_{E \times E} Q_t(z, A) (\delta_x \otimes \pi)(\mathrm{d} z), \end{align*}

where $\delta_x \otimes \pi$ is the product probability measure of $\delta_x$ and $\pi$ . Note that, for all x and t, it holds that $Q_{x, t}$ is a probability measure on $(E \times E, \mathcal{E} \otimes \mathcal{E})$ . Using Fubini’s theorem and the fact that $(Q_t)$ is a coupling of $(P_t)$ with itself, we obtain that for all $A \in \mathcal{E}$ ,

\begin{align*} Q_{x, t}(A \times E) &= \int_E \int_E Q_t((z_1, z_2), A \times E) \delta_x(\mathrm{d} z_1) \pi(\mathrm{d} z_2) \\ &= \int_E \int_E P_t(z_1, A) \delta_x(\mathrm{d} z_1) \pi(\mathrm{d} z_2) \\ &= \int_E P_t(z_1, A) \delta_x (\mathrm{d} z_1) \\ & = P_t(x, A). \end{align*}

Further, since $\pi$ is invariant with respect to $(P_t)$ , analogously we have

\begin{align*} Q_{x, t}(E \times A) = \int_E P_t(z_2, A) \pi(\mathrm{d} z_2) = \pi(A) . \end{align*}

This shows that $Q_{x, t}$ is a coupling of $P_t(x, \cdot)$ and $\pi$ . Note that, by assumption (i), we may express the diagonal as $D = \{(x, x) \mid x \in E\} = \{(x, y) \in E \times E \mid f(x, y) < c\}$ and, since f is measurable, $D \in \mathcal{E} \otimes \mathcal{E}$ . By Lemma 3.3, it suffices to bound $Q_{x, t} (D^c)$ . Applying Markov’s inequality, we get

\begin{align*} \left\lvert P_t(x, \cdot) - \pi \right\rvert_{\mathrm{tv}} \le Q_{x, t}(D^c) \le \frac{1}{c} \int_{E \times E} f(z) Q_{x, t}(\mathrm{d} z) = \frac{1}{c} \int_{E \times E} (Q_t f)(z) (\delta_x \otimes \pi) (\mathrm{d} z). \end{align*}

Finally, since $(Q_t f)(z) \le {e}^{- \delta t} f(z)$ , applying Fubini’s theorem yields

\begin{align*} \left\lvert P_t(x, \cdot) - \pi \right\rvert_{\mathrm{tv}} \le \frac{1}{c} \cdot {e}^{- \delta t} \int_{E \times E} f(z) (\delta_x \otimes \pi) (\mathrm{d} z) = \frac{1}{c} \cdot {e}^{- \delta t} \int_{E} f(x, y) \pi (\mathrm{d} y) , \end{align*}

which concludes the proof.

3.5. Markov process associated with a jump kernel

We proceed by discussing in what sense we can associate a stochastic process (more precisely a Markov jump process) on $(E, \mathcal{E})$ with a given jump kernel. By a stochastic process we mean a tuple $(\Omega, \mathcal{F}, \mathbb{P}, (X_t)_{t \in \mathbb{R}_{\ge 0}})$ where $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space and $(X_t)_{t \in \mathbb{R}_{\ge 0}}$ is a collection of random variables $X_t\,:\, \Omega \to E$ . Since we will always use $\mathbb{R}_{\ge 0}$ as the index set for $(X_t)$ , we will drop it from now on to simplify notation (the same will hold for filtrations as introduced in the following). For every $\omega \in \Omega$ , we call the function $t \mapsto X_t(\omega)$ the sample path of $\omega$ . We call a stochastic process measurable if the map $(t, \omega) \mapsto X_t(\omega)$ is measurable with respect to $\mathcal{B}_{\mathbb{R}_{\ge 0}} \otimes \mathcal{F}$ and $\mathcal{E}$ , where $\mathcal{B}_{\mathbb{R}_{\ge 0}}$ is the Borel $\sigma$ -field on $\mathbb{R}_{\ge 0}$ .

A filtration on $(\Omega, \mathcal{F}, \mathbb{P}, (X_t))$ is a sequence of $\sigma$ -fields $(\mathcal{F}_t)$ such that $\mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F}$ for all $s, t \in \mathbb{R}_{\ge 0}$ , $s \le t$ . We call a random variable $T\,:\, \Omega \to \overline{\mathbb{R}_{\ge 0}}$ a stopping time with respect to a filtration $(\mathcal{F}_t)$ if, for all $t \in \mathbb{R}_{\ge 0}$ , the event $\{T \le t\}$ is measurable with respect to $\mathcal{F}_t$ . Given a stopping time T with respect to some filtration $(\mathcal{F}_t)$ , we define $\mathcal{F}_T \,:\!=\, \{A \in \mathcal{F} \mid \forall \ t \in \mathbb{R}_{\ge 0}\,:\, A \cap \{T \le t\} \in \mathcal{F}_t\}$ . Further, we say that a stochastic process is progressive with respect to a filtration $(\mathcal{F}_t)$ if, for all $t \in \mathbb{R}_{\ge 0}$ , it holds that $(s, \omega) \mapsto X_s(\omega)$ as a function from $[0, t] \times \Omega$ into E is measurable with respect to $\mathcal{B}_{[0, t]} \otimes \mathcal{F}_t$ and $\mathcal{E}$ . A particularly important filtration is the natural filtration $(\mathcal{F}^{X}_{t})$ , which is given by $\mathcal{F}^{X}_t \,:\!=\, \sigma(X_s \mid s \in [0, t])$ .

Finally, given a probability measure $\mu$ and a Markov transition family $(P_t)$ on $(E, \mathcal{E})$ , we say that $(\Omega, \mathcal{F}, \mathbb{P},(X_t))$ is a Markov process with initial distribution $\mu$ and transition function $(P_t)$ if:

1. (i). for all $A \in \mathcal{E}$ , we have $\mathbb{P}[X_0 \in A] = \mu(A)$ ;

2. (ii). for all $t, s \in \mathbb{R}_{\ge 0}$ and all $A \in \mathcal{E}$ , it holds that $P_t(X_s, A)$ is a version of the conditional probability $\mathbb{P}[X_{s + t} \in A \mid \mathcal{F}^{X}_{s}]$ .

Note that, for every $t \in \mathbb{R}_{\ge 0}$ and $A \in \mathcal{E}$ , it holds that $x \mapsto P_t(x, A)$ is measurable. Hence, it also holds that $\omega \mapsto P_t(X_s(\omega), A)$ is measurable with respect to $\sigma(X_s)$ , and thus

\begin{align*} \mathbb{P}[X_{s + t} \in A \mid \mathcal{F}^{X}_{s}] = \mathbb{P}[X_{s + t} \in A \mid X_s] \qquad \quad\text{$\mathbb{P}$-almost surely.} \end{align*}

This is usually referred to as the Markov property.

It turns out that, given a nonexplosive jump kernel K and associated transition family $(P_t)$ , we can construct a particularly nice version of the Markov process with transition function $(P_t)$ for any given initial distribution (see, for example, [Reference Blumenthal and Getoor3, Section 1.12] and [Reference Jansen23]).

We will make use of the following two lemmas about hitting times for the process that is given by Theorem 3.1. The first lemma formalizes the following intuition: If, for two sets $A, B \in \mathcal{E}$ , the rate of jumping from outside of $A \cup B$ into B is 0, then the process must (almost surely) go through A to ever reach B.

