# Largest Component and Sharpness in Subcritical Continuum Percolation

Niclas Küpper  Mathew Penrose

(University of Bath
)

###### Contents

  1. 1 Introduction
  2. 2 Background and Main Result
    1. 2.1 Definitions and Overview
    2. 2.2 Main Theorem
    3. 2.3 Standard Tools
  3. 3 Exponential Decay
    1. 3.1 Notation
    2. 3.2 Sharpness
    3. 3.3 Proof of Theorem 3.1
      1. 3.3.1 Item I
      2. 3.3.2 Item II
  4. 4 Auxiliary Result
  5. 5 Proof of Theorem 2.7
    1. 5.1 Exponential decay in volume
    2. 5.2 Properties of the correlation length
    3. 5.3 Proof of Theorem 2.7

##  1 Introduction

Random geometric graphs are common stochastic models studied in statistical
physics, wireless networking and as interesting mathematical objects in their
own right. Most random geometric graphs of interest are defined on lattices or
in the continuum. In this paper, we study the random connection model (RCM), a
continuum random geometric graph that is built on a Poisson point process. It
is characterized by a connection function $\psi$ that describes the
probability of connecting two points.

To construct the RCM we first sample a Poisson point process in Euclidean
space and then connect two points, independent of the rest of the process,
with probability given by $\psi$. It is well known that this model (and many
other random graphs) can have two distinct phases. By tuning the density of
points $\lambda$, we can either find ourselves in the _subcritical_ phase with
no long paths, or in the _supercritical_ phase with many long paths. There
also exists a special _critical_ point $\lambda_{c}$ on the boundary of the
two, whose properties are less understood.

The goal of this paper is to understand large components of the subcritical
RCM. We want to know the size of the largest component of the RCM restricted
to a large box $\Lambda_{t}:=[-t/2,t/2]^{d}$ in dimensions $d\geq 2$. The
central tool we use to prove this is often referred to as sharpness of the
phase transition. It roughly states that in the subcritical phase large
components become exponentially unlikely. It will allow us to estimate the
size distribution of components.

Figure 1: Two instances of a near critical SRGG with parameters
$\lambda=2.5,p=0.35$ and an observation window of side-length $80$. The
largest connected components are colored black.

Some of the first results on the RCM were proved by Meester and Roy and can be
found in their book on continuum percolation [11]. One key result includes the
uniqueness of the infinite component (when it exists) and the agreement of two
separate definitions of criticality. In the case where
$\psi(\cdot)=1\\{|\cdot|\leq 1\\}$ it is known that as $t\to\infty$ that the
largest component is of the order of the $\log$ of the size of the box (see
[14]). Last and Ziesche showed a connection to the Ornstein-Zernike equation
[9] and later Heydenreich et al. developed a lace expansion for the two point
function [8].

In the case where $\psi$ is radially symmetric and decreasing Penrose showed
that full connectivity is governed by the probability of having isolated
vertices in [15].

In the same setting Penrose showed in [16] that the supercritical
($\lambda>\lambda_{c}$) largest component of the SRGG as a fraction of all
points converges to $\theta(\lambda)$ in probability for $d=2$.

In the special case where $\psi(x)=p1\\{|x|\leq 1\\}$ it was shown in [10]
that the supercritical convergence holds almost surely. They also demonstrate
that the critical parameters $\lambda_{c}(p)$ and $p_{c}(\lambda)$ are
inverses of the other.

[2] prove that under some technical assumptions the ‘triangle condition’ holds
in sufficiently high dimensions. [1] show the existance (and calculate the
values of) certain critical exponents assuming the ‘triangle condition’ holds.
[3] develop an expansion of the critical value $\lambda_{c}$ in the limit as
$d\to\infty$.

In Section 2 we construct the model, state the main theorem and describe the
existing tools we require. In Section 3 we prove sharpness by adapting an
argument from [5] for bond percolation to the continuum setting. Before moving
on to the main theorem we show an auxiliary result in Section 4. Finally, we
prove the main theorem by using the sharpness result of Section 3 to adapt the
proof of [14] to the RCM.

##  2 Background and Main Result

###  2.1 Definitions and Overview

To construct the Random Connection Model, we will use a formalism akin to [9]
and [8]. We introduce a measurable connection function
$\psi:\mathbb{R}^{d}\rightarrow[0,1]$ that is symmetric, i.e.,
$\psi(x)=\psi(-x)$, and whose support is contained within the closed ball of
unit radius $D_{1}$ at the origin, i.e., $\text{supp}(\psi)\subset D_{1}$.

We want to rigorously establish a method to take any point process and turn it
into a Random Connection Model. To that end let $\eta$ be any well-behaved
point process in $\mathbb{R}^{d}$. We assume there exists an enumeration
$(X_{i})_{i}$ of $\eta$. Let $<$ denote any strict total order on
$\mathbb{R}^{d}$, such as the lexicographic ordering. Let
$(U_{i,j})_{i,j\in\mathbb{N}}$ be independent uniform random variables on
$[0,1]$ independent of $\eta$. Then we will define $\xi[\eta]$ by

$\xi[\eta]:=\big{\\{}(\\{X_{i},X_{j}\\},U_{i,j})\mid
X_{i},X_{j}\in\eta,X_{i}<X_{j}\big{\\}}.$ (1)

We call $\xi[\eta]$ an _independent edge marking of $\eta$_. We note that the
independent edge marking $\xi[\eta]$, together with a choice of $\psi$,
contains sufficient information to extract the geometric random graph of
interest where

$\displaystyle(V,E)=(\eta,\big{\\{}\\{X_{i},X_{j}\\}\mid
X_{i}<X_{j},U_{i,j}\leq\psi(X_{i}-X_{j}),\forall i,j\in\mathbb{N}\big{\\}}).$
(2)

See [8] for more information.

