# On the uniqueness of the infinite cluster and the cluster density
in the Poisson driven random connection model

Mikhail<EMAIL_ADDRESS>and Günter
<EMAIL_ADDRESS>
Karlsruhe Institute of Technology, Institute of Stochastics, 76131 Karlsruhe,
Germany.

###### Abstract

We consider a random connection model (RCM) on a general space driven by a
Poisson process whose intensity measure is scaled by a parameter $t\geq 0$. We
say that the infinite clusters are deletion stable if the removal of a Poisson
point cannot split a cluster in two or more infinite clusters. We prove that
this stability together with a natural irreducibility assumption implies
uniqueness of the infinite cluster. Conversely, if the infinite cluster is
unique then this stability property holds. Several criteria for irreducibility
will be established. We also study the analytic properties of expectations of
functions of clusters as a function of $t$. In particular we show that the
position dependent cluster density is differentiable. A significant part of
this paper is devoted to the important case of a stationary marked RCM (in
Euclidean space), containing the Boolean model with general compact grains and
the so-called weighted RCM as special cases. In this case we establish
differentiability and a convexity property of the cluster density $\kappa(t)$.
These properties are crucial for our proof of deletion stability of the
infinite clusters but are also of interest in their own right. It then follows
that an irreducible stationary marked RCM can have at most one infinite
cluster. This extends and unifies several results in the literature.

Keywords: Random connection model, Poisson process, percolation,
Margulis–Russo formula, cluster density, uniqueness of infinite cluster.

AMS MSC 2020: 60K35, 60G55, 60D05.

##  1 Introduction

Let ${\mathbb{X}}$ be a complete separable metric space, denote its
Borel-$\sigma$-field by ${\mathcal{X}}$, and let $\lambda$ be a locally finite
and diffuse measure on ${\mathbb{X}}$. Let $t\in{\mathbb{R}}_{+}:=[0,\infty)$
be an intensity parameter and let $\eta$ be a Poisson process on
${\mathbb{X}}$ with intensity measure $t\lambda$, defined over a probability
space $(\Omega,{\mathcal{F}},\mathbb{P})$. We often write $\mathbb{P}_{t}$
instead of $\mathbb{P}$ and $\mathbb{E}_{t}$ for the associated expectation
operator.

Let $\varphi\colon{\mathbb{X}}^{2}\to[0,1]$ be a measurable and symmetric
function satisfying

$\displaystyle
D_{\varphi}(x):=\int\varphi(x,y)\,\lambda(dy)<\infty,\quad\lambda\text{-a.e.}\
x\in{\mathbb{X}}.$ (1.1)

We refer to $\varphi$ as connection function. The random connection model
(RCM) is the random graph $\xi$ whose vertices are the points of $\eta$ and
where a pair of distinct points $x,y\in\eta$ forms an edge with probability
$\varphi(x,y)$, independently for different pairs. In an Euclidean setting the
RCM was introduced in [34]; see [31] for a textbook treatment. It can be
defined on point processes other than Poisson. The general Poisson version was
studied in [24]. The RCM is a fundamental and versatile example of a spatial
random graph. Of particular interest is the stationary marked case. In this
case we have ${\mathbb{X}}={\mathbb{R}}^{d}\times{\mathbb{M}}$ for some mark
space ${\mathbb{M}}$ and $\lambda$ is proportional to the product of Lebesgue
measure and a given mark distribution. Then the RCM becomes stationary and
ergodic under shifts in the spatial coordinate. This model contains the
Boolean model (see [25, 38]) with general compact grains and the so-called
weighted RCM as special cases and keeps attracting a lot of attention; see
e.g. [5, 10, 12, 18, 20, 27, 35].

Following common terminology of percolation theory we refer to a component of
$\xi$ as cluster. The RCM $\xi$ percolates, if it has an infinite cluster,
that is a component with infinitely many vertices. We say that the infinite
clusters of $\xi$ are deletion stable if the removal of a point cannot split a
cluster in two or more infinite clusters. If the infinite cluster is unique,
then Theorem 6.4 shows deletion stability. In fact, $\xi$ is then almost
surely even $2$-indivisible in the sense of [32]. Our first main result
(Theorem 6.1) says that deletion stability together with irreducibility
implies (almost sure) uniqueness of the infinite cluster. We prove this by a
peculiar addition and removal procedure, which seems to be new. Our method
crucially relies on the properties of the underlying Poisson process.
Irreducibility is a very natural assumption for uniqueness (see Remark 5.8)
and will be discussed in Section 5. Theorem 11.1 shows that the infinite
clusters of the stationary marked RCM are deletion stable. This is the second
main result of this paper. Our proof transfers some of the beautiful ideas
from the seminal paper [1] by Aizenman, Kesten and Newman to the continuum. To
this end we significantly extend and complement the arguments in [23], where
the methods from [1] were used to treat the Gilbert graph with deterministic
balls. Taken together, Theorems 6.1 and 11.1 yield uniqueness of the infinite
cluster of an irreducible stationary marked RCM; see Theorem 12.1. This
extends and unifies several results in the literature. The stationary
(unmarked) RCM was treated in [31] for an isotropic and norm-decreasing
connection function; see also [4]. A special case of the marked RCM was
treated in [21]. The uniqueness of the infinite cluster of the spherical
Boolean model was proved in [30, 31]. As a consequence of Theorem 6.4 we also
obtain that an irreducible stationary marked RCM is $2$-indivisible.

We also establish several analytic properties of cluster expectations, first
in the general and then in the stationary marked case. Since clusters are not
locally determined, the proof of these results requires some efforts. In
particular we show that the position dependent cluster density (given by
(8.6)) is, as a function of $t$, continuously differentiable; see Theorem 8.8.
In the stationary marked case this is true for the cluster density
$\kappa(t)$, defined by (4.6); see Theorem 10.1. Our proofs partially follow
[6], where the Boolean model with deterministic balls was considered. We also
prove that $t\kappa(t)+d_{\varphi}t^{2}/2$ is a convex function of $t$, where
$d_{\varphi}$ is the expected degree of a typical vertex, given by (4.2). This
remarkable property was established in [1, 13] for discrete percolation models
and in [23] for the Boolean model with deterministic balls. This convexity is
crucial for proving deletion stability of the infinite clusters in the
stationary marked case and its proof heavily depends on the (amenability)
properties of Euclidean space; see Remark 11.4.

With the exception of [23], all previous uniqueness proofs in continuum
percolation seem to use the approach in [9]; see also [15]. It is often argued
that this approach is more elegant than the one in [1]. However, our paper
shows that the methods from [1] can be conveniently extended to the continuum,
at least in the case of a Poisson driven RCM. Moreover, this approach provides
a lot of additional information on the clusters, which are valid for all
values of the intensity parameter $t$. And last but not least our general
uniqueness theorem applies to a general state space ${\mathbb{X}}$, without
any structural assumptions.

The paper is organized as follows. In Section 2 we give the formal definition
of the RCM $\xi$, while Section 3 presents the RCM version of the multivariate
Mecke equation and the Margulis–Russo formula. In Section 4 we discuss the
stationary marked RCM, an important special case of the general RCM. In
Section 5 we define a RCM to be irreducible if, roughly speaking, every pair
of Poisson points has a positive probability of being in the same cluster.
Without such a property one cannot expect the infinite cluster (if it exists)
to be unique. For a stationary marked RCM with an atom in the mark space,
Theorem 5.6 characterises irreducibility in terms of the symmetric function
$\int\varphi((0,p),(x,q))\,dx$, which is the expected number of neighbours of
$(0,p)$ with mark $q$. In the general case we provide with our Theorems 5.9
and 5.13 sufficient conditions for irreducibility, which, together with
Theorem 5.6, seem to cover all interesting examples. In Section 6 we prove
that deletion stability of infinite clusters and irreducibility together imply
uniqueness of the infinite cluster; see Theorem 6.1. Section 7 presents a
spatial Markov property. In Section 8 we establish differentiability of
certain cluster expectation, while Section 9 rewrites the derivatives as
Margulis–Russo type formulas. In Section 10 we show that the position
dependent cluster density is continuously differentiable. In Section 11 we
prove that the infinite clusters of the stationary marked RCM are deletion
stable; see Theorem 11.1. The final Section 12 provides several examples of
irreducible stationary marked RCMs.

For the reader’s convenience, we list below our main results separately for
the general and the stationary marked cases.

Main results for the general RCM:

  * •

Theorem 6.1 shows that an irreducible RCM with deletion stable infinite
clusters can have at most one infinite cluster, while Theorem 6.4 shows that
deletion stability is necessary for uniqueness.

  * •

Theorem 8.8 shows continuous differentiability of certain cluster expectation,
while Theorem 9.4 and Remark 9.5 rewrite the derivative as a Margulis-Russo
type formula.

  * •

Theorem 10.1 shows continuous differentiability of the position dependent
cluster density, while Theorem 10.7 shows that this remains true after some
additional integration.

Main results for the stationary marked RCM:

  * •

Theorem 5.6 characterises irreducibility in the case of an atom in the mark
space, while Theorems 5.9 and 5.13 provide sufficient conditions for a general
mark space.

  * •

Theorem 11.1 shows that the infinite clusters of a stationary marked RCM are
deletion stable.

  * •

Theorem 11.7 shows that the cluster density $\kappa(t)$ is continuously
differentiable and that $t\kappa(t)+d_{\varphi}t^{2}/2$ is convex.

##  2 Formal definition of the RCM

It is convenient to model a RCM as a suitable point process. Let
${\mathbf{N}}$ denote the space of all simple locally finite counting measures
on ${\mathbb{X}}$, equipped with the standard $\sigma$-field, see e.g. [25]. A
measure $\nu\in{\mathbf{N}}$ is identified with its support
$\\{x\in{\mathbb{X}}:\nu(\\{x\\})=1\\}$ and describes the set of vertices of a
(deterministic) graph. If $\nu(\\{x\\})=1$ we write $x\in\nu$. Using the Dirac
measure $\delta_{x}$ at point $x\in{\mathbb{X}}$, any $\nu\in{\mathbf{N}}$ can
be written as a finite or infinite sum
$\nu=\delta_{x_{1}}+\delta_{x_{2}}+\cdots$, where the $x_{i}$ are pairwise
distinct and do not accumulate in bounded sets. The space of (undirected)
graphs with vertices from ${\mathbb{X}}$ (and no loops) is described by the
set ${\mathbf{G}}$ of all counting measures $\mu$ on
${\mathbb{X}}\times{\mathbf{N}}$ with the following properties. First we
assume that the measure $V(\mu):=\mu(\cdot\times{\mathbf{N}})$ is locally
finite and simple, that is, an element of ${\mathbf{N}}$. Hence, if $x\in
V(\mu)$ (that is $\mu(\\{x\\}\times{\mathbf{N}})=1$), then there is a unique
$\psi_{x}\in{\mathbf{N}}$ such that $(x,\psi_{x})\in\mu$. We assume that
$x\notin\psi_{x}$. Finally, if $x\in V(\mu)$ and $y\in\psi_{x}$ then we assume
that $(y,\psi_{y})\in\mu$ and $x\in\psi_{y}$. Also ${\mathbf{G}}$ is equipped
with the standard $\sigma$-field. There is an edge between $x,y\in V(\mu)$ if
$y\in\psi_{x}$ (and hence $x\in\psi_{y}$). If $\psi_{x}=0$, then $x$ is
isolated.

We write $|\mu|:=\mu({\mathbb{X}}\times{\mathbf{N}})$ for the cardinality of
$\mu\in{\mathbf{G}}$ and similarly for $\nu\in{\mathbf{N}}$. Hence
$|\mu|=|V(\mu)|$. For $x,y\in V(\mu)$ we write $x\sim y$ (in $\mu$) if there
is an edge between $x$ and $y$ and $x\leftrightarrow y$ (in $\mu$) if there is
a path in $\mu$ leading from $x$ to $y$. For $A\subset{\mathbb{X}}$ we write
$x\sim A$ (in $\mu$) if there exists $y\in A\cap V(\mu)$ such that $x\sim y$.

Let $\mu,\mu^{\prime}\in{\mathbf{G}}$. Then $\mu$ is a subgraph of
$\mu^{\prime}$ if $V(\mu)\leq V(\mu^{\prime})$ (as measures) and if
$(x,\psi)\in\mu$ and $(x,\psi^{\prime})\in\mu^{\prime}$ together imply that
$\psi\leq\psi^{\prime}$. Note that this is not the same as
$\mu\leq\mu^{\prime}$.

Let $\chi$ be a simple point process on ${\mathbb{X}}$, that is a random
element of ${\mathbf{N}}$. The reader should think of a Poisson process
possibly augmented by additional (deterministic) points. By [25, Proposition
6.2] there exist random elements $X_{1},X_{2},\ldots$ of ${\mathbb{X}}$ such
that

$\displaystyle\chi=\sum^{|\chi|}_{n=1}\delta_{X_{n}},$ (2.1)

where $X_{m}\neq X_{n}$ whenever $m\neq n$ and $m,n\leq|\chi|$. Let
$(Z_{m,n})_{m,n\in{\mathbb{N}}}$ be a double sequence of random elements
uniformly distributed on $[0,1]$ such that $Z_{m,n}=Z_{n,m}$ for all
$m,n\in{\mathbb{N}}$ and such that the $Z_{m,n}$, $m<n$, are independent. Then
the RCM (based on $\chi$) is the point process

$\displaystyle\xi:=\sum^{|\chi|}_{m=1}\delta_{(X_{m},\Psi_{m})},$ (2.2)

where

$\displaystyle\Psi_{m}:=\sum^{|\chi|}_{n=1}{\mathbf{1}}\\{n\neq
m,Z_{m,n}\leq\varphi(X_{m},X_{n})\\}\delta_{X_{n}}.$

In this notation we suppress the dependence on the $Z_{m,n}$. While the
definition of $\xi$ depends on the ordering of the points of $\chi$, its
distribution does not.

We now introduce some notation used throughout the paper. For
$\mu,\mu^{\prime}\in{\mathbf{G}}$ we often interpret $\mu+\mu^{\prime}$ as the
measure in ${\mathbf{G}}$ with the same support as $\mu+\mu^{\prime}$. A
similar convention applies to $\nu,\nu^{\prime}\in{\mathbf{N}}$. Let
$\mu\in{\mathbf{G}}$. For $B\in{\mathcal{X}}$ we write
$\mu(B):=\mu(B\times{\mathbf{N}})$. More generally, given a measurable
function $f\colon{\mathbb{X}}\to{\mathbb{R}}$ we write $\int
f(x)\,\mu(dx):=\int f(x)\,\mu(dx\times{\mathbf{N}})$. Similarly, given
$x\in{\mathbb{X}}$, we write $x\in\mu$ instead of $x\in V(\mu)$
(=$\mu(\cdot\times{\mathbf{N}})$). In the same spirit we write
$g(\mu):=g(V(\mu))$, whenever $g$ is a mapping on ${\mathbf{N}}$. These
(slightly abusing) conventions lighten the notation and should not cause any
confusion. For $B\in{\mathcal{X}}$ we denote by $\mu[B]\in{\mathbf{G}}$ the
restriction of $\mu$ to $B$, that is the graph with vertex set $V(\mu)\cap B$
which keeps only those edges from $\mu$ with both end points from $B$. In the
same way we use the notation $\mu[\nu]$ for $\nu\in{\mathbf{N}}$. Similarly
for a measure $\nu$ on ${\mathbb{X}}$ (for instance for $\nu\in{\mathbf{N}}$)
we denote by $\nu_{B}:=\nu(B\cap\cdot)$ the restriction of $\nu$ to a set
$B\in{\mathcal{X}}$. Assume now that $v\in V(\mu)$. For $n\in{\mathbb{N}}_{0}$
let $C^{v}_{n}(\mu)\in{\mathbf{G}}$ denote the graph restricted to the set of
vertices $x\in V(\mu)$ with $d_{\mu}(v,x)=n$, where $d_{\mu}$ denotes distance
within the graph $\mu$. Note that $C^{v}_{0}(\mu)$ is just the isolated vertex
$v$. Slightly abusing our notation we write $C^{v}_{0}(\mu)=\delta_{v}$. For
$v\notin V(\mu)$ we set $C^{v}(\mu):=0$, interpreted as an empty graph (a
graph with no vertices). The cluster $C^{v}(\mu)$ of $v$ in $\mu$ is the graph
$\mu$ restricted to

$\displaystyle\sum^{\infty}_{n=0}V(C^{v}_{n}(\mu)),$

while $C^{v}_{\leq{n}}(\mu)$, $n\in{\mathbb{N}}_{0}$, is the graph $\mu$
restricted to $V(C^{v}_{0}(\mu))+\cdots+V(C^{v}_{n}(\mu))$. For later purposes
it will be convenient to define $C^{v}_{\leq{-1}}(\mu)=C^{v}_{-1}(\mu):=0$ as
the zero measure. For $\mu\in{\mathbf{G}}$ and $x\in{\mathbb{X}}$ we denote by
$\mu-\delta_{x}:=\mu[V(\mu)-\delta_{x}]$ the graph resulting from $\mu$ by
removing the point $x$. If $x\notin V(\mu)$ then $\mu-\delta_{x}=\mu$.

##  3 Basic properties of the RCM

Let $\xi$ be a RCM based on a Poisson process $\eta$ on ${\mathbb{X}}$ with
diffuse intensity measure $\lambda$. Our first crucial tool is a version of
the Mecke equation (see [25, Chapter 4]) for $\xi$. Given $n\in{\mathbb{N}}$
and $x_{1},\ldots,x_{n}\in{\mathbb{X}}$ we denote
$\eta^{x_{1},\ldots,x_{n}}:=\eta+\delta_{x_{1}}+\cdots+\delta_{x_{n}}$
(removing possible multiplicities) and let $\xi^{x_{1},\ldots,x_{n}}$ denote a
RCM based on $\eta^{x_{1},\ldots,x_{n}}$. It is useful to construct
$\xi^{x_{1},\ldots,x_{n}}$ in a specific way as follows. We connect $x_{1}$
with the points in $\eta$ using independent connection decisions which are
independent of $\xi$. We then proceed inductively finally connecting $x_{n}$
to $\eta+\delta_{x_{1}}+\cdots+\delta_{x_{n-1}}$. For $n\in{\mathbb{N}}$ and a
measurable function $f\colon{\mathbb{X}}^{n}\times{\mathbf{G}}\to[0,\infty]$
the Mecke equation for $\xi$ states that

$\displaystyle\mathbb{E}\int
f(x_{1},\ldots,x_{n},\xi)\,\eta^{(n)}(d(x_{1},\ldots,x_{n}))=\mathbb{E}\int
f(x_{1},\ldots,x_{n},\xi^{x_{1},\ldots,x_{n}})\,\lambda^{n}(d(x_{1},\ldots,x_{n})),$
(3.1)

where integration with respect to the factorial measure $\eta^{(n)}$ of $\eta$
means summation over all $n$-tuples of pairwise distinct points from $\eta$. A
convenient way to prove this and related formulas is to introduce a
probability kernel $\Gamma$ from ${\mathbf{N}}$ to ${\mathbf{G}}$, satisfying

$\displaystyle\mathbb{P}((\eta,\xi)\in\cdot)=\mathbb{E}\int{\mathbf{1}}\\{(\eta,\mu)\in\cdot\\}\,\Gamma(\eta,d\mu).$
(3.2)

The kernel $\Gamma$ is just a regular version of the conditional distribution
of $\xi$ given $\eta$ and can be defined explicitly; see Section 2. A crucial
property of this kernel is

$\displaystyle\mathbb{E}\Gamma(\eta^{x_{1},\ldots,x_{n}},\cdot)=\mathbb{P}(\xi^{x_{1},\ldots,x_{n}}\in\cdot),\quad\lambda^{n}\text{-a.e.\
$(x_{1},\ldots,x_{n})\in{\mathbb{X}}^{n}$}.$ (3.3)

It follows from [25, Theorem 4.4] that the left-hand side of (3.1) is given by

$\displaystyle\mathbb{E}\iint
f(x_{1},\ldots,x_{n},\mu)\,\Gamma(\eta^{x_{1},\ldots,x_{n}},d\mu)\,\lambda^{n}(d(x_{1},\ldots,x_{n})).$

Therefore (3.1) follows from (3.3).

Given $v\in{\mathbb{X}}$ we sometimes use (3.1) in the form

$\displaystyle\mathbb{E}\int
f(x_{1},\ldots,x_{n},\xi^{v})\,\eta^{(n)}(d(x_{1},\ldots,x_{n}))=\mathbb{E}\int
f(x_{1},\ldots,x_{n},\xi^{v,x_{1},\ldots,x_{n}})\,\lambda^{n}(d(x_{1},\ldots,x_{n})).$
(3.4)

This can be derived from (3.1) as follows. We can write $\xi^{v}=h(\xi,v,U)$,
where $U$ is a random element of $[0,1]^{\infty}$ with independent and
uniformly distributed components, independent of $\xi$; see the proof of Lemma
6.2 for more detail. It remains to note that $h(\xi^{x_{1},\ldots,x_{n}},v,U)$
has the same distribution as $\xi^{v,x_{1},\ldots,x_{n}}$, provided that
$v,x_{1},\ldots,x_{n}$ are pairwise distinct.

To state another useful version of $\eqref{e:Mecke}$ we recall the notation
$\mu-\delta_{x}=\mu[V(\mu)-\delta_{x}]$ for $\mu\in{\mathbf{G}}$ and
$x\in{\mathbb{X}}$. Given $n\in{\mathbb{N}}$ and
$x_{1},\ldots,x_{n}\in{\mathbb{X}}$ we define
$\mu-\delta_{x_{1}}-\cdots-\delta_{x_{n}}$ inductively. The kernel $\Gamma$
has the property

$\displaystyle\int{\mathbf{1}}\\{\mu-\delta_{x_{1}}-\cdots-\delta_{x_{n}}\in\cdot\\}\,\Gamma(\nu,d\mu)=\Gamma(\nu-\delta_{x_{1}}-\cdots-\delta_{x_{n}},\cdot),\quad\nu\in{\mathbf{N}}.$

Therefore we obtain from [25, Theorem 4.5] for each measurable
$f\colon{\mathbb{X}}^{n}\times{\mathbf{G}}\to[0,\infty]$ that

$\displaystyle\mathbb{E}\int
f(x_{1},\ldots,x_{n},\xi-\delta_{x_{1}}-\cdots-\delta_{x_{n}})\,\eta^{(n)}(d(x_{1},\ldots,x_{n}))=\mathbb{E}\int
f(x_{1},\ldots,x_{n},\xi)\,\lambda^{n}(d(x_{1},\ldots,x_{n})).$ (3.5)

Given $v\in{\mathbb{X}}$ we also have

$\displaystyle\mathbb{E}\int f(x,\xi^{v}-\delta_{x})\,\eta(dx)=\mathbb{E}\int
f(x,\xi^{v})\,\lambda(dx).$ (3.6)

This follows similarly as (3.4). Indeed, given $\eta$ and $x\in\eta$ the
random graph $h(\xi,v,U)-\delta_{x}$ has the same distribution as
$h(\xi-\delta_{x},v,U)$, provided that $v\neq x$.

