# Uniqueness of locally stable Gibbs point processes via spatial birth-death
dynamics

Samuel Baguley, Andreas Göbel and Marcus Pappik  Hasso Plattner Institute,
University of Potsdam, Germany  {samuel.baguley, andreas.goebel,
<EMAIL_ADDRESS>
###### Abstract.

We prove that for every locally stable and tempered pair potential $\phi$ with
bounded range, there exists a unique infinite-volume Gibbs point process on
$\mathbb{R}^{d}$ for every activity
$\lambda<(\textup{e}^{L}\hat{C}_{\phi})^{-1}$, where $L$ is the local
stability constant and
$\hat{C}_{\phi}\coloneqq\sup_{x\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}1-\textup{e}^{-\left\lvert\phi(x,y)\right\rvert}\mathrm{d}y$
is the (weak) temperedness constant. Our result extends the uniqueness regime
that is given by the classical Ruelle–Penrose bound by a factor of at least e,
where the improvements becomes larger as the negative parts of the potential
become more prominent (i.e., for attractive interactions at low temperature).
Our technique is based on the approach of Dyer et al. (Rand. Struct. & Alg.
’04): we show that for any bounded region and any boundary condition, we can
construct a Markov process (in our case spatial birth-death dynamics) that
converges rapidly to the finite-volume Gibbs point process while effects of
the boundary condition propagate sufficiently slowly. As a result, we obtain a
spatial mixing property that implies uniqueness of the infinite-volume Gibbs
measure.

###### 2020 Mathematics Subject Classification:

Primary: 82B21, Secondary: 60G55, 60J76

Keywords: Gibbs point processes $\cdot$ uniqueness of Gibbs measures $\cdot$
spatial birth-death processes $\cdot$ coupling $\cdot$ mixing times

Funding: Andreas Göbel was funded by the project PAGES (project No. 467516565)
of the German Research Foundation (DFG). Marcus Pappik was funded by the HPI
Research School on Data Science and Engineering.

##  1\. Introduction

In statistical physics, point processes play a central role as a mathematical
model for gasses of interacting particles: the gas is modeled as a random
point configuration in a ground space $\mathcal{X}$, where every point in a
configuration represents the location of a particle. A corner stone of this
approach for modelling gasses are Gibbs point processes, which historically
arise from describing the distribution of particles as the solution to a
variational principle that aims at minimizing the mean free excess energy of
the system (see for example [7, Section 1.3]). While these ideas date back all
the way to Gibbs [15] (and even further to the work of Boltzmann and Maxwell),
the theory of Gibbs point processes was substantially developed further
throughout the 60s and 70s by the work of Dobrushin [9], Lanford and Ruelle
[26], Preston [32], Georgii [14] and many others, and it remains an active
area of research till the present days [19, 28, 16, 29, 8, 21, 2] (see also
[35] for a classical and [7, 20] for more modern introductions to the topic).

Given a finite-volume region $\Lambda\subset\mathcal{X}$ of the ground space
$\mathcal{X}$ (for simplicity, think $\mathcal{X}=\mathbb{R}^{d}$), a Gibbs
point process on $\Lambda$ is described by a probability density with respect
to a Poisson point process on $\Lambda$ of intensity 1. This probability
density is specified by an activity $\lambda\in\mathbb{R}_{\geq 0}$ and an
energy function $H$, which assigns a value from $(-\infty,\infty]$ to finite
point sets $\eta\subset\mathcal{X}$. Given $\lambda$ and $H$, every finite
point set $\eta\subset\Lambda$ has a density proportional to
$\lambda^{\left\lvert\eta\right\rvert}\textup{e}^{-H(\eta)}$.

Opposed to this finite-volume setting, statistical physics is usually
interested in large particle systems. Such systems are more accurately modeled
by considering point processes on regions of infinite volume, while the
configuration space is extended to all locally finite point sets (i.e., we
only require finitely many points in every bounded subregion). In this case,
the definition via probability densities often fails as the density cannot be
normalized properly and configurations typically have infinite energy.
Instead, given an activity $\lambda$ and energy function $H$, an infinite-
volume Gibbs point process is defined by the _Dobrushin–Lanford–Ruelle
formalism_ [9, 26] (DLR), which imposes a set of consistency requirements: any
finite-volume projection of an infinite-volume Gibbs point process should be
consistent with its finite-volume definition discussed above (see Definition
2). Due to the non-constructive nature of this approach, two main questions
arise. Given an activity $\lambda$ and an energy function $H$:

  * •

_Existence:_ is there at least one point process on $\mathcal{X}$ that
satisfies the DLR formalism?

  * •

_Uniqueness:_ is there at most one point process on $\mathcal{X}$ that
satisfies the DLR formalism?111Note that usually in mathematics, uniqueness
means that there is exactly one object that satisfies the requirements.
However, for Gibbs point processes, this task is usually split into proving
that there is a most one and at least one Gibbs measure. Hence it will be more
convenient to refer to these two problems as existence and uniqueness
respectively.

While the existence of an infinite-volume Gibbs measures can be proven under
fairly general assumptions, which are usually stated in terms of the energy
function $H$ alone, proving uniqueness is notoriously more difficult [7, 20,
8]. In particular, conditions for uniqueness often involve that the activity
$\lambda$ is sufficiently small, depending on various properties of the energy
function (see for example [17, 29, 28]).

On top of being a genuinely intriguing mathematical question, uniqueness of
infinite-volume Gibbs measures in the sense of DLR gained a lot of attention
in mathematical physics as a probabilistic notion of _absence of phase
transitions_ [20, 29, 28, 19, 2]. As such, it poses an alternative to (much
older) analytic notions, such as analyticity of the infinite-volume pressure
(see [20, Section 4] for a detailed introduction), which are usually proven by
studying the convergence of series expansions [30, 33, 23] or the contraction
integral identities [29, 28, 16]. While in many cases such analytic proof
techniques can be modified to yield uniqueness in the DLR sense (see for
example [21] and [29]) and although both notions of absence of phase
transitions often hold in similar activity regimes, to the best of our
knowledge no general rigorous connection has been proved. We defer a more
detailed discussion to Section 6 and proceed to sketching our main result.

###  1.1. Main result and proof technique

In this paper, we focus on giving an activity regime for uniqueness of the
infinite-volume Gibbs point process for a class of energy functions that
result from pair interactions, that is, the case that there is some pair
potential $\phi:\mathcal{X}^{2}\to(-\infty,\infty]$ such that
$H(\eta)=\sum_{x,y\in\binom{\eta}{2}}\phi(x,y)$, where we sum over unordered
pairs of distinct points in $\eta$. Moreover, we assume the following:

  * •

_Weak temperedness:_ We call $\phi$ weakly tempered if

$\hat{C}_{\phi}\coloneqq\sup_{x\in\mathcal{X}}\int_{\mathcal{X}}1-\textup{e}^{-\left\lvert\phi(x,y)\right\rvert}\mathrm{d}y<\infty,$

and we refer to $\hat{C}_{\phi}$ as the weak temperedness constant. This
condition should be thought of as assuming that the total influence of every
point on the rest of the space is bounded.222While this definition has
appeared before in the literature (e.g., [33]), we are not aware of it having
a name. We call it weak temperedness due to its resemblance to the
temperedness condition
$C_{\phi}\coloneqq\sup_{x\in\mathcal{X}}\int_{\mathcal{X}}\lvert
1-\textup{e}^{-\phi(x,y)}\rvert\mathrm{d}y<\infty$ and the fact that
$\hat{C}_{\phi}\leq C_{\phi}$.

  * •

_Local stability:_ We call $\phi$ locally stable with constant
$L\in\mathbb{R}_{\geq 0}$ if for every finite $\eta\subset\mathcal{X}$ with
$H(\eta)<\infty$ and every $x\in\mathcal{X}$ it holds that

$H(\eta\cup x)-H(\eta)\geq-L.$

This condition should be thought of as assuming that for any realizable
configuration of points $\eta$, adding a point $x$ decreases the energy of the
configuration by at most $L$.

  * •

_Bounded range:_ We say that the has bounded range if there exists some
$R\in\mathbb{R}_{\geq 0}$ such that $\phi(x,y)=0$ for every pair of points
$x,y\in\mathcal{X}$ of distance at least $R$.

Our main result is the following uniqueness statement for Gibbs point
processes in Euclidean space $\mathcal{X}=\mathbb{R}^{d}$.

######  Theorem (Main).

For every bounded-range pair potential $\phi$ on $\mathbb{R}^{d}$ that is
locally stable with constant $L$ and weakly tempered with constant
$\hat{C}_{\phi}$, and every activity
$\lambda<(\textup{e}^{L}\hat{C}_{\phi})^{-1}$, there exists a unique Gibbs
point process on $\mathbb{R}^{d}$ in the sense of the Dobrushin–Lanford–Ruelle
formalism.

The formal version of the above theorem is given in Theorem 32 and Corollary
33. We defer a detailed comparison between our activity bound and the existing
literature to Section 6, where we compare it to various results that prove
absence of a phase transition (in the sense of DLR or analytically) in a
similar setting. However, we remark that, for bounded-range locally stable
potentials, our bound provides an improvement by a factor of
$\textup{e}C_{\phi}/\hat{C}_{\phi}\geq\textup{e}$ over the classical
Penrose–Ruelle bound [34, 30], which implies uniqueness via contraction of the
Kirkwood–Salsburg equations, and which constituted the best known bound in
this setting for more than 50 years. We proceed with discussing how we obtain
our result.

In the setting of our main theorem, existence of the infinite-volume Gibbs
point process follows from known results (see Theorem 3), and our main
contribution is to prove that the infinite-volume Gibbs measure is unique. Our
argument is purely probabilistic and inspired by the work of Dyer, Sinclair,
Vigoda and Weitz [10] for discrete hard-core gasses on a lattice. The main
idea is to construct a Markov process (in our case a spatial birth-death
process as introduced by Preston [31]) for every finite-volume region
$\Lambda$ such that the process is reversible (hence stationary) with respect
to the finite-volume Gibbs measure on $\Lambda$. We then establish two
properties for this family of Markov processes.

Firstly, we prove a _rapid mixing_ property, meaning that these processes
converge rapidly to their stationary distribution in total variation distance.
We show this rapid mixing result for all activities
$\lambda<(\textup{e}^{L}\hat{C}_{\phi})^{-1}$ by arguing that the local
stability and weak temperedness assumptions imply contraction of a suitably
constructed coupling. This step is fairly robust with respect to the
underlying space (it holds on arbitrary complete separable metric spaces) and
does not rely on the bounded range of the potential.

Secondly, we prove _slow disagreement percolation_ , which means that, for
each of these Markov processes, local disagreements between initial
configurations spread slowly. For this step, we use the local stability along
with the bounded range assumption and the fact that Euclidean space “grows
slowly”.

Combining both properties above shows that, under the conditions of the main
theorem, finite-volume Gibbs point processes satisfy a _spatial mixing_
property: different parts of a random point configuration become
asymptotically independent as the distance increases. We use this observation
to show that any pair of point processes satisfying the DLR formalism must
have identical finite-volume projections. By a standard result [6, Corollary
9.1.IX], every point process is uniquely characterized by its finite-volume
projections, and hence there is at most one infinite-volume Gibbs measure.

The proof technique we sketched above was previously applied in the continuous
setting by Helmuth, Perkins and Petti [17], who used a discrete-time Markov
chain to study the special case of uniqueness of the infinite-volume hard-
sphere model. Our work may be seen as a natural extension to more general
potentials. Moreover, spatial birth-death processes have been used before to
study different aspects of Gibbs point processes [1, 36, 25, 24, 12],
sometimes under different name such as _continuum Glauber dynamics_. In
particular the generator and its spectral gap have been studied in both the
finite-volume [1] and infinite-volume setting [25, 24], and the results have
been used to prove absence of a phase transition for various classes of models
[24]. However none of these results imply or are implied by ours. Moreover, as
our result is based on a simple coupling argument, most of the theory on
spatial birth-death processes that we require holds in the setting of jump
processes on general state spaces as presented in the classical texts of
Feller [11] and Blumenthal and Getoor [3]. This allows us to keep the
presentation of our result and our proofs surprisingly elementary.

###  1.2. Outline of the paper

The rest of the paper is structured as follows. In Section 2 we formally
introduce finite- and infinite-volume Gibbs point processes along with several
useful properties. In particular, we present a sufficient condition for
uniqueness of infinite-volume Gibbs measures, which essentially constitutes a
spatial mixing property. As we aim to keep most of our discussions as general
as possible, we introduce point processes in the setting of complete separable
metric spaces using the notion of random counting measures. However, switching
between this perspective and the intuition of random point sets is
straightforward. In Section 3 we give a short introduction to jump processes,
which contain the spatial birth-death processes we aim to use. We further
prove various lemmas that will allow to bound the speed of convergence to the
stationary distribution and to obtain tail bounds on certain hitting times,
the latter of which will be useful for the disagreement percolation. In
Section 4 we then construct the family of spatial birth-death processes that
we study and prove the desired rapid mixing and slow disagreement percolation.
Finally, in Section 5, we prove our main uniqueness result that we sketched
above. We end our paper by discussing our result and compare our bound with
the existing literature in Section 6.

##  2\. Gibbs Point Processes

###  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 $\mathcal{B}_{\text{b}}$ 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 set of functions
$\\{N_{\Lambda}:\mathcal{N}\to\mathbb{N}_{0}\cup\\{\infty\\},\eta\mapsto\eta(\Lambda)\mid\Lambda\in\mathcal{B}\\}$.
Further, we write $\mathcal{N}_{\text{f}}$ for the set of finite counting
measures on $(\mathcal{X},\mathcal{B})$ and equip it with
$\mathcal{R}_{\text{f}}$, the trace of $\mathcal{N}_{\text{f}}$ 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
$\mathcal{N}_{\Lambda}\coloneqq\\{\eta\in\mathcal{N}\mid\eta(\mathcal{X}\setminus\Lambda)=0\\}$
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
$\eta\in\mathcal{N}_{\text{f}}$ or $I=\mathbb{N}$ if
$\eta\in\mathcal{N}\setminus\mathcal{N}_{\text{f}}$ (see [6, Proposition
9.1.III]). As a consequence, we have for every $\eta\in\mathcal{N}$ and every
measurable $f:\mathcal{X}\to\overline{\mathbb{R}_{\geq 0}}$ 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 $\eta\in\mathcal{N}_{\text{f}}$.

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_

$S_{\eta}\coloneqq\\{x\in\mathcal{X}\mid\eta(x)>0\\},$

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 [6, Proposition 9.1.III]).
We can now write $\eta$ as

$\eta=\sum_{x\in S_{\eta}}\eta(x)\cdot\delta_{x}.$

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

$\displaystyle\eta\cap\xi$ $\displaystyle\coloneqq\sum_{x\in S_{\eta}\cap
S_{\xi}}\min(\eta(x),\xi(x))\cdot\delta_{x}$ $\displaystyle\eta\cup\xi$
$\displaystyle\coloneqq\sum_{x\in S_{\eta}\cup
S_{\xi}}\max(\eta(x),\xi(x))\cdot\delta_{x}$ $\displaystyle\eta\setminus\xi$
$\displaystyle\coloneqq\sum_{x\in
S_{\eta}}\max(0,\eta(x)-\xi(x))\cdot\delta_{x}$ $\displaystyle\eta\oplus\xi$
$\displaystyle\coloneqq(\eta\setminus\xi)+(\xi\setminus\eta).$

Thinking of locally finite counting measures as point (multi-)sets, these
operation correspond directly to intersection, union, set difference and
symmetric difference. It is easily checked that
$\eta\cup\xi=(\eta\oplus\xi)+(\eta\cap\xi)$ 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 $P$ on $\mathcal{X}$, 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

$P[\Lambda](A)=\int_{\mathcal{N}}\mathds{1}_{\\{\eta_{\Lambda}\in
A\\}}P(\mathrm{d}\eta).$

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 1 ([6, Corollary 9.1.IX]).