Proof. Note that by Theorem 3.1 (vii) it holds that

\begin{align*} \mathbb{P}_{\mu}[T_B \le T_A, T_B < \infty] &\le \mathbb{P}_{\mu}[\exists m \in \mathbb{N}_0\,:\, \tau_m < \infty, Z_m \in B, \forall k \le m\,:\, Z_k \notin A \cup B] \\ &\le \mathbb{P}_{\mu}[\exists m \in \mathbb{N}\,:\, Z_m \in B, Z_{m-1} \notin A \cup B] + \mathbb{P}_{\mu}[Z_0 \in B] . \end{align*}

By Theorem 3.1 (vi) and the assumption that $\mu(B) = 0$ it holds that $\mathbb{P}_{\mu}[Z_0 \in B] = 0$ and

\begin{align*} \mathbb{P}_{\mu}[Z_m \in B, Z_{m-1} \notin A \cup B] = \int_{(A \cup B)^c} \int_{(A \cup B)^c} \Pi(y, B) \Pi^{m-1}(x, \mathrm{d} y) \mu(\mathrm{d} x). \end{align*}

As we assume $K(y, B) = 0$ for $y \notin A \cup B$ , it holds that, inside the integral, $\Pi(y, B) = 0$ , and taking a union bound over $m \in \mathbb{N}$ concludes the proof.

The second lemma states that, if we consider a sequence of regions $A_1, \dots, A_k \in \mathcal{E}$ , each of which we enter with rate at most $\rho$ , then the probability of hitting those sets in order and before some time t is upper-bounded by the probability that a Poisson random variable with rate $\rho t$ is at least k. Before proving this, we show the following helpful bound on the transition family.

Proof. We show that the statement holds for each of the recursively defined sub-Markov families $(P^{(n)}_t)$ (see (6) and (7)) via induction on n. The result follows then from taking $n \to \infty$ . For $n = 0$ , the statement is trivial as $P^{(0)}_t(x, A) = {e}^{- \kappa(x) \cdot t} \delta_{x}(A) = 0$ for all $x \notin A$ . Suppose the statement holds for some $n \ge 0$ . Then, we use the fact that $P_t^{(n)}$ is sub-stochastic to obtain

\begin{align*} P_t^{(n+1)}(x, A) &= P_t^{(0)}(x, A) + \int_0^t {e}^{- \kappa(x) \cdot s} \int_{E} P_{t-s}^{(n)}(y, A) K(x, \mathrm{d} y) \mathrm{d} s \\ &\le \int_0^t {e}^{- \kappa(x) \cdot s} \cdot [\rho \cdot (t - s) \cdot K(x, A^c) + \rho] \mathrm{d} s \\ &\le \rho \int_0^t {e}^{- \kappa(x) \cdot s} ( 1 - \kappa(x) \cdot s) \mathrm{d} s + \rho \cdot t \int_0^t {e}^{- \kappa(x) \cdot s} \kappa(x) \mathrm{d} s \\ &= \rho \cdot t . \end{align*}

We now use Claim 3.1 to prove the following lemma.

Lemma 3.8. Let K be a nonexplosive jump kernel and $\mu$ be a probability distribution on $(E, \mathcal{E})$ . Let $(\Omega, \mathcal{F}, \mathbb{P}_{\mu}, (X_t))$ be the Markov process associated with K and $\mu$ as above. Suppose $A_1, \dots, A_k$ are such that

1. (i). For all $i \in [k]$ , we have $\mu(A_i) = 0$ .

2. (ii). There is some $\rho \in \mathbb{R}_{\ge 0}$ such that, for all $i \in [k]$ and $x \notin A_i$ , it holds that $K(x, A_i) \le \rho$ .

Set $T_i \,:\!=\, \inf\{t \in \mathbb{R}_{\ge 0} \mid X_t \in A_i\}$ . Then, for all $t \in \mathbb{R}_{\ge 0}$ , it holds that

\begin{align*} \mathbb{P}_{\mu}[T_1 < T_2 < \dots < T_k \le t] \le 1 - F_{\rho t}(k-1), \end{align*}

where $F_{\rho t}$ is the cumulative distribution function of a Poisson distribution with rate $\rho t$ .

Proof. To simplify notation, set $T_0 \,:\!=\, 0$ . For $i \in [k]$ , define

\begin{align*} S_i \,:\!=\, \begin{cases} T_{i} - T_{i - 1} &\text{ if } T_{i} > T_{i-1}, \\ \infty &\text{ otherwise, } \end{cases} \end{align*}

and $\zeta_i = \sum_{j=1}^{i} S_j$ , and note that

\begin{align*} \mathbb{P}_{\mu}[T_1 < T_2 < \dots < T_k \le t] = \mathbb{P}_{\mu}[\zeta_k \le t] . \end{align*}

The following claim is key for proving this lemma.

Next, let $G_{i, \rho}$ be the cumulative distribution function of an Erlang distribution with shape i and rate $\rho$ . We will use Claim 3.2 to argue inductively that $\mathbb{P}_{\mu}[\zeta_i \le t] \le G_{i, \rho}(t)$ . For $i = 1$ , observe that $\zeta_1 = S_1$ and by Claim 3.2 we have

\begin{align*} \mathbb{P}_{\mu}[S_1 \le t] \le 1 - {e}^{-\rho t} = G_{1, \rho}(t). \end{align*}

For the induction step, we use Claim 3.2 and the fact that $\zeta_i$ is $\mathcal{F}^{X}_{T_i}$ -measurable to obtain

\begin{align*} \mathbb{P}_{\mu}[\zeta_{i+1} \le t] = 1 - \mathbb{E}_{\mu}[\mathbb{P}_{\mu}[S_{i+1} > t - \zeta_{i} \mid \zeta_{i}]] \le 1 - \mathbb{E}_{\mu}[\unicode{x1D7D9}_{\{\zeta_i > t\}} + \unicode{x1D7D9}_{\{\zeta_i \le t\}} \cdot {e}^{- \rho (t - \zeta_i)}]. \end{align*}

Further, by the induction hypothesis and the fact that $x \mapsto \unicode{x1D7D9}_{x > t} + \unicode{x1D7D9}_{x \le t} \cdot {e}^{-\rho (t - x)}$ is nondecreasing, we have

\begin{align*} \mathbb{E}_{\mu}[\unicode{x1D7D9}_{\{\zeta_i > t\}} + \unicode{x1D7D9}_{\{\zeta_i \le t\}} \cdot {e}^{- \rho (t - \zeta_i)}] \ge 1 - G_{i, \rho}(t) + {e}^{-\rho t} \int_{0}^{t} {e}^{\rho x} G_{i, \rho}(\mathrm{d} x) = 1 - G_{i+1, \rho}(t). \end{align*}

Hence, we have $\mathbb{P}_{\mu}[\zeta_{i+1} \le t] \le G_{i+1, \rho}(t)$ , which concludes the induction. Finally, observing that $G_{k, \rho}(t) = 1 - F_{\rho t}(k - 1)$ concludes the proof.