We define $\bar{\varphi}:=1-\varphi$ and for $x\in{\mathbb{X}}$,
$\nu\in{\mathbf{N}}$

$\displaystyle\bar{\varphi}(\nu,x):=\prod_{y\in\nu}\bar{\varphi}(x,y),\quad\varphi(\nu,x):=1-\bar{\varphi}(\nu,x),\
\quad\varphi_{\lambda}(\nu):=\int\varphi(\nu,x)\,\lambda(dx).$ (3.7)

We recall our general convention $\varphi(\mu,x):=\varphi(V(\mu),x)$ and
$\varphi_{\lambda}(\mu):=\varphi_{\lambda}(V(\mu))$ for $\mu\in{\mathbf{G}}$.
Throughout we often abbreviate $C^{v}:=C^{v}(\xi^{v})$,
$C^{v}_{n}:=C^{v}_{n}(\xi^{v})$ and $C^{v}_{\leq n}:=C^{v}_{\leq n}(\xi^{v})$.
Moreover we write $C^{v!}:=C^{v}-\delta_{v}$.

We shall need the following consequence of (3.1).

######  Lemma 3.1.

Let $v\in{\mathbb{X}}$ and $h\colon{\mathbf{G}}\to[0,\infty)$ be measurable.
Then

$\displaystyle\mathbb{E}\int
h(\xi^{v}-\delta_{x})\,C^{v!}(dx)=\mathbb{E}h(\xi^{v})\varphi_{\lambda}(C^{v}).$
(3.8)

###### Proof.

Let $I$ denote the left-hand side of the asserted formula. Then

$\displaystyle I$ $\displaystyle=\mathbb{E}\int
h(\xi^{v}-\delta_{x}){\mathbf{1}}\\{x\in
C^{v}(\xi^{v})\\}\,\eta(dx)=\int\mathbb{E}h(\xi^{v,x}-\delta_{x}){\mathbf{1}}\\{x\in
C^{v}(\xi^{v,x})\\}\,\lambda(dx),$

where we have used the Mecke equation (3.4) to obtain the second identity. By
definition we have that $\xi^{v,x}-\delta_{x}=\xi^{v}$ for each
$x\in{\mathbb{X}}$. Hence we obtain that

$\displaystyle I$ $\displaystyle=\int\mathbb{E}h(\xi^{v}){\mathbf{1}}\\{x\in
C^{v}(\xi^{v,x})\\}\,\lambda(dx)=\int\mathbb{E}h(\xi^{v})\mathbb{P}(x\in
C^{v}(\xi^{v,x})\mid\xi^{v})\,\lambda(dx).$

By definition of $\xi^{v,x}$ we have $\mathbb{P}(x\in
C^{v}(\xi^{v,x})\mid\xi^{v})=\varphi(x,C^{v})$, concluding the proof. ∎

Next we turn to the Margulis–Russo formula. Let $\lambda_{1}$ and
$\lambda_{2}$ be two measures on ${\mathbb{X}}$, where $\lambda_{1}$ is
locally finite and $\lambda_{2}$ is finite. Given $t\geq 0$ we consider a RCM
driven by a Poisson process $\eta$ with intensity measure
$\lambda_{1}+t\lambda_{2}$. The associated expectation operator is denoted by
$\mathbb{E}_{t}$. Let $f\colon{\mathbf{G}}\to[-\infty,\infty]$ be a measurable
function and assume that $\mathbb{E}_{t_{0}}|f(\xi)|<\infty$ for some
$t_{0}>0$. From [25, Exercise 3.8] and (3.2) we then obtain that
$\mathbb{E}_{t}|f(\xi)|<\infty$ for all $t\leq t_{0}$. We assert that

$\displaystyle\frac{d}{dt}\mathbb{E}_{t}f(\xi)=\int\mathbb{E}_{t}[f(\xi^{x})-f(\xi)]\,\lambda_{2}(dx),\quad
t\in[0,t_{0}).$ (3.9)

Using the kernel $\Gamma$, this can be seen as follows. From [25, Theorem
19.3] we obtain that

$\displaystyle\frac{d}{dt}\mathbb{E}_{t}f(\xi)=\int\mathbb{E}_{t}[\tilde{f}(\eta^{x})-\tilde{f}(\eta)]\,\lambda_{2}(dx),\quad
t\in[0,t_{0}),$

where $\tilde{f}(\nu):=\int f(\mu)\,\Gamma(\nu,d\mu)$, $\nu\in{\mathbf{N}}$.
Note that $\tilde{f}(\xi)$ is $\mathbb{P}_{t}$-a.s. well-defined.

Take $t\in[0,t_{0})$. Theorem 19.3 in [25] shows that
$\int\mathbb{E}_{t}[|\tilde{f}(\eta^{x})-\tilde{f}(\eta)|]\,\lambda_{2}(dx)<\infty$.
Furthermore we have

$\displaystyle\int\mathbb{E}_{t}[|\tilde{f}(\eta)|]\,\,\lambda_{2}(dx)=\lambda_{2}({\mathbb{X}})\mathbb{E}_{t}[|\tilde{f}(\eta)|]\leq\lambda_{2}({\mathbb{X}})\mathbb{E}_{t}[|f(\xi)|]<\infty,$

where we have used the triangle inequality and (3.2). Therefore we also have
$\int\mathbb{E}_{t}[|\tilde{f}(\eta^{x})|]\,\lambda_{2}(dx)<\infty$. It
follows that

$\displaystyle\frac{d}{dt}\mathbb{E}_{t}f(\xi)=\int\big{(}\mathbb{E}_{t}\tilde{f}(\eta^{x})-\mathbb{E}_{t}\tilde{f}(\eta)\big{)}\,\lambda_{2}(dx)=\int\big{(}\mathbb{E}_{t}f(\xi^{x})-\mathbb{E}_{t}f(\xi)\big{)}\,\lambda_{2}(dx),$

where we have used (3.3). Since the above right-hand side is finite we have
$|\mathbb{E}_{t}f(\xi^{x})|<\infty$ and hence also
$\mathbb{E}_{t}|f(\xi^{x})|<\infty$ for $\lambda_{2}$-a.e. $x$. This implies
(3.9).

##  4 The stationary marked RCM

In this section we introduce an important special case of the general RCM. The
setting is that of [10, 12]. Special cases were studied in [11, 17, 18].

Let ${\mathbb{M}}$ be a complete separable metric space equipped with a
probability measure $\mathbb{Q}$. This is our mark space, while $\mathbb{Q}$
is said to be the mark distribution. In this section we consider the space
${\mathbb{X}}={\mathbb{R}}^{d}\times{\mathbb{M}}$ equipped with the product of
the Borel $\sigma$-field ${\mathcal{B}}({\mathbb{R}}^{d})$ on
${\mathbb{R}}^{d}$ and the Borel $\sigma$-field on ${\mathbb{M}}$. We assume
that $\lambda=t\lambda_{d}\otimes\mathbb{Q}$, where $t\in{\mathbb{R}}_{+}$ and
$\lambda_{d}$ denotes Lebesgue measure on ${\mathbb{R}}^{d}$. If
$(x,p)\in{\mathbb{X}}$ then we call $x$ location of $(x,p)$ and $p$ the mark
of $x$. Instead of ${\mathbf{N}}$ we consider the (smaller set)
${\mathbf{N}}({\mathbb{X}})$ of all counting measures $\chi$ on ${\mathbb{X}}$
such that $\chi(\cdot\times{\mathbb{M}})$ is locally finite (w.r.t. the
Euclidean metric) and simple. The symmetric connection function
$\varphi\colon{\mathbb{R}}^{d}\times{\mathbb{M}}\to[0,1]$ is assumed to
satisfy

$\displaystyle\varphi((x,p),(y,q))=\varphi((0,p),(y-x,q)).$ (4.1)

This allows us to write $\varphi(x,p,q):=\varphi((0,p),(x,q))$, where $0$
denotes the origin in ${\mathbb{R}}^{d}$. We also assume that

$\displaystyle
d_{\varphi}:=\iint\varphi(x,p,q)\,dx\,\mathbb{Q}^{2}(d(p,q))<\infty,$ (4.2)

referring to Remark 4.2 for some comments. Let $t>0$ and let $\eta$ be a
Poisson process on ${\mathbb{X}}$ with intensity measure $t\lambda$. We can
and will assume that $\eta$ is a random element of
${\mathbf{N}}({\mathbb{X}})$. We consider a RCM $\xi$ based on $\eta$ and
connection function $\varphi$.

The RCM $\xi$ is stationary in the sense that $T_{x}\xi\overset{d}{=}\xi$,
$x\in{\mathbb{R}}^{d}$, where for $\mu\in{\mathbf{G}}$, the measure $T_{x}\mu$
is (as usual) defined by

$\displaystyle
T_{x}\mu:=\int{\mathbf{1}}\\{(y-x,q,\nu)\in\cdot\\}\,\mu(d(y,q,\nu)).$

To see this, it is convenient to define $\xi$ in a slightly different way,
without changing its distribution. As at (2.1) we can write

$\displaystyle\eta=\sum^{\infty}_{m=1}\delta_{(X_{m},Q_{m})},$ (4.3)

where $X_{1},X,\ldots$ are pairwise distinct random elements of
${\mathbb{R}}^{d}$ and $Q_{1},Q_{2},\ldots$ are random elements of
${\mathbb{M}}$. Let $Z^{\prime}_{m,n}$, $m,n\in{\mathbb{N}}$, be independent
random variables uniformly distributed on $[0,1]$ and set
$Z^{\prime}_{m}:=(Z^{\prime}_{m,n})_{n\in{\mathbb{N}}}$, $m\in{\mathbb{N}}$.
By the marking theorem (see [25, Theorem 5.6]),

$\displaystyle\eta^{*}:=\sum^{\infty}_{m=1}\delta_{(X_{m},Q_{m},Z^{\prime}_{m})}$
(4.4)

is again a Poisson process. We then connect $(X_{m},Q_{m})$ with
$(X_{n},Q_{n})$ if $X_{m}$ is lexicographically smaller than $X_{n}$ and
$Z^{\prime}_{m,\tau}\leq\varphi(X_{n}-X_{m},Q_{m},Q_{n})$, where the
${\mathbb{N}}$-valued random variable $\tau$ is determined by the fact that
$X_{n}$ is the $\tau$-th nearest neighbour of $X_{m}$ in the set
$\\{X_{k}:k\neq m\\}$, where we can use the lexicographic order to break ties.
Then we have $\xi=F(\eta^{*})$ for a well-defined measurable mapping $F$.
Since the nearest neighbour relation is translation invariant it follows from
(4.1) that $F$ can be assumed to satisfy $T_{x}\xi=F(T_{x}\eta^{*})$ for each
$x\in{\mathbb{R}}^{d}$. Since $T_{x}\eta^{*}\overset{d}{=}\eta^{*}$ it follows
that $\xi$ is stationary. The same argument combined with [25, Exercise 10.1]
shows that $\xi$ is ergodic, i.e. we have $\mathbb{P}(\xi\in A)\in\\{0,1\\}$
for each translation invariant measurable $A\subset{\mathbf{G}}$. If
${\mathbb{M}}$ contains only one element, we identify ${\mathbb{X}}$ with
${\mathbb{R}}^{d}$. In this case $\xi$ is said to be a stationary RCM.

The following consequence of the Mecke equation will be often used to treat
cluster expectations.

######  Lemma 4.1.

Let $B\in{\mathcal{B}}({\mathbb{R}}^{d})$ and
$f\colon{\mathbb{N}}\to{\mathbb{R}}_{+}$. Then

$\displaystyle\mathbb{E}_{t}\int{\mathbf{1}}\\{x\in
B\\}f(|C^{(x,p)}(\xi)|)\,\eta(d(x,p))=t\lambda_{d}(B)\mathbb{E}_{t}\int
f(|C^{(0,p)}|)\,\mathbb{Q}(dp).$ (4.5)

###### Proof.

By the Mecke equation (3.1) the left-hand side of (4.5) equals

$\displaystyle t\,\mathbb{E}_{t}\iint{\mathbf{1}}\\{x\in
B\\}f(|C^{(x,p)}(\xi^{(x,p)})|)\,dx\,\mathbb{Q}(dp)=t\,\mathbb{E}_{t}\iint{\mathbf{1}}\\{x\in
B\\}f(|C^{(0,p)}(T_{x}\xi^{(x,p)})|)\,dx\,\mathbb{Q}(dp),$

where we have used that $|C^{(x,p)}(\mu)|=|C^{(0,p)}(T_{x}\mu)|$ for all
$\mu\in{\mathbf{G}}$. It follows from stationarity of $\xi$ and definition of
$\xi^{(x,p)}$, that $T_{x}\xi^{(x,p)}\overset{d}{=}\xi^{(0,p)}$ for
$\lambda_{d}\otimes\mathbb{Q}$-a.e.
$(x,p)\in{\mathbb{R}}^{d}\times{\mathbb{M}}$. Therefore the result follows. ∎

Let $Q_{0}$ be a random element of ${\mathbb{M}}$ with distribution
$\mathbb{Q}$ which is independent of $\eta^{*}$ given by (4.4). In accordance
with Palm theory we refer to $C^{(0,Q_{0})}(\xi^{(0,Q_{0})})$ as cluster of
the typical vertex (of $\xi$).

######  Remark 4.2.

Let $p\in{\mathbb{M}}$. Then the degree $D_{p}$ of $(0,p)$ (the origin marked
with $p$) in $\xi^{(0,p)}$ has a Poisson distribution with parameter
$t\int\varphi(x,p,q)\,dx\,\mathbb{Q}(dq)$. Our integrability assumption (4.2)
means that $\int\mathbb{E}D_{p}\,\mathbb{Q}(dp)<\infty$. This means that the
expected degree of the typical vertex is finite. Hence (4.2) excludes Pareto
type degree distributions but is still much weaker than the integrability
assumption made in [10].

The function

$\displaystyle\kappa(t):=\int\mathbb{E}_{t}|C^{(0,p)}|^{-1}\,\mathbb{Q}(dp)=\mathbb{E}_{t}|C^{(0,Q_{0})}|^{-1},\quad
t\in{\mathbb{R}}_{+},$ (4.6)

plays a crucial role in Section 11. To interpret it, we introduce a point
process $\eta_{c}\leq\eta(\cdot\times{\mathbb{M}})$ modeling finite clusters
as follows. Let $(x,p)\in\eta$. Then $x\in\eta_{c}$ if
$|C^{(x,p)}(\xi)|<\infty$ and $x$ is the lexicographically smallest spatial
coordinate of the points in $C^{(x,p)}(\xi)$. Since $\xi$ is stationary, it is
easy to see that $\eta_{c}$ is a stationary point process. The following
result shows that $t\kappa(t)$ is the intensity of $\eta_{c}$, that is the
density of finite clusters. With a slight abuse of language we refer to
$\kappa(t)$ as cluster density. In the unmarked case this function is also
called free energy; see [1, 6, 13].

######  Lemma 4.3.

For each $t\in{\mathbb{R}}_{+}$ we have that
$t\kappa(t)=\mathbb{E}_{t}\eta_{c}([0,1]^{d})$.

###### Proof.

The result follows from [27, Proposition 3.1] upon taking there $\eta$ as the
projection of the point process $\\{(x,p)\in\eta:|C^{(x,p)}(\xi)|<\infty\\}$
onto ${\mathbb{R}}^{d}$ and $\xi:=\eta_{c}$. A direct proof can start with

$\displaystyle
t\kappa(t)=\mathbb{E}_{t}\int{\mathbf{1}}\\{x\in[0,1]^{d}\\}|C^{(x,p)}(\xi)|^{-1}\,\eta(d(x,p)),$
(4.7)

a consequence of (4.5). The right-hand side can be written as

$\displaystyle\mathbb{E}_{t}\iint{\mathbf{1}}\\{x\in[0,1]^{d}\\}|C^{(x,p)}(\xi)|^{-1}{\mathbf{1}}\\{\tau(x,p)=y\\}\,\eta(d(x,p))\,\eta_{c}(dy),$

where $\tau(x,p)$ is the lexicographic minimum of the spatial coordinates of
$C^{(x,p)}(\xi)$. The key step is then the application of the refined Campbell
theorem for $\eta_{c}$. ∎

The cluster density can also be obtained as an ergodic limit:

######  Proposition 4.4.

Let $B_{n}\in{\mathcal{B}}({\mathbb{R}}^{d})$, $n\in{\mathbb{N}}$, be an
increasing sequence of compact convex sets whose inradius diverges to
$\infty$. Then

$\displaystyle\lim_{n\to\infty}(\lambda_{d}(B_{n}))^{-1}\int{\mathbf{1}}\\{x\in
B_{n}\\}|C^{(x,p)}(\xi)|^{-1}\,\eta(d(x,p))=t\kappa(t),\quad\mathbb{P}_{t}\text{-a.s.}$

###### Proof.

For each $\mu\in{\mathbf{G}}$

$\displaystyle
M_{\mu}:=\int{\mathbf{1}}\\{x\in\cdot\\}|C^{(x,p)}(\mu)|^{-1}\,\mu(d(x,p))$

is a locally finite measure on ${\mathbb{R}}^{d}$. For
$x,y\in{\mathbb{R}}^{d}$ and $\mu\in{\mathbf{G}}$ we have
$C^{(x,p)}(T_{y}\mu)=T_{y}C^{(x+y,p)}(\mu)$. Therefore, we obtain for
$B\in{\mathcal{B}}({\mathbb{R}}^{d})$ and $y\in{\mathbb{R}}^{d}$

$\displaystyle M_{T_{y}\mu}(B)$ $\displaystyle:=\int{\mathbf{1}}\\{x\in
B\\}|C^{(x+y,p)}(\mu)|^{-1}\,T_{y}\mu(d(x,p))$
$\displaystyle=\int{\mathbf{1}}\\{x-y\in
B\\}|C^{(x,p)}(\mu)|^{-1}\,\mu(d(x,p)).$

This means that $M_{T_{y}\mu}=T_{y}M_{\mu}$. Therefore $M_{\xi}$ is a
stationary and ergodic random measure. By (4.7) it has intensity $t\kappa(t)$.
Hence the result follows from [33, Satz 3]; see also [22, Theorem 30.10]. ∎

We continue with a basic fact from percolation theory. Define

$\displaystyle\theta(t):=\mathbb{P}_{t}\big{(}\big{|}C^{(0,Q_{0})}(\xi^{(0,Q_{0})})\big{|}=\infty\big{)}=\int\mathbb{P}_{t}\big{(}|C^{(0,p)}|=\infty\big{)}\,\mathbb{Q}(dp),\quad
t\geq 0,$ (4.8)

as the probability that the cluster of a typical vertex has infinite size. Let
$C_{\infty}$ denote the set of all $\mu\in{\mathbf{G}}$ such that $\mu$ has an
infinite cluster.

######  Proposition 4.5.

Let $t>0$. Then $\theta(t)>0$ iff $\mathbb{P}_{t}(\xi\in C_{\infty})=1$.

###### Proof.

Let $B\in{\mathcal{B}}({\mathbb{R}}^{d})$ be a Borel set with
$\lambda_{d}(B)\in(0,\infty)$. By (4.5),

$\displaystyle\theta(t)=(t\lambda_{d}(B))^{-1}\mathbb{E}_{t}\int{\mathbf{1}}\\{x\in
B\\}{\mathbf{1}}\\{|C^{(x,p)}(\xi)|=\infty\\}\,\eta(d(x,p)).$

Hence, if $\theta(t)>0$, then the probability that there is some
$(x,p)\in\eta$ with $|C^{(x,p)}(\xi)|=\infty$ must be positive. Since $\xi$ is
ergodic and $C_{\infty}$ is translation invariant, we obtain
$\mathbb{P}_{t}(\xi\in C_{\infty})=1$. If, on the other hand, $\theta(t)=0$,
then the probability that $|C^{(x,p)}(\xi)|=\infty$ for some $(x,p)\in\eta$
with $x\in B$ is zero. Letting $B\uparrow{\mathbb{R}}^{d}$ we obtain that
$\mathbb{P}_{t}(\xi\in C_{\infty})=0$. ∎

The critical intensity (percolation threshold) is defined by

$\displaystyle t_{c}:=\inf\\{t>0:\theta(t)>0\\}.$ (4.9)

If $t<t_{c}$ then $\mathbb{P}_{t}(\xi\in C_{\infty})=0$ and if $t>t_{c}$ then
$\mathbb{P}_{t}(\xi\in C_{\infty})=1$. Under a natural irreducibility
assumption our Theorem 12.1 will show that $\xi$ can have at most one infinite
cluster.

We finish this section with some examples.

######  Example 4.6.

In the unmarked case the connection function $\varphi$ is just a function on
${\mathbb{R}}^{d}$. Under the minimal assumption $d_{\varphi}\in(0,\infty)$ it
was shown in [34] that $t_{c}\in(0,\infty)$.

######  Example 4.7.

Assume that ${\mathbb{M}}={\mathbb{R}}_{+}$ and
$\varphi(x,p,q)={\mathbf{1}}\\{\|x\|\leq p+q\\}$, where $\|x\|$ denotes the
Euclidean norm of $x\in{\mathbb{R}}^{d}$. The RCM $\xi$ is then said to be the
Gilbert graph with radius distribution $\mathbb{Q}$; see e.g. [25, Chapter 16]
for more detail. The integrability assumption (4.2) is then equivalent with
$\int r^{d}\,\mathbb{Q}(dr)<\infty$, which is the minimal assumption for
having a reasonable model. Under the assumption $\mathbb{Q}\\{0\\}<1$ it was
proved in [16, 19] that $t_{c}\in(0,\infty)$.

######  Example 4.8.

Suppose that ${\mathbb{M}}$ equals the space $\mathcal{C}^{d}$ of all non-
empty compact subsets of ${\mathbb{R}}^{d}$, equipped with the Hausdorff
metric cf. [25, 38]. Let
$V\colon\mathcal{C}^{d}\cup\\{\emptyset\\}\to[0,\infty]$ be measurable and
translation invariant with $V(\emptyset)=0$. For instance, $V$ could be the
volume or, if $\mathbb{Q}$ is concentrated on the convex bodies, a linear
combination of the intrinsic volumes; see [38]. Assume that the connection
function is given by

$\displaystyle\varphi((x,K),(y,L))=1-e^{-V((K+x)\cap(L+y))},\quad(x,K),(y,L)\in{\mathbb{R}}^{d}\times\mathcal{C}^{d}.$

Then (4.1) follows from translation invariance of $V$. The case of the Gilbert
graph arises if $\mathbb{Q}$ is concentrated on balls centered at the origin.
A sufficient condition for (4.2) is

$\displaystyle\int D(K)^{d}\,\mathbb{Q}(dK)<\infty,$

where $D(K)$ is the radius of the smallest ball centered at the origin and
containing $K$. This easily follows from

$\displaystyle\varphi(x,K,L)\leq{\mathbf{1}}\\{K\cap(L+x)\neq\emptyset\\}\leq{\mathbf{1}}\\{\|x\|\leq
D(K)+D(L)\\}.$

The random closed set $\bigcup_{(x,K)\in\eta}K+x$ is known as the Boolean
model and a fundamental model of stochastic geometry (see [25, 38]) and
continuum percolation (see [31]). This model corresponds to the choice
$V(K)=\infty\cdot{\mathbf{1}}\\{K\neq\emptyset\\}$. In that case and under
some additional assumptions on $\mathbb{Q}$ it was proved in [19] that
$t_{c}\in(0,\infty)$. The present much more general model is taken from [5]
and is partially motivated by statistical physics.