Let $P,Q\in\mathcal{P}(\mathcal{N},\mathcal{R})$. Suppose for all
$\ell\in\mathbb{N}$,
$\Lambda_{1},\dots,\Lambda_{\ell}\in\mathcal{B}_{\text{b}}$ and
$m_{1},\dots,m_{\ell}\in\mathbb{N}_{0}$ it holds that

$P(N_{\Lambda_{1}}=m_{1},\dots,N_{\Lambda_{\ell}}=m_{\ell})=Q(N_{\Lambda_{1}}=m_{1},\dots,N_{\Lambda_{\ell}}=m_{\ell}).$

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
$S\in\mathbb{R}_{\geq 0}$ such that for all $\eta\in\mathcal{N}_{\text{f}}$ it
holds that

$H(\eta)\coloneqq\frac{1}{2}\int_{\mathcal{X}^{2}}\phi(x,y)\eta^{(2)}(\mathrm{d}x,\mathrm{d}y)\geq-S\cdot\eta(\mathcal{X}).$

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

  1. (1)

It holds that $H(\mathbf{0})=0$ (_non-degenerate_).

  2. (2)

Suppose $\eta,\xi\in\mathcal{N}_{\text{f}}$ are such that $\xi\leq\eta$ 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

$H_{\Lambda}(\eta\mid\xi)\coloneqq
H(\eta)+\int_{\Lambda}\int_{\mathcal{X}\setminus\Lambda}\phi(x,y)\xi(\mathrm{d}y)\eta(\mathrm{d}x),$

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 $\lambda\in\mathbb{R}_{\geq 0}$ by

$\Xi_{\Lambda\mid\xi}(\lambda)\coloneqq
1+\sum_{n=1}^{\infty}\frac{\lambda^{n}}{n!}\int_{\Lambda^{n}}\exp\left(-H_{\Lambda}\Big{(}\sum_{i=1}^{n}\delta_{x_{i}}\Big{|}\xi\Big{)}\right)\nu^{n}(\mathrm{d}\boldsymbol{x}).$

Obviously, it holds that $\Xi_{\Lambda\mid\xi}(\lambda)\geq 1$. If further
$\xi$ 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

$\mu_{\Lambda\mid\xi}(A)\coloneqq\frac{\textup{e}^{\nu(\Lambda)}}{\Xi_{\Lambda\mid\xi}(\lambda)}\int_{A}\lambda^{\eta(\Lambda)}\textup{e}^{-H_{\Lambda}(\eta\mid\xi)}Q_{\Lambda}(\mathrm{d}\eta)$

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_ 333An alternative condition that is frequently considered is
temperedness, which requires that
$C_{\phi}\coloneqq\sup_{x\in\mathcal{X}}\int_{\mathcal{X}}|1-\textup{e}^{-\phi(x,y)}|\nu(\mathrm{d}y)<\infty$.
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 $\hat{C}_{\phi}\leq C_{\phi}$, where
the inequality is strict if $\phi$ is tempered and as a non-trivial negative
part. if

$\hat{C}_{\phi}\coloneqq\sup_{x\in\mathcal{X}}\int_{\mathcal{X}}1-\textup{e}^{-\left\lvert\phi(x,y)\right\rvert}\nu(\mathrm{d}y)<\infty,$

and we call $\hat{C}_{\phi}$ the _weak temperedness constant_ of $\phi$.
Moreover, we say that $\phi$ has bounded range $R\in\mathbb{R}_{\geq 0}$ if
for all $x,y\in\mathcal{X}$ with $d(x,y)\geq R$ it holds that $\phi(x,y)=0$.
Finally, we call $\phi$ _locally stable_ with constant $L\in\mathbb{R}_{\geq
0}$ if, for all $\eta\in\mathcal{N}_{\text{f}}$ with $H(\eta)<\infty$ and all
$x\in\mathcal{X}$, it holds that

$H(\eta+\delta_{x})-H(\eta)\geq-L.$

###  2.6. Infinite-volume Gibbs point process

Given an activity $\lambda\in\mathbb{R}_{\geq 0}$ 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 Dobrushin-Lanford-Ruelle
formalism.

######  Definition 2 (DLR formalism).

Given an activity $\lambda$ and potential $\phi$, we say a point process
$\mu\in\mathcal{P}(\mathcal{N},\mathcal{R})$ satisfies the DLR formalism if
for all $\Lambda\in\mathcal{B}_{\text{b}}$ the following holds:

  1. (1)

for
$\mathcal{C}_{\Lambda}=\\{\xi\in\mathcal{N}\mid\Xi_{\Lambda\mid\xi}(\lambda)<\infty\\}$
we have $\mu(\mathcal{C}_{\Lambda})=1$, and

  2. (2)

for all measurable $f:\mathcal{N}\to\mathbb{R}_{\geq 0}$ we have

$\int_{\mathcal{N}}f(\eta)\mu(\mathrm{d}\eta)=\int_{\mathcal{N}}\int_{\mathcal{N}_{\Lambda}}f(\eta+\xi_{\mathcal{X}\setminus\Lambda})\mu_{\Lambda\mid\xi}(\mathrm{d}\eta)\mu(\mathrm{d}\xi).$

In this case, we call $\mu$ an _infinite-volume Gibbs point process_ (or
infinite-volume Gibbs measure) compatible with $\lambda$ and $\phi$. Further,
we write $\mathcal{G}(\lambda,\phi)$ for the set of such infinite-volume Gibbs
point processes.

As discussed earlier, we will mostly focus on proving conditions for which
$\left\lvert\mathcal{G}(\lambda,\phi)\right\rvert\leq 1$ (i.e., uniqueness of
infinite-volume Gibbs point processes). However, at this point, we would like
to point out that, in the setting studies 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.

######  Theorem 3 (Existence, [20, Theorem 5.6]).

Suppose $\phi$ is locally stable and has bounded range, then for all
$\lambda\in\mathbb{R}_{\geq 0}$ it holds that
$\mathcal{G}(\lambda,\phi)\neq\emptyset$.

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 [7, Section 2] and [20,
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_

$\displaystyle\mathcal{N}_{H}\coloneqq\\{\eta\in\mathcal{N}\mid\text{ for all
$\xi\in\mathcal{N}_{\text{f}}$ with $\xi\leq\eta$ it holds that
$H(\xi)<\infty$}\\}.$ (1)

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, the following lemma allows us to restrict to boundary conditions
from $\mathcal{N}_{H}$ when characterizing infinite-volume Gibbs measures.

######  Lemma 4.

For every pair potential $\phi$ it holds that $\mathcal{N}_{H}\in\mathcal{R}$.
Moreover, for every activity $\lambda\in\mathbb{R}_{\geq 0}$ and all
$\mu\in\mathcal{G}(\lambda,\phi)$ it holds that $\mu(\mathcal{N}_{H})=1$.

###### Proof.

Fix some $x\in\mathcal{X}$ and, for $n\in\mathbb{N}$, let
$\Lambda_{n}=\\{y\in\mathcal{X}\mid d(x,y)<n\\}$. If
$\eta\in\mathcal{N}\setminus\mathcal{N}_{H}$ then, by definition, there is
some $\xi\in\mathcal{N}_{\text{f}}$ with $\xi\leq\eta$ and $H(\xi)=\infty$. By
the fact that $H$ is hereditary and $\xi$ is finite, we can find some
$n\in\mathbb{N}$ such that $H(\eta_{\Lambda_{n}})=\infty$. Moreover, if such
an $n\in\mathbb{N}$ exists, then obviously
$\eta\in\mathcal{N}\setminus\mathcal{N}_{H}$. By setting
$A_{n}=\\{\eta\in\mathcal{N}\mid H(\eta_{\Lambda_{n}})=\infty\\}$ and noting
that $A_{n}\in\mathcal{R}$, we have
$\mathcal{N}\setminus\mathcal{N}_{H}=\bigcup_{n\in\mathbb{N}}A_{n}\in\mathcal{R}$.

Next, suppose that $\mu\in\mathcal{G}(\lambda,\phi)$. By the union bound, it
suffices to show that $\mu(A_{n})=0$ for every $n\in\mathbb{N}$. Observe that
for all $\xi\in\mathcal{N}$ and all $\eta\in A_{n}$ it holds that
$H_{\Lambda_{n}}(\eta\mid\xi)=\infty$. Therefore,
$\mu_{\Lambda_{n}\mid\xi}(A_{n})=0$ for all $\xi$ such that
$\mu_{\Lambda_{n}\mid\xi}$ is well-defined. But then, by the DLR formalism in
Definition 2, it follows that $\mu(A_{n})=0$, which concludes the proof. ∎

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

$W(x,\eta)\coloneqq\int_{\mathcal{X}}\phi(x,y)\eta(\mathrm{d}y)$

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

######  Lemma 5.

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

$\displaystyle
H_{\Lambda}(\eta\mid\xi)=\frac{1}{2}\int_{\Lambda}W(x,\eta-\delta_{x})\eta(\mathrm{d}x)+\int_{\Lambda}W(x,\xi_{\Lambda^{c}})\eta(\mathrm{d}x).$
(2)

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

$\displaystyle
H_{\Lambda}(\eta+\delta_{x}\mid\xi)=H_{\Lambda}(\eta\mid\xi)+W(x,\eta+\xi_{\Lambda^{c}}).$
(3)

###### Proof.

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

$H(\eta)=\frac{1}{2}\int_{\Lambda}W(x,\eta-\delta_{x})\eta(\mathrm{d}x),$

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

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

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 $(\infty,\infty]$, observing
that

$H(\eta+\delta_{x})=H(\eta)+W(x,\eta).$

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

$\displaystyle\int_{\Lambda}\int_{\Lambda^{c}}\phi(y,z)\xi(\mathrm{d}z)(\eta+\delta_{x})(\mathrm{d}y)$
$\displaystyle=\int_{\Lambda}\int_{\Lambda^{c}}\phi(y,z)\xi(\mathrm{d}z)\eta(\mathrm{d}y)+\int_{\Lambda^{c}}\phi(x,z)\xi(\mathrm{d}z)$
$\displaystyle=\int_{\Lambda}\int_{\Lambda^{c}}\phi(y,z)\xi(\mathrm{d}z)\eta(\mathrm{d}y)+W(x,\xi_{\Lambda^{c}})$

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

$\displaystyle H_{\Lambda}(\eta+\delta_{x}\mid\xi)$
$\displaystyle=H(\eta)+W(x,\eta)+\int_{\Lambda}\int_{\Lambda^{c}}\phi(y,z)\xi(\mathrm{d}z)\eta(\mathrm{d}y)+W(x,\xi_{\Lambda^{c}})$
$\displaystyle=H_{\Lambda}(\eta\mid\xi)+W(x,\eta+\xi_{\Lambda^{c}}).$

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.

######  Lemma 6.

Suppose $\phi$ is a locally stable pair potential with constant $L$. For all
$x\in\mathcal{X}$ and $\eta\in\mathcal{N}_{H}$ it holds that
$W(x,\eta)\geq-L$.

###### Proof.

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

$W(x,\eta)=H(\eta+\delta_{x})-H(\eta)\geq-L.$

Next, suppose that $\eta\in\mathcal{N}_{H}\setminus\mathcal{N}_{\text{f}}$,
and let $f:\mathcal{X}\to\overline{\mathbb{R}_{\geq
0}},y\mapsto\max(0,-\phi(x,y))$. Note that to prove our claim, it suffices to
show that $\int f\mathrm{d}\eta\leq L$. 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

$\int_{\mathcal{X}}f(y)\eta(\mathrm{d}y)=\lim_{n\to\infty}\int_{\mathcal{X}}f(y)\eta_{n}(\mathrm{d}y).$

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

$\int_{\mathcal{X}}f(y)\eta_{n}(\mathrm{d}y)=\sum_{i=1}^{n}f(x_{i})=-\sum_{i\in
S_{n}}\phi(x,x_{i})=-(H(\eta_{S_{n}}+\delta_{x})-H(\eta_{S_{n}}))\leq L.$

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.

######  Lemma 7.

Suppose $\phi$ is locally stable with constant $L$. For all
$\Lambda\in\mathcal{B}_{\text{b}}$ and $\xi\in\mathcal{N}_{H}$ it holds that
$\Xi_{\Lambda\mid\xi}(\lambda)<\infty$. In particular, $\mu_{\Lambda\mid\xi}$
is well-defined.

###### Proof.

The main idea is to show that, for all $\eta\in\mathcal{N}_{\Lambda}$, it
holds that $H_{\Lambda}(\eta\mid\xi)\geq-CL\eta(\Lambda)$ for some absolute
constant $C$. If $H(\eta)=\infty$ then $H_{\Lambda}(\eta\mid\xi)=\infty$.
Otherwise, we use identity (2) and Lemma 6 to obtain

$H_{\Lambda}(\eta\mid\xi)=\frac{1}{2}\int_{\Lambda}W(x,\eta-\delta_{x})\eta(\mathrm{d}x)+\int_{\Lambda}W(x,\xi_{\Lambda^{c}})\eta(\mathrm{d}x)\geq-\frac{3}{2}L\eta(\Lambda).$

Thus, we get

$\displaystyle\Xi_{\Lambda\mid\xi}(\lambda)\leq
1+\sum_{n=1}^{\infty}\frac{\lambda^{n}\textup{e}^{\frac{3}{2}Ln}\nu(\Lambda)^{n}}{n!},$

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 [14] and Nguyen and Zessin
[38], and which is hence known as GNZ equation. As some of our definitions
differ slightly from those of the original papers, we briefly reprove it.

######  Lemma 8 (Finite-volume GNZ equation [14, 38]).

Suppose $\phi$ is locally stable with constant $L$, and let
$\Lambda\in\mathcal{B}_{\text{b}}$ and $\xi\in\mathcal{N}_{H}$. For all non-
negative measurable
$F:\mathcal{X}\times\mathcal{N}\to\overline{\mathbb{R}_{\geq 0}}$ it holds
that

$\int_{\mathcal{N}_{\Lambda}}\int_{\Lambda}F(x,\eta)\eta(\mathrm{d}x)\mu_{\Lambda\mid\xi}(\mathrm{d}\eta)=\int_{\Lambda}\int_{\mathcal{N}_{\Lambda}}\lambda\textup{e}^{-W(x,\eta+\xi_{\Lambda^{c}})}F(x,\eta+\delta_{x})\mu_{\Lambda\mid\xi}(\mathrm{d}\eta)\nu(\mathrm{d}x).$

###### Proof.

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

$\displaystyle\int_{\mathcal{N}_{\Lambda}}\int_{\Lambda}F(x,\eta)\eta(\mathrm{d}x)\mu_{\Lambda\mid\xi}(\mathrm{d}\eta)$
$\displaystyle=\frac{\textup{e}^{\nu(\Lambda)}}{\Xi_{\Lambda\mid\xi}(\lambda)}\int_{\mathcal{N}_{\Lambda}}\int_{\Lambda}F(x,\eta)\eta(\mathrm{d}x)\lambda^{\eta(\Lambda)}\textup{e}^{-H_{\Lambda}(\eta\mid\xi)}Q_{\Lambda}(\mathrm{d}\eta)$
$\displaystyle=\int_{\mathcal{N}_{\Lambda}}\int_{\Lambda}G(x,\eta)\eta(\mathrm{d}x)Q_{\Lambda}(\mathrm{d}\eta)$

for
$G(x,\eta)\coloneqq\frac{\textup{e}^{\nu(\Lambda)}}{\Xi_{\Lambda\mid\xi}(\lambda)}F(x,\eta)\lambda^{\eta(\Lambda)}\textup{e}^{-H_{\Lambda}(\eta\mid\xi)}$.
Applying the Mecke equation (see [27, Proposition 4.1]) and the decomposition
in (3) then yields

$\displaystyle\int_{\mathcal{N}_{\Lambda}}\int_{\Lambda}G(x,\eta)\eta(\mathrm{d}x)Q_{\Lambda}(\mathrm{d}\eta)$
$\displaystyle=\int_{\Lambda}\int_{\mathcal{N}_{\Lambda}}G(x,\eta+\delta_{x})Q_{\Lambda}(\mathrm{d}\eta)\nu(\mathrm{d}x)$
$\displaystyle=\int_{\Lambda}\frac{\textup{e}^{\nu(\Lambda)}}{\Xi_{\Lambda\mid\xi}(\lambda)}\int_{\mathcal{N}_{\Lambda}}\lambda\textup{e}^{-W(x,\eta+\xi_{\Lambda^{c}})}F(x,\eta+\delta_{x})\lambda^{\eta(\Lambda)}\textup{e}^{-H_{\Lambda}(\eta\mid\xi)}Q_{\Lambda}(\mathrm{d}\eta)\nu(\mathrm{d}x)$
$\displaystyle=\int_{\Lambda}\int_{\mathcal{N}_{\Lambda}}\lambda\textup{e}^{-W(x,\eta+\xi_{\Lambda^{c}})}F(x,\eta+\delta_{x})\mu_{\Lambda\mid\xi}(\mathrm{d}\eta)\nu(\mathrm{d}x),$

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

$\displaystyle\mathcal{N}_{\Lambda\mid\xi}\coloneqq\\{\eta\in\mathcal{N}_{\Lambda}\mid\eta+\xi_{\Lambda^{c}}\in\mathcal{N}_{H}\\}.$
(4)

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.