Proof of Claim 3.2. Note that

\begin{align*} \mathbb{P}_{\mu}[S_i > s \mid \mathcal{F}^{X}_{T_{i-1}}] = \unicode{x1D7D9}_{\{T_i \le T_{i-1}\}} + \mathbb{P}_{\mu}[T_{i} > T_{i-1} + s \mid \mathcal{F}^{X}_{T_{i-1}}]. \end{align*}

Hence, our main concern is to lower-bound $\mathbb{P}_{\mu}[T_{i} > T_{i-1} + s \mid \mathcal{F}^{X}_{T_{i-1}}]$ . To this end, for every $n \in \mathbb{N}$ , we consider the event $D^{(s)}_n \,:\!=\, \{T_i > T_{i-1}\} \cap \big\{\forall k \in \{0, \dots, n\}\,:\, X_{T_{i-1} + \frac{k s}{n}} \notin A_i\big\}$ . By Theorem 3.1 (i), we know that $(X_t)$ has piecewise constant sample paths. Thus, $\unicode{x1D7D9}_{D^{(s)}_n} \to \unicode{x1D7D9}_{\{T_i > T_{i-1} + s\}}$ pointwise as $n \to \infty$ , and $\lim_{n \to \infty} \mathbb{P}_{\mu}[D^{(s)}_{n} \mid \mathcal{F}^{X}_{T_{i-1}}] = \mathbb{P}_{\mu}[T_{i} > T_{i-1} + s \mid \mathcal{F}^{X}_{T_{i-1}}]$ by dominated convergence. Further, using the strong Markov property from Theorem 3.1 (iv) and Claim 3.1, we obtain

\begin{align*} \mathbb{P}_{\mu}\big[D^{(s)}_{n} \mid \mathcal{F}^{X}_{T_{i-1}} \big] &= \mathbb{E}_{\mu}\big[\unicode{x1D7D9}_{D^{(s - s/n)}_{n-1}} \cdot \mathbb{P}_{\mu}[X_{T_{i-1} + s} \notin A_i \mid \mathcal{F}^{X}_{T_{i-1} + \frac{n-1}{n}s}] \mid \mathcal{F}^{X}_{T_{i-1}}\big] \\ &= \mathbb{E}_{\mu}\big[\unicode{x1D7D9}_{D^{(s - s/n)}_{n-1}} \cdot P_{s/n}(X_{T_{i-1} + \frac{n-1}{n}s}, A_i^c) \mid \mathcal{F}^{X}_{T_{i-1}}\big] \\ &\ge \mathbb{P}_{\mu}\big[D^{(s - s/n)}_{n-1} \mid \mathcal{F}^{X}_{T_{i-1}}\big] \cdot \left(1 - \frac{\rho s}{n}\right) . \end{align*}

Applying this argument recursively and taking the limit $n \to \infty$ yields

\begin{align*} \mathbb{P}_{\mu}[T_{i} > T_{i-1} + s \mid \mathcal{F}^{X}_{T_{i-1}}] \ge \unicode{x1D7D9}_{\{T_i > T_{i - 1}\}} \cdot \lim_{n \to \infty} \left(1 - \frac{\rho s}{n}\right)^n = \unicode{x1D7D9}_{\{T_i > T_{i - 1}\}} \cdot {e}^{- \rho s}. \end{align*}

Hence, we have

\begin{align*} \mathbb{P}_{\mu}[S_i > s \mid \mathcal{F}^{X}_{T_{i-1}}] \ge \unicode{x1D7D9}_{\{T_i \le T_{i-1}\}} + \unicode{x1D7D9}_{\{T_i > T_{i - 1}\}} \cdot {e}^{- \rho s} \ge {e}^{- \rho s} . \end{align*}

4.1. Identity coupling

We start by constructing a coupling for the jump kernels $K_{\Lambda \mid \xi}, K_{\Lambda \mid \zeta}$ for any $\Lambda \in \mathcal{B}_{{b}}$ and $\xi, \zeta \in \mathcal{N}_{H}$ , which we will call the identity coupling. We remark that a similar coupling for spatial birth–death processes has been studied previously [Reference Schuhmacher and Stucki39] for applying Stein’s method to Gibbs point processes.

Lemma 4.2. Let $\Lambda \in \mathcal{B}_{{b}}$ and $\xi, \zeta \in \mathcal{N}_{H}$ , and for every $\eta_1 \in \mathcal{N}_{\Lambda \mid \xi}$ , $ \eta_2 \in \mathcal{N}_{\Lambda \mid \zeta}$ and $A \in \mathcal{R}_{\Lambda \mid \xi} \otimes \mathcal{R}_{\Lambda \mid \zeta}$ define

\begin{align*} D_1((\eta_1, \eta_2), A) &\,:\!=\,\! \int_{\Lambda} \unicode{x1D7D9}_{A}(\eta_1 \setminus \delta_x, \eta_2) (\eta_1 \setminus \eta_2)(\mathrm{d} x), \\ D_2((\eta_1, \eta_2), A) &\,:\!=\,\! \int_{\Lambda} \unicode{x1D7D9}_{A}(\eta_1, \eta_2 \setminus \delta_x) (\eta_2 \setminus \eta_1)(\mathrm{d} x), \\ D_{\cap}((\eta_1, \eta_2), A) &\,:\!=\,\! \int_{\Lambda} \unicode{x1D7D9}_{A}(\eta_1 \setminus \delta_x, \eta_2 \setminus \delta_x) (\eta_1 \cap \eta_2)(\mathrm{d} x), \\ B_{1}((\eta_1, \eta_2), A) &\,:\!=\, \!\lambda \int_{\Lambda} \max\left(0, {e}^{-W(x, \eta_1 + \xi_{\Lambda^c})} - {e}^{-W(x, \eta_2 + \zeta_{\Lambda^c})}\right) \cdot \unicode{x1D7D9}_{A}(\eta_1 + \delta_x, \eta_2)\nu(\mathrm{d} x), \\ B_{2}((\eta_1, \eta_2), A) &\,:\!=\,\! \lambda \int_{\Lambda} \max\left(0, {e}^{-W(x, \eta_2 + \zeta_{\Lambda^c})} - {e}^{-W(x, \eta_1 + \xi_{\Lambda^c})}\right) \cdot \unicode{x1D7D9}_{A}(\eta_1, \eta_2 + \delta_x)\nu(\mathrm{d} x), \\ B_{\cap}((\eta_1, \eta_2), A) &\,:\!=\,\! \lambda \int_{\Lambda} \min\left({e}^{-W(x, \eta_1 + \xi_{\Lambda^c})}, {e}^{-W(x, \eta_2 + \zeta_{\Lambda^c})}\right) \cdot \unicode{x1D7D9}_{A}(\eta_1 + \delta_x, \eta_2 + \delta_x)\nu(\mathrm{d} x) . \end{align*}

Then $K \,:\!=\, D_1 + D_2 + D_{\cap} + B_1 + B_2 + B_{\cap}$ is a coupling of $K_{\Lambda \mid \xi}$ and $K_{\Lambda \mid \zeta}$ in the sense of Definition 3.2.