######  Example 4.9.

Assume that ${\mathbb{M}}=(0,1)$ equipped with Lebesgue measure $\mathbb{Q}$.
Assume that

$\displaystyle\varphi((x,p),(y,q))=\rho(g(p,q)\|x-y\|^{d}),$

for a decreasing function $\rho\colon[0,\infty)\to[0,1]$ and a function
$g\colon(0,1)\times(0,1)\to[0,\infty)$ which is increasing in both arguments.
We assume that $m_{\rho}:=\int\rho(\|x\|^{d})\,dx$ is positive and finite.
This model was studied in [18] under the name weigt-dependent random
connection model. A simple calculation shows that

$\displaystyle d_{\varphi}=m_{\rho}\iint g(p,q)^{-1}\,dp\,dq.$

To ensure (4.2) we have to assume that $g^{-1}$ is integrable. This is the
case in all examples studied in [18], where it is also asserted that
$t_{c}<\infty$. Sufficient conditions for $t_{c}\in(0,\infty)$ can also be
found in [10, 11].

##  5 Irreducibility

In this section, we first consider a general RCM $\xi$ based on a Poisson
process $\eta$ on ${\mathbb{X}}$ with diffuse intensity measure $t\lambda$. We
fix the intensity parameter $t>0$ and therefore drop the lower index $t$ in
$\mathbb{P}_{t}$. To simplify the notation, we take $t=1$. We say that $\xi$
is irreducible if

$\displaystyle\mathbb{P}(\text{$x_{1}\leftrightarrow x_{2}$ in
$\xi^{x_{1},x_{2}}$})>0,\quad\lambda^{2}\text{-a.e.
$(x_{1},x_{2})\in{\mathbb{X}}^{2}$}.$ (5.1)

Given $k\in{\mathbb{N}}$ and random elements $Y_{1},\ldots,Y_{k}$ of
${\mathbb{X}}$ we let $\Xi[Y_{1},\dots,Y_{k}]$ be a RCM based on the point
process $\delta_{Y_{1}}+\cdots+\delta_{Y_{k}}$. Of course we can allow here
some of the $Y_{1},\ldots,Y_{k}$ to be deterministic. Further we define for
each $n\in{\mathbb{N}}$ a measurable function
$\varphi^{(n)}\colon{\mathbb{X}}^{2}\to[0,\infty)$ recursively by
$\varphi^{(1)}:=\varphi$ and

$\displaystyle\varphi^{(n+1)}(x_{1},x_{2}):=\int\varphi^{(n)}(x_{1},z)\varphi(z,x_{2})\,\lambda(dz),\quad
x_{1},x_{2}\in{\mathbb{X}},\,n\in{\mathbb{N}}.$

These functions are symmetric. It follows straight from the Mecke equation
(3.1) that

$\displaystyle\varphi^{(n)}(x_{1},x_{2})=\mathbb{E}\int\prod_{i=1}^{n}{\mathbf{1}}\\{\text{$y_{i-1}\sim
y_{i}$ in $\xi^{x_{1},x_{2}}$}\\}\,\eta^{(n-1)}(d(y_{1},\dots,y_{n-1})),$
(5.2)

where $y_{0}:=x_{1}$ and $y_{n}:=x_{2}$. This is the expected number of paths
of length $n$ from $x_{1}$ to $x_{2}$ in $\xi^{x_{1},x_{2}}$.

######  Proposition 5.1.

The following six statements are equivalent:

  * (i)

$\xi$ is irreducible.

  * (ii)

There exist for $\lambda^{2}$-a.e. $(x_{1},x_{2})\in{\mathbb{X}}^{2}$ a set
$B\in{\mathcal{X}}$ with $\lambda(B)\in(0,\infty)$, an $n\in{\mathbb{N}}_{0}$
and independent random variables $Y_{1},\ldots,Y_{n}$ with distribution
$\lambda_{B}/\lambda(B)$ such that

$\displaystyle\mathbb{P}(\text{$x_{1}\leftrightarrow x_{2}$ in
$\Xi[x_{1},x_{2},Y_{1},\ldots,Y_{n}]$})>0.$ (5.3)

  * (iii)

There exists for $\lambda^{2}$-a.e. $(x_{1},x_{2})\in{\mathbb{X}}^{2}$ an
$n\in{\mathbb{N}}_{0}$ such that

$\displaystyle\int\mathbb{P}(\text{$x_{1}\leftrightarrow x_{2}$ in
$\Xi[x_{1},x_{2},y_{1},\ldots,y_{n}]$})\lambda^{n}(d(y_{1},\dots,y_{n}))>0.$
(5.4)

  * (iv)

For $\lambda^{2}$-a.e. $(x_{1},x_{2})\in{\mathbb{X}}^{2}$ it holds that

$\displaystyle\sup_{n\geq 1}\varphi^{(n)}(x_{1},x_{2})>0.$ (5.5)

  * (v)

For $\lambda^{2}$-a.e. $(x_{1},x_{2})\in{\mathbb{X}}^{2}$ we have

$\displaystyle\sup_{n\geq k}\varphi^{(n)}(x_{1},x_{2})>0,\quad
k\in{\mathbb{N}}.$ (5.6)

  * (vi)

There exist for $\lambda^{2}$-a.e. $(x_{1},x_{2})\in{\mathbb{X}}^{2}$ a set
$B\in{\mathcal{X}}$ with $\lambda(B)\in(0,\infty)$, an $n\in{\mathbb{N}}$ and
independent random variables $Y_{1},\ldots,Y_{n}$ with distribution
$\lambda_{B}/\lambda(B)$ such that

$\displaystyle\mathbb{P}(\text{$x_{1}\leftrightarrow x_{2}$ in
$\Xi^{\prime}[x_{1},x_{2},Y_{1},\ldots,Y_{n}]$})>0,$ (5.7)

where $\Xi^{\prime}[x_{1},x_{2},Y_{1},\ldots,Y_{n}]$ is the graph obtained
from $\Xi[x_{1},x_{2},Y_{1},\ldots,Y_{n}]$ by removing the edge between
$x_{1}$ and $x_{2}$ (if there is one).

###### Proof.

Assume that (i) holds, and take $x_{1},x_{2}\in{\mathbb{X}}$ be such that
$\mathbb{P}(\text{$x_{1}\leftrightarrow x_{2}$ in $\xi^{x_{1},x_{2}}$})>0$.
Let $(B_{m})$ be a sequence of measurable sets of finite $\lambda$-measure
increasing towards ${\mathbb{X}}$. By monotone convergence there exists
$m\in{\mathbb{N}}$ such that $x_{1},x_{2}\in B_{m}$ and

$\displaystyle\mathbb{P}(\text{$x_{1}\leftrightarrow x_{2}$ in
$\xi^{x_{1},x_{2}}[B_{m}]$})>0.$ (5.8)

Let $B:=B_{m}$. Since $\xi^{x_{1},x_{2}}[B]$ is a RCM based on
$\delta_{x_{1}}+\delta_{x_{2}}+\eta_{B}$ we have
$\eta_{B}\overset{d}{=}\sum^{\eta(B)}_{k=0}Y_{k}$, where $Y_{1},Y_{2},\ldots$
are independent with distribution $\lambda_{B}/\lambda(B)$. Splitting the
event $\\{\text{$x_{1}\leftrightarrow x_{2}$ in $\xi^{x_{1},x_{2}}[B]$}\\}$
according to the value of $\eta(B)$ yields (ii).

Assume that (ii) holds, then

$\displaystyle\mathbb{P}(\text{$x_{1}\leftrightarrow x_{2}$ in
$\Xi[x_{1},x_{2},Y_{1},\ldots,Y_{n}]$})=\lambda(B)^{-n}\int\mathbb{P}(\text{$x_{1}\leftrightarrow
x_{2}$ in
$\Xi[x_{1},x_{2},y_{1},\ldots,y_{n}]$})\,\lambda_{B}^{n}(d(y_{1},\dots,y_{n}))>0,$

which implies (iii).

Assume that $x_{1},x_{2}\in{\mathbb{X}}$ satisfy (5.4). If
$x_{1}\leftrightarrow x_{2}$ in $\Xi[x_{1},x_{2},y_{1},\ldots,y_{n}]$ then
there exist $k\in\\{0,\ldots,n\\}$ and pairwise distinct
$i_{1},\ldots,i_{k}\in\\{1,\ldots,n\\}$ with $x_{1}\sim
y_{i_{1}}\sim\cdots\sim y_{i_{k}}\sim x_{2}$ in
$\Xi[x_{1},x_{2},y_{1},\ldots,y_{n}]$. Therefore and by the symmetry of
$\lambda^{n}$

$\displaystyle\sum^{n}_{k=0}\int\prod^{k+1}_{i=1}\varphi(y_{i-1},y_{i})\,\lambda^{k}(d(y_{1},\dots,y_{k}))>0,$

where $y_{0}:=x_{1}$ and $y_{k+1}:=x_{2}$. Hence (iv) follows.

Assume that (iv) holds, then for $\lambda^{2}$-a.e.
$(x_{1},x_{2})\in{\mathbb{X}}^{2}$

$\displaystyle 0<\sum_{m,n=1}^{\infty}\varphi^{(m+n)}(x_{1},x_{2})=$
$\displaystyle\sum_{m,n=1}^{\infty}\int\varphi^{(m)}(x_{1},z)\varphi^{(n)}(z,x_{2})\,\lambda(dz)$
$\displaystyle=$
$\displaystyle\int\left(\sum_{m=1}^{\infty}\varphi^{(m)}(x_{1},z)\right)\left(\sum_{n=1}^{\infty}\varphi^{(n)}(z,x_{2})\right)\,\lambda(dz).$

Therefore $\sup\limits_{n\geq 2}\varphi^{(n)}(x_{1},x_{2})>0$ for
$\lambda^{2}$-a.e. $(x_{1},x_{2})\in{\mathbb{X}}^{2}$. To obtain (5.6) for
general $k\in{\mathbb{N}}$ one has to start with a $k$-fold summation instead
of a double summation.

Assume now that $x_{1},x_{2}\in{\mathbb{X}}$ satisfy (5.6), then there exists
$n\geq 2$ such that $\varphi^{(n)}(x_{1},x_{2})>0$. Therefore there exists
$B\in{\mathcal{X}}$ of finite $\lambda$-measure with

$\displaystyle\int\prod^{n}_{i=1}\varphi(y_{i-1},y_{i})\,\lambda^{n-1}_{B}(d(y_{1},\dots,y_{n-1}))>0,$

where $y_{0}:=x_{1}$ and $y_{n}:=x_{2}$. This implies (vi).

Finally, note that for any $n\in{\mathbb{N}}$

$\displaystyle\mathbb{P}(\text{$x_{1}\leftrightarrow x_{2}$ in
$\Xi^{\prime}[x_{1},x_{2},Y_{1},\ldots,Y_{n}]$})\leq\mathbb{P}$
$\displaystyle(\text{$x_{1}\leftrightarrow x_{2}$ in
$\Xi[x_{1},x_{2},Y_{1},\ldots,Y_{n}]$})$
$\displaystyle=\mathbb{P}(\text{$x_{1}\leftrightarrow x_{2}$ in
$\xi^{x_{1},x_{2}}[B]$}\mid\eta(B)=n).$

Therefore (vi) implies (i). ∎

######  Remark 5.2.

Proposition 5.1 shows that irreducibility does not depend on the intensity
parameter $t$ as long it is positive.

In the remainder of this section we consider the stationary marked RCM as
discussed in Section 4. Recall that without loss of generality, we can take
$t=1$. It is easy to see that for all $n\in{\mathbb{N}}$ and all
$(x,p),(y,q)\in{\mathbb{X}}$

$\displaystyle\varphi^{(n)}((x,p),(y,q))=\varphi^{(n)}((0,p),(y-x,q))=\varphi^{(n)}((0,q),(x-y,p))$
(5.9)

We write $\varphi^{(n)}(x,p,q):=\varphi^{(n)}((0,p),(x,q))$ and note that
$\varphi^{(n)}(x,p,q)=\varphi^{(n)}(-x,q,p)$. In the unmarked case we can
identify ${\mathbb{X}}$ with ${\mathbb{R}}^{d}$. In this case
$\varphi^{(n)}=\varphi^{\ast n}$ is the $n$-fold convolution of $\varphi$,
where $\varphi$ is considered as function on ${\mathbb{R}}^{d}$.

######  Proposition 5.3.

The stationary (unmarked) RCM is irreducible.

The proof of Proposition 5.3 is a quick consequence of the first part of the
following lemma.

######  Lemma 5.4.

Assume that $f\colon{\mathbb{R}}^{d}\to[0,\infty)$ is a bounded measurable
function with $0<\int f(y)\,dy<\infty$ and $f(y)=f(-y)$ for all
$y\in{\mathbb{R}}^{d}$. Let $R>0$.

  1. (i)

There exist $n\in{\mathbb{N}}$ and $\varepsilon>0$ such that $f^{\ast
n}(x)\geq\varepsilon$ whenever $\|x\|\leq R$.

  2. (ii)

Let $g\colon{\mathbb{R}}^{d}\to{\mathbb{R}}_{+}$ be another bounded measurable
function with $\int g(y)\,dy>0$ and let $x\in{\mathbb{R}}^{d}$. Then there
exists $n\in{\mathbb{N}}$ such that $(f^{\ast n}\ast g)(x)>0$.

###### Proof.

(i) The convolution of an integrable and a bounded function is bounded and
uniformly continuous; see [14, Proposition 8.8]. It follows that $f^{\ast 2}$
is bounded and uniformly continuous. Since $\int f^{\ast 2}(x)\,dx=\big{(}\int
f(x)\,dx\big{)}^{2}>0$ there exist a ball $B^{\prime}\subset{\mathbb{R}}^{d}$
with positive radius and $\varepsilon^{\prime}>0$ such that $f^{\ast
2}\geq\varepsilon^{\prime}$ on $B^{\prime}$. Since $f$ is symmetric, $f^{\ast
2}$ is symmetric as well. Hence we can find a ball $B$ with center $0$ and
positive radius and some $\varepsilon>0$ such that $f^{\ast 4}\geq\varepsilon$
on $B$. Finally we find $m\in{\mathbb{N}}$ and $\varepsilon>0$ such that
$f^{\ast 4m}(x)\geq\varepsilon$ whenever $\|x\|\leq R$.

(ii) By assumption $\int g(x){\mathbf{1}}\\{g(x)\geq\varepsilon_{0}\\}\,dx>0$
for some $\varepsilon_{0}>0$. Set $C:=\\{g\geq\varepsilon_{0}\\}$. Then we
have for each $n\in{\mathbb{N}}$ that

$\displaystyle(f^{\ast n}\ast g)(x)\geq\varepsilon_{0}\int f^{\ast
n}(x-z){\mathbf{1}}\\{z\in C\\}\,dz.$

Choose $R>0$ so large that

$\displaystyle\int{\mathbf{1}}\\{\|x-z\|\leq R,z\in C\\}\,dz>0.$

By the first part of the lemma we can find $n\in{\mathbb{N}}$ and
$\varepsilon>0$ such that $f^{\ast n}(y)\geq\varepsilon$ whenever $\|y\|\leq
R$. It follows that

$\displaystyle(f^{\ast n}\ast g)(x)$ $\displaystyle\geq\varepsilon_{0}\int
f^{\ast n}(x-z){\mathbf{1}}\\{\|x-z\|\leq R,z\in C\\}\,dz$
$\displaystyle\geq\varepsilon_{0}\varepsilon\int{\mathbf{1}}\\{\|x-z\|\leq
R,z\in C\\}\,dz.$

By the choice of $R$ this is positive. ∎

###### Proof of Proposition 5.3.

We can use Lemma 5.4 (i) and condition (5.5) from Proposition 5.1 to conclude
the proof. Indeed, given $x_{1},x_{2}\in{\mathbb{X}}$ we find an
$n\in{\mathbb{N}}$ such that $\varphi^{(n)}(x_{1},x_{2})=\varphi^{\ast
n}(x_{2}-x_{1})>0$. ∎

It is natural to characterize irreducibility of the stationary marked RCM in
terms of the functions $d_{\varphi}^{(n)}\colon{\mathbb{M}}^{2}\to[0,\infty]$,
$n\in{\mathbb{N}}$, defined by

$\displaystyle d_{\varphi}^{(n)}(p,q):=\int\varphi^{(n)}(x,p,q)\,dx,\quad
p,q\in{\mathbb{M}}.$

Similarly as at (5.2) we see that $\int
d^{(n)}_{\varphi}(p,q){\mathbf{1}}\\{q\in A\\}\,\mathbb{Q}(dq)$ is the
expected number of paths of length $n$ from $(0,p)$ to some location with mark
in a measurable set $A\subset{\mathbb{M}}$. From the symmetry property of
$\varphi^{(n)}$ we obtain that $d_{\varphi}^{(n)}$ is symmetric. Furthermore,

$\displaystyle
d_{\varphi}^{(n)}(p,q)=\int\prod_{i=1}^{n}d_{\varphi}(q_{i-1},q_{i})\,\mathbb{Q}^{n-1}(d(q_{1},\ldots,q_{n-1})),$

where $d_{\varphi}(\cdot,\cdot):=d_{\varphi}^{(1)}(\cdot,\cdot)$, $q_{0}:=p$,
$q_{n}:=q$. Therefore

$\displaystyle d_{\varphi}^{(m+n)}(p,q)=\int
d_{\varphi}^{(m)}(p,r)d_{\varphi}^{(n)}(r,q)\,\mathbb{Q}(dr),\quad
p,q\in{\mathbb{M}},\,m,n\in{\mathbb{N}}.$ (5.10)

######  Lemma 5.5.

Let $\xi$ be a stationary marked RCM and assume that $\xi$ is irreducible.
Then

$\displaystyle\sup_{n\geq
1}d_{\varphi}^{(n)}(p,q)>0,\quad\mathbb{Q}^{2}\text{-a.e.
$(p,q)\in{\mathbb{M}}^{2}$}.$ (5.11)

###### Proof.

Assume on the contrary that there exists some measurable set
$B\subset{\mathbb{M}}^{2}$ satisfying $\mathbb{Q}^{2}(B)>0$, and
$d_{\varphi}^{(n)}(p,q)=0$ for all $(p,q)\in B$ and for each
$n\in{\mathbb{N}}$. But then $\varphi^{(n-1)}((0,p),(x,q))=0$ for all
$(p,q)\in B$, $n\in{\mathbb{N}}$ and $\lambda_{d}$-a.e.
$x\in{\mathbb{R}}^{d}$. This contradicts Proposition 5.1 (iv). ∎

We continue with the case where $\mathbb{Q}$ has an atom. This covers a
discrete (that is finite or countably infinite) mark space and generalizes
Proposition 5.3.

######  Theorem 5.6.

Let $\xi$ be a stationary marked RCM and assume that there exists
$p_{0}\in{\mathbb{M}}$ with $\mathbb{Q}\\{p_{0}\\}>0$. Then $\xi$ is
irreducible iff

$\displaystyle\sup_{n\geq
1}d_{\varphi}^{(n)}(p_{0},q)>0,\quad\mathbb{Q}\text{-a.e.\
$q\in{\mathbb{M}}$}.$ (5.12)

###### Proof.

By $\mathbb{Q}\\{p_{0}\\}>0$ and (5.12) there exists $n\in{\mathbb{N}}$ such
that $d_{\varphi}^{(n)}(p_{0},p_{0})>0$. Then we have for each
$x\in{\mathbb{R}}^{d}$

$\displaystyle\varphi^{(2n)}(x,p_{0},p_{0})$
$\displaystyle=\iint\varphi^{(n)}(z,p_{0},q)\varphi^{(n)}(x-z,q,p_{0})\,\mathbb{Q}(dq)\,dz$
$\displaystyle\geq\mathbb{Q}\\{p_{0}\\}\int\varphi^{(n)}(z,p_{0},p_{0})\varphi^{(n)}(x-z,p_{0},p_{0})\,dz=:\mathbb{Q}\\{p_{0}\\}\psi_{0}(x).$
(5.13)

By induction we obtain for all $m\in{\mathbb{N}}$ that

$\displaystyle\varphi^{(2mn)}(x,p_{0},p_{0})\geq\mathbb{Q}\\{p_{0}\\}^{2m-1}\psi^{\ast
m}_{0}(x),\quad x\in{\mathbb{R}}^{d}.$ (5.14)

The function $\psi_{0}$ is symmetric and bounded. Moreover,

$\displaystyle\int\psi_{0}(x)\,dx=\Big{(}\int\varphi^{(n)}(z,p_{0},p_{0})\,dz\Big{)}^{2}=(d_{\varphi}^{(n)}(p_{0},p_{0}))^{2}>0.$

Take $p,q\in{\mathbb{M}}$. In view of the assertion and assumption (5.12) we
can assume that there exist $k,l\in{\mathbb{N}}$ such that
$d_{\varphi}^{(k)}(p_{0},p)>0$ and $d_{\varphi}^{(l)}(p_{0},q)>0$. Then we
obtain for each $x\in{\mathbb{R}}^{d}$ and each $m\in{\mathbb{N}}$ that

$\displaystyle\varphi^{(2mn+k+l)}(x,p,q)$
$\displaystyle=\iint\varphi^{(k)}(z,p,r)\varphi^{(2mn)}(w,r,s)\varphi^{(l)}(x-z-w,s,q)\,\mathbb{Q}^{2}(d(r,s))\,d(z,w)$
$\displaystyle\geq\mathbb{Q}\\{p_{0}\\}^{2}\int\varphi^{(k)}(z,p,p_{0})\varphi^{(2mn)}(w,p_{0},p_{0})\varphi^{(l)}(x-z-w,p_{0},q)\,d(z,w).$

By (5.14) this means

$\displaystyle\varphi^{(2mn+k+l)}(x,p,q)$
$\displaystyle\geq\mathbb{Q}\\{p_{0}\\}^{2m+1}\int\varphi^{(k)}(z,p,p_{0})\psi^{\ast
m}_{0}(w)\varphi^{(l)}(x-z-w,p_{0},q)\,d(z,w)$
$\displaystyle=\mathbb{Q}\\{p_{0}\\}^{2m+1}(g\ast\psi^{\ast m}_{0}\ast
h)(x)=\mathbb{Q}\\{p_{0}\\}^{2m+1}(\psi^{\ast m}_{0}\ast g\ast h)(x),$

with the obvious definitions of the functions $g$ and $h$. By the choice of
$k$ and $l$ we have

$\displaystyle\int g\ast h(x)\,dx=\int g(x)\,dx\int h(x)\,dx>0.$

Therefore we obtain from Lemma 5.4 that $(\psi^{\ast m}_{0}\ast g\ast h)(x)$
is positive for some sufficiently large $m$. Hence Proposition 5.1 (iv) yields
the assertion. ∎

######  Remark 5.7.

If $p_{0}$ is an atom of $\mathbb{Q}$, then (5.11) and (5.12) are equivalent.
Indeed, from (5.10) we have for all $m,n\in{\mathbb{N}}$ and all
$p,q\in{\mathbb{M}}$ that

$\displaystyle
d_{\varphi}^{(m+n)}(p,q)\geq\mathbb{Q}\\{p_{0}\\}d_{\varphi}^{(m)}(p,p_{0})d_{\varphi}^{(n)}(p_{0},q).$

This is positive as soon as
$d_{\varphi}^{(m)}(p_{0},p)d_{\varphi}^{(n)}(p_{0},q)>0$.