######  Lemma 9.

Suppose $\phi$ is locally stable with constant $L$, and let
$\Lambda\in\mathcal{B}_{\text{b}}$. For every $\xi\in\mathcal{N}_{H}$ it holds
that $\mu_{\Lambda\mid\xi}(\mathcal{N}_{\Lambda\mid\xi})=1$.

###### 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 $\zeta\in\mathcal{N}_{\text{f}}$,
$\zeta\leq\xi_{\Lambda^{c}}$ such that $H(\eta+\zeta)=\infty$. Since
$\xi\in\mathcal{N}_{H}$, it further holds that $H(\zeta)<\infty$. Next, note
that

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

Hence, there must be some $x\in S_{\eta},y\in S_{\zeta}$ such that
$\phi(x,y)=\infty$. But then, since $\zeta\leq\xi_{\Lambda^{c}}$, 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

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

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
equivalence of infinite-volume Gibbs measures that is derived from the DLR
formalism.

######  Lemma 10.

Let $\mu_{1},\mu_{2}\in\mathcal{G}(\lambda,\phi)$ for an activity
$\lambda\in\mathbb{R}_{\geq 0}$ and potential $\phi$. Suppose there is a non-
decreasing sequence $(\Lambda_{k})_{k\in\mathbb{N}}$ in
$\mathcal{B}_{\text{b}}$, such that

  1. (1)

for every $\Delta\in\mathcal{B}_{\text{b}}$ it holds that
$\Delta\subseteq\Lambda_{k}$ for some $k\in\mathbb{N}$, and

  2. (2)

for every $k\in\mathbb{N}$ it holds that

$\lim_{n\to\infty}\operatorname*{ess\,sup}_{\xi_{1}\sim\mu_{1},\xi_{2}\sim\mu_{2}}\left\lvert\mu_{\Lambda_{k+n}\mid\xi_{1}}[\Lambda_{k}]-\mu_{\Lambda_{k+n}\mid\xi_{2}}[\Lambda_{k}]\right\rvert_{\mathrm{tv}}=0,$

where $\left\lvert\cdot-\cdot\right\rvert_{\mathrm{tv}}$ denotes the total
variation distance, and the essential supremum is taken over pairs
$(\xi_{1},\xi_{2})\in\mathcal{N}\times\mathcal{N}$ and with respect to the
product measure $\mu_{1}\otimes\mu_{2}$.

Then $\mu_{1}=\mu_{2}$.

###### Proof.

By Lemma 1, it suffices to show that $\mu_{1}(A)=\mu_{2}(A)$ for all
$A\in\mathcal{R}$ of the form

$A=\\{N_{\Delta_{1}}=m_{1},\dots,N_{\Delta_{\ell}}=m_{\ell}\\}$

for $\ell\in\mathbb{N}$,
$\Delta_{1},\dots,\Delta_{\ell}\in\mathcal{B}_{\text{b}}$ 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 $1\leq
j\leq\ell$. This implies $\eta(\Delta_{j})=\eta_{\Lambda_{k}}(\Delta_{j})$ for
all $\eta\in\mathcal{N}$ and all $1\leq j\leq\ell$. 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 for all
$n\in\mathbb{N}_{0}$

$\displaystyle\mu_{1}(A)=\int_{\mathcal{N}}\int_{\Lambda_{k+n}}\mathds{1}_{\\{\eta_{\Lambda_{k}}\in
A\\}}\mu_{\Lambda_{k+n}\mid\xi}(\mathrm{d}\eta)\mu_{1}(\mathrm{d}\xi)=\int_{\mathcal{N}}\mu_{\Lambda_{k+n}\mid\xi}[\Lambda_{k}](A)\mu_{1}(\mathrm{d}\xi).$

Applying the same argument to $\mu_{2}$ yields

$\displaystyle\left\lvert\mu_{1}(A)-\mu_{2}(A)\right\rvert$
$\displaystyle\leq\int_{\mathcal{N}}\int_{\mathcal{N}}\left\lvert\mu_{\Lambda_{k+n}\mid\xi_{1}}[\Lambda_{k}](A)-\mu_{\Lambda_{k+n}\mid\xi_{2}}[\Lambda_{k}](A)\right\rvert\mu_{1}(\mathrm{d}\xi_{1})\mu_{2}(\mathrm{d}\xi_{2})$
$\displaystyle\leq\int_{\mathcal{N}}\int_{\mathcal{N}}\left\lvert\mu_{\Lambda_{k+n}\mid\xi_{1}}[\Lambda_{k}]-\mu_{\Lambda_{k+n}\mid\xi_{2}}[\Lambda_{k}]\right\rvert_{\mathrm{tv}}\mu_{1}(\mathrm{d}\xi_{1})\mu_{2}(\mathrm{d}\xi_{2})$
$\displaystyle\leq\operatorname*{ess\,sup}_{\xi_{1}\sim\mu_{1},\xi_{2}\sim\mu_{2}}\left\lvert\mu_{\Lambda_{k+n}\mid\xi_{1}}[\Lambda_{k}]-\mu_{\Lambda_{k+n}\mid\xi_{2}}[\Lambda_{k}]\right\rvert_{\mathrm{tv}}.$

Since this holds for all $n$, taking the limit $n\to\infty$ shows
$\mu_{1}(A)=\mu_{2}(A)$, concluding the proof. ∎

##  3\. Jump processes

###  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
$K:E\times\mathcal{E}\to\overline{\mathbb{R}_{\geq 0}}$ such that:

  1. (1)

For all $x\in E$, $K(x,\cdot)$ is a measure on $(E,\mathcal{E})$.

  2. (2)

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

Further we say a kernel $K$ is

  * •

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

  * •

_sub-stochastic_ if $K(x,E)\leq 1$ 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 non-negative measurable functions $f:E\to\overline{\mathbb{R}_{\geq
0}}$. More precisely, we define the function
$Kf:E\to\overline{\mathbb{R}_{\geq 0}}$ by setting

$(Kf)(x)\coloneqq\int_{E}f(y)K(x,\mathrm{d}y)$

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 $(P_{t})_{t\in\mathbb{R}_{\geq 0}}$ such
that

  1. (1)

For every $t\in\mathbb{R}_{\geq 0}$, $P_{t}$ is a sub-stochastic kernel.

  2. (2)

For all $x\in E$ and $A\in\mathcal{E}$ we have $P_{0}(x,A)=\delta_{x}(A)$.

  3. (3)

$(P_{t})_{t\in\mathbb{R}_{\geq 0}}$ satisfies the Chapman–Kolmogorov equation:
for all $t,s\in\mathbb{R}_{\geq 0}$ and all $x\in E$ , $A\in\mathcal{E}$ it
holds that

$P_{t+s}(x,A)=\int_{E}P_{s}(y,A)P_{t}(x,\mathrm{d}y).$

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

###  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 $K:E\times\mathcal{E}\to\mathbb{R}_{\geq 0}$ is a
jump kernel on $(E,\mathcal{E})$ if:

  1. (1)

$K$ is a finite kernel.

  2. (2)

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

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

$\displaystyle\Pi(x,A)\coloneqq\begin{cases}\frac{1}{\kappa(x)}\cdot
K(x,A)&\text{ if $\kappa(x)>0$}\\\ \delta_{x}(A)&\text{ otherwise}\end{cases}$
(5)

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
$t\in\mathbb{R}_{\geq 0}$ we set

$\displaystyle P_{t}^{(0)}(x,A)=\textup{e}^{-\kappa(x)t}\cdot\delta_{x}(A).$
(6)

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

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

The following properties can easily be checked via induction over $n$ (see for
example [11]):

  1. (1)

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. (2)

For every $x\in E$, $A\in\mathcal{E}$ and $t\in\mathbb{R}_{\geq 0}$, it holds
that $P_{t}^{(n)}(x,A)$ is non-decreasing in $n$.

It immediately follows by monotone convergence that the limit
$P_{t}(x,A)=\lim_{n\to\infty}P_{t}^{(n)}(x,A)$ 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 $t\in\mathbb{R}_{\geq 0}$. In that case,
we say that the jump kernel $K$ is _non-explosive_ and simply call $(P_{t})$
the _transition family associated to_ $K$.

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

######  Lemma 11 ([5, Theorem $16$]).

Let $K$ be a jump kernel on $(E,\mathcal{E})$ and let $\kappa$ denote its rate
function. Suppose there exists a sequence $(E_{n})_{n\in\mathbb{N}}$ with
$E_{n}\in\mathcal{E}$ and a non-negative measurable function
$f:E\to\mathbb{R}_{\geq 0}$ such that the following holds:

  1. (1)

$E_{n}$ converges to $E$ monotonically.

  2. (2)

For all $n\in\mathbb{N}$ we have $\sup_{x\in E_{n}}\kappa(x)<\infty$.

  3. (3)

$\lim_{n\to\infty}\inf_{x\in E\setminus E_{n}}f(x)=\infty$.

  4. (4)

There exists some $c\in\mathbb{R}$ such that for all $x\in E$ we have

$(Kf)(x)\leq(c+\kappa(x))\cdot f(x)$

Then $K$ is non-explosive.

###  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
$t\in\mathbb{R}_{\geq 0}$ and $A\in\mathcal{E}$ it holds that

$\int_{E}P_{t}(x,A)\pi(\mathrm{d}x)=\pi(A).$

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

$\int_{A}P_{t}(x,B)\pi(\mathrm{d}x)=\int_{B}P_{t}(x,A)\pi(\mathrm{d}x),$

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 [4].

######  Lemma 12 ([37, Proposition 2]).

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

$\int_{A}K(x,B)\pi(\mathrm{d}x)=\int_{B}K(x,A)\pi(\mathrm{d}x)$

for all $A,B\in\mathcal{E}$ then $(P_{t})$ is reversible with respect to
$\pi$. Further, if $K$ is non-explosive 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.

######  Lemma 13 ([18, Theorem 2.4]).

Let $\pi_{1},\pi_{2}$ be probability measures on $(E,\mathcal{E})$. Suppose
$(E,\mathcal{E})$ is such that $\mathcal{E}\otimes\mathcal{E}$ contains the
diagonal set $D\coloneqq\\{(x,x)\mid x\in E\\}$ then, for every coupling $\pi$
of $\pi_{1}$ and $\pi_{2}$, it holds that

$\left\lvert\pi_{1}-\pi_{2}\right\rvert_{\mathrm{tv}}\leq 2\cdot\pi(D^{c}),$

where
$\left\lvert\pi_{1}-\pi_{2}\right\rvert_{\mathrm{tv}}\coloneqq\sup_{A\in\mathcal{E}}\left\lvert\pi_{1}(A)-\pi_{2}(A)\right\rvert$
denotes the total variation distance between $\pi_{1}$ and $\pi_{2}$.

The notion of a coupling extends directly to Markov transition families as
follows.

######  Definition 14.

Suppose $(P_{t})$, $(P^{\prime}_{t})$ are Markov transition families on
$(E_{1},\mathcal{E}_{1})$ and $(E_{2},\mathcal{E}_{2})$. We call a Markov
transition family $(Q_{t})$ on $(E_{1}\times
E_{2},\mathcal{E}_{1}\otimes\mathcal{E}_{2})$ a coupling of $(P_{t})$,
$(P^{\prime}_{t})$ if for all $x\in E_{1},y\in E_{2}$ and
$t\in\mathbb{R}_{\geq 0}$ it holds that $Q_{t}((x,y),\cdot)$ is a coupling of
$P_{t}(x,\cdot)$ and $P^{\prime}_{t}(y,\cdot)$.

We wish to construct couplings between (jump) Markov transition families based
on their jump kernels. Following the ideas in Chen [5], we introduce the
following definition.

######  Definition 15.

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 non-negative bounded measurable functions
$f_{1}:E_{1}\to\mathbb{R}_{\geq 0},f_{2}:E_{2}\to\mathbb{R}_{\geq 0}$ and all
$x_{1}\in E,x_{2}\in E_{2}$ it holds that

$\displaystyle\int_{E_{1}\times
E_{2}}(f_{1}(y_{1})-f_{1}(x_{1}))K((x_{1},x_{2}),\mathrm{d}y_{1}\times\mathrm{d}y_{2})=(K_{1}f_{1})(x_{1})-\kappa_{1}(x_{1})\cdot
f_{1}(x_{1})$ $\displaystyle\int_{E_{1}\times
E_{2}}(f_{2}(y_{2})-f_{2}(x_{2}))K((x_{1},x_{2}),\mathrm{d}y_{1}\times\mathrm{d}y_{2})=(K_{2}f_{2})(x_{2})-\kappa_{2}(x_{2})\cdot
f_{2}(x_{2}).$

The following result relates the two notions of a coupling that are given in
Definitions 14 and 15.

######  Lemma 16 ([5, Theorems $13$ and $25$]).

Let $K_{1},K_{2}$ be non-explosive jump kernels on $(E_{1},\mathcal{E}_{1})$
and $(E_{2},\mathcal{E}_{2})$. If $K$ is a coupling of $K_{1},K_{2}$ in the
sense of Definition 15 then $K$ is non-explosive, and the Markov transition
family associated to $K$ is a coupling of the Markov transition families
associated to $K_{1},K_{2}$ in the sense of Definition 14.

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 17.

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

$(Kf)(x)\leq(\kappa(x)-\delta)\cdot f(x).$

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

$(P_{t}f)(x)\leq\textup{e}^{-\delta t}f(x).$

###### Proof.

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

$(P_{t}^{(n)}f)(x)\leq\textup{e}^{-\delta t}f(x),$

where $(P^{(n)}_{t})$ is as defined in (6) and (7). The desired result then
follows from the fact that, for all $t\in\mathbb{R}_{\geq 0}$ 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

$(P_{t}^{(0)}f)(x)=\textup{e}^{-\kappa(x)t}f(x).$

Hence, we need to show $\textup{e}^{-\kappa(x)t}f(x)\leq\textup{e}^{-\delta
t}f(x)$ 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 non-negative, our assumptions
imply $0\leq(Kf)(x)\leq(\kappa(x)-\delta)\cdot f(x)$. Thus, we have
$\delta\leq\kappa(x)$ 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

$\displaystyle(P_{t}^{(n)}f)(x)$ $\displaystyle=\textup{e}^{-\kappa(x)t}\cdot
f(x)+\int_{0}^{t}\textup{e}^{-\kappa(x)s}\int_{E}(P_{t-s}^{(n-1)}f)(y)K(x,\mathrm{d}y)\mathrm{d}s$
$\displaystyle\leq\textup{e}^{-\kappa(x)t}\cdot
f(x)+\int_{0}^{t}\textup{e}^{-\kappa(x)s}\int_{E}\textup{e}^{-\delta\cdot(t-s)}f(y)K(x,\mathrm{d}y)\mathrm{d}s$
$\displaystyle=\textup{e}^{-\kappa(x)t}\cdot f(x)+\textup{e}^{-\delta
t}\int_{0}^{t}\textup{e}^{-(\kappa(x)-\delta)\cdot s}(Kf)(x)\mathrm{d}s.$

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

$\displaystyle(P_{t}^{(n)}f)(x)$
$\displaystyle\leq\left(\textup{e}^{-\kappa(x)t}+\textup{e}^{-\delta
t}\int_{0}^{t}\textup{e}^{-(\kappa(x)-\delta)\cdot
s}(\kappa(x)-\delta)\mathrm{d}s\right)\cdot f(x)$
$\displaystyle=\left(\textup{e}^{-\kappa(x)t}+\textup{e}^{-\delta
t}\left(1-\textup{e}^{-(\kappa(x)-\delta)\cdot t}\right)\right)\cdot f(x)$
$\displaystyle=\textup{e}^{-\delta t}f(x),$

which concludes the induction step and proves the claim. ∎

We can now combine Lemma 17 with a suitable coupling of a non-explosive jump
kernel to derive information about the speed of convergence of the associated
Markov transition family to its stationary distribution.