Let $\mathcal{P}_{\lambda}$ be a homogeneous Poisson point process on
$\mathbb{R}^{d}$ with intensity $\lambda>0$. We will write
$\xi_{\lambda}:=\xi[\mathcal{P}_{\lambda}]$ to simplify notation. We will
require a coupling of the RCM across varying $\lambda$. Note that using the
standard coupling of $\lambda$ will not work, as we change the order of the
edges in $\xi[\mathcal{P}_{\lambda}]$ by increasing the number of Poisson
points. This can be achieved by choosing some maximum $\lambda_{M}$ and taking
the coupling across $\mathcal{P}_{\lambda}$ in the usual way. To fix this, we
label the Poisson points for $\mathcal{P}_{\lambda_{M}}$, and fix this
labeling for all $\mathcal{P}_{\lambda}$ for $\lambda\in[0,\lambda_{M}]$.

Next, we require a method to add additional points to the Poisson point
process without disturbing the already existing edges. This can be achieved by
sampling additional random variables $(U_{-1,n})_{n\in\mathbb{N}}$, and
assigning these to the edges between the added point and all other points, in
effect viewing the additional point as before all others in the total order.
If for some $x\in\mathbb{R}^{d}$ we wish to add $x$ to $\xi_{\lambda}$ we
write $\xi_{\lambda}^{x}:=\xi[\mathcal{P}_{\lambda}\cup\\{x\\}]$, by some
abuse of notation. For further additional points we simply iterate the first
index of $U_{i,j}$ down by one.

Finally we define connected components in the usual way for the graph induced
by the independent edge marking $\xi_{\lambda}$. We write
$C_{x}:=C_{x}(\lambda):=C(x,\xi^{x}_{\lambda})$ for the connected component of
$x\in\mathbb{R}^{d}$.

The object of interest is the _largest connected component_ of this graph when
restricted to a large box (see Definition 2.5). To this end, let $\Lambda_{l}$
denote the box $[-l/2,l/2]^{d}$ with side-length $l$. We also define the
shifted set $\Lambda_{l}(x):=\Lambda_{l}+x$ for all $x\in\mathbb{R}^{d}$. The
largest connected component within this box is represented as
$L_{1}(\xi_{\lambda}\cap\Lambda_{l})$. The number of nodes in this component
is denoted by $|L_{1}(\xi_{\lambda}\cap\Lambda_{l})|$. More generally, we use
$|\cdot|$ to indicate the number of vertices in a graph.

To study the largest component we will require the tools of percolation
theory. For the purposes of this theory, it will be convenient to introduce an
extra point into our Poisson process. Let $o$ denote the origin of
$\mathbb{R}^{d}$. We define $\xi_{\lambda}^{o}$ as the random geometric graph
described earlier, but with the inclusion of an additional point at the
origin. The edges between $o$ and all other points are added in the same
manner as before, independently of everything else. We begin our discussion by
defining a connection between points and sets.

######  Definition 2.1 (Connection).

Let $x,y\in\mathcal{P}_{\lambda}$. We say $x$ and $y$ are _connected_ if a
finite sequence of edges $(e_{i})_{i\leq n}$ exists in $\xi_{\lambda}$ such
that $e_{i}=\\{z_{i},z_{i+1}\\}$ where $z_{0}=x$ and $z_{n+1}=y$. We denote
this event as $\\{x\leftrightarrow y\text{ in }\xi_{\lambda}\\}$. We call two
sets $A,B\subset\mathbb{R}^{d}$ _connected_ if there exist points $x\in
A\cap\mathcal{P}_{\lambda}$ and $y\in B\cap\mathcal{P}_{\lambda}$ such that
$x\leftrightarrow y$. We denote this event as $A\leftrightarrow B$. Lastly,
for the event $\\{x\\}\leftrightarrow A$ in $\xi_{\lambda}^{x}$, we simply
write $x\leftrightarrow A$ in $\xi_{\lambda}^{x}$.

######  Definition 2.2 (Percolation Probability).

Let $\Lambda_{t}^{c}=\mathbb{R}^{d}\setminus\Lambda_{t}$. Given
$\lambda\in(0,\infty)$ and given a connection function $\psi$, the
$t$-percolation probability is defined as

$\theta_{t}(\lambda)=\theta_{t}(\lambda,\psi)=\mathbb{P}[o\leftrightarrow\Lambda_{t}^{c}\text{
in }\xi_{\lambda}^{o}].$ (3)

We define the percolation event as
$\\{o\leftrightarrow\infty\\}:=\bigcap_{t>0}\\{o\leftrightarrow\Lambda_{t}^{c}\\}$.
We define the percolation probability as

$\theta(\lambda)=\mathbb{P}_{\lambda}[o\leftrightarrow\infty\text{ in
}\xi_{\lambda}^{o}]=\lim_{t\to\infty}\theta_{t}(\lambda).$ (4)

######  Remark 2.3.

The quantity $\theta(\lambda)$ is well-defined. The $t$-percolation
probability is monotonically decreasing in $t$ and bounded from below by $0$,
ensuring the limit exists. The quantity $\theta(\lambda)$ can also be
interpreted as the probability that the origin is connected to the unique
infinite component, should it exist. Uniqueness of the infinite component is
established in [11].