The next remark shows the relevance of irreducibility for the uniqueness of
the infinite cluster.

######  Remark 5.8.

Assume that ${\mathbb{M}}$ is discrete and that $\mathbb{Q}\\{p\\}>0$ for each
$p\in{\mathbb{M}}$. Given $p,q\in{\mathbb{M}}$ we write $p\simeq q$ if either
$p=q$ or $\sup_{n\geq 1}d_{\varphi}^{(n)}(p,q)>0$. It follows from (5.10) that
$\simeq$ is an equivalence relation. Let $[p]:=\\{q\in{\mathbb{M}}:p\simeq
q\\}$ be the equivalence class of $p\in{\mathbb{M}}$. Then
$\eta_{[p]}:=\\{(x,q)\in\eta:q\in[p]\\}$ are for different equivalence classes
independent Poisson processes with intensity measures
$\lambda_{d}\otimes\mathbb{Q}([p]\cap\cdot)$. Assume now that there exist some
marks $p,q\in{\mathbb{M}}$ such that $[p]\cap[q]=\emptyset$. We assert that
$\xi[\eta_{[p]}]$ and $\xi[\eta_{[q]}]$ are vertex disjoint, that is, there is
no edge in $\xi$ between $\eta_{[p]}$ and $\eta_{[q]}$. To see this, we take a
bounded Borel set $B\subset{\mathbb{R}}^{d}$ and let $A$ denote the event that
there exist $x\in B$ and $y\in{\mathbb{R}}^{d}$ such that $(x,p),(y,q)\in\eta$
and $(x,p)\leftrightarrow(y,q)$ in $\xi$. Similarly, as in previous
calculations, we obtain

$\displaystyle\mathbb{P}(A)\leq\sum^{\infty}_{n=1}\lambda_{d}(B)\mathbb{Q}\\{p\\}\mathbb{Q}\\{q\\}d^{(n)}_{\varphi}(p,q)$

which comes to zero. If $\xi[\eta_{[p]}]$ and $\xi[\eta_{[q]}]$ both
percolate, then $\xi$ has at least two infinite clusters.

We now turn to a general mark space. Under a suitable minorization assumption
for the connection function the proof of Theorem 5.6 still works.

######  Theorem 5.9.

Assume that there exist a measurable set $A\subset{\mathbb{M}}$ with
$\mathbb{Q}(A)>0$ and some $p_{0}\in{\mathbb{M}}$ satisfying

$\displaystyle\varphi(x,p,q)\geq\varphi(x,p,p_{0}),\quad\lambda_{d}\otimes\mathbb{Q}^{2}\text{-a.e.
$(x,p,q)\in{\mathbb{R}}^{d}\times{\mathbb{M}}\times A$}.$ (5.15)

Assume also that (5.12) holds. Then the RCM $\xi$ is irreducible.

###### Proof.

We first show that (5.12) implies

$\displaystyle\sup_{n\geq 1}d_{\varphi}^{(n)}(p_{0},p_{0})>0.$ (5.16)

By (5.10) we have

$\displaystyle\sum_{m,n=1}^{\infty}d_{\varphi}^{(m+n)}(p_{0},p_{0})$
$\displaystyle=\sum_{m,n=1}^{\infty}\int
d_{\varphi}^{(m)}(p_{0},q)d_{\varphi}^{(n)}(q,p_{0})\,\mathbb{Q}(dq)$
$\displaystyle=\int\bigg{(}\sum^{\infty}_{m=1}d_{\varphi}^{(m)}(p_{0},q)\bigg{)}\bigg{(}\sum^{\infty}_{n=1}d_{\varphi}^{(n)}(p_{0},q)\bigg{)}\,\mathbb{Q}(dq).$

By (5.12) the above integrand is positive for $\mathbb{Q}$-a.e.
$q\in{\mathbb{M}}$. Hence the integral is positive and (5.16) follows.

It follows from (5.15) and the recursive structure of $\varphi^{(m)}$ that

$\displaystyle\varphi^{(m)}(x,p,q)\geq\varphi^{(m)}(x,p,p_{0}),\quad\lambda_{d}\otimes\mathbb{Q}^{2}\text{-a.e.\
$(x,p,q)\in{\mathbb{R}}^{d}\times{\mathbb{M}}\times A$},\,m\in{\mathbb{N}}.$
(5.17)

By (5.16) there exists $n\in{\mathbb{N}}$ such that
$d_{\varphi}^{(n)}(p_{0},p_{0})>0$. By (5.17) it follows as at (5) that

$\displaystyle\varphi^{(2n)}(x,p_{0},p_{0})\geq\mathbb{Q}(A)\psi_{0}(x),\quad\lambda_{d}\text{-a.e.}\
x\in{\mathbb{R}}^{d},$

where the function $\psi_{0}$ is defined as before. As at (5.14) it follows by
induction that

$\displaystyle\varphi^{(2mn)}(x,p_{0},p_{0})\geq\mathbb{Q}(A)^{2m-1}\psi^{\ast
m}_{0}(x),\quad\lambda_{d}\text{-a.e.\
$x\in{\mathbb{R}}^{d}$},\,m\in{\mathbb{N}}.$ (5.18)

Take $p,q\in{\mathbb{M}}$ and $k,l\in{\mathbb{N}}$ such that
$d_{\varphi}^{(k)}(p_{0},p)d_{\varphi}^{(l)}(p_{0},q)>0$. As in the proof of
Theorem 5.6 we obtain for $\lambda_{d}$-a.e. $x\in{\mathbb{R}}^{d}$ and each
$m\in{\mathbb{N}}$ that

$\displaystyle\varphi^{(2mn+k+l)}(x,p,q)\geq\mathbb{Q}(A)^{2m+1}(\psi^{\ast
m}_{0}\ast g\ast h)(x).$

Hence we can finish the proof as before. ∎

######  Remark 5.10.

Let the assumptions of Theorem 5.9 be satisfied. Then we have for all
$m,n\in{\mathbb{N}}$ that

$\displaystyle
d_{\varphi}^{(m+n)}(p,q)\geq\mathbb{Q}(A)d_{\varphi}^{(m)}(p,p_{0})d_{\varphi}^{(n)}(p_{0},q),\quad
p,q\in{\mathbb{M}}.$

Therefore (5.12) implies (5.11).

######  Remark 5.11.

Natural candidates for the pair $(A,p_{0})$ in (5.15) can be obtained as
follows. Assume that $A\subset{\mathbb{M}}$ is a measurable set such that
$\mathbb{Q}(A)>0$ and $\varphi(x,p,\cdot)$ is for all
$(x,p)\in{\mathbb{R}}^{d}\times{\mathbb{M}}$ increasing on $A$ w.r.t. some
partial order. We can then take $x_{0}$ as a minimal element of $A$, if it
exists.

######  Remark 5.12.

Remark 5.10 shows that the assumptions of Theorem 5.9 imply (5.11). We do not
know whether condition (5.11) alone implies that $\xi$ is irreducible. This
condition does not provide information on the final space location of paths.
In the proof of Theorem 5.9 we use (5.15) to decouple space locations and
marks. Condition (5.11) is much weaker than condition (D.2) in [10]. Condition
(5.15) does not occur in [10]. However, our examples in Section 12 indicate
that either this condition or the assumptions of the forthcoming Theorem 5.13
should cover all interesting examples.

A minimal assumption for irreducibility could be

$\displaystyle\int
d_{\varphi}(p,q)\,\mathbb{Q}(dq)>0,\quad\mathbb{Q}\text{-a.e.\
$p\in{\mathbb{M}}$}.$ (5.19)

If $\int d_{\varphi}(p,q)\,\mathbb{Q}(dq)=0$ for some $p\in{\mathbb{M}}$ then
(5.10) would imply that $\int d_{\varphi}^{(n)}(p,q)\,\mathbb{Q}(dq)=0$ for
all $n\in{\mathbb{N}}$. By Lemma 5.5 an irreducible $\xi$ cannot have this
property for all $p$ in a set of positive $\mathbb{Q}$-measure. Under suitable
assumptions on $\mathbb{Q}$ and $\varphi$ we shall show with Theorem 5.13 that
(5.19) is also sufficient for irreducibility.

In Theorem 5.13 we will consider a partial ordering $\preceq$ on
${\mathbb{M}}$ which is measurable, that is $\\{(p,q):p\preceq q\\}$ is a
measurable subset of ${\mathbb{M}}^{2}$. Slightly generalizing [28] we say
that ${\mathbb{M}}$ is POP space. A real-valued function $f$ on ${\mathbb{M}}$
is said to be non-decreasing if $x\preceq y$ implies $f(x)\leq f(y)$. The
probability measure $\mathbb{Q}$ is called (positively) associated if

$\displaystyle\int fg\,d\mathbb{Q}\geq\int f\,d\mathbb{Q}\int g\,d\mathbb{Q}$
(5.20)

for all non-decreasing measurable $f,g\colon{\mathbb{M}}\to{\mathbb{R}}$ for
which the integrals make sense. Our next result provides assumptions on
$\varphi$ and $\mathbb{Q}$, under which the minimal assumption (5.19) implies
irreducibility. Corollary 5.14 and Example 12.7 will demonstrate the
usefulness of this result.

######  Theorem 5.13.

Assume that ${\mathbb{M}}$ is a POP space and that $\mathbb{Q}$ is associated.
Assume also that $\varphi(x,p,\cdot)$ is non-decreasing for all
$(x,p)\in{\mathbb{R}}^{d}\times{\mathbb{M}}$. Then the RCM $\xi$ is
irreducible iff (5.19) holds.

###### Proof.

As argued above, condition (5.19) is necessary for irreducibility.

In the remainder of the proof we assume that (5.19) holds. We aim at checking
condition (iv) of Proposition (5.1). By assumption on $\varphi$ and the
symmetry properties of $\varphi$ we have that $\varphi(x,\cdot,\cdot)$ is for
each $x\in{\mathbb{R}}^{d}$ non-decreasing in both arguments. By the recursive
structure of $\varphi^{(n)}$ this property generalizes to $\varphi^{(n)}$ for
each $n\in{\mathbb{N}}$. We define
$\psi^{(n)}\colon{\mathbb{R}}^{d}\to[0,\infty]$ for each $n\in{\mathbb{N}}$ by

$\displaystyle\psi^{(n)}(x):=\int\varphi^{(n)}(x,p,q)\,\mathbb{Q}^{2}(d(p,q)),\quad
x\in{\mathbb{R}}^{d}.$

The function $\psi:=\psi^{(1)}$ is symmetric, bounded and integrable; see
(4.2). For each $n\in{\mathbb{N}}$ we obtain from (5.20) that

$\displaystyle\psi^{(n+1)}(x)$
$\displaystyle=\int\varphi^{(n)}(z,p,r)\varphi(x-z,r,q)\,\mathbb{Q}^{3}(d(p,q,r))\,dz$
$\displaystyle\geq\int\bigg{[}\int\varphi^{(n)}(z,p,r)\,\mathbb{Q}^{2}(d(p,r))\int\varphi(x-z,r,q)\,\mathbb{Q}^{2}(d(r,q))\bigg{]}\,dz.$

Therefore it follows by induction that

$\displaystyle\psi^{(n)}(x)\geq\psi^{\ast n}(x),\quad
x\in{\mathbb{R}}^{d},\,n\in{\mathbb{N}}.$ (5.21)

Take $p,q\in{\mathbb{M}}$. In view of our goal and assumption (5.19) we can
assume that $\int d_{\varphi}(p,q)\,\mathbb{Q}(dq)>0$ and $\int
d_{\varphi}(p,q)\,\mathbb{Q}(dp)>0$. From (5.20) we obtain for each
$x\in{\mathbb{R}}^{d}$ and each $n\in{\mathbb{N}}$ that

$\displaystyle\varphi^{(n+2)}(x,p,q)$
$\displaystyle=\iint\varphi(z,p,r)\varphi^{(n)}(w,r,s)\varphi(x-z-w,s,q)\,\mathbb{Q}^{2}(d(r,s))\,d(z,w),$
$\displaystyle\geq\int g(z)\psi^{(n)}(w)h(x-z-w)\,d(z,w),$

where $g(z):=\int\varphi(z,p,r)\,\mathbb{Q}(dr)$ and
$h(z):=\int\varphi(z,r,q)\,\mathbb{Q}(dr)$. Therefore we obtain from (5.21)
that

$\displaystyle\varphi^{(n+2)}(x,p,q)\geq(g\ast\psi^{\ast n}\ast
h)(x)=(\psi^{\ast n}\ast g\ast h)(x).$

By the choice of $p,q$ we have $\int g(x)\,dx>0$ and $\int h(x)\,dx>0$. Hence
we deduce from Lemma 5.4 that $(\psi^{\ast m}\ast g\ast h)(x)$ is positive for
some sufficiently large $m$. Hence Proposition 5.1 (iv) yields the assertion.
∎

######  Corollary 5.14.

Assume that ${\mathbb{M}}\subset{\mathbb{R}}$ is an interval and that
$\varphi(x,p,\cdot)$ is non-decreasing for all
$(x,p)\in{\mathbb{R}}^{d}\times{\mathbb{M}}$. Then the RCM $\xi$ is
irreducible iff (5.19) holds.

###### Proof.

Since any probability measure on ${\mathbb{M}}$ is associated (see e.g. [28]),
the result follows from Theorem 5.13. ∎

##  6 Deletion stability and uniqueness

In this section, we consider a general RCM $\xi$ based on a Poisson process
$\eta$ on ${\mathbb{X}}$ with diffuse intensity measure $\lambda$. Given
$(x,\mu)\in{\mathbb{X}}\times{\mathbf{G}}$ we let $N^{\infty}(x,\mu)$ denote
the number of infinite clusters in $C^{x}(\mu)-\delta_{x}$. We say that the
infinite clusters in $\xi$ are deletion stable if

$\displaystyle\mathbb{P}(N^{\infty}(x,\xi^{x})\geq
2)=0,\quad\lambda\text{-a.e.\ $x\in{\mathbb{X}}$}.$ (6.1)

Using the Mecke equation it is not difficult to see that the infinite clusters
in $\xi$ are deletion stable if $N_{ds}=0$ a.s., where

$\displaystyle N_{ds}:=\int{\mathbf{1}}\\{N^{\infty}(x,\xi)\geq
2\\}\,\eta(dx).$ (6.2)

######  Theorem 6.1.

Assume that $\xi$ is irreducible and that the infinite clusters of $\xi$ are
deletion stable. Then $\xi$ has $\mathbb{P}$-almost surely at most one
infinite cluster.

We prove Theorem 6.1 in stages. Let $Y_{1},\ldots,Y_{n}$ be random elements of
${\mathbb{X}}$, which are a.s. pairwise distinct. In accordance with Section 3
we define a random connection model $\xi^{Y_{1},\ldots,Y_{n}}$ based on the
point process $\eta+\delta_{Y_{1}}+\cdots+\delta_{Y_{n}}$ as follows. We
connect $Y_{1}$ with the points in $\eta$ using independent connection
decisions which are independent of $\xi$. We then proceed inductively finally
connecting $Y_{n}$ to $\eta+\delta_{Y_{1}}+\cdots+\delta_{Y_{n-1}}$.

######  Lemma 6.2.

Suppose that $B\in{\mathcal{X}}$ with $\lambda(B)\in(0,\infty)$ and let
$Y_{1},\ldots,Y_{n}$ be independent random variables with distribution
$\lambda_{B}/\lambda(B)$, independent of $\xi$. Assume that the infinite
clusters of $\xi$ are deletion stable, then

$\displaystyle\int\mathbb{P}(N^{\infty}(Y_{n},\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{n}})\geq
2)\,\lambda^{2}(d(x_{1},x_{2}))=0.$ (6.3)

###### Proof.

It is useful to add a point $x\in{\mathbb{X}}$ to a graph $\mu\in{\mathbf{G}}$
in the following explicit way. There are measurable mappings
$\pi_{n}\colon{\mathbf{N}}\to{\mathbb{R}}^{d}$ such that
$\mu(\cdot\times{\mathbf{N}})=\sum_{n=1}^{|\mu|}\delta_{\pi_{n}(\mu)}$, for
each $\mu\in{\mathbf{G}}$. Let $(\mu,x)\in{\mathbf{G}}\times{\mathbb{X}}$ and
$u=(u_{n})_{n\geq 1}\in[0,1]^{\infty}$. Define $\mu^{x}_{u}\in{\mathbf{G}}$ as
the graph with vertex measure $V(\mu)+\delta_{x}$, edges from $\mu$ and
further edges between $\pi_{n}(\mu)$ and $x$ if $\varphi(\pi_{n}(\mu),x)\geq
u_{n}$. Define $h(x,\mu,u):={\mathbf{1}}\\{N^{\infty}(x,\mu^{x}_{u})\geq
2\\}$. Assume that $U$ is a random element of $[0,1]^{\infty}$ with
independent and uniformly distributed components, independent of $\xi$. Then
${\mathbf{1}}\\{N^{\infty}(x,\xi^{x})\geq 2\\}$ has the same distribution as
$h(x,\xi,U)$ and deletion stability means that

$\displaystyle\iiint h(x,\mu,u)\,\mathbb{P}(\xi\in
d\mu)\,\lambda(dx)\,\mathbb{P}(U\in du)=0.$ (6.4)

Given $x_{1},x_{2}\in{\mathbb{X}}$ we also have

$\displaystyle{\mathbf{1}}\\{N^{\infty}(Y_{n},\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{n}})\geq
2\\}\overset{d}{=}h(Y_{n},\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{n-1}},U_{n}),$

where $U_{n}$ is independent of the pair
$(Y_{n},\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{n-1}})$ and has the same distribution
as $U$. Therefore

$\displaystyle\iint\mathbb{P}$
$\displaystyle(N^{\infty}(Y_{n},\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{n}})\geq
2)\,\lambda^{2}(d(x_{1},x_{2}))$
$\displaystyle=(\lambda(B))^{-1}\iiint\mathbb{E}h(y_{n},\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{n-1}},u)\,\lambda_{B}(dy_{n})\,\mathbb{P}(U\in
du)\,\lambda^{2}(d(x_{1},x_{2}))$
$\displaystyle=(\lambda(B))^{-n}\iiint\mathbb{E}h(y_{n},\xi^{x_{1},x_{2},y_{1},\ldots,y_{n-1}},u)\,\mathbb{P}(U\in
du)\lambda_{B}^{n}(d(y_{1},\ldots,y_{n}))\,\lambda^{2}(d(x_{1},x_{2})),$

where we have used the definition of $\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{n-1}}$.
From the Mecke equation we obtain that the above equals

$\displaystyle(\lambda(B))^{-n}\mathbb{E}\iiint
h(y_{n},\xi,u){\mathbf{1}}\\{y_{1},\ldots,y_{n-1}\in
B\\}\,\lambda_{B}(dy_{n})\,\mathbb{P}(U\in
du)\,\eta^{(n+1)}(d(x_{1},x_{2},y_{1},\ldots,y_{n-1})).$

By (6.4), the integral $\iint h(y,\xi,u)\,\lambda_{B}(dy)\,\mathbb{P}(U\in
du)$ does almost surely vanish. This concludes the proof. ∎

For given $x_{1},x_{2}\in{\mathbb{X}}$ let $A(x_{1},x_{2})$ be the event that
the clusters $C^{x_{1}}(\xi^{x_{1},x_{2}})$ and $C^{x_{2}}(\xi^{x_{1},x_{2}})$
are infinite and not connected. Further, for $n\in{\mathbb{N}}_{0}$ let
$B_{n}(x_{1},x_{2})$ be the event that $x_{1}$ and $x_{2}$ are connected in
$\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{n}}$, where $Y_{1},\dots,Y_{n}$ are defined
in Lemma 6.2.

######  Lemma 6.3.

Let the assumptions of Lemma 6.2 be in force. Then for a given
$n\in{\mathbb{N}}_{0}$

$\displaystyle\int\mathbb{P}(A(x_{1},x_{2})\cap
B_{n}(x_{1},x_{2}))\,\lambda^{2}(d(x_{1},x_{2}))=0.$ (6.5)

###### Proof.

We can remove the points $Y_{n},\ldots,Y_{1}$ from
$\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{n}}$ one by one. Each time we can apply
Lemma 6.2. Hence removing $Y_{i}$ (for $i\leq n$) cannot split the cluster of
$Y_{i}$ in $\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{i}}$ into more than one infinite
cluster. Take $x_{1},x_{2}\in{\mathbb{X}}$ and $n\in{\mathbb{N}}_{0}$ such
that $B_{n}(x_{1},x_{2})$ holds and assume for the sake of contradiction that
$A(x_{1},x_{2})$ holds. In particular $C^{x_{1}}(\xi^{x_{1},x_{2}})$ and
$C^{x_{2}}(\xi^{x_{1},x_{2}})$ are vertex disjoint, so that there must be an
$i\in\\{1,\ldots,n\\}$ such that $x_{1},x_{2}$ are connected in
$\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{i}}$ but not in
$\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{i-1}}$. Hence, the removal of $Y_{i}$ would
split the cluster of $Y_{i}$ in $\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{i}}$ into
two infinite clusters. This is a contradiction, showing that almost surely
$B_{n}(x_{1},x_{2})\subset A^{c}(x_{1},x_{2})$ for $\lambda^{2}$-a.e.
$(x_{1},x_{2})\in{\mathbb{X}}^{2}$. ∎

###### Proof of Theorem 6.1.

We need to show that almost surely two points of $\eta$ cannot belong to two
different infinite clusters. By the Mecke equation (3.1) for $n=2$ the latter
is equivalent to

$\displaystyle\int\mathbb{P}(A(x_{1},x_{2}))\,\lambda^{2}(d(x_{1},x_{2}))=0.$
(6.6)

The following arguments apply to $\lambda^{2}$-a.e.
$(x_{1},x_{2})\in{\mathbb{X}}^{2}$. By Proposition 5.1 (vi) there exist a set
$B\in{\mathcal{X}}$ with $0<\lambda(B)<\infty$, an $n\in{\mathbb{N}}$ and
random variables $Y_{1},\ldots,Y_{n}$ with distribution
$\lambda_{B}/\lambda(B)$ such that
$\mathbb{P}(B^{\prime}_{n}(x_{1},x_{2}))>0$, where

$\displaystyle B^{\prime}_{n}(x_{1},x_{2}):=\\{x_{1}\leftrightarrow
x_{2}\text{ in }\Xi^{\prime}[x_{1},x_{2},Y_{1},\ldots,Y_{n}]\\}.$

We can couple the random graphs $\xi^{x_{1},x_{2},Y_{1},\ldots,Y_{n}}$ and
$\Xi^{\prime}[x_{1},x_{2},Y_{1},\ldots,Y_{n}]$ in such a way that
$\xi^{x_{1},x_{2}}$ and $\Xi^{\prime}[x_{1},x_{2},Y_{1},\ldots,Y_{n}]$ are
independent and every edge in the latter graph is also present in the former.
Then $B^{\prime}_{n}(x_{1},x_{2})$ implies $B_{n}(x_{1},x_{2})$ and we obtain
from Lemma 6.3 that

$\displaystyle\mathbb{P}(A(x_{1},x_{2})\cap B^{\prime}_{n}(x_{1},x_{2}))=0.$

By the above coupling the events $A(x_{1},x_{2})$ and
$B^{\prime}_{n}(x_{1},x_{2})$ are independent. Hence
$\mathbb{P}(A(x_{1},x_{2}))=0$, as required. ∎

In Theorem 6.4 below we will see that deletion stability of infinite clusters
is in fact necessary for uniqueness of the infinite cluster. In fact,
uniqueness implies an even stronger property. Following [32] we call a graph
$\mu\in{\mathbf{G}}$ $2$-indivisible if the removal of a finite number of
vertices results in at most one infinite cluster. This means that the graph
$\mu[V(\mu)-\nu]$ has at most one infinite component for each finite $\nu\leq
V(\mu)$. If $\xi$ is almost surely $2$-indivisible, then, in particular, there
is at most one infinite cluster.