######  Lemma 18.

Let $K$ be a non-explosive jump kernel on $(E,\mathcal{E})$ and assume 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 15) and a non-negative measurable function $f:E\times
E\to\mathbb{R}_{\geq 0}$ such that the following hold:

  1. (1)

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

  2. (2)

There exists some $\delta>0$ such that for all $x,y\in E$

$(\tilde{K}f)(x,y)\leq(\tilde{\kappa}(x,y)-\delta)\cdot f(x,y),$

where $\tilde{\kappa}$ is the rate function of $\tilde{K}$.

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

$\left\lvert
P_{t}(x,\cdot)-\pi\right\rvert_{\mathrm{tv}}\leq\frac{1}{c}\cdot\textup{e}^{-\delta
t}\int_{E}f(x,y)\pi(\mathrm{d}y).$

###### Proof.

By Lemma 16 we know that $\tilde{K}$ is non-explosive and that the associated
Markov transition family $(Q_{t})$ is a coupling of $(P_{t})$ with itself in
the sense of Definition 14. Further, by (2) and Lemma 17, we have for all
$x,y\in E$ and all $t\in\mathbb{R}_{\geq 0}$ that
$(Q_{t}f)(x,y)\leq\textup{e}^{-\delta t}f(x,y)$.

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

$Q_{x,t}(A)\coloneqq\int_{E\times
E}Q_{t}(z,A)(\delta_{x}\otimes\pi)(\mathrm{d}z),$

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 for all $A\in\mathcal{E}$

$\displaystyle Q_{x,t}(A\times E)$
$\displaystyle=\int_{E}\int_{E}Q_{t}((z_{1},z_{2}),A\times
E)\delta_{x}(\mathrm{d}z_{1})\pi(\mathrm{d}z_{2})$
$\displaystyle=\int_{E}\int_{E}P_{t}(z_{1},A)\delta_{x}(\mathrm{d}z_{1})\pi(\mathrm{d}z_{2})$
$\displaystyle=\int_{E}P_{t}(z_{1},A)\delta_{x}(\mathrm{d}z_{1})$
$\displaystyle=P_{t}(x,A).$

Further, since $\pi$ is invariant with respect to $(P_{t})$, we have
analogously

$Q_{x,t}(E\times A)=\int_{E}P_{t}(z_{2},A)\pi(\mathrm{d}z_{2})=\pi(A).$

This shows that $Q_{x,t}$ is a coupling of $P_{t}(x,\cdot)$ and $\pi$. Note
that by (1) 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 13, it suffices to bound
$Q_{x,t}(D^{c})$. Applying Markov’s inequality, we get

$\left\lvert P_{t}(x,\cdot)-\pi\right\rvert_{\mathrm{tv}}\leq
Q_{x,t}(D^{c})\leq\frac{1}{c}\int_{E\times
E}f(z)Q_{x,t}(\mathrm{d}z)=\frac{1}{c}\int_{E\times
E}(Q_{t}f)(z)(\delta_{x}\otimes\pi)(\mathrm{d}z).$

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

$\left\lvert
P_{t}(x,\cdot)-\pi\right\rvert_{\mathrm{tv}}\leq\frac{1}{c}\cdot\textup{e}^{-\delta
t}\int_{E\times
E}f(z)(\delta_{x}\otimes\pi)(\mathrm{d}z)=\frac{1}{c}\cdot\textup{e}^{-\delta
t}\int_{E}f(x,y)\pi(\mathrm{d}y),$

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
$(\Omega,\mathcal{F},\mathbb{P},(X_{t})_{t\in\mathbb{R}_{\geq 0}})$ where
$(\Omega,\mathcal{F},\mathbb{P})$ is a probability space and
$(X_{t})_{t\in\mathbb{R}_{\geq 0}}$ is a collection of random variables
$X_{t}:\Omega\to E$. Since we will always use $\mathbb{R}_{\geq 0}$ 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 below). 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
$\mathcal{B}_{\mathbb{R}_{\geq 0}}\otimes\mathcal{F}$ and $\mathcal{E}$, where
$\mathcal{B}_{\mathbb{R}_{\geq 0}}$ is the Borel $\sigma$-field on
$\mathbb{R}_{\geq 0}$.

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
$s,t\in\mathbb{R}_{\geq 0}$, $s\leq t$. We call a random variable
$T:\Omega\to\overline{\mathbb{R}_{\geq 0}}$ a stopping time with respect to a
filtration $(\mathcal{F}_{t})$ if, for all $t\in\mathbb{R}_{\geq 0}$, the
event $\\{T\leq t\\}$ is measurable with respect to $\mathcal{F}_{t}$. Given a
stopping time $T$ with respect to some filtration $(\mathcal{F}_{t})$, we
define $\mathcal{F}_{T}\coloneqq\\{A\in\mathcal{F}\mid\forall
t\in\mathbb{R}_{\geq 0}:A\cap\\{T\leq t\\}\in\mathcal{F}_{t}\\}$. Further, say
that a stochastic process is _progressive_ with respect to a filtration
$(\mathcal{F}_{t})$ if for all $t\in\mathbb{R}_{\geq 0}$ 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
$\mathcal{F}^{X}_{t}\coloneqq\sigma(X_{s}\mid s\in[0,t])$.

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. (1)

for all $A\in\mathcal{E}$ we have $\mathbb{P}[X_{0}\in A]=\mu(A)$, and

  2. (2)

for all $t,s\in\mathbb{R}_{\geq 0}$ 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 $t\in\mathbb{R}_{\geq 0}$ 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

$\mathbb{P}[X_{s+t}\in A\mid\mathcal{F}^{X}_{s}]=\mathbb{P}[X_{s+t}\in A\mid
X_{s}]\text{\hskip 30.00005pt $\mathbb{P}$-almost surely.}$

This is usually referred to as the Markov property.

It turns out that, given a non-explosive 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 [3, §1.12] and [22]).

######  Theorem 19.

Let $K$ be a non-explosive jump kernel and let $\mu$ be a probability measure
on $(E,\mathcal{E})$. There is a stochastic process
$(\Omega,\mathcal{F},\mathbb{P}_{\mu},(X_{t}))$ with values in
$(E,\mathcal{E})$ such that:

  1. (1)

The sample paths of $(X_{t})$ are piecewise constant. That is, for every
$\omega\in\Omega$ and $s\in\mathbb{R}_{\geq 0}$ there is some $h>0$ such that
the map $t\mapsto X_{t}(\omega)$ is constant on $[s,s+h)$.

  2. (2)

For every $A\in\mathcal{E}$, $T_{A}\coloneqq\inf\\{t\in\mathbb{R}_{\geq 0}\mid
X_{t}\in A\\}$ is a stopping time with respect to $(\mathcal{F}^{X}_{t})$.

  3. (3)

$(\Omega,\mathcal{F},\mathbb{P}_{\mu},(X_{t}))$ is a Markov process with
initial distribution $\mu$ and transition function $(P_{t})$.

  4. (4)

The process satisfies the following version of the _strong Markov property_ :
Suppose $T$ is a stopping time with respect to $(\mathcal{F}^{X}_{t})$, then
for every $t\in\mathbb{R}_{\geq 0}$ and $A\in\mathcal{E}$

$\mathbb{P}_{\mu}[T<\infty,X_{T+t}\in
A\mid\mathcal{F}^{X}_{T}]=\mathds{1}_{\\{T<\infty\\}}\cdot
P_{t}(X_{T},A)\text{ $\mathbb{P}_{\mu}$-a.s.}$

Moreover, on the same probability space
$(\Omega,\mathcal{F},\mathbb{P}_{\mu})$ we can construct random variables
$(Z_{n})_{n\in\mathbb{N}_{0}}$ with values in $(E,\mathcal{E})$ and
$(\tau_{n})_{n\in\mathbb{N}_{0}}$ with values in $\overline{\mathbb{R}_{>0}}$
such that

  1. (5)

It holds that $\tau_{0}=0$ and $n\mapsto\tau_{n}$ is strictly increasing.

  2. (6)

$(Z_{n})$ is a Markov chain on $(E,\mathcal{E})$ with transition kernel $\Pi$
(see (5)) and initial law $\mu$.

  3. (7)

For all $n\in\mathbb{N}_{0}$ and $t\in\mathbb{R}_{\geq 0}$ with $\tau_{n}\leq
t<\tau_{n+1}$ it holds that $X_{t}=Z_{n}$.

We will make use of the following two lemmas about hitting times for the
process that is given by Theorem 19. 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 $B$.

######  Lemma 20.

Let $K$ be a non-explosive 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
$A,B\in\mathcal{E}$ are such that $\mu(B)=0$ and $K(x,B)=0$ for all $x\notin
A\cup B$, and define $T_{A}\coloneqq\inf\\{t\in\mathbb{R}_{\geq 0}\mid
X_{t}\in A\\}$ and $T_{B}\coloneqq\inf\\{t\in\mathbb{R}_{\geq 0}\mid X_{t}\in
B\\}$. It holds that $\mathbb{P}_{\mu}[T_{B}\leq T_{A},T_{B}<\infty]=0$.

###### Proof.

Note that by Theorem 19 (7) it holds that

$\displaystyle\mathbb{P}_{\mu}[T_{B}\leq T_{A},T_{B}<\infty]$
$\displaystyle\leq\mathbb{P}_{\mu}[\exists
m\in\mathbb{N}_{0}:\tau_{m}<\infty,Z_{m}\in B,\forall k\leq m:Z_{k}\notin
A\cup B]$ $\displaystyle\leq\mathbb{P}_{\mu}[\exists m\in\mathbb{N}:Z_{m}\in
B,Z_{m-1}\notin A\cup B]+\mathbb{P}_{\mu}[Z_{0}\in B].$

By Theorem 19 (6) and the assumption that $\mu(B)=0$ it holds that
$\mathbb{P}_{\mu}[Z_{0}\in B]=0$ and

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

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 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.

######  Claim 21.

Let $K$ be a jump kernel on $(E,\mathcal{E})$ with associated (sub-)Markov
transition family $(P_{t})$. Let $A\in\mathcal{E}$ be such that there is some
$\rho\in\mathbb{R}_{\geq 0}$ such that $K(x,A)$ for $x\notin A$. For all
$t\in\mathbb{R}_{\geq 0}$ and $x\notin A$ it holds that
$P_{t}(x,A)\leq\rho\cdot t$.

###### 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 $P^{(0)}_{t}(x,A)=\textup{e}^{-\kappa(x)\cdot t}\delta_{x}(A)=0$
for all $x\notin A$. Suppose the statement holds for some $n\geq 0$, we use
the fact that $P_{t}^{(n)}$ is sub-stochastic to obtain

$\displaystyle P_{t}^{(n+1)}(x,A)$
$\displaystyle=P_{t}^{(0)}(x,A)+\int_{0}^{t}\textup{e}^{-\kappa(x)\cdot
s}\int_{E}P_{t-s}^{(n)}(x,A)K(x,\mathrm{d}y)\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}\textup{e}^{-\kappa(x)\cdot
s}\cdot[\rho\cdot(t-s)\cdot K(x,A^{c})+\rho]\mathrm{d}s$
$\displaystyle\leq\rho\int_{0}^{t}\textup{e}^{-\kappa(x)\cdot
s}(1-\kappa(x)\cdot s)\mathrm{d}s+\rho\cdot
t\int_{0}^{t}\textup{e}^{-\kappa(x)\cdot s}\kappa(x)\mathrm{d}s$
$\displaystyle=\rho\cdot t.\qed$

We now use 21 to prove the following Lemma.

######  Lemma 22.

Let $K$ be a non-explosive 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. (1)

For all $i\in[k]$ we have $\mu(A_{i})=0$.

  2. (2)

There is some $\rho\in\mathbb{R}_{\geq 0}$ such that for all $i\in[k]$ and
$x\notin A_{i}$ it holds that $K(x,A_{i})\leq\rho$.

Set $T_{i}\coloneqq\inf\\{t\in\mathbb{R}_{\geq 0}\mid X_{t}\in A_{i}\\}$. For
all $t\in\mathbb{R}_{\geq 0}$ it holds that

$\mathbb{P}_{\mu}[T_{1}<T_{2}<\dots<T_{k}\leq t]\leq 1-F_{\rho t}(k-1),$

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

###### Proof.

To simplify notation, set $T_{0}\coloneqq 0$. For $i\in[k]$, define

$S_{i}\coloneqq\begin{cases}T_{i}-T_{i-1}&\text{ if }T_{i}>T_{i-1},\\\
\infty&\text{ otherwise, }\end{cases}$

and $\zeta_{i}=\sum_{j=1}^{i}S_{j}$, and note that

$\mathbb{P}_{\mu}[T_{1}<T_{2}<\dots<T_{k}\leq
t]=\mathbb{P}_{\mu}[\zeta_{k}\leq t].$

The following claim is key for proving this lemma.

######  Claim 23.

For every $s\in\mathbb{R}_{\geq 0}$ and every $i\in[k]$ it holds that
$\mathbb{P}_{\mu}[S_{i}>s\mid\mathcal{F}^{X}_{T_{i-1}}]\geq\textup{e}^{-\rho
s}$.