Proof of Lemma 4.2. Checking that K is indeed a jump kernel can be done similarly to the proof of Lemma 4.1. To show that K is a coupling of $K_{\Lambda \mid \xi}$ and $K_{\Lambda \mid \zeta}$ , let $f\,:\, \mathcal{N}_{\Lambda \mid \xi} \to \mathbb{R}_{\ge 0}$ be bounded and measurable, and observe that

\begin{align*} \int_{\mathcal{N}_{\Lambda \mid \xi} \times \mathcal{N}_{\Lambda \mid \zeta}} (f(\tau_1) - f(\eta_1)) D_2((\eta_1, \eta_2), \mathrm{d} \tau_1 \times \mathrm{d} \tau_2) = \int_{\Lambda} (f(\eta_1) - f(\eta_1)) (\eta_2 \setminus \eta_1)(\mathrm{d} x) = 0 \end{align*}

and, analogously,

\begin{align*} \int_{\mathcal{N}_{\Lambda \mid \xi} \times \mathcal{N}_{\Lambda \mid \zeta}} (f(\tau_1) - f(\eta_1)) B_2((\eta_1, \eta_2), \mathrm{d} \tau_1 \times \mathrm{d} \tau_2) = 0 . \end{align*}

Further, using the fact that $\eta_1 = (\eta_1 \setminus \eta_2) + (\eta_1 \cap \eta_2)$ we have

\begin{align*} \int_{\mathcal{N}_{\Lambda \mid \xi} \times \mathcal{N}_{\Lambda \mid \zeta}} (f(\tau_1) - f(\eta_1)) (D_1 + D_{\cap})((\eta_1, \eta_2), \mathrm{d} \tau_1 \times \mathrm{d} \tau_2) = \int_{\Lambda} (f(\eta_1 \setminus \delta_x) - f(\eta_1)) \eta_1(\mathrm{d} x), \end{align*}

and, since $a = \max(0, a-b) + \min(a, b)$ for all $a, b \in \mathbb{R}_{\ge 0}$ ,

\begin{align*} \int_{\mathcal{N}_{\Lambda \mid \xi} \times \mathcal{N}_{\Lambda \mid \zeta}} (f(\tau_1) - f(\eta_1)) (B_1 + B_{\cap})&((\eta_1, \eta_2), \mathrm{d} \tau_1 \times \mathrm{d} \tau_2)\\ = &\lambda \int_{\Lambda} {e}^{-W(x, \eta_1 + \xi_{\Lambda^c})} \cdot (f(\eta_1 + \delta_x) - f(\eta_1)) \nu(\mathrm{d} x). \end{align*}

In fact, the integral on the right-hand side should only be over $\{x \in \Lambda \mid \eta + \delta_x \in \mathcal{N}_{\Lambda \mid \xi}\}$ . However, since $\eta \in \mathcal{N}_{\Lambda \mid \xi}$ , it can be shown that all $x \in \Lambda$ with $\eta + \delta_x \notin \mathcal{N}_{\Lambda \mid \xi}$ satisfy $W(x, \eta + \xi_{\Lambda^c}) = \infty$ , justifying this expression. Hence, it holds that

\begin{align*} \int_{\mathcal{N}_{\Lambda \mid \xi} \times \mathcal{N}_{\Lambda \mid \zeta}} (f(\tau_1) - f(\eta_1))& K((\eta_1, \eta_2), \mathrm{d} \tau_1 \times \mathrm{d} \tau_2)\\ &= \int_{\mathcal{N}_{\Lambda \mid \xi}} (f(\tau_1) - f(\eta_1)) K_{\Lambda \mid \xi}(\eta_1, \mathrm{d} \tau_1) \\ &= (K_{\Lambda \mid \xi} f)(\eta_1) - K_{\Lambda \mid \xi}(\eta_1, \mathcal{N}_{\Lambda \mid \xi}) \cdot f(\eta_1) . \end{align*}

By symmetry, it then follows that K is indeed a coupling in the sense of Definition 3.2.

4.2. Rapid mixing

We proceed by showing that, for sufficiently small activity $\lambda$ , the Markov transition family associated to $K_{\Lambda \mid \xi}$ converges rapidly to $\mu_{\Lambda \mid \xi}$ (noting that $\mu_{\Lambda \mid \xi}$ is well defined by Lemma 2.5). We start by identifying the finite volume Gibbs measure $\mu_{\Lambda \mid \xi}$ as a stationary distribution of the Markov transition family associated to $K_{\Lambda \mid \xi}$ . More precisely, recall that by Lemma 2.7 it holds that $\mu_{\Lambda \mid \xi}$ is only supported on $\mathcal{N}_{\Lambda \mid \xi}$ . Thus, we can uniquely identify $\mu_{\Lambda \mid \xi}$ with a distribution $\mu' \in \mathcal{P}(\mathcal{N}_{\Lambda \mid \xi}, \mathcal{R}_{\Lambda \mid \xi})$ with $\mu'(A) = \mu_{\Lambda \mid \xi}(A)$ for all $A \in \mathcal{R}_{\Lambda \mid \xi}$ . With some abuse of terminology and notation, we may not differentiate between $\mu_{\Lambda \mid \xi}$ and the associated distribution on $(\mathcal{N}_{\Lambda \mid \xi}, \mathcal{R}_{\Lambda \mid \xi})$ in the following statements.

Proof. First, note that by Lemma 2.5 it holds that $\mu_{\Lambda \mid \xi}$ is well defined. Using Lemma 3.2 we only need to check that

\begin{align*} \int_{A} K_{\Lambda \mid \xi}(\eta, B) \mu_{\Lambda \mid \xi}(\mathrm{d} \eta) = \int_{B} K_{\Lambda \mid \xi}(\eta, A) \mu_{\Lambda \mid \xi}(\mathrm{d} \eta) \end{align*}

for all $A, B \in \mathcal{R}_{\Lambda \mid \xi}$ . Writing both sides explicitly shows that it suffices to show that

\begin{align*} \int_{A} \int_{\Lambda} \unicode{x1D7D9}_{B}(\eta \setminus \delta_x) \eta(\mathrm{d} x) \mu_{\Lambda \mid \xi}(\mathrm{d} \eta) = \int_{B} \lambda \int_{\Lambda} {e}^{- W(x, \eta + \xi_{\Lambda^c})} \unicode{x1D7D9}_{A}(\eta + \delta_x) \eta(\mathrm{d} x) \mu_{\Lambda \mid \xi}(\mathrm{d} \eta) , \end{align*}

which follows immediately from applying the GNZ equation from Lemma 2.6 to the function $(x, \eta) \mapsto \unicode{x1D7D9}_{A}(\eta) \cdot \unicode{x1D7D9}_{B}(\eta \setminus \delta_x)$ .

Next, we bound the speed of convergence by using Lemma 3.6 and the identity coupling from Lemma 4.2.

Proposition 4.1. Let $\phi$ be locally stable with constant L and weakly tempered with constant $\hat{C}_{\phi}$ . For all $\lambda < \frac{1}{\hat{C}_{\phi} {e}^{L}}$ , there is some $\delta > 0$ such that, for all $\Lambda \in \mathcal{B}_{{b}}$ and $\xi \in \mathcal{N}_{H}$ , the following holds: For $K_{\Lambda \mid \xi}$ as in (8), the associated Markov transition family $(P_t)$ satisfies

\begin{align*} \left\lvert P_t(\eta, \cdot) - \mu_{\Lambda \mid \xi} \right\rvert_{\mathrm{tv}} \le {e}^{- \delta t} \cdot (\lambda {e}^{L} \nu(\Lambda) + \eta(\Lambda)) \end{align*}

for all $\eta \in \mathcal{N}_{\Lambda \mid \xi}$ and all $t \in \mathbb{R}_{\ge 0}$ . In particular, $\delta$ may depend on $\hat{C}_{\phi}$ , L and $\lambda$ , but is independent of $\Lambda$ , $\eta$ , $\xi$ and t.