######  Theorem 6.4.

Assume that $\xi$ has almost surely at most one infinite cluster. Then $\xi$
is almost surely $2$-indivisible. In particular the infinite cluster of $\xi$
is deletion stable.

###### Proof.

Let $A_{\infty}$ denote set of all $\mu\in{\mathbf{G}}$ such that $\mu$ has at
least two infinite clusters. Let $n\in{\mathbb{N}}$. By the uniqueness
assumption and the Mecke equation (3.5) we have

$\displaystyle 0=\int\mathbb{P}(\xi\in
A_{\infty})\,\lambda^{n}(d(x_{1},\ldots,x_{n}))=\mathbb{E}\int{\mathbf{1}}\\{\xi-\delta_{x_{1}}-\cdots-\delta_{x_{n}}\in
A_{\infty}\\}\,\eta^{(n)}(d(x_{1},\ldots,x_{n})).$

Since $n\in{\mathbb{N}}$ is arbitrary, this proves the first result.

If $x\in\eta$ satisfies $N^{\infty}(x,\xi)\geq 2$ then $\xi-\delta_{x}\in
A_{\infty}$. Therefore by the Mecke equation (3.1) we obtain the second
assertion. ∎

######  Remark 6.5.

In accordance with the physics literature (see e.g. [8]) we might call a point
$x\in\eta$ red, if any doubly infinite path in $\xi$ has to use $x$. If $\xi$
has a unique infinite cluster Theorem 6.4 says in particular that $\xi$ cannot
have red points. More generally, we may call a subset of $\eta$ red, if any
doubly infinite path in $\xi$ contains at least one point from this set.
Theorem 6.4 says that $\xi$ cannot have a finite red set.

######  Remark 6.6.

The authors of [7] studied random connection models on finite point processes
in an asymptotic setting. Under a natural irreducibility assumption (similar
to Proposition 5.1 (iv)) they proved uniqueness of the giant component; see
Theorem 3.6 and Example 4.9 in [7].

##  7 A spatial Markov property

We again consider a general RCM $\xi$ based on a Poisson process $\eta$ on
${\mathbb{X}}$ with diffuse intensity measure $\lambda$. Let
$v\in{\mathbb{X}}$. In the next section we shall establish and exploit a
useful explicit change of measure for the distribution of
$C^{v}=C^{v}(\xi^{v})$. This is possible since for $n\in{\mathbb{N}}_{0}$ the
conditional distribution of $C^{v}_{n+1}$ given $C^{v}_{\leq n}$ can be
described in terms of a RCM driven by Poisson process with a thinned intensity
measure. In this section we derive a general version of this spatial Markov
property.

Let $\nu$ be a locally finite and diffuse measure on ${\mathbb{X}}$. Then we
denote by $\Pi_{\nu}$ the distribution of a Poisson process with this
intensity measure. We define a kernel $K_{\nu}$ from
${\mathbf{N}}\times{\mathbf{N}}$ to ${\mathbb{X}}$, by

$\displaystyle
K_{\nu}(\mu,\mu^{\prime},dx):=\bar{\varphi}(\mu,x)\varphi(\mu^{\prime},x)\nu(dx),$
(7.1)

where we recall the definitions (3.7). Proposition 7.2 will provide an
interpretation of this kernel. Denoting by $0$ the zero measure, we note that

$\displaystyle K_{\nu}(0,\mu^{\prime},dx)=\varphi(\mu^{\prime},x)\nu(dx),\quad
K_{\nu}(\mu,0,dx)=\bar{\varphi}(\mu,x)\nu(dx).$ (7.2)

We write $K_{\nu}(\mu,\mu^{\prime}):=K_{\nu}(\mu,\mu^{\prime},\cdot)$. Note
that $K_{\lambda}(0,\mu,{\mathbb{X}})=\varphi_{\lambda}(\mu)$; see (3.7).

For $n\in{\mathbb{N}}_{0}$, $\mu\in{\mathbf{G}}$ and $v\in{\mathbb{X}}$ let
$\Gamma^{v}_{n}(\lambda,\mu,\cdot)$ denote the distribution of a random graph
$\xi_{n}$ defined as follows. Let $\xi_{n}^{\prime}$ be a RCM based on
$\eta_{n}+C^{v}_{n}(\mu)$, where $\eta_{n}$ is a Poisson process with
intensity measure $K_{\lambda}(C^{v}_{\leq{n-1}}(\mu),0)$, and where we recall
that $C^{v}_{\leq{-1}}:=0$. Remove in $\xi_{n}^{\prime}$ all edges between
vertices from $C^{v}_{n}(\mu)$ to obtain a random graph
$\xi_{n}^{\prime\prime}$. Finally set $\xi_{n}:=C^{v}_{\leq
n}(\mu)\oplus\xi_{n}^{\prime\prime}$, with an obvious definition of the
operation $\oplus$. We set $C^{v}_{\leq 0}(\mu):=\delta_{v}$, which is the
graph with vertex set $\\{v\\}$ and no edges.

######  Theorem 7.1.

Let $v\in{\mathbb{X}}$ and $n\in{\mathbb{N}}_{0}$. Then,

$\displaystyle\mathbb{P}(\xi^{v}\in\cdot\mid C^{v}_{\leq
n})=\Gamma^{v}_{n}(\lambda,C^{v}_{\leq n},\cdot),\quad\mathbb{P}\text{-a.s.}$
(7.3)

###### Proof.

This follows from the proof of [20, Lemma 3.3]; see also Proposition 2 in
[29]. Essentially the assertion is equivalent to equation (3.6) in this proof.
The arguments given there apply to a RCM on a general state space
${\mathbb{X}}$ and not only to ${\mathbb{R}}^{d}$. ∎

A quick consequence of Theorem 7.1 is that
$\\{(V(C^{v}_{\leq{n-1}}),V(C^{v}_{n}))\\}_{n\in{\mathbb{N}}_{0}}$ is a Markov
process.

######  Proposition 7.2.

The sequence
$\\{(V(C^{v}_{\leq{n-1}}),V(C^{v}_{n}))\\}_{n\in{\mathbb{N}}_{0}}$ is a Markov
process with transition kernel

$\displaystyle(\mu,\mu^{\prime})\mapsto\int{\mathbf{1}}\\{(\mu+\mu^{\prime},\psi)\in\cdot\\}\,\Pi_{K_{\lambda}(\mu,\mu^{\prime})}(d\psi).$

We also note that

$\displaystyle
K_{\lambda}(\mu,\mu^{\prime},{\mathbb{X}})\leq\int\varphi(\mu^{\prime},x)\,\lambda(dx)\leq\iint\varphi(y,x)\,\mu^{\prime}(dy)\,\lambda(dx),$

where we have used the Bernoulli inequality. Hence

$\displaystyle K_{\lambda}(\mu,\mu^{\prime},{\mathbb{X}})\leq\int
D_{\varphi}(y)\,\mu^{\prime}(dy).$ (7.4)

######  Corollary 7.3.

Let $n\in{\mathbb{N}}_{0}$. Then we have for $\lambda$-a.e. $v\in{\mathbb{X}}$
that $\mathbb{P}(|C^{v}_{n}|<\infty)=1$.

###### Proof.

We can proceed by induction. For $n=0$ the assertion is trivial. Assume that
$\mathbb{P}(|C^{v}_{n}|<\infty)=1$ for some $n\in{\mathbb{N}}_{0}$. From
Proposition 7.2 we know that the conditional distribution of $V(C^{v}_{n+1})$
given $(V(C^{v}_{\leq{n-1}}),V(C_{n}^{v}))$ is that of a Poisson process with
intensity measure $K_{\lambda}(V(C^{v}_{\leq{n-1}}),V(C_{n}^{v}))$. By (7.4)
we obtain that

$\displaystyle\mathbb{E}[|C^{v}_{n+1}|\mid(V(C^{v}_{\leq{n-1}}),V(C_{n}^{v}))]\leq\int
D_{\varphi}(y)\,C_{n}^{v}(dy)$

which is for $\lambda$-a.e. $v\in{\mathbb{X}}$ a.s. finite by our general
assumption (1.1) and induction hypothesis. ∎

The following useful property of the kernel $K_{\lambda}$ can easily be proved
by induction.

######  Lemma 7.4.

Let $n\in{\mathbb{N}}$ and $\mu_{0},\ldots,\mu_{n}\in{\mathbf{N}}$. Then

$\displaystyle
K_{\lambda}(0,\mu_{0})+K_{\lambda}(\mu_{0},\mu_{1})+\cdots+K_{\lambda}(\mu_{0}+\cdots+\mu_{n-1},\mu_{n})=K_{\lambda}(0,\mu_{0}+\cdots+\mu_{n}).$

##  8 Perturbation formulas

In the next sections we vary the intensity measure $\lambda$ and consider
$t\lambda$ for $t\in{\mathbb{R}}_{+}$. We fix $v\in{\mathbb{X}}$ and let
$\mathbb{P}_{t}$ be a probability measure governing a RCM $\xi$ based on
$\eta$, where $\eta$ is a Poisson process with intensity measure $t\lambda$.
The associated expectation is denoted by $\mathbb{E}_{t}$. Recall the
definition (3.7).

######  Lemma 8.1.

Let $\tilde{\xi}$ be a RCM based on a Poisson process $\tilde{\eta}$ with
finite intensity measure $\nu$. Let $f\colon{\mathbf{G}}\to[0,\infty)$. Then

$\displaystyle\mathbb{E}_{t}f(\tilde{\xi})=\mathbb{E}_{1}f(\tilde{\xi})t^{|\tilde{\eta}|}e^{(1-t)\nu({\mathbb{X}})}$

###### Proof.

It is well-known that

$\displaystyle\Pi_{t\nu}=\int{\mathbf{1}}\\{\mu\in\cdot\\}t^{|\mu|}e^{(1-t)\nu({\mathbb{X}})}\,\Pi_{\nu}(d\mu),\quad
t\geq 0.$ (8.1)

This follows, for instance from [25, Exercise 3.7] and an easy calculation.
The assertion then follows by conditioning, using the kernel $\Gamma$ in
(3.2). ∎

######  Proposition 8.2.

Let $v\in{\mathbb{X}}$, $t\in{\mathbb{R}}_{+}$, $n\in{\mathbb{N}}$ and
$t_{0}>0$. Then

$\displaystyle\mathbb{P}_{t}(C^{v}_{\leq
n}\in\cdot)=\mathbb{E}_{t_{0}}{\mathbf{1}}\\{C^{v}_{\leq
n}\in\cdot\\}(t/t_{0})^{\big{|}C^{v}_{\leq
n}\big{|}-1}e^{(t_{0}-t)\varphi_{\lambda}\big{(}C^{v}_{\leq{n-1}}\big{)}}.$

###### Proof.

It is sufficient to consider the special case $t_{0}=1$. The general case can
be proved similarly or can be derived from the special case. We omit the
dependence on $v$ in our notation by writing $C_{n}:=C^{v}_{n}$, and $C_{\leq
n}:=C^{v}_{\leq n}$. Given $\mu\in{\mathbf{G}}$ we let $C_{n}^{+}(\mu)$ denote
the graph $\mu[V(C_{n-1}(\mu))+V(C_{n}(\mu))]$ with the edges between vertices
of $C_{n-1}(\mu)$ removed.

Let $f\colon{\mathbf{G}}\to[0,\infty)$ be measurable. By Theorem 7.1,

$\displaystyle\mathbb{E}_{t}f(C_{\leq n})=\mathbb{E}_{t}\int
f(C_{\leq{n-1}}\oplus
C_{n}^{+}(\mu))\,\Gamma_{n-1}(t\lambda,C_{\leq{n-1}},d\mu).$

By (7.4) we have

$\displaystyle K_{t\lambda}(C_{\leq{n-2}},C_{n-1},{\mathbb{X}})\leq t\int
D_{\varphi}(y)\,C_{n-1}(dy)$

which is almost surely finite by Corollary 7.3. By definition of $\Gamma_{n}$
and the thinning properties of a Poisson process, the distribution of
$C^{+}_{n}(\cdot)$ under $\Gamma_{n-1}(t\lambda,C_{\leq{n-1}},\cdot)$ is that
of a RCM driven by a Poisson process with intensity measure
$K_{t\lambda}(C_{\leq{n-2}},C_{n-1})$ with additional independent connections
to $V(C_{n-1})$; see also Proposition 7.2. Therefore we obtain from Lemma 8.1
that

$\displaystyle\mathbb{E}_{t}f(C_{\leq n})=\mathbb{E}_{t}\int
f(C_{\leq{n-1}}\oplus
C_{n}^{+}(\mu))e^{(1-t)K_{\lambda}(C_{\leq{n-2}},C_{n-1},{\mathbb{X}})}t^{|\mu|}\,\Gamma_{n-1}(\lambda,C_{\leq{n-1}},d\mu).$

Iterating this identity yields that the above equals

$\displaystyle\idotsint$ $\displaystyle f(C_{1}^{+}(\mu_{1})\oplus\cdots\oplus
C_{n}^{+}(\mu_{n}))e^{(1-t)K_{\lambda}(\delta_{v}+\mu_{1}+\cdots+\mu_{n-2},\mu_{n-1},{\mathbb{X}})}\cdots
e^{(1-t)K_{\lambda}(\delta_{v},\mu_{1},{\mathbb{X}})}t^{|\mu_{n}|}\cdots
t^{|\mu_{1}|}$
$\displaystyle\Gamma_{n-1}(\lambda,C_{1}^{+}(\mu_{1})\oplus\cdots\oplus
C_{n-1}^{+}(\mu_{n-1}),d\mu_{n})\cdots\Gamma_{0}(\lambda,\delta_{v},d\mu_{1}).$

By Lemma 7.4 this equals

$\displaystyle\idotsint$ $\displaystyle f(C_{1}^{+}(\mu_{1})\oplus\cdots\oplus
C_{n}^{+}(\mu_{n}))e^{(1-t)K_{\lambda}(0,\delta_{v}+\mu_{1}+\cdots+\mu_{n-1},{\mathbb{X}})}t^{|\mu_{1}|+\cdots+|\mu_{n}|}$
$\displaystyle\Gamma_{n-1}(\lambda,C_{1}^{+}(\mu_{1})\oplus\cdots\oplus
C_{n-1}^{+}(\mu_{n-1}),d\mu_{n})\cdots\Gamma_{0}(\lambda,[\delta_{v}],d\mu_{1}).$

By Theorem 7.1 we obtain

$\displaystyle\mathbb{E}_{t}f(C_{\leq n})=\mathbb{E}_{1}f(C_{\leq
n})t^{|C_{\leq n}|-1}e^{(1-t)\varphi_{\lambda}(C_{\leq{n-1}})}$

and hence the assertion. ∎

######  Theorem 8.3.

Let $v\in{\mathbb{X}}$, $t\in{\mathbb{R}}_{+}$ and $t_{0}>0$. Then

$\displaystyle\mathbb{P}_{t}(C^{v}\in\cdot,|C^{v}|<\infty)=\mathbb{E}_{t_{0}}{\mathbf{1}}\\{C^{v}\in\cdot,|C^{v}|<\infty\\}(t/t_{0})^{|C^{v}|-1}e^{(t_{0}-t)\varphi_{\lambda}(C^{v})}.$
(8.2)

###### Proof.

Again it is sufficient to consider the special case $t_{0}=1$. By Proposition
8.2 the distribution $\mathbb{P}_{t}(C^{v}_{n}\in\cdot)$ is absolutely
continuous w.r.t. $\mathbb{P}_{1}(C^{v}_{n}\in\cdot)$ with Radon–Nikodym
derivative $M^{v}_{n}:=t^{|C^{v}_{\leq
n}|-1}e^{(1-t)\varphi_{\lambda}(C^{v}_{\leq{n-1}})}$. In particular
$\\{M^{v}_{n}\\}_{n\in{\mathbb{N}}_{0}}$ is a (non-negative) martingale with
respect to $\\{\sigma(C^{v}_{\leq n})\\}_{n\in{\mathbb{N}}_{0}}$ and converges
therefore a.s. towards $M^{v}_{\infty}:=\limsup_{n\to\infty}M^{v}_{n}$. By
[37, Theorem VII.6.1] we have

$\displaystyle\mathbb{P}_{t}(C^{v}\in\cdot)=\mathbb{E}_{1}{\mathbf{1}}\\{C^{v}\in\cdot\\}M^{v}_{\infty}+\mathbb{E}_{t}{\mathbf{1}}\\{C^{v}\in\cdot,M^{v}_{\infty}=\infty\\}.$

On the event $\\{|C^{v}|<\infty\\}$ we clearly have

$\displaystyle M^{v}_{\infty}=t^{|C^{v}|-1}e^{(1-t)\varphi_{\lambda}(C^{v})}$

which is finite. This concludes the assertion. ∎

Let $f\colon{\mathbf{G}}\to{\mathbb{R}}$ be a measurable mapping. Define for
$v\in{\mathbb{X}}$, $n\in{\mathbb{N}}$, and $t\in{\mathbb{R}}_{+}$

$\displaystyle F_{n}(\xi^{v}):=f(C^{v}){\mathbf{1}}\\{|C^{v}|=n\\},\ $
$\displaystyle\ f^{v}_{n}(t):=\mathbb{E}_{t}F_{n}(\xi^{v}),$ (8.3)
$\displaystyle F_{\leq n}(\xi^{v}):=f(C^{v}){\mathbf{1}}\\{|C^{v}|\leq n\\},\
$ $\displaystyle\ f^{v}_{\leq n}(t):=\mathbb{E}_{t}F_{\leq n}(\xi^{v}),$ (8.4)
$\displaystyle F(\xi^{v}):=f(C^{v}){\mathbf{1}}\\{|C^{v}|<\infty\\},\ $
$\displaystyle\ f^{v}(t):=\mathbb{E}_{t}F(\xi^{v}).$ (8.5)

We also write $|f|^{v}(t):=\mathbb{E}_{t}|F(\xi^{v})|$ and define
$|f|^{v}_{n}(t)$ and $|f|^{v}_{\leq n}(t)$ similarly. We are interested in the
analytic properties of the function $f^{v}(t)$ under the assumption
$|f|^{v}(t)<\infty$. A key example is the position dependent cluster density

$\displaystyle\kappa^{v}(t):=\mathbb{E}_{t}|C^{v}|^{-1},\quad
t\in{\mathbb{R}}_{+}.$ (8.6)

Our terminology is motivated by the stationary marked case (see Lemma 4.3) and
also supported by the Mecke equation, implying

$\displaystyle\int
t\kappa^{v}(t)\,\lambda_{B}(dv)=\mathbb{E}_{t}\int|C^{v}(\xi)|^{-1}\,\eta_{B}(dv),\quad
B\in{\mathcal{X}}.$

Suppose that $|f|^{v}_{n}(t_{0})<\infty$ for some $t_{0}>0$ and
$n\in{\mathbb{N}}$, then by Theorem 8.3

$\displaystyle
f^{v}_{n}(t)=\left(\frac{t}{t_{0}}\right)^{n-1}\int_{0}^{\infty}e^{-tu}\,\nu_{f,_{n},t_{0}}(du),$
(8.7)

where the signed measure $\nu_{f,n,t_{0}}$ is defined by

$\displaystyle\nu_{f,n,t_{0}}(\cdot):=\mathbb{E}_{t_{0}}{\mathbf{1}}\\{\varphi_{\lambda}(C^{v})\in\cdot\\}e^{t_{0}\varphi_{\lambda}(C^{v})}F_{n}(\xi^{v}).$
(8.8)

By Corollary 7.3 this is a locally finite signed measure on
${\mathbb{R}}_{+}$. It follows from Theorem 8.3 that the function
$|f|^{v}_{n}(t)/t^{n-1}$ is monotone decreasing on $(0,\infty)$, so that

$\displaystyle|f|^{v}_{n}(t)\leq\left(\frac{t}{t_{0}}\right)^{n-1}|f|^{v}_{n}(t_{0}),\quad
t\geq t_{0}.$

######  Lemma 8.4.

Let $v\in{\mathbb{X}}$, $n\in{\mathbb{N}}$ and $t_{0}>0$. If
$|f|^{v}_{n}(t_{0})<\infty$, then for $t\geq t_{0}$

$\displaystyle
f_{n}^{v}(t)=\frac{t^{n}}{t_{0}^{n-1}}\int_{0}^{\infty}\nu_{f,n,t_{0}}[0,u]e^{-tu}\,du.$

###### Proof.

We obtain from (8.7) that

$\displaystyle f^{v}_{n}(t)=\frac{t^{n}}{t_{0}^{n-1}}\iint{\mathbf{1}}\\{u\leq
s\\}e^{-ts}\,ds\,\nu_{f,n,t_{0}}(du).$ (8.9)

Since $\nu_{f,n,t_{0}}$ is locally finite, we can apply Fubini’s theorem to
obtain the assertion. ∎