Next, let $G_{i,\rho}$ be the cumulative distribution function of an Erlang
distribution with shape $i$ and rate $\rho$. We will use 23 to argue
inductively that $\mathbb{P}_{\mu}[\zeta_{i}\leq t]\leq G_{i,\rho}(t)$. For
$i=1$, observe that $\zeta_{1}=S_{1}$ and by 23 we have

$\mathbb{P}_{\mu}[S_{1}\leq t]\leq 1-\textup{e}^{-\rho t}=G_{1,\rho}(t).$

For the induction step, we use 23 and the fact that $\zeta_{i}$ is
$\mathcal{F}^{X}_{T_{i}}$-measurable to obtain

$\mathbb{P}_{\mu}[\zeta_{i+1}\leq t]=\mathbb{E}[\mathbb{P}_{\mu}[S_{i+1}\leq
t-\zeta_{i}\mid\zeta_{i}]]\leq
1-\mathbb{E}[\mathds{1}_{\\{\zeta_{i}>t\\}}+\mathds{1}_{\\{\zeta_{i}\leq
t\\}}\cdot\textup{e}^{-\rho(t-\zeta_{i})}].$

Further, by the induction hypothesis and the fact that
$x\mapsto\mathds{1}_{\\{x>t\\}}+\mathds{1}_{\\{x\leq
t\\}}\cdot\textup{e}^{-\rho(t-x)}$ is non-decreasing, we have

$\mathbb{E}[\mathds{1}_{\\{\zeta_{i}>t\\}}+\mathds{1}_{\\{\zeta_{i}\leq
t\\}}\cdot\textup{e}^{-\rho(t-\zeta_{i})}]\geq
1-G_{i,\rho}(t)+\textup{e}^{-\rho t}\int_{0}^{t}\textup{e}^{\rho
x}G_{i,\rho}(\mathrm{d}x)=1-G_{i+1,\rho}(t)$

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

###### Proof of 23.

Note that

$\mathbb{P}_{\mu}[S_{i}>s\mid\mathcal{F}^{X}_{T_{i-1}}]=\mathds{1}_{\\{T_{i}\leq
T_{i-1}\\}}+\mathbb{P}_{\mu}[T_{i}>T_{i-1}+s\mid\mathcal{F}^{X}_{T_{i-1}}].$

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
$D^{(s)}_{n}\coloneqq\\{T_{i}>T_{i-1}\\}\cap\\{\forall
k\in\\{0,\dots,n-1\\}:X_{T_{i-1}+\frac{ks}{n}}\notin A_{i}\\}$. By Theorem 19
(1), we know that $(X_{t})$ has piecewise constant sample paths. Thus,
$\mathds{1}_{\\{D^{(s)}_{n}\\}}\to\mathds{1}_{\\{T_{i}>T_{i-1}+s\\}}$
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 19 (4) and 21, we obtain

$\displaystyle\mathbb{P}_{\mu}[D^{(s)}_{n}\mid\mathcal{F}^{X}_{T_{i-1}}]$
$\displaystyle=\mathbb{E}[\mathds{1}_{\\{D^{(s-s/n)}_{n-1}\\}}\cdot\mathbb{P}_{\mu}[X_{T_{i-1}+s}\notin
A_{i}\mid\mathcal{F}^{X}_{T_{i-1}+\frac{n-1}{n}s}]\mid\mathcal{F}^{X}_{T_{i-1}}]$
$\displaystyle=\mathbb{E}[\mathds{1}_{\\{D^{(s-s/n)}_{n-1}\\}}\cdot
P_{s/n}(X_{T_{i-1}+\frac{n-1}{n}s},A_{i}^{c})\mid\mathcal{F}^{X}_{T_{i-1}}]$
$\displaystyle\geq\mathbb{P}_{\mu}[D^{(s-s/n)}_{n-1}\mid\mathcal{F}^{X}_{T_{i-1}}]\cdot\left(1-\frac{\rho
s}{n}\right).$

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

$\mathbb{P}_{\mu}[T_{i}>T_{i-1}+s\mid\mathcal{F}^{X}_{T_{i-1}}]\geq\mathds{1}_{\\{T_{i}>T_{i-1}\\}}\cdot\lim_{n\to\infty}\left(1-\frac{\rho
s}{n}\right)^{n}=\mathds{1}_{\\{T_{i}>T_{i-1}\\}}\cdot\textup{e}^{-\rho s}.$

Hence, we have

$\mathbb{P}_{\mu}[S_{i}>s\mid\mathcal{F}^{X}_{T_{i-1}}]\geq\mathds{1}_{\\{T_{i}\leq
T_{i-1}\\}}+\mathds{1}_{\\{T_{i}>T_{i-1}\\}}\cdot\textup{e}^{-\rho
s}\geq\textup{e}^{-\rho s}.\qed$

##  4\. Constructing spatial birth-death dynamics for Gibbs point processes

In this section we construct the jump process that we will use to prove
uniqueness of infinite volume Gibbs measures. To this end, fix some activity
$\lambda\in\mathbb{R}_{\geq 0}$ and let $\phi$ be a pair potential that is
locally stable with constant $L\in\mathbb{R}_{\geq 0}$. For every
$\xi\in\mathcal{N}_{H}$ and every $\Lambda\in\mathcal{B}_{\text{b}}$ we define
$K_{\Lambda\mid\xi}:\mathcal{N}_{\Lambda\mid\xi}\times\mathcal{R}_{\Lambda\mid\xi}\to\mathbb{R}_{\geq
0}$ via

$\displaystyle
K_{\Lambda\mid\xi}(\eta,A)\coloneqq\int_{\Lambda}\mathds{1}_{\\{A\\}}(\eta\setminus\delta_{x})\eta(\mathrm{d}x)+\lambda\int_{\Lambda}\mathds{1}_{\\{A\\}}(\eta+\delta_{x})\cdot\textup{e}^{-W(x,\eta+\xi_{\Lambda^{c}})}\nu(\mathrm{d}x).$
(8)

######  Lemma 24.

Let $\phi$ be locally stable, $\Lambda\in\mathcal{B}_{\text{b}}$ and
$\xi\in\mathcal{N}_{H}$. Then $K_{\Lambda\mid\xi}$, defined as in (8), is a
non-explosive jump kernel on
$(\mathcal{N}_{\Lambda\mid\xi},\mathcal{R}_{\Lambda\mid\xi})$.

###### Proof.

Using the lower bound on $W$ from Lemma 6 and the fact that
$\eta+\xi_{\Lambda^{c}}\in\mathcal{N}_{H}$ for all
$\eta\in\mathcal{N}_{\Lambda\mid\xi}$, we have

$K_{\Lambda\mid\xi}(\eta,\mathcal{N}_{\Lambda})\leq\eta(\Lambda)+\lambda\textup{e}^{L}\cdot\nu(\Lambda)<\infty,$

proving that $K_{\Lambda\mid\xi}$ is indeed finite.

To prove that $K_{\Lambda\mid\xi}$ is non-explosive, we are going to use Lemma
11 with

$E_{n}=\\{\eta\in\mathcal{N}_{\Lambda\mid\xi}\mid\eta(\Lambda)\leq n\\},$

and with $f:\mathcal{N}_{\Lambda\mid\xi}\to\mathbb{R}_{\geq
0},\eta\mapsto\eta(\Lambda)+1$. We check that this choice of $(E_{n})$ and $f$
satisfies the requirements of Lemma 11. Obviously, we have that $E_{n}$
converges monotonically to $\mathcal{N}_{\Lambda\mid\xi}$. Further, by Lemma 6
and the fact that $\eta+\xi_{\Lambda^{c}}\in\mathcal{N}_{H}$ for all
$\eta\in\mathcal{N}_{\Lambda\mid\xi}$, it holds that for every
$n\in\mathbb{N}$

$\sup_{\eta\in E_{n}}K_{\Lambda\mid\xi}(\eta,\mathcal{N}_{\Lambda\mid\xi})\leq
n+\lambda\textup{e}^{L}\cdot\nu(\Lambda)<\infty.$

Next, observe that $\inf_{\eta\in\mathcal{N}_{\Lambda\mid\xi}\setminus
E_{n}}f(\eta)=n+2$ and therefore
$\lim_{n\to\infty}\inf_{\eta\in\mathcal{N}_{\Lambda\mid\xi}\setminus
E_{n}}f(\eta)=\infty$. Finally, we have for all
$\eta\in\mathcal{N}_{\Lambda\mid\xi}$ that

$\displaystyle(K_{\Lambda\mid\xi}f)(\eta)$
$\displaystyle\leq\eta(\Lambda)^{2}+(\eta(\Lambda)+2)\cdot\lambda\textup{e}^{L}\cdot\nu(\Lambda)$
$\displaystyle\leq(\eta(\Lambda)+2\lambda\textup{e}^{L}\cdot\nu(\Lambda))\cdot
f(\eta)$
$\displaystyle\leq(K_{\Lambda\mid\xi}(\eta,\mathcal{N}_{\Lambda\mid\xi})+2\lambda\textup{e}^{L}\cdot\nu(\Lambda))\cdot
f(\eta).$

Hence, setting $c=2\lambda\textup{e}^{L}\cdot\nu(\Lambda)$ and applying Lemma
11 shows that $K_{\Lambda\mid\xi}$ is non-explosive and concludes the proof. ∎

Throughout the following subsections, we study two important properties of the
transition family $(P_{t})$ that is associated with a jump kernel
$K_{\Lambda\mid\xi}$. Firstly, we derive conditions such that
$P_{t}(\eta,\cdot)$ converges rapidly to $\mu_{\Lambda\mid\xi}$ (viewed as a
distribution on $(\mathcal{N}_{\Lambda\mid\xi},\mathcal{R}_{\Lambda\mid\xi})$)
in total variation distance as $t\to\infty$. Secondly, we give conditions such
that, for two transitions families $(P^{\xi}_{t})$ and $(P^{\zeta}_{t})$
associated with $K_{\Lambda\mid\xi}$ and $K_{\Lambda\mid\zeta}$ for boundary
configurations $\xi,\zeta\in\mathcal{N}_{H}$, it holds that the total
variation distance between $P^{\xi}_{t}(\eta,\cdot)$ and
$P^{\zeta}_{t}(\eta,\cdot)$ when projected on some subregion
$\Lambda^{\prime}\subset\Lambda$ that is sufficiently far from $\Lambda^{c}$
only increases slowly as a function of $t$. Combining both properties with
Lemma 10 will yield the desired uniqueness result for infinite-volume Gibbs
measures.

###  4.1. Identity coupling

We start by constructing a coupling for the jump kernels
$K_{\Lambda\mid\xi},K_{\Lambda\mid\zeta}$ for any
$\Lambda\in\mathcal{B}_{\text{b}}$ 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 before by Schuhmacher and
Stucki [36] for applying Stein’s method to Gibbs point processes.

######  Lemma 25.

Let $\Lambda\in\mathcal{B}_{\text{b}}$ 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

$\displaystyle D_{1}((\eta_{1},\eta_{2}),A)$
$\displaystyle\coloneqq\int_{\Lambda}\mathds{1}_{\\{A\\}}(\eta_{1}\setminus\delta_{x},\eta_{2})(\eta_{1}\setminus\eta_{2})(\mathrm{d}x)$
$\displaystyle D_{2}((\eta_{1},\eta_{2}),A)$
$\displaystyle\coloneqq\int_{\Lambda}\mathds{1}_{\\{A\\}}(\eta_{1},\eta_{2}\setminus\delta_{x})(\eta_{2}\setminus\eta_{1})(\mathrm{d}x)$
$\displaystyle D_{\cap}((\eta_{1},\eta_{2}),A)$
$\displaystyle\coloneqq\int_{\Lambda}\mathds{1}_{\\{A\\}}(\eta_{1}\setminus\delta_{x},\eta_{2}\setminus\delta_{x})(\eta_{1}\cap\eta_{2})(\mathrm{d}x)$
$\displaystyle B_{1}((\eta_{1},\eta_{2}),A)$
$\displaystyle\coloneqq\lambda\int_{\Lambda}\max\left(0,\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})}-\textup{e}^{-W(x,\eta_{2}+\zeta_{\Lambda^{c}})}\right)\cdot\mathds{1}_{\\{A\\}}(\eta_{1}+\delta_{x},\eta_{2})\nu(\mathrm{d}x)$
$\displaystyle B_{2}((\eta_{1},\eta_{2}),A)$
$\displaystyle\coloneqq\lambda\int_{\Lambda}\max\left(0,\textup{e}^{-W(x,\eta_{2}+\zeta_{\Lambda^{c}})}-\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})}\right)\cdot\mathds{1}_{\\{A\\}}(\eta_{1},\eta_{2}+\delta_{x})\nu(\mathrm{d}x)$
$\displaystyle B_{\cap}((\eta_{1},\eta_{2}),A)$
$\displaystyle\coloneqq\lambda\int_{\Lambda}\min\left(\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})},\textup{e}^{-W(x,\eta_{2}+\zeta_{\Lambda^{c}})}\right)\cdot\mathds{1}_{\\{A\\}}(\eta_{1}+\delta_{x},\eta_{2}+\delta_{x})\nu(\mathrm{d}x).$

Then $K\coloneqq D_{1}+D_{2}+D_{\cap}+B_{1}+B_{2}+B_{\cap}$ is a coupling of
$K_{\Lambda\mid\xi}$ and $K_{\Lambda\mid\zeta}$ in the sense of Definition 15.

######  Remark 1.

We should think of the way that Lemma 25 presents the construction of the
coupled jump kernel $K$ in terms of its associated Markov process
$(X_{t}^{(1)},X_{t}^{(2)})_{t\in\mathbb{R}_{\geq 0}}$ on
$\mathcal{N}_{\Lambda\mid\xi}\times\mathcal{N}_{\Lambda\mid\zeta}$ as follows:
The kernels $D_{1}$ and $D_{2}$ represent the rate at which only $X_{t}^{(1)}$
or only $X_{t}^{(2)}$ remove a point (i.e., $D_{i}$ is the death rate that is
exclusive to $X_{t}^{(i)}$). In contrast, $D_{\cap}$ is the rate at which both
processes remove a point that they have in common (i.e., $D_{\cap}$ is the
shared death rate on $X_{t}^{(1)}\cap X_{t}^{(2)}$). Analogously, $B_{1}$ and
$B_{2}$ represent the rate at which a point is added to only $X_{t}^{(1)}$ or
only $X_{t}^{(2)}$ (i.e., birth rate exclusive to one process), while
$B_{\cap}$ is the rate at which both processes add the same point (i.e.,
shared birth rate).

###### Proof of Lemma 25.

Checking that $K$ is indeed a jump kernels can be done similarly to the proof
of Lemma 24. To show that $K$ is a coupling of $K_{\Lambda\mid\xi}$ and
$K_{\Lambda\mid\zeta}$, let $f:\mathcal{N}_{\Lambda\mid\xi}\to\mathbb{R}_{\geq
0}$ be bounded and measurable, and observe that

$\int_{\mathcal{N}_{\Lambda\mid\xi}\times\mathcal{N}_{\Lambda\mid\zeta}}(f(\tau_{1})-f(\eta_{1}))D_{2}((\eta_{1},\eta_{2}),\mathrm{d}\tau_{1}\times\mathrm{d}\tau_{2})=\int_{\Lambda}(f(\eta_{1})-f(\eta_{1}))(\eta_{2}\setminus\eta_{1})(\mathrm{d}x)=0$

and analogously

$\int_{\mathcal{N}_{\Lambda\mid\xi}\times\mathcal{N}_{\Lambda\mid\zeta}}(f(\tau_{1})-f(\eta_{1}))B_{2}((\eta_{1},\eta_{2}),\mathrm{d}\tau_{1}\times\mathrm{d}\tau_{2})=0.$

Further, using the fact that
$\eta_{1}=(\eta_{1}\setminus\eta_{2})+(\eta_{1}\cap\eta_{2})$ we have

$\int_{\mathcal{N}_{\Lambda\mid\xi}\times\mathcal{N}_{\Lambda\mid\zeta}}(f(\tau_{1})-f(\eta_{1}))(D_{1}+D_{\cap})((\eta_{1},\eta_{2}),\mathrm{d}\tau_{1}\times\mathrm{d}\tau_{2})=\int_{\Lambda}(f(\eta_{1}\setminus\delta_{x})-f(\eta_{1}))\eta_{1}(\mathrm{d}x),$

and, since $a=\max(0,a-b)+\min(a,b)$ for all $a,b\in\mathbb{R}_{\geq 0}$,444In
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.

$\int_{\mathcal{N}_{\Lambda\mid\xi}\times\mathcal{N}_{\Lambda\mid\zeta}}(f(\tau_{1})-f(\eta_{1}))(B_{1}+B_{\cap})((\eta_{1},\eta_{2}),\mathrm{d}\tau_{1}\times\mathrm{d}\tau_{2})=\lambda\int_{\Lambda}\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})}\cdot(f(\eta_{1}+\delta_{x})-f(\eta_{1}))\nu(\mathrm{d}x).$

Hence, it holds that

$\displaystyle\int_{\mathcal{N}_{\Lambda\mid\xi}\times\mathcal{N}_{\Lambda\mid\zeta}}(f(\tau_{1})-f(\eta_{1}))K((\eta_{1},\eta_{2}),\mathrm{d}\tau_{1}\times\mathrm{d}\tau_{2})$
$\displaystyle=\int_{\mathcal{N}_{\Lambda\mid\xi}}(f(\tau_{1})-f(\eta_{1}))K_{\Lambda\mid\xi}(\eta_{1},\mathrm{d}\tau_{1})$
$\displaystyle=(K_{\Lambda\mid\xi}f)(\eta_{1})-K_{\Lambda\mid\xi}(\eta_{1},\mathcal{N}_{\Lambda\mid\xi})\cdot
f(\eta_{1}).$

By symmetry, it then follows that $K$ is indeed a coupling in the sense of
Definition 15. ∎

###  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 7). 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 9 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
$\mu^{\prime}\in\mathcal{P}(\mathcal{N}_{\Lambda\mid\xi},\mathcal{R}_{\Lambda\mid\xi})$
with $\mu^{\prime}(A)=\mu_{\Lambda\mid\xi}(A)$ 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.