######  Definition 2.4 (Critical Parameters).

Given a connection function $\psi$, the critical parameter $\lambda_{c}$ is
defined as

$\lambda_{c}=\lambda_{c}(\psi)=\inf\\{\lambda\mid\theta(\lambda)>0\\}.$ (5)

A value of $\lambda$ is said to be _subcritical_ if $\lambda<\lambda_{c}$. We
define $\inf\varnothing=\infty$, so if $\psi\equiv 0$, then
$\lambda_{c}=\infty$.

######  Definition 2.5 (Largest Component).

The largest component, as measured by the number of vertices, in a bounded
Borel set $B\subset\mathbb{R}^{d}$ is denoted as $L_{1}(\xi_{\lambda}\cap B)$.
In cases of multiple largest components, the one containing the largest point
with respect to lexicographic ordering $<$ is chosen. The order of this
component is denoted by $|L_{1}(\xi_{\lambda}\cap B)|$. When $B$ and $\xi$ are
clear from the context, we will simply write $L_{1}$.

######  Definition 2.6 (Correlation length).

The correlation length for parameters $(\lambda,\psi)$ is defined as

$\displaystyle\zeta(\lambda,\psi)$
$\displaystyle:=\lim_{n\to\infty}-\frac{1}{n}\log\mathbb{P}[|C_{o}|=n\text{ in
}\xi_{\lambda}^{o}]$ (6)
$\displaystyle=\lim_{n\to\infty}-\frac{1}{n}\log\mathbb{P}[n\leq|C_{o}|<\infty\text{
in }\xi_{\lambda}^{o}].$ (7)

We show in Section 5.2 that the limits (6) and (7) are well defined, finite
and equivalent when $\lambda<\lambda_{c}$. Note that $\log$ will always refer
to the natural logarithm. Note also, that the literature sometimes defines
$\zeta(\lambda,\psi)^{-1}$ to be the correlation length.

###  2.2 Main Theorem

######  Theorem 2.7 (Main Theorem: Large Components).

Consider the RCM with connection function $\psi$ having bounded support and
subcritical intensity $\lambda\in(0,\lambda_{c})$. Then

$\frac{|L_{1}(\xi_{\lambda}\cap\Lambda_{s})|}{\log
s}\to\frac{d}{\zeta(\lambda,\psi)}\textup{ in probability},$

as $s\rightarrow\infty$.

######  Remark 2.8.

Note that a weaker version of this result is more straightforward to prove.
Consider the following limit

$\frac{|L_{1}(\xi_{\lambda}\cap\Lambda_{s})|}{s^{d}}\to 0\quad\text{in
probability.}$ (8)

Then, by a simple application of Markov’s inequality together with the Mecke
formula (see Subsection 2.3) we find that

$\displaystyle\mathbb{P}[|L_{1}(\xi_{\lambda}\cap\Lambda_{s})|\geq\varepsilon
s^{d}]$ $\displaystyle\leq\frac{1}{\varepsilon
s^{d}}\mathbb{E}[\sum_{x\in\mathcal{P}_{\lambda}\cap\Lambda_{s}}\mathbf{1}\\{|C_{x}\cap\Lambda_{s}|\geq\varepsilon
s^{d}\\}]$
$\displaystyle\leq\frac{\lambda}{\varepsilon}\mathbb{P}[|C_{o}|\geq\varepsilon
s^{d}]$ $\displaystyle\longrightarrow 0\text{ by subcriticality.}$

To prove the main result, we will first need to prove that the $t$-percolation
probability decays exponentially fast in $t$. This is done in Section 3, where
we will prove a stronger statement commonly called _sharpness_ of the phase
transition. Sharpness refers to an exponential upper-bound on the probability
from the origin $o$ to a distance $t$ and a lower bound on the supercritical
percolation probability.

###  2.3 Standard Tools

Next we discuss the standard tools we can use with this formalism. These tools
were adapted to this context in [9] and [8].

##### The Mecke Equation.

The first crucial tool is the Mecke equation, which allows us to calculate the
expected values of sums. Using their formalism, it was proven in [8]. Given an
$m\in\mathbb{N}$ and a measurable function $f$ of the correct domain mapping
to $\mathbb{R}_{\geq 0}$

and an independent edge marking $\xi_{\lambda}$, the Mecke equation for
$\xi_{\lambda}$ states

$\mathbb{E}\bigg{[}\sum_{\vec{x}\in\mathcal{P}_{\lambda}^{(m)}}f(\xi_{\lambda},\vec{x})\bigg{]}=\lambda^{m}\int_{(\mathbb{R}^{d})^{m}}\mathbb{E}\bigg{[}f(\xi_{\lambda}^{x_{1},\dots,x_{m}},\vec{x})\bigg{]}\mathrm{d}\vec{x},$

where $\mathcal{P}_{\lambda}^{(m)}=\\{\vec{x}\ |\
x_{i}\in\mathcal{P}_{\lambda},:x_{i}\neq x_{j},\forall i\neq j\\}$ and
$\vec{x}=(x_{1},\dots,x_{m})$. This will mostly be used in the case where
$m=1$, in which case we obtain the simpler form