######  Lemma 8.5.

Let $v\in{\mathbb{X}}$, $n\in{\mathbb{N}}$ and $t_{0}>0$. If
$|f|^{v}_{n}(t_{0})<\infty$, then the function $f^{v}_{n}$ is analytic on
$(t_{0},\infty)$ and for $t>t_{0}$

$\displaystyle\frac{d}{dt}f_{n}^{v}(t)=\frac{nt^{n-1}}{t_{0}^{n-1}}\int_{0}^{\infty}\nu_{f,n,t_{0}}[0,u]e^{-tu}\,du-\frac{t^{n}}{t_{0}^{n-1}}\int_{0}^{\infty}u\nu_{f,n,t_{0}}[0,u]e^{-tu}\,du.$
(8.10)

###### Proof.

Let $\Omega_{t_{0}}:=\\{z\in\mathbb{C}:\Re(z)>t_{0}\\}$, and extend
$f^{v}_{n}$ to $\Omega_{t_{0}}$ by setting

$\displaystyle
f^{v}_{n}(z):=\frac{z^{n}}{t_{0}^{n-1}}\int_{0}^{\infty}\nu_{f,n,t_{0}}[0,u]e^{-zu}\,du,\quad
z\in\Omega_{t_{0}}.$

By (8.8) we have

$\displaystyle|\nu_{f,n,t_{0}}[0,u]|\leq e^{t_{0}u}|f|^{v}_{n}(t_{0}).$

Since $|f|^{v}_{n}(t_{0})<\infty$ this implies that $f^{v}_{n}$ is a complex
analytic function on $\Omega_{t_{0}}$. Since $(t_{0},\infty)\subset$
$\Omega_{t_{0}}\cap{\mathbb{R}}$, the restriction of this function to
$(t_{0},\infty)$ is real analytic. The formula (8.10) follows from Lemma 8.4
the product rule of calculus and the Leibniz rule for differentiating
integrals. The latter can be applied since for each $\varepsilon>0$ and all
$u>0$

$\displaystyle u|\nu_{f,n,t_{0}}[0,u]|e^{-tu}\leq
ue^{(t_{0}-t)u}|f|^{v}_{n}(t_{0})\leq ue^{-\varepsilon u}|f|^{v}_{n}(t_{0})$

uniformly for $t\geq t_{0}+\varepsilon$. ∎

To rewrite Lemma 8.5 in a different way, we define

$\displaystyle M^{v}_{t}:=|C^{v}|-1-t\varphi_{\lambda}(C^{v}),\quad
t\in{\mathbb{R}}_{+},\,v\in{\mathbb{X}}.$ (8.11)

######  Lemma 8.6.

Let $v\in{\mathbb{X}}$, $n\in{\mathbb{N}}$ and $t_{0}>0$. If
$|f|^{v}_{n}(t_{0})<\infty$, then function $f^{v}_{n}$ is analytic on
$(t_{0},\infty)$ and for $t>t_{0}$

$\displaystyle\frac{d}{dt}f_{n}^{v}(t)=t^{-1}\mathbb{E}_{t}\big{[}M^{v}_{t}F_{n}(\xi^{v})\big{]}.$
(8.12)

###### Proof.

By Theorem 8.3,

$\displaystyle
f_{n}^{v}(t)=\left(\frac{t}{t_{0}}\right)^{n-1}\mathbb{E}_{t_{0}}\big{[}F_{n}(\xi^{v})e^{(t_{0}-t)\varphi_{\lambda}(C^{v})}\big{]}.$

Hence the result follows from Lemma 8.5 and calculus, where the application of
the Leibniz differentiation rule can be justified as in the proof of Lemma
8.5. ∎

######  Lemma 8.7.

Let $v\in{\mathbb{X}}$, $n\in{\mathbb{N}}$ and $t_{0}>0$. If $|f|^{v}_{\leq
n}(t_{0})<\infty$, then function $f^{v}_{\leq n}$ is analytic on
$(t_{0},\infty)$ and for $t>t_{0}$

$\displaystyle\frac{d}{dt}f_{\leq
n}^{v}(t)=t^{-1}\mathbb{E}_{t}\big{[}M^{v}_{t}F_{\leq n}(\xi^{v})\big{]}.$
(8.13)

###### Proof.

The result follows from the definition of $f^{v}_{\leq n}$ and Lemma 8.6,
since $|f|^{v}_{\leq n}(t_{0})=\sum\nolimits_{k=1}^{n}|f|^{v}_{k}(t_{0})$. ∎

######  Theorem 8.8.

Let $0<t_{0}<t_{1}<\infty$. Assume for each $t\in[t_{0},t_{1}]$ that
$|f|^{v}(t)<\infty$. Assume moreover that for each $\varepsilon>0$,

$\displaystyle\lim_{n\to\infty}\sup_{t\in[t_{0}+\varepsilon,t_{1}]}\left|\sum_{k>n}\frac{d}{dt}f_{k}^{v}(t)\right|=0.$
(8.14)

Then $f^{v}$ is continuously differentiable on $(t_{0},t_{1}]$ with derivative
given by

$\displaystyle\frac{d}{dt}f^{v}(t)=\lim_{n\to\infty}t^{-1}\frac{d}{dt}f^{v}_{\leq
n}(t)=t^{-1}\sum_{n=1}^{\infty}\frac{d}{dt}f^{v}_{n}(t).$ (8.15)

###### Proof.

Let $n\in{\mathbb{N}}$. Since $|f|^{v}_{\leq
n}(t_{0})\leq|f|^{v}(t_{0})<\infty$, we can apply Lemma 8.7 to obtain that the
function $f^{v}_{\leq n}$ is analytic on $(t_{0},\infty)$, with derivative

$\displaystyle\frac{d}{dt}f_{\leq
n}^{v}(t)=t^{-1}\sum_{k=1}^{n}\frac{d}{dt}f_{n}^{v}(t)=t^{-1}\mathbb{E}_{t}\big{[}M^{v}_{t}F_{\leq
n}(\xi^{v})\big{]},\quad t>t_{0}.$

By dominated convergence

$\displaystyle\lim_{n\to\infty}f^{v}_{\leq n}(t)=f^{v}(t).$

Furthermore we have

$\displaystyle\left|\frac{d}{dt}f^{v}(t)-\frac{d}{dt}f_{\leq
n}^{v}(t)\right|=t^{-1}\left|\sum_{k>n}\frac{d}{dt}f^{v}_{k}(t)\right|.$

By assumption (8.14) this tends to zero uniformly in
$t\in[t_{0}+\varepsilon,t_{1}]$ for each $\varepsilon>0$. A standard result of
analysis gives us that $f^{v}$ is continuously differentiable on
$(t_{0},t_{1}]$ with derivative given by the right-hand side of (8.15). ∎

######  Theorem 8.9.

Let $0<t_{0}<t_{1}<\infty$. Assume for each $t\in[t_{0},t_{1}]$ that
$|f|^{v}(t)<\infty$. Assume moreover that for each $\varepsilon>0$,

$\displaystyle\lim_{n\to\infty}\sup_{t\in[t_{0}+\varepsilon,t_{1}]}\sum_{k>n}\mathbb{E}_{t}\big{|}M^{v}_{t}F_{k}(\xi^{v})\big{|}=0.$
(8.16)

Then $f^{v}$ is continuously differentiable on $(t_{0},t_{1}]$ with derivative
given by

$\displaystyle\frac{d}{dt}f^{v}(t)=t^{-1}\mathbb{E}_{t}\big{[}M^{v}_{t}F(\xi^{v})\big{]}.$
(8.17)

###### Proof.

Let $n\in{\mathbb{N}}$. Since $|f|^{v}_{n}(t_{0})\leq|f|^{v}_{\leq
n}(t_{0})\leq|f|^{v}(t_{0})<\infty$, then by Lemma 8.5 the function
$f^{v}_{n}$ is analytic on $(t_{0},\infty)$, with derivative

$\displaystyle\left|\frac{d}{dt}f_{n}^{v}(t)\right|=t^{-1}\left|\mathbb{E}_{t}\big{[}M^{v}_{t}F_{n}(\xi^{v})\big{]}\right|\leq
t^{-1}\mathbb{E}_{t}\big{|}M^{v}_{t}F_{n}(\xi^{v})\big{|},\quad t>t_{0}.$

Hence

$\displaystyle\left|\sum_{k>n}\frac{d}{dt}f_{k}^{v}(t)\right|\leq
t^{-1}\sum_{k>n}\mathbb{E}_{t}\big{|}M^{v}_{t}F_{n}(\xi^{v})\big{|}.$

By assumption (8.16) this tends to zero uniformly in
$t\in[t_{0}+\varepsilon,t_{1}]$ for each $\varepsilon>0$. Therefore by Theorem
8.8 the function $f^{v}(t)$ is continuously differentiable on $(t_{0},t_{1}]$
with derivative

$\displaystyle\frac{d}{dt}f^{v}(t)=t^{-1}\sum_{n=1}^{\infty}\frac{d}{dt}f^{v}_{n}(t)=t^{-1}\mathbb{E}_{t}\big{[}M^{v}_{t}F(\xi^{v})\big{]},$

where the last equality we get from Fubini’s theorem, since by assumption
(8.16) we have that for $t\in(t_{0},t_{1}]$

$\displaystyle\mathbb{E}_{t}\big{|}M^{v}_{t}F(\xi^{v})\big{|}=\sum_{n\geq
1}\mathbb{E}_{t}\big{|}M^{v}_{t}F_{n}(\xi^{v})\big{|}<\infty.$ ∎

The following theorem provides a large class of functions satisfying the
assumptions of Theorem 8.9, covering the cluster density (8.6). We shall prove
it in Section 10.

######  Theorem 8.10.

Let $f\colon{\mathbf{G}}\to{\mathbb{R}}$ be a measurable mapping satisfying
$|f(\mu)|\leq|\tilde{f}(|V(\mu)|)|$ for each $\mu\in{\mathbf{G}}$, where
$\tilde{f}:{\mathbb{N}}\to{\mathbb{R}}$ satisfies

$\displaystyle\lim_{n\to\infty}\tilde{f}(n)\sqrt{n\log n}=0.$ (8.18)

Then $f^{v}$ is for each $v\in{\mathbb{X}}$ continuously differentiable on
$(0,\infty)$ with derivative given by (8.17).

##  9 Difference operators

In this section we shall rewrite Theorem 8.8 and Theorem 8.9 in the form of a
Margulis–Russo formula. Recall that $\mu-\delta_{x}:=\mu[V(\mu)-\delta_{x}]$
is the graph resulting from $\mu$ by removing the point $x$ (if $x\in V(\mu)$)
along with all edges with vertex $x$ for $\mu\in{\mathbf{G}}$ and
$x\in{\mathbb{X}}$. Given a measurable function
$f\colon{\mathbf{G}}\to{\mathbb{R}}$ and $x\in{\mathbb{X}}$ we define
$\nabla_{x}f\colon{\mathbf{G}}\to{\mathbb{R}}$ by

$\displaystyle\nabla_{x}f(\mu):=f(\mu)-f(\mu-\delta_{x}).$ (9.1)

######  Theorem 9.1.

Let the assumptions of the Theorem 8.8 be satisfied. Then the function $f^{v}$
is continuously differentiable on $(t_{0},t_{1}]$ with derivative given by

$\displaystyle\frac{d}{dt}f^{v}(t)=\lim_{n\to\infty}t^{-1}\mathbb{E}_{t}\int\nabla_{x}F_{\leq
n}(\xi^{v})\,C^{v!}(dx)=\sum_{n\geq
1}t^{-1}\mathbb{E}_{t}\int\nabla_{x}F_{n}(\xi^{v})\,C^{v!}(dx).$ (9.2)

We start the proof with the counterpart of Lemma 8.5 and Lemma 8.7.

######  Lemma 9.2.

Let $v\in{\mathbb{X}}$, $n\in{\mathbb{N}}$ and $t_{0}>0$. If
$|f|^{v}_{n}(t_{0})<\infty$, then function $f^{v}_{n}$ is analytic on
$(t_{0},\infty)$ and for $t>t_{0}$

$\displaystyle\frac{d}{dt}f_{n}^{v}(t)=t^{-1}\mathbb{E}_{t}\int\nabla_{x}F_{n}(\xi^{v})\,C^{v!}(dx).$

###### Proof.

Let $t>t_{0}$. We wish to apply Lemma 8.6. By definition we have

$\displaystyle\mathbb{E}_{t}|F_{n}(\xi^{v})||C^{v}|=n|f|^{v}_{n}(t)\leq
n\left(\frac{t}{t_{0}}\right)^{n-1}|f|^{v}_{n}(t_{0})$

which is finite by assumption. Therefore we obtain from Theorem 8.3, (8.8)
(with $|f|$ instead of $f$) and Fubini’s theorem

$\displaystyle\mathbb{E}_{t}|F_{n}(\xi^{v})|\varphi_{\lambda}(C^{v})=$
$\displaystyle\left(\frac{t}{t_{0}}\right)^{n-1}\int_{0}^{\infty}ue^{-tu}\,\nu_{|f|,n,t_{0}}(du)=\left(\frac{t}{t_{0}}\right)^{n-1}\int_{0}^{\infty}\int_{u}^{\infty}(ts-1)e^{-ts}\,ds\,\nu_{|f|,n,t_{0}}(du)$
$\displaystyle=$
$\displaystyle\left(\frac{t}{t_{0}}\right)^{n-1}\int_{0}^{\infty}\nu_{|f|,n,t_{0}}[0,s](ts-1)e^{-ts}\,ds$
$\displaystyle\leq$
$\displaystyle\,|f|^{v}_{n}(t_{0})\frac{t^{n}}{t_{0}^{n-1}}\int_{0}^{\infty}se^{(t_{0}-t)s}\,ds<\infty,$

where we have used that $\nu_{|f|,n,t_{0}}[0,s]\leq
e^{t_{0}s}|f|^{v}_{n}(t_{0})$. Hence we obtain from Lemma 3.1 that

$\displaystyle
t\,\mathbb{E}_{t}F_{n}(\xi^{v})\varphi_{\lambda}(C^{v})=\mathbb{E}_{t}\int
F_{n}(\xi^{v}-\delta_{x})\,C^{v!}(dx).$

Now the assertion follows from Lemma 8.6. ∎

######  Lemma 9.3.

Let $v\in{\mathbb{X}}$, $n\in{\mathbb{N}}$ and $t_{0}>0$. If $|f|^{v}_{\leq
n}(t_{0})<\infty$, then function $f^{v}_{\leq n}$ is analytic on
$(t_{0},\infty)$ and for $t>t_{0}$

$\displaystyle\frac{d}{dt}f_{\leq
n}^{v}(t)=t^{-1}\mathbb{E}_{t}\int\nabla_{x}F_{\leq n}(\xi^{v})\,C^{v!}(dx).$

###### Proof.

The result follows from the definition of $f^{v}_{\leq n}$, Lemma 8.7 and
Lemma 9.2. ∎

###### Proof of Theorem 9.1.

By Theorem 8.8 we have that $f^{v}$ is continuously differentiable on
$(t_{0},t_{1}]$ with derivative given by (8.15). Hence we can apply Lemma 9.2
and Lemma 9.3 to obtain the assertion. ∎

######  Theorem 9.4.

Let the assumptions of the Theorem 8.9 be satisfied. Assume moreover that for
each $t\in[t_{0},t_{1}]$

$\displaystyle\mathbb{E}_{t}\big{[}|F(\xi^{v})|\left(|C^{v}|+\varphi_{\lambda}(C^{v})\right)\big{]}<\infty.$
(9.3)

Then $f^{v}$ is continuously differentiable on $(t_{0},t_{1}]$ with derivative
given by

$\displaystyle\frac{d}{dt}f^{v}(t)=t^{-1}\mathbb{E}_{t}\int\nabla_{x}F(\xi^{v})\,C^{v!}(dx).$
(9.4)

###### Proof.

Let $t>t_{0}$. Theorem 8.9 states that $f^{v}$ is continuously differentiable
on $(t_{0},t_{1}]$ with derivative given by (8.17). The assertion follows from
(9.3) and Lemma 3.1, since splitting $f$ into its negative and positive part
we can apply Lemma 3.1 to get

$\displaystyle
t\,\mathbb{E}_{t}F(\xi^{v})\varphi_{\lambda}(C^{v})=\mathbb{E}_{t}\int
F(\xi^{v}-\delta_{x})\,C^{v!}(dx).$

The result follows. ∎

######  Remark 9.5.

Let the assumptions of Theorem 9.4 be satisfied. By the Mecke equation (3.6)
we have

$\displaystyle\mathbb{E}_{t}\int\nabla_{x}F(\xi^{v})\,C^{v!}(dx)=t\,\mathbb{E}_{t}\int(F(\xi^{v,x})-F(\xi^{v})){\mathbf{1}}\\{\text{$v\leftrightarrow
x$ in $\xi^{v,x}$}\\}\,\lambda(dx).$

If $v$ and $x$ are not connected in $\xi^{v,x}$, then
$F(\xi^{v,x})=F(\xi^{v})$. Therefore we can rewrite (9.4) as

$\displaystyle\frac{d}{dt}f^{v}(t)=\mathbb{E}_{t}\int(F(\xi^{v,x})-F(\xi^{v}))\,\lambda(dx).$
(9.5)

##  10 Differentiability of the cluster density

In this section we prove in particular that the position dependent cluster
density (given by (8.6)) is continuously differentiable on $(0,\infty)$.

######  Theorem 10.1.

Suppose that $f\colon{\mathbb{N}}\to{\mathbb{R}}$ is a function satisfying

$\displaystyle\lim_{n\to\infty}f(n)\sqrt{n\log n}=0.$ (10.1)

Then $t\mapsto\mathbb{E}_{t}f(|C^{v}|)$ is for each $v\in{\mathbb{X}}$
continuously differentiable on $(0,\infty)$ with derivative given by (8.17).

We prove the theorem via some lemmas, partially following the proof of [6,
(LP) (3.6)]. Let $v\in{\mathbb{X}}$. For $t>0$ and $n\in{\mathbb{N}}$ we
define

$\displaystyle p^{v}_{n}(t):=\mathbb{P}_{t}(|C^{v}|=n).$

Specializing definition (8.8) in the case $f\equiv 1$ we set

$\displaystyle\nu^{v}_{n}(\cdot):=\mathbb{E}_{1}{\mathbf{1}}\\{\varphi_{\lambda}(C^{v})\in\cdot\\}{\mathbf{1}}\\{|C^{v}|=n\\}e^{\varphi_{\lambda}(C^{v})}.$
(10.2)

Then we obtain from (8.7) in the case $t_{0}=1$ that

$\displaystyle p^{v}_{n}(t)=t^{n-1}\int_{0}^{\infty}e^{-tu}\,\nu^{v}_{n}(du).$
(10.3)

Since $p^{v}_{1}(t)=e^{-tD_{\varphi}(v)}$ we have

$\displaystyle\nu^{v}_{1}=\delta_{D_{\varphi}(v)},\quad v\in{\mathbb{X}}.$
(10.4)

######  Lemma 10.2.

Let $v\in{\mathbb{X}}$, $n\in{\mathbb{N}}$ and $u>0$. Then

$\displaystyle\nu^{v}_{n}[0,u]\leq\left(\frac{eu}{n-1}\right)^{n-1},$ (10.5)

where the right-hand side has to be interpreted as $1$ if $n=1$.

###### Proof.

In view of (10.4) we can assume that $n\geq 2$. Since $p^{v}_{n}(t)\leq 1$ for
$t>0$, we have that

$\displaystyle t^{-n+1}\geq\int_{0}^{u}e^{-tu}\nu^{v}_{n}(du)\geq
e^{-tu}\nu^{v}_{n}[0,u).$

Optimizing over $t\in(0,\infty)$ yields the assertion. ∎

######  Lemma 10.3.

Let $v\in{\mathbb{X}}$, $n\in{\mathbb{N}}$ and $t>0$. Then

$\displaystyle
p_{n}^{v}(t)=t^{n}\int_{0}^{\infty}\nu^{v}_{n}[0,u]e^{-tu}\,du.$

###### Proof.

The assertion follows from Lemma 8.4. ∎

######  Lemma 10.4.

Let $v\in{\mathbb{X}}$, $n\in{\mathbb{N}}$. Then $t\mapsto p_{n}^{v}(t)$ is
analytic on $(0,\infty)$ with derivative given by

$\displaystyle\frac{d}{dt}p_{n}^{v}(t)=nt^{n-1}\int_{0}^{\infty}\nu^{v}_{n}[0,u]e^{-tu}\,du-t^{n}\int_{0}^{\infty}u\nu^{v}_{n}[0,u]e^{-tu}\,du$
(10.6)

###### Proof.

The assertion follows from Lemma 8.5. ∎

Lemma 10.4 implies

$\displaystyle\Big{|}\frac{d}{dt}p_{n}^{v}(t)\Big{|}\leq\frac{n}{t}\int_{0}^{\infty}\nu^{v}_{n}[0,u]\left|1-\frac{ut}{n}\right|t^{n}e^{-tu}\,du.$
(10.7)

The next lemma provides a bound for the above right-hand side.

######  Lemma 10.5.

Let $v\in{\mathbb{X}}$, $n\geq 2$ and $\delta\in(0,1)$. Then we have for all
$t>0$ that

$\displaystyle\int_{0}^{\infty}\nu^{v}_{n}[0,u]\left|1-\frac{ut}{n}\right|t^{n}e^{-tu}du\leq\delta
p^{v}_{n}(t)+(1-\delta)^{n}e^{\delta n}+(1+\delta)^{n}e^{-\delta n}.$ (10.8)

###### Proof.

By Lemma 10.3 we have

$\displaystyle\int_{0}^{\infty}\nu^{v}_{n}[0,u]\left|1-\frac{ut}{n}\right|t^{n}e^{-tu}du$
$\displaystyle\leq\delta
p^{v}_{n}(t)+\int_{\left|1-\frac{tu}{n}\right|>\delta}\nu^{v}_{n}[0,u]t^{n}e^{-tu}\left|1-\frac{tu}{n}\right|du$
$\displaystyle\leq\delta
p^{v}_{n}(t)+\left(\frac{e}{n-1}\right)^{n-1}\int_{\left|1-\frac{tu}{n}\right|>\delta}t^{n}u^{n-1}e^{-tu}\left|1-\frac{tu}{n}\right|\,du.$

Changing variables yields that the above equals

$\displaystyle\delta
p^{v}_{n}(t)+\left(\frac{e}{n-1}\right)^{n-1}\int_{\left|1-\frac{u}{n}\right|>\delta}u^{n-1}e^{-u}\left|1-\frac{u}{n}\right|\,du.$

Splitting the integral on the above right-hand side into two pieces
corresponding to $tu<n(1-\delta)$ and $tu>n(1+\delta)$ yields

$\displaystyle\begin{split}\int_{0}^{n(1-\delta)}u^{n-1}e^{-u}\left(1-\frac{u}{n}\right)du&=n^{n-1}(1-\delta)^{n}e^{-n(1-\delta)},\\\
\int_{n(1+\delta)}^{\infty}u^{n-1}e^{-u}\left(\frac{u}{n}-1\right)du&=n^{n-1}(1+\delta)^{n}e^{-n(1-\delta)}.\end{split}$

Since $(1+1/(n-1))^{n-1}<e$ for all $n\geq 2$, we obtain the assertion (10.8).
∎

Let $f$ be as in Theorem 10.1 and $v\in{\mathbb{X}}$. Then $f$ is bounded and

$\displaystyle\mathbb{E}_{t}f(|C^{v}|)=\mathbb{E}_{t}F(\xi^{v})=\sum_{n=1}^{\infty}f(n)p^{v}_{n}(t).$

In order to prove Theorem 10.1 we will check the condition (8.16) on
$[t_{0},\infty)$ for each $t_{0}>0$. This is achieved by the previous and the
following lemma.