######  Lemma 26.

Let $\phi$ be locally stable, $\Lambda\in\mathcal{B}_{\text{b}}$,
$\xi\in\mathcal{N}_{H}$ and let $K_{\Lambda\mid\xi}$ be defined as in (8). The
Markov transition family associated to $K_{\Lambda\mid\xi}$ is reversible with
respect to $\mu_{\Lambda\mid\xi}$.

###### Proof.

First, note that by Lemma 7 it holds that $\mu_{\Lambda\mid\xi}$ is well-
defined. Using Lemma 12 we only need to check that

$\int_{A}K_{\Lambda\mid\xi}(\eta,B)\mu_{\Lambda\mid\xi}(\mathrm{d}\eta)=\int_{B}K_{\Lambda\mid\xi}(\eta,A)\mu_{\Lambda\mid\xi}(\mathrm{d}\eta)$

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

$\int_{A}\int_{\Lambda}\mathds{1}_{\\{B\\}}(\eta\setminus\delta_{x})\eta(\mathrm{d}x)\mu_{\Lambda\mid\xi}(\mathrm{d}\eta)=\int_{B}\lambda\int_{\Lambda}\textup{e}^{-W(x,\eta+\xi_{\Lambda^{c}})}\mathds{1}_{\\{A\\}}(\eta+\delta_{x})\eta(\mathrm{d}x)\mu_{\Lambda\mid\xi}(\mathrm{d}\eta),$

which follows immediately from applying the GNZ equation from Lemma 8 to the
function
$(x,\eta)\mapsto\mathds{1}_{\\{A\\}}(\eta)\cdot\mathds{1}_{\\{B\\}}(\eta\setminus\delta_{x})$.
∎

Next, we bound the speed of convergence by using Lemma 18 and the identity
coupling from Lemma 25.

######  Lemma 27.

Let $\phi$ be locally stable with constant $L$ and weakly tempered with
constant $\hat{C}_{\phi}$. For all
$\lambda<\frac{1}{\hat{C}_{\phi}\textup{e}^{L}}$ there is some $\delta>0$ such
that for all $\Lambda\in\mathcal{B}_{\text{b}}$ 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

$\left\lvert
P_{t}(\eta,\cdot)-\mu_{\Lambda\mid\xi}\right\rvert_{\mathrm{tv}}\leq\textup{e}^{-\delta
t}\cdot(\lambda\textup{e}^{L}\nu(\Lambda)+\eta(\Lambda))$

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

###### Proof.

We aim to prove this statement using Lemma 18. By Lemma 24 we know that
$K_{\Lambda\mid\xi}$ is non-explosive, and Lemma 26 implies that
$\mu_{\Lambda\mid\xi}$ is a stationary distribution of the associated Markov
transition family $(P_{t})$.

Let $K$ be the identity coupling of $K_{\Lambda\mid\xi}$ with itself as
provided by Lemma 25 (i.e., the coupling given by Lemma 25 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
$f:\mathcal{N}_{\Lambda\mid\xi}\times\mathcal{N}_{\Lambda\mid\xi}\to\mathbb{R}_{\geq
0}$ such that the conditions (1) and (2) of Lemma 18 are satisfied for
suitable constants $c,\delta>0$. To this end, we set
$f(\eta_{1},\eta_{2})\coloneqq(\eta_{1}\oplus\eta_{2})(\Lambda)$. 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
$f(\eta_{1},\eta_{2})\geq 1$ if and only if $\eta_{1}\neq\eta_{2}$. Thus,
condition (1) of Lemma 18 is satisfied by $f$ for $c=1$.

To verify condition (2) of Lemma 18, 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

$(D_{1}f)(\eta_{1},\eta_{2})=(f(\eta_{1},\eta_{2})-1)\cdot(\eta_{1}\setminus\eta_{2})(\Lambda).$

  * •

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

$(D_{2}f)(\eta_{1},\eta_{2})=(f(\eta_{1},\eta_{2})-1)\cdot(\eta_{2}\setminus\eta_{1})(\Lambda).$

  * •

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

$(D_{\cap}f)(\eta_{1},\eta_{2})=f(\eta_{1},\eta_{2})\cdot(\eta_{1}\cap\eta_{2})(\Lambda).$

  * •

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

$\displaystyle(B_{1}f)(\eta_{1},\eta_{2})$
$\displaystyle=(f(\eta_{1},\eta_{2})+1)\cdot\lambda\int_{\Lambda}\max\left(0,\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})}-\textup{e}^{-W(x,\eta_{2}+\xi_{\Lambda^{c}})}\right)\nu(\mathrm{d}x),$
$\displaystyle(B_{2}f)(\eta_{1},\eta_{2})$
$\displaystyle=(f(\eta_{1},\eta_{2})+1)\cdot\lambda\int_{\Lambda}\max\left(0,\textup{e}^{-W(x,\eta_{2}+\xi_{\Lambda^{c}})}-\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})}\right)\nu(\mathrm{d}x),$
$\displaystyle(B_{\cap}f)(\eta_{1},\eta_{2})$
$\displaystyle=f(\eta_{1},\eta_{2})\cdot\lambda\int_{\Lambda}\min\left(\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})},\textup{e}^{-W(x,\eta_{2}+\xi_{\Lambda^{c}})}\right)\nu(\mathrm{d}x).$

Further, since
$\eta_{1}\cup\eta_{2}=\eta_{1}\oplus\eta_{2}+\eta_{1}\cap\eta_{2}$ 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

$\kappa(\eta_{1},\eta_{2})\coloneqq
K((\eta_{1},\eta_{2}),\mathcal{N}_{\Lambda\mid\xi}\times\mathcal{N}_{\Lambda\mid\xi})=(\eta_{1}\cup\eta_{2})(\Lambda)+\lambda\int_{\Lambda}\max\left(\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})},\textup{e}^{-W(x,\eta_{2}+\xi_{\Lambda^{c}})}\right)\nu(\mathrm{d}x).$

Combining these observations yields

$(Kf)(\eta_{1},\eta_{2})=\kappa(\eta_{1},\eta_{2})\cdot
f(\eta_{1},\eta_{2})-f(\eta_{1},\eta_{2})+\lambda\int_{\Lambda}\left\lvert\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})}-\textup{e}^{-W(x,\eta_{2}+\xi_{\Lambda^{c}})}\right\rvert\nu(\mathrm{d}x).$

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

######  Claim 28.

There exists some $\delta>0$, independent of $\Lambda$, such that, for all
$\eta_{1},\eta_{2}\in\mathcal{N}_{\Lambda}$, it holds that

$\lambda\int_{\Lambda}\left\lvert\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})}-\textup{e}^{-W(x,\eta_{2}+\xi_{\Lambda^{c}})}\right\rvert\nu(\mathrm{d}x)\leq(1-\delta)\cdot(\eta_{1}\oplus\eta_{2})(\Lambda).$

Provided 28, we have

$(Kf)(\eta_{1},\eta_{2})\leq\left(\kappa(\eta_{1},\eta_{2})-\delta\right)\cdot
f(\eta_{1},\eta_{2})$

for some $\delta>0$ independent of $\Lambda$ and $\eta_{1},\eta_{2}$. Thus,
applying Lemma 18 yields

$\left\lvert
P_{t}(\eta,\cdot)-\mu_{\Lambda\mid\xi}\right\rvert_{\mathrm{tv}}\leq\textup{e}^{-\delta\cdot
t}\int_{\mathcal{N}_{\Lambda\mid\xi}}f(\eta,\zeta)\mu_{\Lambda\mid\xi}(\mathrm{d}\zeta)$

for $\delta>0$ as desired. Finally, observe that
$f(\eta,\zeta)=(\eta\oplus\zeta)(\Lambda)\leq\eta(\Lambda)+\zeta(\Lambda)$.
Hence, applying the GNZ equation from Lemma 8 with the function
$(x,\zeta)\mapsto\mathds{1}_{\\{\mathcal{N}_{\Lambda\mid\xi}\\}}(\zeta)$ and
using the bound on the influence from Lemma 6 yields

$\int_{\mathcal{N}_{\Lambda\mid\xi}}f(\eta,\zeta)\mu_{\Lambda\mid\xi}(\mathrm{d}\zeta)\leq\eta(\Lambda)+\lambda\textup{e}^{L}\nu(\Lambda),$

which concludes the proof. ∎

###### Proof of 28.

Let $k=(\eta_{1}\oplus\eta_{2})(\Lambda)$. 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 $1\leq i\leq k$ it
holds that $\zeta_{i-1}\oplus\zeta_{i}=\delta_{z}$ 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 $0\leq i\leq
k_{1}$ set
$\zeta_{i}=(\eta_{1}\cap\eta_{2})+\sum_{j=1}^{k_{1}-i}\delta_{y_{j}}$, and for
$k_{1}+1\leq i\leq k_{1}+k_{2}$ 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
$\zeta_{i}\leq\eta_{1}$ or $\zeta_{i}\leq\eta_{2}$, 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

$\lambda\int_{\Lambda}\left\lvert\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})}-\textup{e}^{-W(x,\eta_{2}+\xi_{\Lambda^{c}})}\right\rvert\nu(\mathrm{d}x)\leq\sum_{i=1}^{k}\lambda\int_{\Lambda}\left\lvert\textup{e}^{-W(x,\zeta_{i}+\xi_{\Lambda^{c}})}-\textup{e}^{-W(x,\zeta_{i-1}+\xi_{\Lambda^{c}})}\right\rvert\nu(\mathrm{d}x).$

Therefore, it is sufficient to show that

$\lambda\int_{\Lambda}\left\lvert\textup{e}^{-W(x,\zeta_{i}+\xi_{\Lambda^{c}})}-\textup{e}^{-W(x,\zeta_{i-1}+\xi_{\Lambda^{c}})}\right\rvert\nu(\mathrm{d}x)\leq
1-\delta$

for every $1\leq i\leq k$. 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

$W(x,\zeta_{i}+\xi_{\Lambda^{c}})=W(x,\zeta_{i-1}+\xi_{\Lambda^{c}})+\phi(x,z).$

Thus, by local stability and Lemma 6 we have

$\displaystyle\left\lvert\textup{e}^{-W(x,\zeta_{i}+\xi_{\Lambda^{c}})}-\textup{e}^{-W(x,\zeta_{i-1}+\xi_{\Lambda^{c}})}\right\rvert$
$\displaystyle\leq\min\left(\textup{e}^{-W(x,\zeta_{i-1}+\xi_{\Lambda^{c}})}\cdot\left\lvert\textup{e}^{-\phi(x,z)}-1\right\rvert,\textup{e}^{-W(x,\zeta_{i}+\xi_{\Lambda^{c}})}\cdot\left\lvert
1-\textup{e}^{\phi(x,z)}\right\rvert\right)$
$\displaystyle\leq\textup{e}^{L}\cdot\min\left(\left\lvert\textup{e}^{-\phi(x,z)}-1\right\rvert,\left\lvert
1-\textup{e}^{\phi(x,z)}\right\rvert\right)$
$\displaystyle\leq\textup{e}^{L}\cdot\left(1-\textup{e}^{-\left\lvert\phi(x,z)\right\rvert}\right).$

Combining this with weak temperedness, we obtain

$\lambda\int_{\Lambda}\left\lvert\textup{e}^{-W(x,\zeta_{i}+\xi_{\Lambda^{c}})}-\textup{e}^{-W(x,\zeta_{i-1}+\xi_{\Lambda^{c}})}\right\rvert\nu(\mathrm{d}x)\leq\lambda\textup{e}^{L}\int_{\Lambda}1-\textup{e}^{-\left\lvert\phi(x,z)\right\rvert}\nu(\mathrm{d}x)\leq\lambda\textup{e}^{L}\hat{C}_{\phi}.$

Since we assume $\lambda<\frac{1}{\textup{e}^{L}\hat{C}_{\phi}}$, 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. Firstly, for any
two point processes $P,Q\in\mathcal{P}(\mathcal{N},\mathcal{R})$ and some
region $\Lambda\in\mathcal{B}_{\text{b}}$ we write $\left\lvert
P-Q\right\rvert_{\Lambda}$ for the projected total variation distance
$\left\lvert P[\Lambda]-Q[\Lambda]\right\rvert_{\mathrm{tv}}$. Further, for
every tuple $\boldsymbol{k}=(k_{1},\dots,k_{d})\in\mathbb{Z}^{d}$ we set

$\Lambda_{\boldsymbol{k}}\coloneqq\left[(k_{1}-\frac{1}{2})R,(k_{1}+\frac{1}{2})R\right]\times\dots\times\left[(k_{d}-\frac{1}{2})R,(k_{d}+\frac{1}{2})R\right].$

For $n\in\mathbb{N}$ we set $V_{n}\coloneqq(\mathbb{Z}\cap[-n,n])^{d}$ and

$\Lambda^{(n)}\coloneqq\bigcup_{\boldsymbol{k}\in
V_{n}}\Lambda_{\boldsymbol{k}}=[-n-1/2,n+1/2]^{d}.$

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
$E_{n}\coloneqq\\{\\{\boldsymbol{j},\boldsymbol{k}\\}\in\binom{V_{n}}{2}\mid\lVert\boldsymbol{j}-\boldsymbol{k}\rVert_{\infty}=1\\}$.
In accordance with this graph construction, we denote by
$\Gamma_{n}(\boldsymbol{k})\coloneqq\\{\boldsymbol{j}\in
V_{n}\mid\\{\boldsymbol{k},\boldsymbol{j}\\}\in E_{n}\\}$ the neighborhood of
$\boldsymbol{k}\in V_{n}$ in $G_{n}$. Further, we partition the vertex set
$V_{n}$ into _inner vertices_
$V^{\textup{in}}_{n}\coloneqq\\{\boldsymbol{k}\in
V_{n}\mid\lVert\boldsymbol{k}\rVert_{\infty}<n\\}$ and _outer vertices_
$V^{\textup{out}}_{n}\coloneqq V_{n}\setminus
V^{\textup{in}}_{n}=\\{\boldsymbol{k}\in
V_{n}\mid\lVert\boldsymbol{k}\rVert_{\infty}=n\\}$.

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
$\xi,\zeta\in\mathcal{N}_{H}$

######  Lemma 29.

Let $n\in\mathbb{N}$ and $\xi,\zeta\in\mathcal{N}_{H}$, and let $K$ be the
identity coupling of $K_{n\mid\xi}$ and $K_{n\mid\zeta}$. For every
$\boldsymbol{k}\in V_{n}$ set
$\Delta_{\boldsymbol{k}}\coloneqq\\{(\eta_{1},\eta_{2})\in\mathcal{N}_{n\mid\xi}\times\mathcal{N}_{n\mid\zeta}\mid(\eta_{1}\oplus\eta_{2})_{\Lambda_{\boldsymbol{k}}}>0\\}$.
Then $K$ satisfies the following properties:

  1. (1)

For all $\boldsymbol{k}\in V^{\textup{in}}_{n}$ and every
$(\eta_{1},\eta_{2})\notin\Delta_{\boldsymbol{k}}\cup\bigcup_{\boldsymbol{j}\in\Gamma_{n}(\boldsymbol{k})}\Delta_{\boldsymbol{j}}$
it holds that $K((\eta_{1},\eta_{2}),\Delta_{\boldsymbol{k}})=0$.

  2. (2)

For all $\boldsymbol{k}\in V_{n}$ and every
$(\eta_{1},\eta_{2})\notin\Delta_{\boldsymbol{k}}$ it holds that
$K((\eta_{1},\eta_{2}),\Delta_{\boldsymbol{k}})\leq\lambda\textup{e}^{L}R^{d}$.