$\mathbb{E}\bigg{[}\sum_{x\in\mathcal{P}_{\lambda}}f(\xi_{\lambda},x)\bigg{]}=\lambda\int_{\mathbb{R}^{d}}\mathbb{E}\bigg{[}f(\xi_{\lambda}^{x},x)\bigg{]}\mathrm{d}x,$

Note that if we force additional points into our point process we have to
modify the Mecke equation as follows

$\displaystyle\mathbb{E}\bigg{[}\sum_{x\in\mathcal{P}_{\lambda}^{y}}f(\xi_{\lambda}^{y},x)\bigg{]}=\mathbb{E}\bigg{[}f(\xi_{\lambda}^{y},y)\bigg{]}+\lambda\int\mathbb{E}\bigg{[}f(\xi_{\lambda}^{x,y},x)\bigg{]}\mathrm{d}x,$

where $y\in\mathbb{R}^{d}$.

##### Margulis-Russo formula.

This formula is standard on lattice models, and has been expanded to the RCM
in [9]. Let $f$ be a function with $\xi[\eta]$ as input. For any Borel set
$\Lambda\subset\mathbb{R}^{d}$ and an edge marking $\xi[\eta]$ of $\eta$ we
define the restriction of $\xi[\eta]$ to the set $\Lambda$ by

$\xi[\eta\cap\Lambda]:=\big{\\{}(\\{x,y\\},u)\in\xi\mid\\{x,y\\}\subset\eta\cap\Lambda\big{\\}}.$

We say a function $f$ lives on a Borel set $\Lambda\subset\mathbb{R}^{d}$ if
$f(\xi[\eta\cap\Lambda])=f(\xi[\eta])$ almost surely. Now assume $f$ lives on
some bounded $\Lambda$ and there exists some $\lambda_{0}>0$ such that
$\mathbb{E}[f(\xi_{\lambda_{0}})]<\infty$. Then the Margulis-Russo formula
states that for every $\lambda\leq\lambda_{0}$ we have

$\frac{\partial}{\partial\lambda}\mathbb{E}[f(\xi_{\lambda})]=\int_{\Lambda}\mathbb{E}[f(\xi^{x}_{\lambda})-f(\xi_{\lambda})]\mathrm{d}x.$

##  3 Exponential Decay

###  3.1 Notation

Recall that we write $C_{x}=C(x,\xi_{\lambda}^{x})$ for the cluster of $x$,
that is, the collection of all $y\in\mathcal{P}_{\lambda}$ such that
$x\leftrightarrow y$ in $\xi_{\lambda}^{x}$. To be clear, we might write that
some event $A$ happens in $\xi[C(x,\xi_{\lambda}^{x})]$, in which case we _do
not_ resample the edges between vertices in $C(x,\xi_{\lambda}^{x})$.

We call a measurable function $f:\mathbb{R}^{d}\to[0,1]$ a _thinning function_
if it has bounded support. Let $\mathcal{T}$ denote the set of all such
functions. Given a point process $\eta$ with (independent) markings
$U_{x}\sim\text{Unif}([0,1])$ we define

$f_{*}\eta:=\\{x\in\eta\mid U_{x}\leq f(x)\\}$ (9)

to be the $f$-thinning of $\eta$. For an independent edge marking $\xi$ and
some choice of $\lambda>0$ we let $U_{x}:=\lambda_{x}/\lambda$.

Figure 2: The $y$-axis describes the intensity of each point and the $x$-axis
represents space. For some function $f$ and a Poisson point process $\eta$ of
intensity $\lambda$. The points marked in blue (below the curve) are the
$f$-thinning of $\eta$.

###  3.2 Sharpness

The term sharpness of the phase transition originates from the coincidence of
two different definitions of criticality. The first is $\lambda_{c}$ as we
have defined it. The second $\lambda^{E}_{c}$ is the point where the quantity
$\mathbb{E}_{\lambda}[|C_{o}|]$ first becomes infinite. It is immediate that
$\lambda_{c}\geq\lambda_{c}^{E}$, since an infinite path from the origin
implies an infinite component. Theorem 3.1 below immediately gives
$\lambda_{c}=\lambda_{c}^{E}$, and is now commonly referred to as sharpness in
the literature.

Our approach to proving sharpness relies heavily on the method developed in
[5]. Similarly to that paper, we define a functional that takes a thinning
function as an input and returns a real number:

$\varphi_{\lambda}(f):=\lambda\int_{\mathbb{R}^{d}}(1-f(y))\mathbb{P}[o\leftrightarrow
y\text{ in }\xi[f_{*}\mathcal{P}_{\lambda}\cup\\{o,y\\}]]\mathrm{d}y.$ (10)

This value can be interpreted as the expected number of points in
$\mathcal{P}_{\lambda}\setminus f_{*}\mathcal{P}_{\lambda}$ that can be
reached from the origin only using points in $f_{*}\mathcal{P}_{\lambda}$. We
can now define a new critical parameter

$\tilde{\lambda}_{c}:=\sup\\{\lambda\mid\exists
f\in\mathcal{T}:\varphi_{\lambda}(f)<1\\}.$ (11)

######  Theorem 3.1 (Sharpness).

For any $d\geq 1$ it holds that $\tilde{\lambda}_{c}=\lambda_{c}$ and

  1. (I)

For all $\lambda<\lambda_{c}$ there exists some $c>0$ such that for all $t$

$\theta_{t}(\lambda,\psi)\leq e^{-ct}.$

  2. (II)