######  Lemma 10.6.

Suppose that $f\colon{\mathbb{N}}\to{\mathbb{R}}$. Then

$\displaystyle\mathbb{E}_{t}|M^{v}_{t}F_{n}(\xi^{v})|\leq|f(n)|\int_{0}^{\infty}\nu^{v}_{n}[0,u]\left|n-ut\right|t^{n}e^{-tu}du.$
(10.9)

###### Proof.

It is easy to see the following identities which follow from integration by
parts

$\displaystyle\int_{u}^{(n-1)/t}(n-ts)e^{-ts}ds$
$\displaystyle=-t^{-1}(n-ts)e^{-ts}\Big{|}_{u}^{(n-1)/t}-\int_{u}^{(n-1)/t}e^{-ts}ds$
$\displaystyle=t^{-1}\left((n-tu)e^{-tu}-e^{-(n-1)}+e^{-(n-1)}-e^{-tu}\right)=t^{-1}(n-1-tu)e^{-tu},$
$\displaystyle\int_{u}^{\infty}(n-ts)e^{-ts}ds$
$\displaystyle=t^{-1}(n-tu)e^{-tu}-\int_{u}^{\infty}e^{-ts}ds=t^{-1}(n-1-tu)e^{-tu}.$

Since $\nu^{v}_{n}$ is locally finite, we can apply Fubini’s theorem to obtain
that

$\displaystyle\mathbb{E}_{t}|M^{v}_{t}F_{n}(\xi^{v})|$
$\displaystyle=|f(n)|\mathbb{E}_{t}\big{[}|n-1-t\varphi_{\lambda}(C^{v})|{\mathbf{1}}\\{|C^{v}|=n\\}\big{]}=|f(n)|t^{n-1}\int_{0}^{\infty}|n-1-tu|e^{-tu}\nu^{v}_{n}(du)$
$\displaystyle=|f(n)|t^{n-1}\left(\int_{0}^{(n-1)/t}(n-1-tu)e^{-tu}\nu^{v}_{n}(du)-\int_{(n-1)/t}^{\infty}(n-1-tu)e^{-tu}\nu^{v}_{n}(du)\right)$
$\displaystyle=|f(n)|t^{n}\left(\int_{0}^{(n-1)/t}\int_{u}^{(n-1)/t}(n-ts)e^{-ts}ds\,\nu^{v}_{n}(du)-\int_{(n-1)/t}^{\infty}\int_{u}^{\infty}(n-ts)e^{-ts}ds\,\nu^{v}_{n}(du)\right)$
$\displaystyle=|f(n)|t^{n}\left(\int_{0}^{(n-1)/t}\nu_{n}^{v}[0,s](n-ts)e^{-ts}ds-\int_{(n-1)/t}^{\infty}\nu_{n}^{v}[0,s](n-ts)e^{-ts}ds\right)$
$\displaystyle=|f(n)|\int_{0}^{\infty}\nu^{v}_{n}[0,s]\left|n-st\right|t^{n}e^{-ts}\,ds-2|f(n)|\int_{(n-1)/t}^{n/t}\nu_{n}^{v}[0,s](n-ts)t^{n}e^{-ts}ds.$

∎

###### Proof of Theorem 10.1.

Let $v\in{\mathbb{X}}$, $t_{0}>0$ and $n\geq 2$. We need to check the
condition (8.16). To do so, we start with inequality (10.9). In (10.8) we
choose $\delta\equiv\delta_{n}$ by $\delta_{n}:=\sqrt{9\log n/n}$. We use the
inequalities $(1-r)e^{r}\leq e^{-r^{2}/2}$ which holds for all $r\geq 0$ and
$(1+r)e^{-r}\leq e^{-r^{2}/3}$ which holds for all $r\in[0,1/3)$. Then we
obtain for all sufficiently large $n\in{\mathbb{N}}$ that
$(1-\delta_{n})^{n}e^{-n\delta_{n}}\leq n^{-9/2}$ and
$(1-\delta_{n})^{n}e^{-n\delta_{n}}\leq n^{-3}$. Hence there exist
$n_{0}\in{\mathbb{N}}$ such that for each $t\geq t_{0}$

$\displaystyle\sum_{n=n_{0}}^{\infty}t^{-1}|f(n)|\int_{0}^{\infty}\nu^{v}_{n}[0,u]\left|n-ut\right|t^{n}e^{-tu}du\leq\frac{\sqrt{9}}{t_{0}}\sum_{n=n_{0}}^{\infty}|f(n)|\sqrt{n\log
n}p^{v}_{n}(t)+\frac{2}{t_{0}}\sum_{n=n_{0}}^{\infty}n^{-2}.$ (10.10)

Let $\varepsilon>0$ and choose $n_{1}\geq n_{0}$ such that $|f(n)|\sqrt{n\log
n}\leq\varepsilon$ for each $n\geq n_{1}$. Then

$\sum_{n=n_{1}}^{\infty}|f(n)|\sqrt{n\log n}p^{v}_{n}(t)\leq\varepsilon,$

finishing the proof. ∎

###### Proof of Theorem 8.10.

We check condition (8.16). By assumption (8.18) it suffices to show that

$\displaystyle\lim_{n\to\infty}\sup_{t\geq
t_{0}}\sum_{k>n}|\tilde{f}(k)|\mathbb{E}_{t}\big{|}M^{v}_{t}\big{|}{\mathbf{1}}\\{|C^{v}|=k\\})=0,$

for any $t_{0}>0$. This follows from (10.9) and the proof of Theorem 10.1. ∎

Later we shall need the following integrated version of Theorem 10.1

######  Theorem 10.7.

Assume that ${\mathbb{X}}={\mathbb{Y}}\times{\mathbb{M}}$ is the product of
two complete separable metric spaces and let $\mathbb{Q}$ be a finite measure
$\mathbb{Q}$ on ${\mathbb{M}}$. Suppose that
$f\colon{\mathbb{N}}\to{\mathbb{R}}$ is a function satisfying (10.1). Then
$t\mapsto\int\mathbb{E}_{t}f(|C^{(y,q)}|)\,\mathbb{Q}(dq)$ is for each
$y\in{\mathbb{Y}}$ continuously differentiable on $(0,\infty)$.

###### Proof.

Let $y\in{\mathbb{Y}}$ and $t_{0}>0$. We know from Theorem 10.1 that
$t\mapsto\mathbb{E}_{t}f(|C^{(y,q)}|)$ is for each
$(y,q)\in{\mathbb{Y}}\times{\mathbb{M}}$ continuously differentiable. The
assertion follows from the Leibniz differentiation rule once we can show that

$\displaystyle\sum_{n=1}^{\infty}|f(n)|\int_{0}^{\infty}\nu^{(y,q)}_{n}[0,u]\left|n-ut\right|t^{n}e^{-tu}du\leq
c,\quad t\geq t_{0},\,q\in{\mathbb{M}},$ (10.11)

for some $c>0$. Since $f(n)\sqrt{n\log n}$ is bounded, we see from (10.10)
that the above series, starting from $n=n_{0}$ is bounded in
$q\in{\mathbb{M}}$ and $t\geq t_{0}$. The remaining terms in the series can be
bounded by (10.8). Similarly as in the proof of Lemma 8.5 one can show that
$\int f^{(y,q)}_{n}(t)\,\mathbb{Q}(dq)$ is an analytic function on
$(0,\infty)$. Therefore the continuity of the derivative follows from (10.10),
since $\sum\limits_{n\geq n_{0}}|f(n)|\sqrt{n\log n}\int
p_{n}^{(y,q)}(t)\,\mathbb{Q}(dq)\to 0$ as $n_{0}\to\infty$ uniformly in
$t\in{\mathbb{R}}_{+}$. ∎

Strengthening the assumption on the function $f$ in Theorem 10.7, we can write
the derivative as a Margulis–Russo type formula.

######  Theorem 10.8.

Suppose that $f\colon{\mathbb{N}}\to{\mathbb{R}}$ is a function satisfying

$\displaystyle\sup\limits_{n\geq 1}|f(n)|n<\infty.$ (10.12)

Then $t\mapsto\mathbb{E}_{t}f(|C^{v}|)$ is for each $v\in{\mathbb{X}}$
continuously differentiable on $(0,\infty)$ with derivative given by (9.4).

###### Proof.

It is enough to check condition (9.3) on $[t_{0},\infty)$ for each $t_{0}>0$.
Condition (10.12) implies that

$\displaystyle\mathbb{E}_{t}|f(|C^{v}|)||C^{v}|<\infty.$

It follows from Fubini’s theorem and Lemma 10.2 that for $n\geq 2$

$\displaystyle\mathbb{E}_{t}|F_{n}(\xi^{v})|\varphi_{\lambda}(C^{v})$
$\displaystyle=|f(n)|t^{n-1}\int_{0}^{\infty}ue^{-tu}\nu_{n}^{v}(du)=|f(n)|t^{n-1}\int_{0}^{\infty}\int_{u}^{\infty}(ts-1)e^{-ts}\,ds\,\nu_{n}^{v}(du)$
$\displaystyle=|f(n)|t^{n-1}\int_{0}^{\infty}\nu_{n}^{v}[0,s](ts-1)e^{-ts}\,ds<|f(n)|t^{n}\int_{0}^{\infty}\nu_{n}^{v}[0,s]se^{-ts}\,ds$
$\displaystyle\leq|f(n)|\left(2np^{v}_{n}(t)+t^{n}\int_{2n}^{\infty}\nu_{n}^{v}[0,s]se^{-ts}\,ds\right)$
$\displaystyle\leq|f(n)|\left(2np^{v}_{n}(t)+t^{-1}\left(\frac{e}{n-1}\right)^{n-1}\int_{2n}^{\infty}u^{n}e^{-u}\,du\right)$
$\displaystyle=|f(n)|2np^{v}_{n}(t)+t^{-1}|f(n)|\left(\frac{e}{n-1}\right)^{n-1}n!\
\mathbb{P}(X_{n}\leq n),$

where $X_{n}$ has a Poisson distribution with parameter $2n$. By assumption
(10.12) the sum over the first terms is converging. By a rather elementary
concentration inequality we have $\mathbb{P}(X_{n}\leq n)\leq e^{-n}$ for each
$n\in{\mathbb{N}}$. Therefore the sum over the second terms is converging too.
∎

######  Remark 10.9.

The position dependent cluster density satisfies the condition (10.12) and its
derivative can be represented by (9.4), i.e.

$\displaystyle\frac{d}{dt}\kappa^{v}(t)$
$\displaystyle=t^{-1}\left(\mathbb{P}_{t}(|C^{v}|<\infty)-\kappa^{v}(t)-\mathbb{E}_{t}\int|C^{v}-\delta_{x}|^{-1}\,C^{v!}(dx)\right)$
$\displaystyle=\mathbb{E}_{t}\int(|C^{v}(\xi^{v,x})|^{-1}-|C^{v}|^{-1}){\mathbf{1}}\\{x\in
C^{v}(\xi^{v,x})\\}\,\lambda(dx).$

##  11 Deletion stability of the stationary marked RCM

In this section we consider the stationary marked RCM as introduced in Section
4. Hence we take a Poisson process $\eta$ on
${\mathbb{R}}^{d}\times{\mathbb{M}}$ with intensity measure
$t\lambda=t\lambda_{d}\otimes\mathbb{Q}$ and consider the random connection
model $\xi$ based on $\eta$ and a fixed connection function
$\varphi\colon({\mathbb{R}}^{d}\times{\mathbb{M}})^{2}\to[0,1]$ satisfying
(4.1) and (4.2).

######  Theorem 11.1.

The infinite clusters of a stationary marked random connection model are
deletion stable.

Our proof of the theorem partially follows the seminal paper [1]. It requires
a significant extension of some of the arguments in [23] treating the Gilbert
graph with deterministic radii; see Example 4.8.

We need to introduce some further notation. For $\mu\in{\mathbf{G}}$ and
$(x,p)\in V(\mu)$ we denote by $N^{0}(x,p,\mu)$ the number of clusters in
$C^{(x,p)}(\mu)-\delta_{(x,p)}$. Hence $N^{0}(x,p,\mu)$ is the number of
clusters in $\mu-\delta_{(x,p)}$ which are connected in $\mu$ with $(x,p)$. We
then define $N^{+}(x,p,\mu)$ similarly to $N^{0}(x,p,\mu)$, except that at
most one infinite cluster is counted, i.e.

$\displaystyle
N^{+}(x,p,\mu):=N^{0}(x,p,\mu)-{\mathbf{1}}\\{N^{\infty}(x,p,\mu)\geq
1\\}(N^{\infty}(x,p,\mu)-1).$

Given $B\in{\mathcal{B}}({\mathbb{R}}^{d})$ and a measure $\nu$ on
${\mathbb{R}}^{d}\times{\mathbb{M}}$ it will be convenient to write
$\nu_{B}:=\nu_{B\times{\mathbb{M}}}$ for the restriction of $\nu$ to
$B\times{\mathbb{M}}$.

We fix some arbitrary $t_{0}>0$. It is then no restriction of generality to
assume that $t\in(0,t_{0}]$. Let $(B_{n})_{n\in{\mathbb{N}}}$ be an increasing
sequence of convex and compact sets with union ${\mathbb{R}}^{d}$. Our proofs
require a specific coupling of the RCM $\xi$ with two random graphs
$\xi_{n,0}$ and $\xi_{n,+}$, $n\in{\mathbb{N}}$, according two different
boundary conditions: free and wired. To this end we let $\tilde{\xi}$ be a RCM
based on a Poisson process $\tilde{\eta}$ with intensity measure
$t_{0}\lambda$. We can assume without loss of generality that $\eta$ is
$t/t_{0}$-thinning of $\tilde{\eta}$ (see [25, Corollary 5.9]) and that $\xi$
is given as the restriction $\tilde{\xi}[\eta]$ of $\tilde{\xi}$ to the
vertices from $\eta$. Let us first set
$\xi_{n}:=\tilde{\xi}[\eta_{B_{n}}+\tilde{\eta}_{B^{c}_{n}}]$. This is a RCM
driven by the Poisson process $\eta_{B_{n}}+\tilde{\eta}_{B^{c}_{n}}$ which
has intensity measure $t\lambda_{B_{n}}+t_{0}\lambda_{B^{c}_{n}}$. We define
$\xi_{n,0}$ as the restriction $\xi[\eta_{B_{n}}]=\xi_{n}[\eta_{B_{n}}]$ of
$\xi$ to $B_{n}\times{\mathbb{M}}$. This is a RCM driven by $\eta_{B_{n}}$. We
let $\xi_{n,+}$ be the random graph resulting from $\xi_{n}$ by connecting all
point from $\tilde{\eta}_{B^{c}_{n}}$. The reader should keep in mind that
$\xi_{n,+}$ is a very simple function of the RCM $\xi_{n}$. In fact,
$\xi_{n,+}$ is also a RCM with the connection function to be suitably
modified. An important property of this coupling is that $\xi_{n,0}$ is a
subgraph of $\xi$, while $\xi$ is a subgraph of $\xi_{n,+}$ (in fact of
$\xi_{n}$).

For $(x,p)\in\eta_{B_{n}}$ we define $C^{(x,p)}_{n,0}:=C^{(x,p)}(\xi_{n,0})$
and $C^{(x,p)}_{n,+}:=C^{(x,p)}(\xi_{n,+})$ noting that

$V(C^{(x,p)}_{n,+})=V(C^{(x,p)}_{n,0})+{\mathbf{1}}\\{(x,p)\leftrightarrow\tilde{\eta}_{B^{c}_{n}}\text{in
$\xi_{n}$}\\}\tilde{\eta}_{B^{c}_{n}}.$

Note also that if $(x,p)\notin\eta_{B_{n}}$, then
$C^{(x,p)}_{n,0}=C^{(x,p)}_{n,+}=0$. Note also that $C^{(x,p)}_{n,+}$ is
infinite iff $(x,p)$ is connected (in $\xi_{n}$) to
$\tilde{\eta}_{B^{c}_{n}}$. Otherwise it is finite and coincides (by the
coupling construction) with $C^{(x,p)}_{n,0}=C^{(x,p)}(\xi)$.

######  Lemma 11.2.

Let $(x,p)\in{\mathbb{R}}^{d}\times{\mathbb{M}}$. Then, almost surely,
$V(C^{(x,p)}_{n,0})\uparrow V(C^{(x,p)}(\xi))$ and
$V(C^{(x,p)}_{n,+})\downarrow V(C^{(x,p)}(\xi))$ as $n\to\infty$.

###### Proof.

Let $(x,p)\in\eta$ otherwise the statement is trivial. There exists
$m\in{\mathbb{N}}$ such that $x\in B_{m}$. We shall always take $n\geq m$. The
second assertion has to be interpreted this way. Clearly $C^{(x,p)}_{n,0}$ is
a subgraph of $C^{(x,p)}(\xi)$. Assume that $(y,q)\in C^{(x,p)}(\xi)$. Then
there exists $n$ such that $(x,p)$ is connected to $(y,q)$ within $\xi_{n,0}$.
This proves the first assertion. Next we note that $C^{(x,p)}(\xi)$ is a
subgraph of $C^{(x,p)}_{n+1,+}$ while $C^{(x,p)}_{n+1,+}$ is a subgraph of
$C^{(x,p)}_{n,+}$. Assume that $(y,q)\in C^{(x,p)}_{n,+}$ for each $n\geq m$.
For large enough $n$ we then have $y\in B_{n}$ and hence $(y,q)\in
C^{(x,p)}(\xi)$. ∎

For each $n\in{\mathbb{N}}$ we define

$\displaystyle
M_{n,\star}:=\int|C^{(x,p)}_{n,\star}|^{-1}\,\eta_{B_{n}}(d(x,p)),$

where we use a star to denote either $0$ or $+$. A simple counting argument
shows that $M_{n,0}$ is the number of clusters (finite) in $\xi_{n,0}$ while
$M_{n,+}$ is the number of finite clusters in $\xi_{n,+}$. Moreover, we have
that

$\displaystyle M_{n,+}\leq M_{n}\leq M_{n,0},$ (11.1)

where (see also Proposition 4.4)

$\displaystyle M_{n}:=\int|C^{(x,p)}(\xi)|^{-1}\,\eta_{B_{n}}(d(x,p)).$ (11.2)

Recalling the definition (4.6) of the cluster density $\kappa(t)$, we have the
following lemma.

######  Lemma 11.3.

Let $t\in[0,t_{0}]$. Then
$(\lambda_{d}(B_{n}))^{-1}\mathbb{E}_{t}M_{n,\star}\to t\kappa(t)$ as
$n\to\infty$.

###### Proof.

By Lemma 4.1,

$\displaystyle\mathbb{E}_{t}M_{n}$ $\displaystyle=t\iint{\mathbf{1}}\\{x\in
B_{n}\\}\mathbb{E}_{t}|C^{(x,p)}|^{-1}\,dx\,\mathbb{Q}(dp)=\lambda_{d}(B_{n})t\kappa(t).$

Almost surely $M_{n,0}-M_{n}$ is less than the number of clusters with points
from $\eta_{B_{n}}$ which are connected in $\xi$ with $\eta_{B_{n}^{c}}$, and
therefore less than the number of points from $\eta_{B_{n}}$ which are
directly connected in $\xi$ with $\eta_{B_{n}^{c}}$. Analogously,
$M_{n}-M_{n,+}$ is less than number of clusters with points from
$\eta_{B_{n}}$ which are connected in $\xi_{n}$ with
$\tilde{\eta}_{B_{n}^{c}}$, and therefore less than the number of points from
$\eta_{B_{n}}$ which are directly connected in $\xi_{n}$ with
$\tilde{\eta}_{B_{n}^{c}}$. Then with probability one, we have

$\displaystyle\begin{split}M_{n,0}-\int{\mathbf{1}}\\{(x,p)\sim\tilde{\eta}_{B^{c}_{n}}\textrm{
in $\xi_{n}$}\\}\eta_{B_{n}}(d(x,p))&\leq
M_{n,0}-\int{\mathbf{1}}\\{(x,p)\sim\eta_{B^{c}_{n}}\textrm{ in
$\xi$}\\}\eta_{B_{n}}(d(x,p))\\\ &\leq M_{n}\leq
M_{n,+}+\int{\mathbf{1}}\\{(x,p)\sim\tilde{\eta}_{B^{c}_{n}}\textrm{ in
$\xi_{n}$}\\}\eta_{B_{n}}(d(x,p)).\end{split}$

By the Mecke equation, we have

$\displaystyle\mathbb{E}_{t}\int{\mathbf{1}}\\{(x,p)\sim\tilde{\eta}_{B^{c}_{n}}\textrm{
in $\xi_{n}$}\\}\eta_{B_{n}}(d(x,p))$ $\displaystyle=t\iint{\mathbf{1}}\\{x\in
B_{n}\\}\left(1-e^{-t_{0}\int{\mathbf{1}}\\{y\in
B^{c}_{n}\\}\varphi(y-x,p,q)\,dy\,\mathbb{Q}(dq)}\right)\,dx\,\mathbb{Q}(dp)$
$\displaystyle\leq t_{0}t\iiint{\mathbf{1}}\\{x\in B_{n},y\in
B^{c}_{n}\\}\varphi(y-x,p,q)\,dx\,dy\,\mathbb{Q}^{2}(d(p,q))$
$\displaystyle=t_{0}t\iint{\mathbf{1}}\\{x\in B_{n},y\in
B^{c}_{n}\\}\psi(y-x)\,dx\,dy,$

where

$\displaystyle\psi(x):=\int\varphi(x,p,q)\,\,\mathbb{Q}^{2}(d(p,q)),\quad
x\in{\mathbb{R}}^{d}.$ (11.3)

By assumption (4.2), $\psi$ is integrable.