###### Proof.

Let $D_{1},D_{2},D_{\cap},B_{1},B_{2},B_{\cap}$ be as in Lemma 25. Set
$\Lambda=\Lambda^{(n)}$ to simplify notation, and observe that for all
$\boldsymbol{k}\in V_{n}$ and $\eta_{1},\eta_{2}\notin\Delta_{\boldsymbol{k}}$
it holds that

$\displaystyle K((\eta_{1},\eta_{2}),\Delta_{\boldsymbol{k}})$
$\displaystyle=B_{1}((\eta_{1},\eta_{2}),\Delta_{\boldsymbol{k}})+B_{2}((\eta_{1},\eta_{2}),\Delta_{\boldsymbol{k}})$
$\displaystyle=\lambda\int_{\Lambda_{\boldsymbol{k}}}\left\lvert\textup{e}^{-W(x,\eta_{1}+\xi_{\Lambda^{c}})}-\textup{e}^{-W(x,\eta_{2}+\zeta_{\Lambda^{c}})}\right\rvert\nu(\mathrm{d}x).$

Using Lemma 6, the fact that $\phi$ is locally stable and that
$\nu(\Lambda_{\boldsymbol{k}})=R^{d}$ proves (2).

For (1), note that, if $\boldsymbol{k}\in V^{\textup{in}}_{n}$, it holds that
$d(x,y)\geq R$ for any $x\in\Lambda_{\boldsymbol{k}}$ and
$y\notin\Lambda_{\boldsymbol{k}}\cup\bigcup_{\boldsymbol{j}\in\Gamma_{n}(\boldsymbol{k})}\Lambda_{\boldsymbol{j}}$.
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
$(\eta_{1},\eta_{2})\notin\Delta_{\boldsymbol{k}}\cup\bigcup_{\boldsymbol{j}\in\Gamma_{n}(\boldsymbol{k})}\Delta_{\boldsymbol{j}}$,
and consequently $K((\eta_{1},\eta_{2}),\Delta_{\boldsymbol{k}})=0$ as
claimed. ∎

We now combine Lemma 29 with the process construction in Theorem 19 and Lemmas
20 and 22 to prove the following lemma.

######  Lemma 30.

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
$t\in\mathbb{R}_{\geq 0}$ with
$t<\frac{n-m}{\textup{e}^{2}(6m+3)^{d}R^{d}\lambda\textup{e}^{L}}$ it holds
that

$\left\lvert
P_{t}(\boldsymbol{0},\cdot)-Q_{t}(\boldsymbol{0},\cdot)\right\rvert_{\Lambda^{(m)}}\leq\textup{e}^{-(n-m)}.$

###### Proof.

Let $K$ be the identity coupling of $K_{n\mid\xi}$ and $K_{n\mid\zeta}$ as
given by Lemma 25. By Lemmas 16 and 24, we know that $K$ is non-explosive, and
its associated transition family is a coupling of $(P_{t})$ and $(Q_{t})$ in
the sense of Definition 14. Next, let
$(\Omega,\mathcal{F},\mathbb{P},(X_{t}))$ be the Markov process associated
with the jump kernel $K$ and the starting distribution
$\delta_{\boldsymbol{0}}\otimes\delta_{\boldsymbol{0}}$ as given by Theorem
19. For $\boldsymbol{k}\in V_{n}$, define $\Delta_{\boldsymbol{k}}$ as in
Lemma 29 and set $T_{\boldsymbol{k}}\coloneqq\inf\\{t\in\mathbb{R}_{\geq
0}\mid X_{t}\in\Delta_{\boldsymbol{k}}\\}$. Using Lemma 13 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
$\delta_{\boldsymbol{0}}\otimes\delta_{\boldsymbol{0}}$, we obtain

$\left\lvert
P_{t}(\boldsymbol{0},\cdot)-Q_{t}(\boldsymbol{0},\cdot)\right\rvert_{\Lambda^{(m)}}\leq\sum_{\boldsymbol{k}\in
V_{m}}\mathbb{P}[X_{t}\in\Delta_{\boldsymbol{k}}]\leq\sum_{\boldsymbol{k}\in
V_{m}}\mathbb{P}[T_{\boldsymbol{k}}\leq t].$

We proceed by deriving a uniform upper bound on
$\mathbb{P}[T_{\boldsymbol{k}}\leq t]$ for every $\boldsymbol{k}\in V_{m}$. 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
$(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{\ell})\in V_{n}^{\ell}$ a _path of
length $\ell$_ in $G_{n}$ if for all $2\leq i\leq\ell$ it holds that
$\boldsymbol{k}_{i}\in\Gamma_{n}(\boldsymbol{k}_{i-1})$ and for all $1\leq
i<j\leq\ell$ it holds that $\boldsymbol{k}_{i}\neq\boldsymbol{k}_{j}$ (i.e.,
$(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{\ell})$ 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

$\mathfrak{P}_{\ell}(W_{1},W_{2})\coloneqq\\{(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{\ell})\in\mathfrak{P}^{\ell}\mid\boldsymbol{k}_{1}\in
W_{1},\boldsymbol{k}_{\ell}\in W_{2}\\},$

and we write
$\mathfrak{P}(W_{1},W_{2})\coloneqq\bigcup_{\ell\in\mathbb{N}}\mathfrak{P}_{\ell}(W_{1},W_{2})$
for all paths from $W_{1}$ to $W_{2}$ regardless of their length.

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 $\Lambda_{\boldsymbol{k}}$ for $\boldsymbol{k}\in V_{n}$
can only be generated if there already was a disagreement in any box that is
adjacent to $\boldsymbol{k}$ in the graph $G_{n}$. In particular, if there is
a disagreement in some box $\boldsymbol{k}\in V_{m}$ at time $t$ (i.e.,
$X_{t}\in\Delta_{\boldsymbol{k}}$), then there must be some path
$(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{\ell})$ such that
$\boldsymbol{k}_{1}=\boldsymbol{k}$ and $\boldsymbol{k}_{\ell}\in
V^{\textup{out}}_{n}$ that this disagreement can be traced back to. Formally,
it must hold that the respective hitting times satisfy
$T_{\boldsymbol{k}_{\ell}}<T_{\boldsymbol{k}_{\ell-1}}<\dots<T_{\boldsymbol{k}_{1}}\leq
t$. We use Lemma 20 and part (1) of Lemma 29 to formalize this idea.

######  Claim 31.

For all $\boldsymbol{k}\in V_{m}$ and all $t\in\mathbb{R}_{\geq 0}$ it holds
that

$\mathbb{P}[T_{\boldsymbol{k}}\leq
t]\leq\sum_{(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{\ell})\in\mathfrak{P}(\boldsymbol{k},V^{\textup{out}}_{n})}\mathbb{P}[T_{\boldsymbol{k}_{\ell}}<T_{\boldsymbol{k}_{\ell-1}}<\dots<T_{\boldsymbol{k}_{1}}\leq
t].$

Combining 31 with Lemma 22 and part (2) of Lemma 29, we get

$\mathbb{P}[T_{\boldsymbol{k}}\leq
t]\leq\sum_{\ell\in\mathbb{N}}\left\lvert\mathfrak{P}_{\ell}(\boldsymbol{k},V^{\textup{out}}_{n})\right\rvert\cdot(1-F_{\rho}(\ell-1)),$

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

$\mathbb{P}[T_{\boldsymbol{k}}\leq
t]\leq\sum_{\ell>n-m}\textup{e}^{-\ell}(2m+1)^{-d\ell}\leq\left\lvert
V_{m}\right\rvert^{-1}\cdot\textup{e}^{-(n-m)}$

and summing over $\boldsymbol{k}\in V_{m}$ concludes the proof. ∎

###### Proof of 31.

For $\ell\in\mathbb{N}$ define

$\displaystyle S^{\text{f}}_{\ell}(\boldsymbol{k})$
$\displaystyle\coloneqq\\{(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{j})\in\mathfrak{P}_{j}\mid
j\leq\ell,\boldsymbol{k}_{1}=\boldsymbol{k},\boldsymbol{k}_{j}\in
V^{\textup{out}}_{n},\forall i<j:\boldsymbol{k}_{i}\in V^{\textup{in}}_{n}\\}$
$\displaystyle S^{\text{o}}_{\ell}(\boldsymbol{k})$
$\displaystyle\coloneqq\\{(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{\ell})\in\mathfrak{P}_{\ell}\mid\boldsymbol{k}_{1}=\boldsymbol{k},\forall
i\leq\ell:\boldsymbol{k}_{i}\in V^{\textup{in}}_{n}\\}$ $\displaystyle
S_{\ell}(\boldsymbol{k})$ $\displaystyle\coloneqq S^{\text{f}}_{\ell}\cup
S^{\text{f}}_{\ell}.$

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

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

$\displaystyle\mathbb{P}[T_{\boldsymbol{k}}\leq
t]\leq\sum_{(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{j})\in
S_{\ell}(\boldsymbol{k})}\mathbb{P}[T_{\boldsymbol{k}_{j}}<T_{\boldsymbol{k}_{j-1}}<\dots<T_{\boldsymbol{k}_{1}}\leq
t].$ (9)

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

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

$\displaystyle\sum_{(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{\ell})\in
S^{\text{o}}_{\ell}(\boldsymbol{k})}\mathbb{P}[T_{\boldsymbol{k}_{\ell}}<T_{\boldsymbol{k}_{\ell-1}}<\dots<T_{\boldsymbol{k}_{1}}\leq
t]$ $\displaystyle\hskip
40.00006pt\leq\sum_{(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{\ell+1})\in
S^{\text{o}}_{\ell+1}(\boldsymbol{k})}\mathbb{P}[T_{\boldsymbol{k}_{\ell+1}}<T_{\boldsymbol{k}_{\ell}}<\dots<T_{\boldsymbol{k}_{1}}\leq
t]$ $\displaystyle\hskip
40.00006pt+\sum_{(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{\ell+1})\in
S^{\text{f}}_{\ell+1}(\boldsymbol{k})\setminus
S^{\text{f}}_{\ell}(\boldsymbol{k})}\mathbb{P}[T_{\boldsymbol{k}_{\ell+1}}<T_{\boldsymbol{k}_{\ell}}<\dots<T_{\boldsymbol{k}_{1}}\leq
t].$

To this end, fix some $(\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{\ell})\in
S^{\text{o}}_{\ell}(\boldsymbol{k})$ and note that, by definition,
$\boldsymbol{k}_{\ell}\in V^{\textup{in}}_{n}$. Set
$A\coloneqq\\{T_{\boldsymbol{k}_{\ell}}<T_{\boldsymbol{k}_{\ell-1}}<\dots<T_{\boldsymbol{k}_{1}}\leq
t\\}$ and note that $A\subseteq\\{T_{\boldsymbol{k}_{\ell}}<\infty\\}$.
Further, define $T_{\Gamma}\coloneqq\inf\\{s\in\mathbb{R}_{\geq 0}\mid
X_{s}\in\bigcup_{\boldsymbol{j}\in\Gamma_{n}(\boldsymbol{k}_{\ell})}\Delta_{\boldsymbol{j}}\\}$.
By Lemma 20 and part (1) of Lemma 29, it holds that

$\mathbb{P}[A]=\mathbb{P}[T_{\Gamma}<T_{\boldsymbol{k}_{\ell}},A]\leq\sum_{\boldsymbol{j}\in\Gamma_{n}(\boldsymbol{k}_{\ell})}\mathbb{P}[T_{\boldsymbol{j}}<T_{\boldsymbol{k}_{\ell}},A].$

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

##  5\. Uniqueness of infinite-volume Gibbs measures

We will now prove our main uniqueness result for infinite-volume Gibbs
measures.

######  Theorem 32.

Suppose $(\mathcal{X},d)$ is the $d$-dimensional Euclidean space for some
$d\in\mathbb{N}$, equipped with the respective Lebesgue measure. Let $\phi$ be
a finite-range pair potential that is locally stable with constant $L$ and
weakly tempered with constant $\hat{C}_{\phi}$. For all
$\lambda<(\textup{e}^{L}\hat{C}_{\phi})^{-1}$ it holds that
$\left\lvert\mathcal{G}(\lambda,\phi)\right\rvert\leq 1$.

###### Proof.

Suppose $\mathcal{G}(\lambda,\phi)$ is non-empty. Our goal is to use Lemma 10
to argue that, for all $\mu_{1},\mu_{2}\in\mathcal{G}(\lambda,\phi)$, it holds
that $\mu_{1}=\mu_{2}$. To this end, set
$\Lambda_{k}=[-(k+\frac{1}{2})R,(k+\frac{1}{2})R]^{d}$ where $R$ is an upper-
bound on the range of $\phi$, and note that the sequence
$(\Lambda_{k})_{k\in\mathbb{N}}$ is non-decreasing and satisfies assumption
(1) of Lemma 10. Moreover, note that by Lemma 4, it holds that
$\mu_{1}(\mathcal{N}_{H})=1$ and $\mu_{2}(\mathcal{N}_{H})=1$. Hence, it
suffices to show that for every $k\in\mathbb{N}$

$\lim_{n\to\infty}\sup_{\xi,\zeta\in\mathcal{N}_{H}}\left\lvert\mu_{\Lambda_{k+n}\mid\xi}-\mu_{\Lambda_{k+n}\mid\zeta}\right\rvert_{\Lambda_{k}}=0.$

Let $\delta>0$ be as in Lemma 18 and note that for every $\varepsilon>0$, we
can choose $n\geq-\ln\varepsilon$ large enough, depending on $\varepsilon$,
$\lambda$, $L$, $R$, $d$, $\hat{C}_{\phi}$ and $k$, such that there is some
$t\in\mathbb{R}_{\geq 0}$ with

$\displaystyle\delta^{-1}\ln\left(\frac{\lambda(2n+2k+1)^{d}R^{d}\textup{e}^{L}}{\varepsilon}\right)\leq
t<\frac{n}{\textup{e}^{2}(6k+3)^{d}R^{d}\lambda\textup{e}^{L}}.$ (10)

Fix any such $n$ and let $\xi,\zeta\in\mathcal{N}_{H}$. Let $(P_{t})$ and
$(Q_{t})$ be the transition families associated with the jump kernels
$K_{\Lambda_{k+n}\mid\xi}$ and $K_{\Lambda_{k+n}\mid\zeta}$ as in (8). By the
triangle inequality, we have

$\left\lvert\mu_{\Lambda_{k+n}\mid\xi}-\mu_{\Lambda_{k+n}\mid\zeta}\right\rvert_{\Lambda_{k}}\leq\left\lvert\mu_{\Lambda_{k+n}\mid\xi}-P_{t}(\boldsymbol{0},\cdot)\right\rvert_{\Lambda_{k}}+\left\lvert
P_{t}(\boldsymbol{0},\cdot)-Q_{t}(\boldsymbol{0},\cdot)\right\rvert_{\Lambda_{k}}+\left\lvert
Q_{t}(\boldsymbol{0},\cdot)-\mu_{\Lambda_{k+n}\mid\zeta}\right\rvert_{\Lambda_{k}}$

for every $t\in\mathbb{R}_{\geq 0}$. Choosing any $t$ as in (10), Lemma 18
yields

$\left\lvert\mu_{\Lambda_{k+n}\mid\xi}-P_{t}(\boldsymbol{0},\cdot)\right\rvert_{\Lambda_{k}}\leq\left\lvert\mu_{\Lambda_{k+n}\mid\xi}-P_{t}(\boldsymbol{0},\cdot)\right\rvert_{\mathrm{tv}}\leq\varepsilon$

and analogously

$\left\lvert
Q_{t}(\boldsymbol{0},\cdot)-\mu_{\Lambda_{k+n}\mid\zeta}\right\rvert_{\Lambda_{k}}\leq\left\lvert
Q_{t}(\boldsymbol{0},\cdot)-\mu_{\Lambda_{k+n}\mid\zeta}\right\rvert_{\mathrm{tv}}\leq\varepsilon.$

Further, for the same $t$, Lemma 30 and $n\geq-\ln\varepsilon$ yield

$\left\lvert
P_{t}(\boldsymbol{0},\cdot)-Q_{t}(\boldsymbol{0},\cdot)\right\rvert_{\Lambda_{k}}\leq\textup{e}^{-n}\leq\varepsilon.$

Since our choice of $n$ does not depend on $\xi$ and $\zeta$, we get

$\sup_{\xi,\zeta\in\mathcal{N}_{H}}\left\lvert\mu_{\Lambda_{k+n}\mid\xi}-\mu_{\Lambda_{k+n}\mid\zeta}\right\rvert_{\Lambda_{k}}\leq
3\varepsilon$

for every $n$ large enough, which proves the claim. ∎

Combining Theorem 32 with known existence results for infinite-volume Gibbs
measures, such as Theorem 3, we obtain the following corollary.