For all $\lambda>\lambda_{c}$ we have

$\theta(\lambda,\psi)\geq\frac{\lambda-\lambda_{c}}{\lambda}.$

######  Remark 3.2.

By using the definition of $\varphi_{t}$ we can recover a standard bound. By
choosing $f\equiv 0$ we get that

$\varphi_{\lambda}(f)=\lambda\int\psi(y)\mathrm{d}y\quad\text{ and hence
}\quad\lambda_{c}\geq\frac{\mathbf{1}}{\int\psi(y)\mathrm{d}y}.$ (12)

###  3.3 Proof of Theorem 3.1

We show items I and II for $\tilde{\lambda}_{c}$. It follows immediately from
I and II that $\tilde{\lambda}_{c}=\lambda_{c}$. We fix
$\lambda\in(0,\lambda_{c})$ and set $\eta=\mathcal{P}_{\lambda}$.

For any given $t$ we identify $\Lambda_{t+2}^{c}$ with an abstract ‘ghost
vertex’ $g_{t}$ which is added to $\xi$ in the same way as other added
vertices. We add an edge between $x\in\eta\cap\Lambda_{t}$ and $g_{t}$ with
probability $\psi(x,g_{t})=1\\{x\in\Lambda_{t+2}\setminus\Lambda_{t}\\}$. Then

$\theta_{t}(\lambda,\psi)=\mathbb{P}[o\leftrightarrow\Lambda_{t}^{c}]=\mathbb{P}[o\leftrightarrow
g_{t}],$ (13)

since any vertex in $\Lambda_{t}^{c}\cap\Lambda_{t+2}$ is by definition of
$\psi(\cdot,g_{t})$ connected to $g_{t}$.

We will require the following lemma from [8, Corollary 3.4].

######  Lemma 3.3 (Stopping Set Lemma).

Let $\lambda>0$. Let $v,u,x\in\mathbb{R}^{d}$ be distinct. Then, for
$\mathbb{P}[C(v,\xi_{\lambda}^{v})\in\cdot]$-a.e. $A$,

$\mathbb{P}\big{[}u\leftrightarrow x\textup{ in }\xi[(\eta\setminus
A)\cup\\{u,x\\}]\mid C(v,\xi^{v})=A\big{]}=\mathbb{P}\big{[}u\leftrightarrow
x\textup{ in }\xi[f^{A}_{*}\eta\cup\\{u,x\\}]\big{]},$ (14)

where $f^{A}(\cdot)=\prod_{z\in A}(1-\psi(z-\cdot))$ is the probability of not
connecting to $A$.

Note that for general applications $f^{A}$ might not have bounded support.

####  3.3.1 Item I

Assume that $\lambda<\tilde{\lambda}_{c}$. Then, by definition, we can and do
choose a thinning function $f$ such that $\varphi_{\lambda}(f)<1$. Choose
$L>1$ such that $\text{supp}f\subset\Lambda_{L-2}$. This implies that all
vertices of $f_{*}\eta$ are in $\Lambda_{L-2}$. We write
$\mathcal{C}:=C(o,\xi[f_{*}\eta\cup\\{o\\}])$ for the component of the origin
of the thinned process.

We remark that $\\{o\leftrightarrow x\text{ in
}\xi[C\cup\\{x\\}]\\}\cap\\{\mathcal{C}=C\\}$ is exactly the same event as
$\\{C\sim x\\}\cap\\{\mathcal{C}=C\\}$, in particular, we remind the reader
that we do not resample the edges when we write $\xi[C]$.

Let $k\in\mathbb{N}$. The event $\\{o\leftrightarrow\Lambda_{kL}^{c}\textup{
in }\xi_{\lambda}^{o}\\}$ holds if and only if there exists some
$x\in\eta\setminus f_{*}\eta$ such that $o\leftrightarrow x$ in
$\xi[f_{*}\eta\cup\\{o,x\\}]$ and $x\leftrightarrow\Lambda_{kL}^{c}$ off
$\mathcal{C}$. Then, by applying the Markov inequality followed by the Mecke
equation we find that

$\displaystyle\theta_{kL}(\lambda)$
$\displaystyle\leq\mathbb{E}\left[\sum_{x\in\eta\setminus
f_{*}\eta}1\\{o\leftrightarrow x\text{ in
}\xi[f_{*}\eta\cup\\{o,x\\}]\\}1\\{x\leftrightarrow\Lambda_{kL}^{c}\text{ off
}\mathcal{C}\\}\right]$ (15)
$\displaystyle=\lambda\int_{\mathbb{R}^{d}}\mathbb{P}[o\leftrightarrow x\text{
in }\xi[f_{*}\eta\cup\\{o,x\\}],x\leftrightarrow\Lambda_{kL}^{c}\text{ off
}\mathcal{C}](1-f(x))\mathrm{d}x$
$\displaystyle=\lambda\int_{\mathbb{R}^{d}}\int\mathbb{P}[o\leftrightarrow
x\text{ in }\xi[C\cup\\{x\\}],x\leftrightarrow\Lambda_{kL}^{c}\text{ in
}\xi[(\eta\setminus C)\cup\\{x\\}]\mid\mathcal{C}=C]$
$\displaystyle\qquad\mathbb{P}[\mathcal{C}\in\mathrm{d}C](1-f(x))\mathrm{d}x.$