Let K be the identity coupling of $K_{\Lambda \mid \xi}$ with itself as provided by Lemma 4.2 (i.e. the coupling given by Lemma 4.2 for $\zeta = \xi$ ), and let $D_1, D_2, D_{\cap}, B_1, B_2, B_{\cap}$ be defined accordingly. To conclude our proof, it remains to find a measurable function $f\,:\, \mathcal{N}_{\Lambda \mid \xi} \times \mathcal{N}_{\Lambda \mid \xi} \to \mathbb{R}_{\ge 0}$ such that conditions (i) and (ii) of Lemma 3.6 are satisfied for suitable constants $c, \delta > 0$ . To this end, we set $f(\eta_1, \eta_2) \,:\!=\, (\eta_1 \ominus \eta_2)(\Lambda)$ . Note that f can equivalently be expressed as $\sup_{\Delta \in \mathcal{B}_{\Lambda}} \left\lvert \eta_1(\Delta) - \eta_2(\Delta) \right\rvert$ , showing that it is indeed measurable. Further, it holds that $f(\eta_1, \eta_2) \ge 1$ if and only if $\eta_1 \neq \eta_2$ . Thus, condition (i) of Lemma 3.6 is satisfied by f for $c = 1$ .

To verify condition (ii) of Lemma 3.6, we first observe the following.

• • For all $x \in S_{\eta_1 \setminus \eta_2}$ , we have $f(\eta_1 \setminus \delta_x, \eta_2) = f(\eta_1, \eta_2) - 1$ . Hence, it holds that

  \begin{align*} (D_1 f)(\eta_1, \eta_2) = (f(\eta_1, \eta_2) - 1) \cdot (\eta_1 \setminus \eta_2)(\Lambda). \end{align*}

• • For all $x \in S_{\eta_2 \setminus \eta_1}$ , we have $f(\eta_1, \eta_2 \setminus \delta_x) = f(\eta_1, \eta_2) - 1$ . Hence, it holds that

  \begin{align*} (D_2 f)(\eta_1, \eta_2) = (f(\eta_1, \eta_2) - 1) \cdot (\eta_2 \setminus \eta_1)(\Lambda). \end{align*}

• • For all $x \in S_{\eta_2 \cap \eta_1}$ , we have $f(\eta_1 \setminus \delta_x, \eta_2 \setminus \delta_x) = f(\eta_1, \eta_2)$ . Hence, it holds that

  \begin{align*} (D_{\cap} f)(\eta_1, \eta_2) = f(\eta_1, \eta_2) \cdot (\eta_1 \cap \eta_2)(\Lambda). \end{align*}

• • For all $x \in \Lambda$ , we have $f(\eta_1 + \delta_x, \eta_2) = f(\eta_1, \eta_2) + 1$ , $f(\eta_1, \eta_2 + \delta_x) = f(\eta_1, \eta_2) + 1$ and $f(\eta_1 + \delta_x, \eta_2 + \delta_x) = f(\eta_1, \eta_2)$ . Hence, it holds that

  \begin{align*} (B_1 f)(\eta_1, \eta_2) &= (f(\eta_1, \eta_2) + 1)\! \cdot \! \lambda\! \int_{\Lambda} \max\left(0, {e}^{-W(x, \eta_1 + \xi_{\Lambda^c})}\! - {e}^{-W(x, \eta_2 + \xi_{\Lambda^c})}\right)\! \nu(\mathrm{d} x) , \\ (B_2 f)(\eta_1, \eta_2) &= (f(\eta_1, \eta_2) + 1)\! \cdot \! \lambda \!\int_{\Lambda} \max\left(0, {e}^{-W(x, \eta_2 + \xi_{\Lambda^c})}\! - {e}^{-W(x, \eta_1 + \xi_{\Lambda^c})}\right)\! \nu(\mathrm{d} x) , \\ (B_{\cap} f)(\eta_1, \eta_2) &= f(\eta_1, \eta_2)\! \cdot \! \lambda\! \int_{\Lambda} \min\left({e}^{-W(x, \eta_1 + \xi_{\Lambda^c})}, {e}^{-W(x, \eta_2 + \xi_{\Lambda^c})}\right)\! \nu(\mathrm{d} x) . \end{align*}

Further, since $\eta_1 \cup \eta_2 = \eta_1 \ominus \eta_2 + \eta_1 \cap \eta_2$ and $\max(0, a - b) + \max(0, b - a) + \min(a, b) = \max(a, b)$ for every $a, b \in \mathbb{R}$ , we can write the rate function of K as

\begin{align*} \kappa(\eta_1, \eta_2) &\,:\!=\, K((\eta_1, \eta_2), \mathcal{N}_{\Lambda \mid \xi} \times \mathcal{N}_{\Lambda \mid \xi})\\ & \, = (\eta_1 \cup \eta_2)(\Lambda) + \lambda \int_{\Lambda} \max\left({e}^{-W(x, \eta_1 + \xi_{\Lambda^c})}, {e}^{-W(x, \eta_2 + \xi_{\Lambda^c})}\right) \nu(\mathrm{d} x) . \end{align*}

Combining these observations yields

\begin{align*} (Kf)(\eta_1, \eta_2) = \kappa(\eta_1, \eta_2) f(\eta_1, \eta_2) - f(\eta_1, \eta_2) + \lambda \int_{\Lambda} \left\lvert {e}^{-W(x, \eta_1 + \xi_{\Lambda^c})} - {e}^{-W(x, \eta_2 + \xi_{\Lambda^c})} \right\rvert \nu(\mathrm{d} x). \end{align*}

Next, we use the following claim that we prove below.

Claim 4.1. There exists some $\delta > 0$ , independent of $\Lambda$ , such that, for all $\eta_1, \eta_2 \in \mathcal{N}_{\Lambda}$ , it holds that

Provided Claim 4.1, we have

for some $\delta > 0$ independent of $\Lambda$ and $\eta_1, \eta_2$ . Thus, applying Lemma 3.6 yields

for $\delta > 0$ , as desired. Finally, observe that $f(\eta, \zeta) = (\eta \ominus \zeta)(\Lambda) \le \eta(\Lambda) + \zeta(\Lambda)$ . Hence, applying the GNZ equation from Lemma 2.6 with the function $(x, \zeta) \mapsto \unicode{x1D7D9}_{\mathcal{N}_{\Lambda \mid \xi}} (\zeta)$ and using the bound on the influence from Lemma 2.4 yields

\begin{align*} \int_{\mathcal{N}_{\Lambda \mid \xi}} f(\eta, \zeta) \mu_{\Lambda \mid \xi}(\mathrm{d} \zeta) \le \eta(\Lambda) + \lambda {e}^{L} \nu(\Lambda), \end{align*}

which concludes the proof.

Proof of Claim 4.1. Let $k = (\eta_1 \ominus \eta_2)(\Lambda)$ . We start by constructing a sequence $\zeta_0, \dots, \zeta_k \in \mathcal{N}_{\Lambda \mid \xi}$ such that $\zeta_0 = \eta_1$ , $\zeta_k = \eta_2$ and, for all $1 \le i \le k$ , it holds that $\zeta_{i - 1} \ominus \zeta_{i} = \delta_z$ for some $z \in \Lambda$ . To this end, let $k_1 = (\eta_1 \setminus \eta_2)(\Lambda)$ and $k_2 = (\eta_2 \setminus \eta_1)(\Lambda)$ , and note that $k = k_1 + k_2$ . Let $\eta_1 \setminus \eta_2 = \sum_{j = 1}^{k_1} \delta_{y_j}$ and let $\eta_2 \setminus \eta_1 = \sum_{j = k_1 + 1}^{k_1 + k_2} \delta_{y_j}$ for some sequence $y_1, \dots, y_{k_1 + k_2} \in \Lambda$ . Now, for $0 \le i \le k_1$ set $\zeta_i = (\eta_1 \cap \eta_2) + \sum_{j = 1}^{k_1 - i} \delta_{y_j}$ , and for $k_1 + 1 \le i \le k_1 + k_2$ set $\zeta_i = (\eta_1 \cap \eta_2) + \sum_{j = k_1 + 1}^{i} \delta_{y_j}$ . By construction, $\zeta_{i}$ can be obtained from $\zeta_{i-1}$ by adding or subtracting a Dirac measure. Using the fact that $\zeta_{k_1} = \eta_1 \cap \eta_2$ and that $\eta_1 = (\eta_1 \cap \eta_2) + (\eta_1 \setminus \eta_2)$ and $\eta_2 = (\eta_1 \cap \eta_2) + (\eta_2 \setminus \eta_1)$ , we further see that $\zeta_0 = \eta_1$ and $\zeta_{k} = \eta_2$ . Moreover, note that $\zeta_i \le \eta_1$ or $\zeta_i \le \eta_2$ ; hence $\zeta_i \in \mathcal{N}_{\Lambda \mid \xi}$ , and the sequence $\zeta_0, \dots, \zeta_k$ satisfies the desired properties.