Let $\varepsilon>0$ and choose $r>0$ so large that
$\int{\mathbf{1}}\\{|z|>r\\}\psi(z)\,dz\leq\varepsilon$. Then

$\displaystyle\int{\mathbf{1}}\\{x\in B_{n},y\in
B^{c}_{n},|y-x|>r\\}\psi(y-x)\,dx\,dy\leq\varepsilon\lambda_{d}(B_{n}).$

Further

$\displaystyle\frac{1}{\lambda_{d}(B_{n})}\int{\mathbf{1}}\\{x\in B_{n},y\in
B^{c}_{n},|y-x|\leq r\\}\psi(y-x)\,dx\,dy\leq
d_{\varphi}\frac{\lambda_{d}((B_{n})_{\ominus
r})}{\lambda_{d}(B_{n})}\overset{n\to\infty}{\longrightarrow}0,$ (11.4)

where, for a bounded set $B\subset{\mathbb{R}}^{d}$, $B_{\ominus r}:=\\{x\in
B:d(x,\partial B)\leq r\\}$ and $\partial B$ denotes the boundary of $B$.
Therefore

$\displaystyle\limsup_{n\to\infty}\frac{\mathbb{E}_{t}M_{n,0}}{\lambda_{d}(B_{n})}-\varepsilon
t_{0}t$ $\displaystyle\leq
t\kappa(t)\leq\liminf_{n\to\infty}\frac{\mathbb{E}_{t}M_{n,+}}{\lambda_{d}(B_{n})}+\varepsilon
t_{0}t$

Taking into account (11.1), this yields the assertion. ∎

######  Remark 11.4.

The convergence on the right-hand side of (11.4) is crucial for the proof of
Lemma 11.3. This amenability property of Euclidean space is also important for
Lemma 11.8.

Let $n\in{\mathbb{N}}$. We will now explore the derivatives of
$t\mapsto\mathbb{E}_{t}M_{n,\star}$. For $(x,p)\in B_{n}\times{\mathbb{M}}$ we
define $N^{\star}_{n}(x,p):=N^{0}(x,p,\xi^{(x,p)}_{n,\star})$, the finite
volume counterparts of $N^{0}(x,p,\xi^{(x,p)})$ and $N^{+}(x,p,\xi^{(x,p)})$.
By this definition $N^{0}_{n}(x,p)$ is the number of (finite) clusters in
$\xi_{n,0}$ which are connected to $(x,p)$ in $\xi^{(x,p)}_{n,0}$, and
$N^{+}_{n}(x,p)$ is the number of clusters (with at most one infinite) in
$\xi_{n,+}$ which are connected to $(x,p)$ in $\xi^{(x,p)}_{n,+}$.

######  Lemma 11.5.

For any $n\in{\mathbb{N}}$ and either choice of boundary conditions the
function $t\mapsto\mathbb{E}_{t}M_{n,\star}$ is differentiable on $[0,t_{0})$
and the derivative is given by

$\displaystyle\frac{d}{dt}\mathbb{E}_{t}M_{n,\star}=\lambda_{d}(B_{n})-\mathbb{E}_{t}\iint{\mathbf{1}}\\{x\in
B_{n}\\}N^{\star}_{n}(x,p)\,dx\,\mathbb{Q}(dp).$

###### Proof.

Since $M_{n,\star}\leq\eta(B_{n})$ we have $\mathbb{E}_{t}M_{n,\star}<\infty$
for all $t>0$. We now apply the Margulis-Russo formula (3.9), where
$\lambda_{2}=(\lambda_{d})_{B_{n}}\otimes\mathbb{Q}$ and $\lambda_{1}=0$ for
the free boundary condition ($\star=0$) and
$\lambda_{1}=t_{0}(\lambda_{d})_{B^{c}_{n}}\otimes\mathbb{Q}$ for the wired
boundary condition ($\star=+$). Hence $\mathbb{E}_{t}M_{n,\star}$ is a
differentiable function of $t$ and

$\displaystyle\frac{d}{dt}\mathbb{E}_{t}M_{n,\star}=\mathbb{E}_{t}\iint{\mathbf{1}}\\{x\in
B_{n}\\}\big{(}M_{n,\star}(\xi_{n,\star}^{(x,p)})-M_{n,\star}(\xi_{n,\star})\big{)}\,dx\,\mathbb{Q}(dp).$

Let $(x,p)\in B_{n}\times{\mathbb{M}}$. If $N^{\star}_{n}(x,p)=0$, then with
probability one
$M_{n,\star}(\xi_{n,\star}^{(x,p)})-M_{n,\star}(\xi_{n,\star})=1$. Otherwise
the removal of $(x,p)$ from $\xi^{(x,p)}_{n,\star}$ results in
$M_{n,\star}(\xi_{n,\star}^{(x,p)})-M_{n,\star}(\xi_{n,\star})=1-N^{\star}_{n}(x,p)$
a.s., proving the result. ∎

######  Lemma 11.6.

For any $n\in{\mathbb{N}}$ and either choice of boundary conditions
$\mathbb{E}_{t}M_{n,\star}+\lambda_{d}(B_{n})d_{\varphi}t^{2}/2$ is a convex
function of $t$ on $[0,t_{0})$.

###### Proof.

For $(x,p)\in{\mathbb{R}}^{d}\times{\mathbb{M}}$ we let $\Psi(x,p)$ denote the
point process of the Poisson neighbours of $(x,p)$ in $\xi^{(x,p)}$, that is
the points in $\eta$ which are directly connected to $(x,p)$ in $\xi^{(x,p)}$.
For a Borel set $B\subset{\mathbb{R}}^{d}$ we let $\Psi_{B}(x,p)$ denote the
restriction of $\Psi(x,p)$ to $B\times{\mathbb{M}}$. We further denote
$d_{\varphi}(p):=\int d(p,q)\,\mathbb{Q}(dq)$ so that $d_{\varphi}=\int
d_{\varphi}(p)\,\mathbb{Q}(dp)$. By Lemma 11.5,

$\displaystyle\frac{d}{dt}\bigg{[}\mathbb{E}_{t}M_{n,\star}$
$\displaystyle+\lambda_{d}(B_{n})d_{\varphi}\frac{t^{2}}{2}\bigg{]}$
$\displaystyle=\lambda_{d}(B_{n})-\mathbb{E}_{t}\iint{\mathbf{1}}\\{x\in
B_{n}\\}N^{\star}_{n}(x,p)\,dx\,\mathbb{Q}(dp)+t\lambda_{d}(B_{n})d_{\varphi}$
$\displaystyle=\lambda_{d}(B_{n})+\iint{\mathbf{1}}\\{x\in
B_{n}\\}\big{(}td_{\varphi}(p)-\mathbb{E}_{t}N^{\star}_{n}(x,p)\big{)}\,dx\,\mathbb{Q}(dp)$
$\displaystyle=\lambda_{d}(B_{n})+\iint{\mathbf{1}}\\{x\in
B_{n}\\}\big{(}\mathbb{E}_{t}|\Psi(x,p)|-\mathbb{E}_{t}N^{\star}_{n}(x,p)\big{)}\,dx\,\mathbb{Q}(dp)$
$\displaystyle=\lambda_{d}(B_{n})+\iint{\mathbf{1}}\\{x\in
B_{n}\\}\left(\mathbb{E}_{t}|\Psi_{B^{c}_{n}}(x,p)|+\mathbb{E}_{t}\big{(}|\Psi_{B_{n}}(x,p)|-N^{\star}_{n}(x,p)\big{)}\right)\,dx\,\mathbb{Q}(dp),$

Clearly $\mathbb{E}_{t}|\Psi_{B^{c}_{n}}(x,p)|$ is increasing in $t$. We shall
now argue that
$\mathbb{E}_{t}\big{[}\Psi_{B_{n}}(x,p)-N^{\star}_{n}(x,p)\big{]}$ is
increasing in $t$. Applying the Margulis-Russo formula (3.9) similarly as in
the proof of Lemma 11.5, we see that it is sufficient to check that
$\Psi_{B_{n}}(x,p)-N^{\star}_{n}(x,p)$ cannot strictly decrease when adding a
point $(y,q)\in B_{n}\times{\mathbb{M}}$ to $\eta$. Assume first that $(y,q)$
is not directly connected to $(x,p)$. Then $\Psi_{B_{n}}(x,p)$ does not change
while $N^{\star}_{n}(x,p)$ can only decrease (namely by connecting some of the
clusters in $\xi_{n,\star}$ which are connected to $(x,p)$ in
$\xi^{(x,p)}_{n,\star}$). Assume now that $(y,q)$ is directly connected to
$(x,p)$, so that $\Psi_{B_{n}}(x,p)$ increases by one. In that case
$N^{\star}_{n}(x,p)$ can increase by at most $1$, namely if some of the
clusters in $\xi_{n,\star}$ which are not connected to $(x,p)$ in
$\xi^{(x,p)}_{n,\star}$ get connected to the new point $(y,q)$ while none of
the clusters in $\xi_{n,\star}$ which are connected to $(x,p)$ in
$\xi^{(x,p)}_{n,\star}$ are connected by $(y,q)$. This proves the asserted
monotonicity and hence the convexity assertion. ∎

Now we are in the position to prove the first main result in this section.

######  Theorem 11.7.

The function $t\mapsto t\kappa(t)+d_{\varphi}t^{2}/2$ is continuously
differentiable on $(0,\infty)$, convex on ${\mathbb{R}}_{+}$ and right
differentiable at zero.

###### Proof.

The first assertion follows from Theorem 10.7 while the second follows from
Lemmas 11.3 and 11.6 and the (elementary) fact that the limit of a sequence of
convex functions is convex. The function is right differentiable at zero since
$\kappa$ is a monotone function. ∎

In the final step of the proof of Theorem 11.1 we need to identify the limits
of the derivatives in Lemma 11.5.

######  Lemma 11.8.

Let $t\in[0,t_{0}]$. Then

$\displaystyle\liminf_{n\to\infty}\,(\lambda_{d}(B_{n}))^{-1}\mathbb{E}_{t}\iint{\mathbf{1}}\\{x\in
B_{n}\\}N^{0}_{n}(x,p)\,dx\,\mathbb{Q}(dp)$
$\displaystyle\geq\int\mathbb{E}_{t}N^{0}(0,p,\xi^{(0,p)})\,\mathbb{Q}(dp),$
$\displaystyle\limsup_{n\to\infty}\,(\lambda_{d}(B_{n}))^{-1}\mathbb{E}_{t}\iint{\mathbf{1}}\\{x\in
B_{n}\\}N^{+}_{n}(x,p)\,dx\,\mathbb{Q}(dp)$
$\displaystyle\leq\int\mathbb{E}_{t}N^{+}(0,p,\xi^{(0,p)})\,\mathbb{Q}(dp).$

###### Proof.

Similarly as in the proof of Proposition 4.5 by stationarity, we have

$\displaystyle\int\mathbb{E}_{t}N^{\star}(0,p,\xi^{(0,p)})\,\mathbb{Q}(dp)=(\lambda_{d}(B_{n}))^{-1}\mathbb{E}_{t}\iint{\mathbf{1}}\\{x\in
B_{n}\\}N^{\star}(x,p,\xi^{(x,p)})\,dx\,\mathbb{Q}(dp),\quad
n\in{\mathbb{N}}.$

Hence our task is to show that $N^{\star}(x,p,\xi^{(x,p)})$ is well
approximated by $N^{\star}_{n}(x,p)$. For a given Borel set
$B\subset{\mathbb{R}}^{d}$ and $(x,p)\in{\mathbb{R}}^{d}\times{\mathbb{M}}$ we
denote by $N^{0}_{B}(x,p)$ the number of clusters in $\xi$ to which the
Poisson neighbors of $(x,p)$ in $B\times{\mathbb{M}}$ belong. Note that a.s.

$\displaystyle N^{0}_{B}(x,p)+N^{0}_{B^{c}}(x,p)\geq N^{0}(x,p,\xi^{(x,p)}).$

It is, moreover, easy to see that $N^{0}_{n}(x,p)\geq N^{0}_{B_{n}}(x,p)$ a.s.
for each $n\in{\mathbb{N}}$. It follows that a.s.

$\displaystyle N^{0}_{n}(x,p)\geq
N^{0}(x,p,\xi^{(x,p)})-N^{0}_{B_{n}^{c}}(x,p).$ (11.5)

Obviously $N^{0}_{B_{n}^{c}}(x,p)$ is dominated by the number of points from
$\eta_{B_{n}^{c}}$ which are directly connected to $(x,p)$ in $\xi^{(x,p)}$.
Therefore

$\displaystyle\mathbb{E}_{t}N^{0}_{B_{n}^{c}}(x,p)\leq
t\iint{\mathbf{1}}\\{y\in B^{c}_{n}\\}\varphi(y-x,p,q)\,dy\,\mathbb{Q}(dq).$

It now follows from (11.5) and exactly as in the proof of Lemma 11.3 that for
each $\varepsilon>0$

$\displaystyle\liminf_{n\to\infty}\,(\lambda_{d}(B_{n}))^{-1}\mathbb{E}_{t}\iint{\mathbf{1}}\\{x\in
B_{n}\\}N^{0}_{n}(x,p)\,dx\,\mathbb{Q}(dp)$
$\displaystyle\geq\int\mathbb{E}_{t}N^{0}(0,p,\xi^{(0,p)})\,\mathbb{Q}(dp)-\varepsilon
t.$

This implies the first asserted inequality. The second follows from
$N^{+}_{n}(x,p)\leq N^{+}(x,p,\xi^{(x,p)})$ a.s.. ∎

###### Proof of Theorem 11.1.

The convex function in Theorem 11.7 is differentiable and approximated by the
differentiable convex functions
$(\lambda_{d}(B_{n}))^{-1}\mathbb{E}_{t}M_{n,\star}+d_{\varphi}t^{2}/2$; see
Lemmas 11.3 and 11.5. A classical result from convex analysis (see [36,
Theorem 25.7]) implies that

$\displaystyle\lim_{n\to\infty}(\lambda_{d}(B_{n}))^{-1}\frac{d}{dt}\mathbb{E}_{t}M_{n,\star}=\frac{d}{dt}t\kappa(t).$

Therefore we obtain from Lemma 11.5 that the limes inferior in Lemma 11.8
coincides with the limes superior. Hence Lemma 11.8 yields

$\displaystyle\int\mathbb{E}_{t}N^{0}(0,p,\xi^{(0,p)})\,\mathbb{Q}(dp)\leq\int\mathbb{E}_{t}N^{+}(0,p,\xi^{(0,p)})\,\mathbb{Q}(dp),$

or

$\displaystyle\int\mathbb{E}_{t}(N^{0}(0,p,\xi^{(0,p)})-N^{+}(0,p,\xi^{(0,p)}))\,\mathbb{Q}(dp)\leq
0.$

Since

$\displaystyle
N^{0}(0,p,\xi^{(0,p)})-N^{+}(0,p,\xi^{(0,p)})={\mathbf{1}}\\{N^{\infty}(0,p,\xi^{(0,p)})\geq
1\\}(N^{\infty}(0,p,\xi^{(0,p)})-1),$

we obtain

$\displaystyle\int\mathbb{P}_{t}(N^{\infty}(0,p,\xi^{(0,p)})\geq
2)\,\mathbb{Q}(dp)=0.$ (11.6)

Using stationarity as in the proof of Lemma 4.1 we see, that this is
equivalent to the assertion. ∎

The preceding proof yields the following corollary.

######  Corollary 11.9.

The cluster density is continuously differentiable function and

$\displaystyle\frac{d}{dt}(t\kappa(t))=1-\int\mathbb{E}_{t}N^{0}(0,p,\xi^{(0,p)})\,\mathbb{Q}(dp).$

######  Remark 11.10.

Assume that $\varphi(x,p,q)=\tilde{\varphi}(\|x\|,p,q)$ and $t=1$, for a
measurable function
$\tilde{\varphi}\colon[0,\infty)\times{\mathbb{M}}\times{\mathbb{M}}\to[0,1]$
which is decreasing and right-continuous in the first coordinate. Using the
notation at (2.2) and (4.3) we define

$\displaystyle
W_{m,n}:=\frac{\|X_{m}-X_{n}\|}{\tilde{\varphi}^{-1}(Z_{m,n},Q_{m},Q_{n})},\quad
m,n\in{\mathbb{N}},$ (11.7)

where $\tilde{\varphi}^{-1}(s,p,q):=\inf\\{r\geq 0:\varphi(r,p,q)\leq s\\}$,
$(s,p,q)\in[0,1]\times{\mathbb{M}}\times{\mathbb{M}}$. Given $r>0$ we define a
RCM $\xi_{r}$ with vertex set $\eta$ by connecting $X_{m}$ with $X_{n}$ if
$W_{m,n}\leq r$. Note that $W_{m,n}\leq r$ iff

$\displaystyle Z_{m,n}\leq\tilde{\varphi}(r^{-1}\|X_{m}-X_{n}\|,Q_{m},Q_{n}).$

Since $\sum^{\infty}_{n=1}\delta_{(r^{-1}X_{n},Q_{n})}$ is under
$\mathbb{P}_{1}$ a Poisson process with intensity measure
$r^{d}\lambda_{d}\otimes\mathbb{Q}$, we hence have

$\displaystyle\mathbb{P}_{1}(\xi_{r}\in\cdot)=\mathbb{P}_{r^{d}}(\xi\in\cdot),\quad
r>0,$ (11.8)

i.e. a joint coupling of the RCMs with different intensity parameters. In the
unmarked case this construction can be found in [4, Example 1.3].

######  Remark 11.11.

Consider the setting of Remark 11.10 and the complete graph with vertex set
$\eta$. We can interpret the random variable (11.7) as weight of the edge
between $(X_{m},Q_{m})$ and $(X_{n},Q_{n})$. As in [3] we define the
associated minimal spanning forest $T$ as the forest (a graph without cycles)
with vertex set $\eta$ and an edge between $(X_{m},Q_{m})$ and $(X_{n},Q_{n})$
if there is no path between these points with weights strictly less than
$W_{m,n}$. In special cases it was observed in [2, 3, 4, 6] that there is a
close relationship between the RCM $\xi_{r}$ and $T$. For instance it was
proved in [3] that the trees (clusters) of $T$ are all infinite and can only
have one or two ends. Two-ended trees $T$ can only occur if $r$ equals the
percolation threshold in which case $T$ contains all points of the infinite
clusters (should they exist). It would be interesting to explore the
consequences of deletion stability of $\xi_{r}$ for $T$.

##  12 The stationary marked RCM: irreducibility and uniqueness

In this section we consider a stationary marked RCM $\xi$ as introduced in
Section 4. When combined with Theorem 11.1, Theorem 6.1 immediately yields the
following result.

######  Theorem 12.1.

An irreducible stationary marked random connection model can almost surely
have at most one infinite cluster.

######  Remark 12.2.

Theorems 12.1 and 6.4 show that an irreducible stationary marked RCM is
$2$-indivisible. In particular this holds at the critical intensity $t_{c}$.
This provides some evidence for the absence of doubly-infinite paths at
criticality. In fact, it is a common belief that in Euclidean space there is
no infinite cluster in the critical phase.

We now present several examples, starting with the classical stationary RCM;
see Example 4.6.

######  Example 12.3.

By Theorem 12.1 and Proposition 5.3 the (unmarked) stationary RCM can have at
most one infinite component. This generalizes [31, Theorem 6.3], where it is
assumed that $\varphi(x)=\tilde{\varphi}(\|x\|)$, $x\in{\mathbb{R}}^{d}$, for
a decreasing function $\tilde{\varphi}\colon[0,\infty)\to[0,1]$. The proof
there is based on an extension of the approach from [9] to the continuum and
is very different from ours.

Next we treat the simple case, where the connection factorizes; see also [10,
Section 1.2].

######  Example 12.4.

Let $\psi\colon{\mathbb{R}}^{d}\to[0,1]$ be a symmetric function with
$0<m_{\psi}:=\int\psi(x)\,dx<\infty$ and let $K\colon{\mathbb{M}}^{2}\to[0,1]$
be measurable and symmetric. Assume that $\varphi(x,p,q)=\psi(x)K(p,q)$,
$(x,p,q)\in{\mathbb{R}}^{d}\times{\mathbb{M}}^{2}$. Then

$\displaystyle\varphi^{(n)}(x,p,q)=\psi^{\ast
n}(x)K^{n}(p,q),\quad(x,p,q)\in{\mathbb{R}}^{d}\times{\mathbb{M}}^{2},$ (12.1)

where $K^{1}=K$ and $K^{n}(p,q)=\int K^{n-1}(p,r)K(r,q)\,\mathbb{Q}(dr)$,
$n\geq 2$. By Proposition 5.1 (iv) and Lemma 5.4 (i) $\xi$ is irreducible iff

$\displaystyle\sup_{n\geq 1}K^{n}(p,q)>0,\quad\mathbb{Q}^{2}\text{-a.e.\
$(p,q)\in{\mathbb{M}}^{2}$}.$

We continue with the examples from Section 4.

######  Example 12.5.

Let us consider Example 4.8. Assume that there exists
$x_{0}\in{\mathbb{R}}^{d}$ (for instance $x_{0}=0$) such for $\mathbb{Q}$-a.e.
$K\in\mathcal{C}^{d}$ there exists $\varepsilon>0$ such that
$B(x_{0},\varepsilon)\subset K$, where $B(x_{0},\varepsilon)$ denotes the ball
with center $x_{0}$ and radius $\varepsilon$. Assume also that the function
$V$ is increasing w.r.t. set inclusion and that $V(K)>0$ if $K\neq\emptyset$.
Under these assumptions the model is irreducible. As point $p_{0}$ in (5.15)
we can take the ball $B(x_{0},\varepsilon_{0})$, where $\varepsilon_{0}>0$ is
chosen such that $\mathbb{Q}(\\{K:B(x_{0},\varepsilon_{0})\subset K\\})>0$. To
check (5.12) we take $K\in\mathcal{C}^{d}$ such that
$B(x_{0},\varepsilon)\subset K$ for some $\varepsilon\in(0,\varepsilon_{0}]$.
We have

$\displaystyle
d_{\varphi}(B(x_{0},\varepsilon_{0}),K)=\int\big{(}1-e^{-V(B(x_{0},\varepsilon_{0})\cap(K+x))}\big{)}\,dx\geq\int\big{(}1-e^{-V(B(x_{0},\varepsilon_{0})\cap(B(x_{0}+x,\varepsilon))}\big{)}\,dx.$

By assumption on $V$ this is positive, since

$\displaystyle\int{\mathbf{1}}\\{B(x_{0},\varepsilon_{0})\cap
B(x_{0}+x,\varepsilon)\neq\emptyset\\}\,dx=\int{\mathbf{1}}\\{B(0,\varepsilon_{0})\cap
B(x,\varepsilon)\neq\emptyset\\}\,dx>0.$

We can now apply Theorem 12.1 to conclude that the infinite cluster is unique.
For the spherical Boolean model this result can be found as Theorem 3.6 in
[31]. For general Boolean models (i.e.
$\varphi(x,K,L)={\mathbf{1}}\\{K\cap(L+x)\neq\emptyset\\}$) the result seems
to be new.

######  Example 12.6.

The weighted RCM from Example 4.9 is irreducible. Indeed we can take any
$p_{0}\in(0,1)$ and $A=[p_{0},1)$. Furthermore we have for all $p,q\in(0,1)$
that $d_{\varphi}(p,q)=m_{\rho}g(p,q)^{-1}$, which is positive. By Theorem
12.1 the infinite cluster is unique. This was asserted in [18] without