Notice that when conditioning on $\mathcal{C}=C$ the two events become
independent. The former depends only on the edges between $C$ and $x$, while
the latter only depends on $\eta\setminus C$ and the edges between itself and
$x$. If $x\not\in\Lambda_{L}$ we have $\mathbb{P}[x\sim C]=0$. By using Lemma
3.3, translation invariance and observing that for a path starting at $x$ to
reach $\Lambda_{kL}^{c}$ it must first reach $\Lambda_{(k-1)L}(x)^{c}$ we
bound

$\mathbb{P}[x\leftrightarrow\Lambda_{kL}^{c}\text{ in }\xi[(\eta\setminus
C)\cup\\{x\\}]\mid\mathcal{C}=C]\leq\mathbb{P}[o\leftrightarrow\Lambda_{(k-1)L}\text{
in }\xi^{o}].$ (16)

Applying this to the inequality we find that

$\displaystyle\theta_{kL}(\lambda)$
$\displaystyle\leq\lambda\iint\mathbb{P}[o\leftrightarrow x\text{ in
}\xi[C\cup\\{x\\}]\mid\mathcal{C}=C]$ (17)
$\displaystyle\qquad\times\mathbb{P}[x\leftrightarrow\Lambda_{kL}^{c}\text{ in
}\xi[(\eta\setminus
C)\cup\\{x\\}]\mid\mathcal{C}=C]\mathbb{P}[\mathcal{C}\in\mathrm{d}C](1-f(x))\mathrm{d}x$
$\displaystyle\leq\lambda\int\mathbb{P}[o\leftrightarrow x\text{ in
}\xi[\mathcal{C}\cup\\{x\\}]](1-f(x))\mathrm{d}x\times\mathbb{P}[o\leftrightarrow\Lambda_{(k-1)L}^{c}\text{
in }\xi^{o}]$ $\displaystyle=\varphi_{\lambda}(f)\theta_{(k-1)L}(\lambda).$

The final line is true simply by definition of $\mathcal{C}$ and
$\varphi_{\lambda}$. By iteration we find that

$\theta_{kL}(\lambda)\leq\varphi_{\lambda}(f)^{k}.$ (18)

Together with the fact that $\theta_{t}$ is decreasing in $t$ we have that
item I holds for all $\lambda<\widetilde{\lambda_{c}}$.

####  3.3.2 Item II

We will prove item II by first showing for all $t\geq 1$ the lower bound

$\frac{\mathrm{d}}{\mathrm{d}\lambda}\theta_{t}(\lambda)\geq\frac{1}{\lambda}\big{(}\inf_{f\in\mathcal{T}}\varphi(f)\big{)}(1-\theta_{t}(\lambda)).$
(19)

Write $u\leftrightarrow v$ _through_ $x$ if _every_ possible path from $u$ to
$v$ passes through $x$. We apply Lemma 3.3 to the ghost-vertex to get the
‘outside component’ (see equation (13)). Write
$\mathcal{D}=C(g_{t},\xi^{o,g_{t}})$ for the outside component (without $x$).
The first equality in the next display comes from the Margulis-Russo formula.
We then marginalize over possible configurations of $\mathcal{D}$. Finally, we
see that for $o\leftrightarrow g_{t}$ through $x$ we require every path
$o\leftrightarrow x$ to avoid $\mathcal{D}$, and for $x$ to connect to
$\mathcal{D}$. In symbols

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\lambda}\theta_{t}(\lambda,\psi)$
$\displaystyle=\int_{\Lambda_{t+2}}\mathbb{P}[o\leftrightarrow g_{t}\text{
through }x\text{ in }\xi[\eta\cup\\{o,x,g_{t}\\}]\mathrm{d}x$ (20)
$\displaystyle=\int_{\Lambda_{t+2}}\int\mathbb{P}[o\leftrightarrow g_{t}\text{
through }x\text{ in
}\xi^{o,x,g_{t}}\mid\mathcal{D}=C]\mathbb{P}[\mathcal{D}\in\mathrm{d}C]\mathrm{d}x$
$\displaystyle=\int_{\Lambda_{t+2}}\int\mathbb{P}[o\leftrightarrow x\text{ in
}\xi[(\eta\setminus C)\cup\\{o,x\\}],x\sim C\mid\mathcal{D}=C]$
$\displaystyle\qquad\qquad 1\\{o\not\in
C\\}\mathbb{P}[\mathcal{D}\in\mathrm{d}C]\mathrm{d}x$

Next, we observe that $o\leftrightarrow x$ and $x\sim C$ are conditionally
independent, as they rely on a disjoint set of edges. Then we will be able to
apply Lemma 3.3. We first define

$\mathcal{T}_{t}:=\\{f:\mathbb{R}^{d}\to[0,1]\text{
measurable}\mid\text{supp}(f)\subset\Lambda_{t}\\},$ (21)