By the triangle inequality, we now have

Therefore, it is sufficient to show that

for every $1 \le i \le k$ . By symmetry, we may assume without loss of generality that $\zeta_{i} = \zeta_{i-1} + \delta_z$ for some $z \in \Lambda$ . It then holds that

Thus, by local stability and Lemma 2.4 we have

\begin{align*} &\left\lvert {e}^{-W(x, \zeta_{i} + \xi_{\Lambda^c})} - {e}^{-W(x, \zeta_{i-1} + \xi_{\Lambda^c})} \right\rvert\\ &\qquad\qquad\qquad\quad\le \min\left({e}^{-W(x, \zeta_{i-1} + \xi_{\Lambda^c})} \cdot \left\lvert {e}^{-\phi(x, z)} - 1 \right\rvert, {e}^{-W(x, \zeta_{i} + \xi_{\Lambda^c})} \cdot \left\lvert 1 - {e}^{\phi(x, z)} \right\rvert\right) \\ &\qquad\qquad\qquad\quad\le {e}^{L} \cdot \min\left(\left\lvert {e}^{-\phi(x, z)} - 1 \right\rvert, \left\lvert 1 - {e}^{\phi(x, z)} \right\rvert\right) \\ &\qquad\qquad\qquad\quad\le {e}^{L} \cdot \left(1 - {e}^{-\left\lvert \phi(x, z) \right\rvert}\right) . \end{align*}

Combining this with weak temperedness, we obtain

Since we assume $\lambda < \frac{1}{{e}^{L} \hat{C}_{\phi}}$ , this proves the claim.

4.3. Slow percolation of disagreements

Throughout this section, we will use slightly stronger assumptions than in Section 4.2, namely:

• • We assume that $\phi$ is locally stable with constant L and has range at most R. We may assume $R > 0$ to avoid trivialities.

• • We assume that the underlying space $(\mathcal{X}, d)$ is the d-dimensional Euclidean space for some fixed dimension $d \in \mathbb{N}$ , and that $\nu$ is the respective Lebesgue measure.

Moreover, we introduce the following notational conventions. First, for every tuple $\pmb{k} = (k_1, \dots, k_{d}) \in \mathbb{Z}^{d}$ we set

\begin{align*} \Lambda_{\pmb{k}} \,:\!=\, \left[\left(k_1 - \frac{1}{2}\right) R, \left(k_1 + \frac{1}{2}\right) R\right] \times \dots \times \left[\left(k_{d} - \frac{1}{2}\right) R, \left(k_{d} + \frac{1}{2}\right) R\right]. \end{align*}

For $n \in \mathbb{N}$ we set $V_n \,:\!=\, (\mathbb{Z} \cap [-n, n])^{d}$ and

\begin{align*} \Lambda^{(n)} \,:\!=\, \bigcup_{\pmb{k} \in V_n} \Lambda_{\pmb{k}} = [-n - 1/2, n + 1/2]^{d} . \end{align*}

For every $\xi \in \mathcal{N}_{H}$ , we simplify notation and write $K_{n \mid \xi}$ for $K_{\Lambda^{(n)} \mid \xi}$ as given in (8), $\mathcal{N}_{n \mid \xi}$ for $\mathcal{N}_{\Lambda^{(n)} \mid \xi}$ and $\mathcal{R}_{n \mid \xi}$ for $\mathcal{R}_{\Lambda^{(n)} \mid \xi}$ .

Throughout the section, it will be useful to consider an undirected graph $G_n$ on the vertex set $V_n$ with edges $E_n \,:\!=\, \{\{\pmb{j}, \pmb{k}\} \subseteq V_n \mid \lVert \pmb{j} - \pmb{k} \rVert_{\infty} = 1 \}$ . In accordance with this graph construction, we denote by $\Gamma_n(\pmb{k}) \,:\!=\, \{\pmb{j} \in V_n \mid \{\pmb{k}, \pmb{j}\} \in E_n\}$ the neighborhood of $\pmb{k} \in V_n$ in $G_n$ . Further, we partition the vertex set $V_n$ into inner vertices $V^{{in}}_n \,:\!=\, \{\pmb{k} \in V_n \mid \lVert \pmb{k} \rVert_{\infty} < n\}$ and outer vertices $V^{{out}}_n \,:\!=\, V_n \setminus V^{{in}}_n = \{\pmb{k} \in V_n \mid \lVert \pmb{k} \rVert_{\infty} = n\}$ .

Our first lemma is a simple observation about the jump rates of the identity coupling of $K_{n \mid \xi}$ and $K_{n \mid \zeta}$ for any pair $\xi, \zeta \in \mathcal{N}_{H}$ .

Proof. Let $D_1, D_2, D_{\cap}, B_1, B_2, B_{\cap}$ be as in Lemma 4.2. Set $\Lambda = \Lambda^{(n)}$ to simplify notation, and observe that, for all $\pmb{k} \in V_n$ and $(\eta_1, \eta_2) \notin \Delta_{\pmb{k}}$ , it holds that

\begin{align*} K((\eta_1, \eta_2), \Delta_{\pmb{k}}) &= B_1((\eta_1, \eta_2), \Delta_{\pmb{k}}) + B_2((\eta_1, \eta_2), \Delta_{\pmb{k}}) \\ &= \lambda \int_{\Lambda_{\pmb{k}}} \left\lvert {e}^{-W(x, \eta_1 + \xi_{\Lambda^c})} - {e}^{-W(x, \eta_2 + \zeta_{\Lambda^c})} \right\rvert \nu(\mathrm{d} x) . \end{align*}

Using Lemma 2.4, the fact that $\phi$ is locally stable and that $\nu(\Lambda_{\pmb{k}}) = R^{d}$ proves (ii).

For (i), note that, if $\pmb{k} \in V^{{in}}_n$ , it holds that $d(x, y) \ge R$ for any $x \in \Lambda_{\pmb{k}}$ and $y \notin \Lambda_{\pmb{k}} \cup \bigcup_{\pmb{j} \in \Gamma_n(\pmb{k})} \Lambda_{\pmb{j}}$ . In particular, since the range of $\phi$ is bounded by R, we have $\phi(x, y) = 0$ for any such pair of points. Hence, it is easily checked that $W(x, \eta_1 + \xi_{\Lambda^c}) = W(x, \eta_2 + \zeta_{\Lambda^c})$ for every $(\eta_1, \eta_2) \notin \Delta_{\pmb{k}} \cup \bigcup_{\pmb{j} \in \Gamma_n(\pmb{k})} \Delta_{\pmb{j}}$ , and consequently $K((\eta_1, \eta_2), \Delta_{\pmb{k}}) = 0$ as claimed.