######  Corollary 33.

In the setting of Theorem 32, it holds that
$\left\lvert\mathcal{G}(\lambda,\phi)\right\rvert=1$ for all
$\lambda<(\textup{e}^{L}\hat{C}_{\phi})^{-1}$.

##  6\. Discussion

We conclude the paper by comparing our result to the existing literature. We
further give a detailed discussion of our assumptions and potential ways to
extend our results in future work.

###  6.1. Comparison to existing bounds for absence of phase transitions

To compare our results to the existing literature, it is useful to introduce
the inverse temperature parameter $\beta\geq 0$. For any fixed pair potential
$\phi$ on $\mathbb{R}^{d}$, we denote by $\phi_{\beta}\coloneqq\beta\phi$ the
pair potential at inverse temperature $\beta$. Intuitively, changing $\beta$
scales the strength of interactions between particles: in the small
temperature limit ($\beta\to\infty$) interactions between particles become
more relevant for the behavior of the system, whereas in the large temperature
limit ($\beta\to 0$) only hard-core interactions (cases where
$\phi(\cdot,\cdot)=\infty$) prevail. We write
$\hat{C}_{\beta}\coloneqq\hat{C}_{\phi_{\beta}}$ for the associated weak
temperedness constant and
$C_{\beta}\coloneqq\sup_{x\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}|1-\textup{e}^{-\beta\phi(x,y)}|\mathrm{d}y$
for the associated temperedness constant. Note that, if $\phi$ is (weakly)
tempered, the same applies to $\phi_{\beta}$ for all $\beta$ (see for example
[20, Exercise 5.3]). Moreover, if $\phi$ is locally stable with constant $L$,
then $\phi_{\beta}$ is locally stable with constant $\beta L$. Our bound for
uniqueness of the infinite-volume Gibbs measure then translates into
$\lambda<(\textup{e}^{\beta L}\hat{C}_{\beta})^{-1}$ with the additional
assumption that $\phi$ has bounded range.

We start by comparing our result with the classical Penrose–Ruelle bound of
$\lambda<(\textup{e}^{2\beta S+1}C_{\beta})^{-1}$ for stable pair potentials,
where $S$ is the stability constant of $\phi$, which first appeared in [30,
34]. Within this regime of $\lambda$, uniqueness of infinite-volume Gibbs
measures can be established by considering the Kirkwood–Salsburg equations,
which describe the point density functions of any infinite-volume Gibbs point
process as a fix point of a linear operator. Arguing that this linear operator
contracts under a suitable metric proves uniqueness of that fixed point and by
extension shows that there is at most one Gibbs measure (see [20, Chapters
5.6-5.8] for a detailed discussion). In the setting of locally stable
potentials, this approach yields uniqueness of infinite-volume Gibbs measures
for $\lambda<(\textup{e}^{\beta L+1}C_{\beta})^{-1}$. Hence, our result yields
an improvement by a factor of $\textup{e}C_{\beta}/\hat{C}_{\beta}$ at the
cost of relying on a bounded-range assumption. In particular, this improvement
is lower-bounded by a factor of e, and if $\phi$ has some non-trivial negative
part (i.e., attractive interactions), it becomes more substantial for large
$\beta$ (i.e., low temperature).

The second bound we want to compare our result to is a recently obtained
result by Qidong He [16]. They showed that for locally stable, tempered, and
translation and rotation invariant pair potentials, the (empty-boundary)
infinite-volume pressure
$\lim_{n\to\infty}\nu(\Lambda_{n})^{-1}\log(\Xi_{\Lambda\mid\boldsymbol{0}}(\lambda))$
is an analytic function555Here, the limit is taken along a suitable sequence
$\Lambda_{n}\nearrow\mathbb{R}^{d}$. See [20] for a detailed discussion. for
any
$\lambda<\textup{e}^{2(1-W(\textup{e}A_{\beta}/C_{\beta}))}(\textup{e}^{\beta
L+1}C_{\beta})^{-1}$, where $W$ denotes Lambert’s W-function and
$A_{\beta}\coloneqq\int_{\mathbb{R}^{d}}\mathds{1}_{\\{\phi(0,x)<0\\}}\cdot(\textup{e}^{-\beta\phi(0,x)}-1)\mathrm{d}x$.
The result is obtained by adapting a technique introduced by Michelen and
Perkins [29] for repulsive potentials (i.e., $\phi\geq 0$), which relies on
showing contraction of a recursive integral identity for the (generalized
complex) point density. We point out that this result aims at a different
notion of absence of a phase transition, and, to our knowledge, does not a
priori imply uniqueness of the infinite-volume Gibbs measure in the DLR sense.
However, we believe that their proof can be modified to show uniqueness up to
the same activity threshold via similar arguments as in [29, Section 4], while
handling the effect of the boundary condition as part of the modulation of the
activity function in the sense of [16]. With this in mind, we observe that the
relation of their bound to ours heavily depends on the inverse temperature
$\beta$ and the potential in question. In particular, if the potential is
purely repulsive or the temperature is high (i.e., small $\beta$), their bound
is up to a factor of e larger than ours. On the other hand, for potentials
with a non-trivial attractive parts and low temperature (i.e., large $\beta$)
our bound performs better as $W(\textup{e}A_{\beta}/C_{\beta})\in[0,1)$ while
$C_{\beta}/\hat{C}_{\beta}\to\infty$ for $\beta\to\infty$. Again, this
improvement comes at the cost that we require the potential to have bounded
range.

Lastly, we compare our result to the bound for analyticity of the infinite-
volume pressure (again with empty boundary) by Procacci and Yuhjtman [33].
They show that for stable and weakly tempered potentials, the infinite-volume
pressure is analytic for all $\lambda<(\textup{e}^{\beta
S+1}\hat{C}_{\beta})^{-1}$ by proving absolute convergence of the cluster
expansion, a series expansion of
$\log(\Xi_{\Lambda\mid\boldsymbol{0}}(\lambda))$. In the setting of locally
stable potentials, this translates to a bound of $\lambda<(\textup{e}^{\beta
L/2+1}\hat{C}_{\beta})^{-1}$. Again, this approach aims at a different notion
of phase transition. However, as opposed to the result we discussed before, it
is not known whether uniqueness of the infinite-volume Gibbs measure can be
proven for the same activity regime. On the one hand, the precise relation
between conditions for convergence of the cluster expansion and uniqueness of
Gibbs measures is elusive (see [21] for a more comprehensive discussion). On
the other hand, it is not clear whether the activity bound is preserved when
accounting for the effect of boundary conditions on the convergence criterion
(see [13] for a similar observation). If uniqueness of the infinite-volume
Gibbs measure were shown to hold up to the convergence bound from [33], then
this would give an improvement over our result in the high-temperature regime
($\beta<2/L$), while our result would still be stronger at low temperature.
However, a proof of this fact would be non-trivial, and it is not clear
whether it should hold at all. It would make for an interesting contrast to
our previous comparison to [16], where our bound was better at low-
temperature.

We finish our discussion of related uniqueness results by noting that for the
special case of purely non-negative potentials, stronger results are known due
to Michelen and Perkins [29, 28], which improve our uniqueness regime by a
factor of at least e. It remains open whether such an improvement carries over
to potentials with non-trivial negative part. As discussed before, a promising
step towards this was presented in [16], which maintains this improvement at
small $\beta$.

###  6.2. Assumptions and extensions

The first assumption we discuss is that $\phi$ needs to have bounded range.
This is in fact only required in the disagreement percolation argument in
Section 4.3, where we use it to make sure that disagreements between any pair
of birth-death processes only spread locally. In contrast, our second main
ingredient, which is that each birth-death process rapidly converges to a
finite-volume Gibbs distribution, does not rely on it. A similar observation
is true for the fact that we restrict our main result to Euclidean space.
While the mixing result in Lemma 18 holds in any complete separable metric
space, the disagreement percolation in Lemma 30 requires additional
information on the growth rate of the underlying space. Both of these
assumptions are inherited from the discrete origin of our proof technique
[10], which is in studying lattice gasses that naturally satisfy such
conditions. However, one could hope to remove the bounded-range assumption by
applying a suitable truncation of the pair potential at some range (maybe
requiring additional assumptions on the decay rate of the pair potential).

The second main assumption we discuss is local stability. This assumption is
relevant throughout most of our proofs: we use it for proving non-
explosiveness of the jump kernel (in Lemma 24), to bound the rate at which the
Markov transition family converges to its stationary distribution (in Lemma 18
and 28) and to bound the overall rate at which two coupled processes spawn a
disagreement (in Lemma 30 and Lemma 29). All of these applications make use of
the fact that local stability permits a bounding of the rate at which new
points are generated uniformly over the configurations space of the process.
Interestingly, other approaches to prove uniqueness, such as proving
contraction of the Kirkwood–Salsburg operator, encounter similar difficulties;
there the problem is solved by observing that stability implies that at least
one point in any configuration interacts in a “locally stable manner” with the
rest of the configuration (see [35, Chapter 4.2] for details). However, it is
not clear how a similar strategy can be implemented here.

Finally, we would like to take a more detailed look at our obtained activity
bound of $\lambda<(\textup{e}^{L}\hat{C}_{\phi})^{-1}$. This bound is solely
due to the activity regime for which we can prove that each birth-death
process converges rapidly to its stationary distribution. Therefore any
improvement on the activity bound for rapid mixing would immediately improve
the uniqueness regime. The main candidate for approaching this is to use a
more suitable metric $f$ on the configuration space in the proof of Lemma 18
(see also Lemma 27) to show contraction of the coupling. For the hard-sphere
model (i.e., $\phi(x,y)=\infty\cdot\mathds{1}_{\\{d(x,y)<r\\}}$ for some fixed
$r\in\mathbb{R}_{\geq 0}$), this approach was used by Helmuth, Perkins and
Petti [17] to gain an additional factor of $2$ on the activity bound. The
intuition behind their metric was to weight pairs of configurations based on
how likely they are to spawn disagreements. In particular, this approach
exploits the fact that, in the hard-sphere model, adding a point only
decreases this likelihood. While this idea generalizes nicely for other
repulsive potentials (i.e., $\phi\geq 0$), it becomes much harder to apply
with attractive interactions, where adding a point might make it more likely
to spawn new disagreements. However, focusing on more specific models could be
a reasonable step towards finding better conditions for rapid convergence of
the associated birth-death dynamics.

## References

  * [1] L. Bertini, N. Cancrini, and F. Cesi.  The spectral gap for a Glauber-type dynamics in a continuous gas.  Annales de l’Institut Henri Poincare (B) Probability and Statistics, 38(1):91–108, 2002.
  * [2] S. Betsch and G. Last.  On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential.  Annales de l’Institut Henri Poincare (B) Probability and Statistics, 59(2):706–725, 2023.
  * [3] R. Blumenthal and R. Getoor.  Markov Processes and Potential Theory.  Pure and Applied Mathematics. Academic Press, 1968.
  * [4] M. F. Chen.  Reversible Markov processes in abstract space.  Chinese Ann. of Math., 1:3–10, 1980 (in Chinese).
  * [5] M. F. Chen.  Coupling for jump processes.  Acta Mathematica Sinica, 2(2):123–136, 1986.
  * [6] D. J. Daley and D. Vere-Jones.  An introduction to the theory of point processes: volume II: general theory and structure.  Springer Science & Business Media, 2007.
  * [7] D. Dereudre.  Introduction to the theory of Gibbs point processes.  In Stochastic Geometry, pages 181–229. Springer, 2019.
  * [8] D. Dereudre and T. Vasseur.  Existence of Gibbs point processes with stable infinite range interaction.  Journal of Applied Probability, 57(3):775–791, 2020.
  * [9] P. Dobruschin.  The description of a random field by means of conditional probabilities and conditions of its regularity.  Theory of Probability & Its Applications, 13(2):197–224, 1968\.
  * [10] M. Dyer, A. Sinclair, E. Vigoda, and D. Weitz.  Mixing in time and space for lattice spin systems: A combinatorial view.  Random Structures & Algorithms, 24(4):461–479, 2004.
  * [11] W. Feller.  An introduction to probability theory and its applications, Volume 2, volume 81.  John Wiley & Sons, 1991.
  * [12] R. Fernández, P. Groisman, and S. Saglietti.  Stability of gas measures under perturbations and discretizations.  Reviews in Mathematical Physics, 28(10):1650022, 2016.
  * [13] P. Fialho, B. N. de Lima, and A. Procacci.  Cluster expansion for continuous particle systems interacting via an attractive pair potential and subjected to high density boundary conditions.  arXiv preprint arXiv:2102.02782, 2021.
  * [14] H.-O. Georgii.  Canonical and grand canonical Gibbs states for continuum systems.  Communications in Mathematical Physics, 48:31–51, 1976.
  * [15] J. W. Gibbs.  Elementary principles in statistical mechanics: developed with especial reference to the rational foundations of thermodynamics.  C. Scribner’s sons, 1902.
  * [16] Q. He.  Analyticity for classical hard-core gases via recursion, 2024.
  * [17] T. Helmuth, W. Perkins, and S. Petti.  Correlation decay for hard spheres via Markov chains.  The Annals of Applied Probability, 32(3):2063–2082, 2022.
  * [18] F. den Hollander.  Probability theory: The coupling method.  Lecture notes available online at
https://prob.math.leidenuniv.nl/lecturenotes/CouplingLectures.pdf, 2012\.

  * [19] P. Houdebert and A. Zass.  An explicit Dobrushin uniqueness region for gibbs point processes with repulsive interactions.  Journal of Applied Probability, 59(2):541–555, 2022.
  * [20] S. Jansen.  Gibbsian point processes.  Lecture notes available online at
https://www.mathematik.uni-muenchen.de/personen/professoren/jansen/index.html,
2018\.

  * [21] S. Jansen.  Cluster expansions for Gibbs point processes.  Advances in Applied Probability, 51(4):1129–1178, 2019.
  * [22] S. Jansen.  Markov jump processes.  Lecture notes available online at
https://www.mathematik.uni-muenchen.de/personen/professoren/jansen/index.html,
2020\.

  * [23] S. Jansen and L. Kolesnikov.  Cluster expansions: Necessary and sufficient convergence conditions.  Journal of Statistical Physics, 189(3):33, 2022.
  * [24] Y. Kondratiev, T. Kuna, and N. Ohlerich.  Spectral gap for Glauber type dynamics for a special class of potentials.  Electronic Journal Of Probability, 18, 2013.
  * [25] Y. Kondratiev and E. Lytvynov.  Glauber dynamics of continuous particle systems.  Annales de l’Institut Henri Poincare (B) Probability and Statistics, 41(4):685–702, 2005.
  * [26] O. E. Lanford III and D. Ruelle.  Observables at infinity and states with short range correlations in statistical mechanics.  Communications in Mathematical Physics, 13(3):194–215, 1969.
  * [27] G. Last and M. Penrose.  Lectures on the Poisson process, volume 7.  Cambridge University Press, 2017.
  * [28] M. Michelen and W. Perkins.  Potential-weighted connective constants and uniqueness of Gibbs measures.  arXiv preprint arXiv:2109.01094, 2021.
  * [29] M. Michelen and W. Perkins.  Analyticity for classical gasses via recursion.  Communications in Mathematical Physics, pages 1–22, 2022.
  * [30] O. Penrose.  Convergence of fugacity expansions for fluids and lattice gases.  Journal of Mathematical Physics, 4(10):1312–1320, 1963.
  * [31] C. Preston.  Spatial birth and death processes.  Advances in applied probability, 7(3):465–466, 1975.