over which we will take an infimum. By definition $g_{t}$ is always in
$\mathcal{D}$. For any possible configuration $C$ of $\mathcal{D}$ we have
$\mathbb{P}[x\sim C]\geq\psi(x,g_{t})$, and in particular
supp$(f^{C})\subset\Lambda_{t}$, so $f^{C}\in\mathcal{T}_{t}$. Hence,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\lambda}\mathbb{P}[o\leftrightarrow
g_{t}]$ $\displaystyle=\int_{\Lambda_{t+2}}\int\mathbb{P}[o\leftrightarrow
x\text{ in }\xi[(\eta\setminus C)\cup\\{o,x\\}]\mid\mathcal{D}=C]$ (22)
$\displaystyle\qquad\qquad P[x\sim C]1\\{o\not\in
C\\}\mathbb{P}[\mathcal{D}\in\mathrm{d}C]\mathrm{d}x$ (23)
$\displaystyle=\int_{\Lambda_{t+2}}\int\mathbb{P}[o\leftrightarrow x\text{ in
}\xi[f_{*}^{C}\eta\cup\\{o,x\\}]](1-f^{C}(x))1\\{o\not\in
C\\}\mathbb{P}[\mathcal{D}\in\mathrm{d}C]\mathrm{d}x$ (24)
$\displaystyle\geq\inf_{f\in\mathcal{T}_{t}}\int_{\Lambda_{t+2}}\int\mathbb{P}[o\leftrightarrow
x\text{ in }\xi[f_{*}\eta\cup\\{o,x\\}]](1-f(x))1\\{o\not\in
C\\}\mathbb{P}[\mathcal{D}\in\mathrm{d}C]\mathrm{d}x.$ (25)

We can expand the domain of integration to all of $\mathbb{R}^{d}$, since the
integrand is $0$ outside of $\Lambda_{t+2}$. Since
$\mathcal{T}_{t}\subset\mathcal{T}$, we can bound the
$\inf_{f\in\mathcal{T}_{t}}$ by $\inf_{f\in\mathcal{T}}$ from below. We find
that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\lambda}\mathbb{P}[o\leftrightarrow
g_{t}]$
$\displaystyle\geq\frac{1}{\lambda}\inf_{f\in\mathcal{T}_{t}}\lambda\int_{\mathbb{R}^{d}}\mathbb{P}[o\leftrightarrow
x\text{ in }\xi[f_{*}\eta\cup\\{o,x\\}]](1-f(x))\mathrm{d}x$ (26)
$\displaystyle\qquad\qquad\times\int 1\\{o\not\in
C\\}\mathbb{P}[\mathcal{D}\in\mathrm{d}C]$ (27)
$\displaystyle\geq\frac{1}{\lambda}\inf_{f\in\mathcal{T}}\varphi_{\lambda}(f)\int
1\\{o\not\in C\\}\mathbb{P}[\mathcal{D}\in\mathrm{d}C]$ (28)
$\displaystyle=\frac{1}{\lambda}\inf_{f\in\mathcal{T}}\varphi_{\lambda}(f)\mathbb{P}[o\nleftrightarrow
g_{t}],$ (29)

where to get (28) from the previous line we apply the definition of $\varphi$
and take the $\inf$ over the larger set $\mathcal{T}$. This gives us (19).

We now derive item II from (19). Note that by definition of
$\tilde{\lambda}_{c}$ we know that for all $\lambda>\tilde{\lambda}_{c}$ and
all $f\in\mathcal{T}$ we have $\varphi_{\lambda}(f)\geq 1$. Then

$\frac{\mathrm{d}}{\mathrm{d}\lambda}\theta_{t}(\lambda)\geq\frac{1}{\lambda}(1-\theta_{t}(\lambda)).\\\
$ (30)

We divide both sides by $1-\theta_{t}$ and integrate from
$\tilde{\lambda}_{c}$ to $\lambda$ to obtain

$\int_{\tilde{\lambda}_{c}}^{\lambda}\frac{\theta^{\prime}_{t}(\lambda^{\prime})}{1-\theta_{t}(\lambda^{\prime})}\mathrm{d}\lambda^{\prime}\geq\int_{\tilde{\lambda}_{c}}^{\lambda}\frac{1}{\lambda^{\prime}}\mathrm{d}\lambda^{\prime}.$
(31)

By $u$-substitution with $u=1-\theta_{t}(\lambda^{\prime})$ we recover

$-\log(1-\theta_{t}(\lambda))+\log(1-\theta_{t}(\tilde{\lambda}_{c}))\geq\log(\frac{\lambda}{\tilde{\lambda}_{c}}).$
(32)

By rearranging we get
$\tilde{\lambda}_{c}(1-\theta_{t}(\tilde{\lambda}_{c}))\geq\lambda(1-\theta_{t}(\lambda))$,
and hence

$\theta_{t}(\lambda)\geq\frac{\lambda-\lambda_{c}(1-\theta_{t}(\tilde{\lambda}_{c}))}{\lambda}.$
(33)

Now we can let $t\to\infty$ and using the fact that
$(1-\theta(\tilde{\lambda}_{c}))\leq 1$ we recover

$\theta(\lambda)\geq\frac{\lambda-\tilde{\lambda}_{c}}{\lambda}.$ (34)

##  4 Auxiliary Result

While the following result is not required to prove the main theorem, it might
still be of interest to the reader.