We now combine Lemma 4.4 with the process construction in Theorem 3.1 and Lemmas 3.7 and 3.8 to prove the following proposition.

Proposition 4.2. Let $n \in \mathbb{N}$ , let $\xi, \zeta \in \mathcal{N}_{H}$ and let $(P_t), (Q_t)$ be the transition families associated with $K_{n \mid \xi}$ and $K_{n \mid \zeta}$ . For every $m \in \mathbb{N}_0$ with $m < n$ and all $t \in \mathbb{R}_{\ge 0}$ with $t < \frac{n - m}{{e}^2 (6m + 3)^{d} R^{d} \lambda {e}^{L}}$ it holds that

\begin{align*} \left\lvert P_t(\pmb{0}, \cdot) - Q_t(\pmb{0}, \cdot) \right\rvert_{\Lambda^{(m)}} \le {e}^{- (n - m)} . \end{align*}

Proof. Let K be the identity coupling of $K_{n \mid \xi}$ and $K_{n \mid \zeta}$ as given by Lemma 4.2. By Lemmas 3.4 and 4.1, we know that K is nonexplosive, and its associated transition family is a coupling of $(P_t)$ and $(Q_t)$ in the sense of Definition 3.1. Next, let $(\Omega, \mathcal{F}, \mathbb{P}, (X_t))$ be the Markov process associated with the jump kernel K and the starting distribution $\delta_{\pmb{0}} \otimes \delta_{\pmb{0}}$ as given by Theorem 3.1, where $\delta_{\pmb{0}}$ denotes the probability distribution concentrated on the empty configuration $\pmb{0}$ . For $\pmb{k} \in V_n$ , define $\Delta_{\pmb{k}}$ as in Lemma 4.4 and set $T_{\pmb{k}} \,:\!=\, \inf\{t \in \mathbb{R}_{\ge 0} \mid X_t \in \Delta_{\pmb{k}}\}$ . Using Lemma 3.3 and the fact that $(X_t)$ is a Markov process with a transition function that is a coupling of $(P_t)$ and $(Q_t)$ , and with initial law $\delta_{\pmb{0}} \otimes \delta_{\pmb{0}}$ , we obtain

\begin{align*} \left\lvert P_t(\pmb{0}, \cdot) - Q_t(\pmb{0}, \cdot) \right\rvert_{\Lambda^{(m)}} \le \sum_{\pmb{k} \in V_{m}} \mathbb{P}[X_t \in \Delta_{\pmb{k}}] \le \sum_{\pmb{k} \in V_{m}} \mathbb{P}[T_{\pmb{k}} \le t]. \end{align*}

We proceed by deriving a uniform upper bound on $\mathbb{P}[T_{\pmb{k}} \le t]$ for every $\pmb{k} \in V_{m}$ . To this end, we first need to introduce some additional terminology and notation. Recall the definition of the graph $G_n = (V_n, E_n)$ . For $\ell \in \mathbb{N}$ , we call a sequence of vertices $(\pmb{k}_{1}, \dots, \pmb{k}_{\ell}) \in V_n^{\ell}$ a path of length $\ell$ in $G_n$ if, for all $2 \le i \le \ell$ , it holds that $\pmb{k}_i \in \Gamma_n(\pmb{k}_{i-1})$ , and, for all $1 \le i < j \le \ell$ , it holds that $\pmb{k}_i \neq \pmb{k}_j$ (i.e. $(\pmb{k}_{1}, \dots, \pmb{k}_{\ell})$ is a simple path in $G_n$ in the graph theoretic sense). We write $\mathfrak{P}_{\ell}$ for the set of all paths of length $\ell$ in $G_n$ . Moreover, for two sets $W_1, W_2 \subseteq V_n$ , we denote the set of paths of length $\ell$ from $W_1$ to $W_2$ by

\begin{align*} \mathfrak{P}_{\ell}(W_1, W_2) \,:\!=\, \{(\pmb{k}_{1}, \dots, \pmb{k}_{\ell}) \in \mathfrak{P}^{\ell} \mid \pmb{k}_1 \in W_1, \pmb{k}_{\ell} \in W_2\}, \end{align*}

and we write $\mathfrak{P}(W_1, W_2) \,:\!=\, \bigcup_{\ell \in \mathbb{N}} \mathfrak{P}_{\ell}(W_1, W_2)$ for all paths from $W_1$ to $W_2$ regardless of their length. Moreover, if $W_1 = \{\pmb{k}\}$ for some $\pmb{k} \in V_n$ , we drop the set notation and simply write $\mathfrak{P}_{\ell}(\pmb{k}, W_2)$ and $\mathfrak{P}(\pmb{k}, W_2)$ .

We proceed by stating the claim that formalizes the following intuition: Due to the bounded range of the potential, a disagreement between the two Markov processes in some box $\Lambda_{\pmb{k}}$ for $\pmb{k} \in V_n$ can only be generated if there already was a disagreement in any box that is adjacent to $\pmb{k}$ in the graph $G_n$ . In particular, if there is a disagreement in some box $\pmb{k} \in V_m$ at time t (i.e. $X_t \in \Delta_{\pmb{k}}$ ), then there must be some path $(\pmb{k}_1, \dots, \pmb{k}_{\ell})$ such that $\pmb{k}_{1} = \pmb{k}$ and $\pmb{k}_{\ell} \in V^{{out}}_n$ that this disagreement can be traced back to. Formally, it must hold that the respective hitting times satisfy $T_{\pmb{k}_{\ell}} < T_{\pmb{k}_{\ell - 1}} < \dots < T_{\pmb{k}_1} \le t$ . We use Lemma 3.7 and part (i) of Lemma 4.4 to formalize this idea.

Claim 4.2. For all $\pmb{k} \in V_m$ and all $t \in \mathbb{R}_{\ge 0}$ , it holds that

\begin{align*} \mathbb{P}[T_{\pmb{k}} \le t] \le \sum_{(\pmb{k}_1, \dots, \pmb{k}_{\ell}) \in \mathfrak{P}(\pmb{k}, V^{{out}}_n)} \mathbb{P}[T_{\pmb{k}_{\ell}} < T_{\pmb{k}_{\ell - 1}} < \dots < T_{\pmb{k}_1} \le t]. \end{align*}

Combining Claim 4.2 with Lemma 3.8 and part (ii) of Lemma 4.4, we get

\begin{align*} \mathbb{P}[T_{\pmb{k}} \le t] \le \sum_{\ell \in \mathbb{N}} \left\lvert \mathfrak{P}_{\ell}(\pmb{k}, V^{{out}}_n) \right\rvert \cdot (1 - F_{\rho t}(\ell - 1)) \text{ for } \rho = \lambda {e}^{d} R^{d}, \end{align*}

where $F_{\rho t}$ is the cumulative distribution function of a Poisson random variable with rate $\rho t$ . Next, note that $\left\lvert \mathfrak{P}_{\ell}(\pmb{k}, V^{{out}}_n) \right\rvert = 0$ for all $\ell \le n - m$ , and $\left\lvert \mathfrak{P}_{\ell}(\pmb{k} , V^{{out}}_n) \right\rvert \le 3^{d \ell}$ for all $\ell > n - m$ . Further, using standard tail bounds for the Poisson distribution, we get that, for all $\ell \ge {e}^{c + 1} \rho t$ with $c \ge d \ln(6 m + 3) + 1$ , it holds that $1 - F_{\rho t}(\ell - 1) \le {e}^{- c \ell} = {e}^{-\ell} 3^{-d \ell} (2m + 1)^{-d \ell}$ . Hence, we have

\begin{align*} \mathbb{P}[T_{\pmb{k}} \le t] \le \sum_{\ell > n - m} {e}^{- \ell} (2m+1)^{- d \ell} \le \left\lvert V_m \right\rvert^{-1} \cdot {e}^{- (n - m)}, \end{align*}

and summing over $\pmb{k} \in V_m$ concludes the proof.

Proof of Claim 4.2. For $\ell \in \mathbb{N}$ define

\begin{align*} S^{\text{f}}_{\ell} (\pmb{k}) &\,:\!=\, \{(\pmb{k}_1, \dots, \pmb{k}_j) \in \mathfrak{P}_j \mid j \le \ell, \pmb{k}_1 = \pmb{k}, \pmb{k}_j \in V^{{out}}_n, \forall i < j\,:\, \pmb{k}_i \in V^{{in}}_n \}, \\ S^{\text{o}}_{\ell} (\pmb{k}) &\,:\!=\, \{(\pmb{k}_1, \dots, \pmb{k}_\ell) \in \mathfrak{P}_{\ell} \mid \pmb{k}_1 = \pmb{k}, \forall i \le \ell\,:\, \pmb{k}_i \in V^{{in}}_n \}, \\ S_{\ell}(\pmb{k}) &\,:\!=\, S^{\text{f}}_{\ell} \cup S^{\text{o}}_{\ell}. \end{align*}

In prose: $S^{\text{f}}_{\ell}(\pmb{k})$ is the set of all paths of length at most $\ell$ that start with $\pmb{k}$ , end in $V^{{out}}_n$ and only go via $V^{{in}}_n$ in-between; $S^{\text{o}}_{\ell} (\pmb{k})$ is the set of paths of length exactly $\ell$ that start in $\pmb{k}$ and never visit $V^{{out}}$ ; and $S_{\ell}(\pmb{k})$ is the union of these disjoint sets.

We will prove our claim by showing inductively that, for all $\ell \in \mathbb{N}$ ,

(9) \begin{align} \mathbb{P}[T_{\pmb{k}} \le t] \le \sum_{(\pmb{k}_1, \dots, \pmb{k}_{j}) \in S_{\ell}(\pmb{k})} \mathbb{P}[T_{\pmb{k}_{j}} < T_{\pmb{k}_{j - 1}} < \dots < T_{\pmb{k}_1} \le t]. \end{align}

To see that this indeed proves the claim, note that $S^{\text{f}}_{\ell}(\pmb{k}) \subseteq \mathfrak{P}(\pmb{k}, V^{{out}}_n)$ for all $\ell \in \mathbb{N}$ , and that $S^{\text{o}}_{\ell}(\pmb{k}) = \emptyset$ for $\ell > \left\lvert V^{{in}}_n \right\rvert$ .

For the induction base, note that the statement holds trivially for $\ell = 1$ since $S_{1}(\pmb{k}) = \{\pmb{k}\}$ . Suppose (9) holds for some $\ell \in \mathbb{N}$ . By splitting up the sum in (9) into a sum for $S^{\text{o}}_{\ell}(\pmb{k})$ and $S^{\text{f}}_{\ell}(\pmb{k})$ and noting that $S^{\text{f}}_{\ell}(\pmb{k}) \subseteq S^{\text{f}}_{\ell+1}(\pmb{k})$ , we see that it suffices to show that

\begin{align*} &\sum_{(\pmb{k}_1, \dots, \pmb{k}_{\ell}) \in S^{\text{o}}_{\ell}(\pmb{k})} \mathbb{P}[T_{\pmb{k}_{\ell}} < T_{\pmb{k}_{\ell - 1}} < \dots < T_{\pmb{k}_1} \le t] \\ &\hspace{4em} \le \sum_{(\pmb{k}_1, \dots, \pmb{k}_{\ell + 1}) \in S^{\text{o}}_{\ell + 1}(\pmb{k})} \mathbb{P}[T_{\pmb{k}_{\ell + 1}} < T_{\pmb{k}_{\ell}} < \dots < T_{\pmb{k}_1} \le t] \\ &\hspace{4em}\hspace{4em}+ \sum_{(\pmb{k}_1, \dots, \pmb{k}_{\ell + 1}) \in S^{\text{f}}_{\ell + 1}(\pmb{k}) \setminus S^{\text{f}}_{\ell}(\pmb{k})} \mathbb{P}[T_{\pmb{k}_{\ell + 1}} < T_{\pmb{k}_{\ell}} < \dots < T_{\pmb{k}_1} \le t] . \end{align*}

To this end, fix some $(\pmb{k}_1, \dots, \pmb{k}_{\ell}) \in S^{\text{o}}_{\ell}(\pmb{k})$ and note that, by definition, $\pmb{k}_{\ell} \in V^{{in}}_n$ . Set $A \,:\!=\, \{T_{\pmb{k}_{\ell}} < T_{\pmb{k}_{\ell - 1}} < \dots < T_{\pmb{k}_1} \le t\}$ and note that $A \subseteq \{T_{\pmb{k}_{\ell}} < \infty\}$ . Further, define $T_{\Gamma} \,:\!=\, \inf\{s \in \mathbb{R}_{\ge 0} \mid X_{s} \in \bigcup_{\pmb{j} \in \Gamma_n(\pmb{k}_{\ell})} \Delta_{\pmb{j}}\}$ . By Lemma 3.7 and part (i) of Lemma 4.4, it holds that

\begin{align*} \mathbb{P}[A] = \mathbb{P}[T_{\Gamma} < T_{\pmb{k}_{\ell}}, A] \le \sum_{\pmb{j} \in \Gamma_n(\pmb{k}_{\ell})} \mathbb{P}[T_{\pmb{j}} < T_{\pmb{k}_{\ell}}, A]. \end{align*}

Now, observe that, if $\pmb{j} = \pmb{k}_i$ for any $1 \le i \le \ell$ , then $\{T_{\pmb{j}} < T_{\pmb{k}_{\ell}}, A\} \subseteq \{T_{\pmb{j}} < T_{\pmb{j}}\}$ , and hence $\mathbb{P}[T_{\pmb{j}} < T_{\pmb{k}_{\ell}}, A] = 0$ . Thus, we only need to consider $\pmb{j} \in \Gamma_n(\pmb{k}_{\ell})$ that extend $(\pmb{k}_1, \dots, \pmb{k}_{\ell})$ to a path of length $\ell + 1$ (i.e. $(\pmb{k}_1, \dots, \pmb{k}_{\ell}, \pmb{j}) \in \mathfrak{P}_{\ell+1}$ ). If $\pmb{j} \in V^{{out}}_n$ then $(\pmb{k}_1, \dots, \pmb{k}_{\ell}, \pmb{j}) \in S^{\text{f}}_{\ell + 1}(\pmb{k}) \setminus S^{\text{f}}_{\ell}(\pmb{k})$ . Otherwise, we have $(\pmb{k}_1, \dots, \pmb{k}_{\ell}, \pmb{j}) \in S^{\text{o}}_{\ell + 1}(\pmb{k})$ . Since distinct paths from $S^{\text{o}}_{\ell}(\pmb{k})$ also produce distinct extensions, this concludes the induction step and proves the claim.