######  Theorem 4.1.

The percolation probability $\theta(\lambda)$ is right-continuous in
$\lambda.$

###### Proof.

Let $\varepsilon>0$. We have the following estimate for all $N\in\mathbb{N}$:

$1-\theta(\lambda)=\sum_{i\geq 1}\mathbb{P}[|C_{o}|=i\text{ in
}\xi_{\lambda}^{o}]\geq\sum_{i=1}^{N}\mathbb{P}[|C_{o}|=i\text{ in
}\xi_{\lambda}^{o}].$ (35)

We choose $N$ such that $\mathbb{P}[|C_{o}|\leq
N]>1-\theta(\lambda)-\varepsilon/2$. Every term in the above sum is
differentiable, and in particular continuous, in $\lambda$ by [12]. Hence we
can choose some $\delta>0$ so that for every
$\lambda^{\prime}\in(\lambda,\lambda+\delta)$ we find

$\sum_{i=1}^{N}\mathbb{P}[|C_{o}|=i\text{ in
}\xi_{\lambda^{\prime}}^{o}]\geq\sum_{i=1}^{N}\mathbb{P}[|C_{o}|=i\text{ in
}\xi_{\lambda}^{o}]-\varepsilon/2\geq 1-\theta(\lambda)-\varepsilon.$

By our definition of $N$,

$\displaystyle 1-\theta(\lambda^{\prime})=\mathbb{P}[|C_{o}|<\infty\text{ in
}\xi_{\lambda^{\prime}}^{o}]\geq\sum_{i=1}^{N}\mathbb{P}[|C_{o}|=i\text{ in
}\xi_{\lambda^{\prime}}^{o}]>1-\theta(\lambda)-\varepsilon.$ (36)

It follows that
$\theta(\lambda)\leq\theta(\lambda^{\prime})<\theta(\lambda)+\varepsilon$.
Thus $\theta(\lambda)$ is right-continuous everywhere in $\lambda$.

∎

##  5 Proof of Theorem 2.7

We now have the tools to show Theorem 2.7. Remember we are trying to show for
$\lambda\in(0,\lambda_{c})$ that

$\frac{|L_{1}(\xi_{\lambda}\cap\Lambda_{t})|}{\log
t}\longrightarrow\frac{d}{\zeta(\lambda)}\ \text{ in probability.}$

We will first use exponential decay of the $t$-percolation probability to
prove exponential decay of the number of vertices in the component containing
the origin. We then use this fact to show $\zeta(\lambda)$ is well defined,
continuous and decreasing.

###  5.1 Exponential decay in volume

To get a hold of the relevant facts about the correlation length, we first
need an exponential bound on the number of vertices in the component
containing the origin. We adapt the following arguments from [4] to the RCM.
We start with a simple lemma.

######  Lemma 5.1.

Suppose $\lambda<\lambda_{c}$. Let $b\geq a\geq 2$. Then

$\mathbb{P}_{\lambda}[\Lambda_{a}\leftrightarrow\Lambda_{b}^{c}]\leq\lambda
a^{d}\exp(-c(b-a))$

where $c=c(\lambda)$ is the same constant as in Theorem 3.1.

###### Proof.

We apply the Markov inequality followed by the Mecke equation. We find that

$\displaystyle\mathbb{P}_{\lambda}[\Lambda_{a}\leftrightarrow\Lambda_{b}^{c}]$
$\displaystyle\leq\mathbb{E}_{\lambda}[\sum_{x\in\mathcal{P}_{\lambda}(\Lambda_{a})}1_{x\leftrightarrow\Lambda_{b}^{c}}]$
$\displaystyle=\lambda\int_{\Lambda_{a}}\mathbb{P}[x\leftrightarrow\Lambda_{b}^{c}]\mathrm{d}x$
$\displaystyle\leq\lambda\text{Leb}(\Lambda_{a})\mathbb{P}_{\lambda}[o\leftrightarrow\Lambda_{b-a}^{c}]$
$\displaystyle\leq\lambda a^{d}\exp(-c(b-a)).$

∎

######  Lemma 5.2 (Exponential decay of $|C_{o}|$).

For all $\lambda\in(0,\lambda_{c})$ there exist constants
$\tilde{C},\tilde{c}>0$ such that for $n$ sufficiently large we find

$\mathbb{P}[|C_{o}|\geq n]\leq\tilde{C}\exp(-\tilde{c}n).$ (37)

###### Proof.

We follow the proof given in [4, Theorem 3.7]. Recall the notation
$\Lambda_{k}(x):=x+[-\frac{k}{2},\frac{k}{2}]^{d}$. We remind the reader that
the notation $C_{x}$ refers to the connected component of $\xi_{\lambda}^{x}$
containing $x$.

Let $\kappa>4$, to be chosen later. We define a new graph which we will call
$\mathcal{H}_{\kappa}$ with vertex set $\kappa\mathbb{Z}^{d}$ and edges
between vertices $x,y\in\mathcal{H}_{\kappa}$ if and only if
$\|x-y\|_{\infty}\leq\kappa$; this graph has degree $D=3^{d}-1$, where in
particular $D$ is independent of $\kappa$.

We call a finite connected set of vertices of $\mathcal{H}_{\kappa}$ an
_animal_. We denote by $\mathcal{A}(m)$ the set of animals that contain the
origin and are of cardinality $m$. For each animal $A\in\mathcal{A}(m)$ let
$T(A)$ be a maximal stable set of sites in $A$. That is the largest collection
of $x\in A$ such that no two $x,y\in T(A)$ share an edge. In the case of
multiple possible such sets any tie breaker will work.

We shall say that a vertex $x$ in $\mathcal{H}_{\kappa}$ is _good_ if the
event $\\{\Lambda_{\kappa}(x)\leftrightarrow\Lambda_{3\kappa/2}(x)^{c}\text{
in }\xi_{\lambda}^{o}\\}$ occurs. We shall say an animal $A$ is good if every
$x\in A$ is good.

Let $c^{\prime}\in(0,1)$, to be chosen later. We set
$m:=m(n):=\lfloor\frac{c^{\prime}n}{\lambda\kappa^{d}}\rfloor$. Let $F_{m}$ be
the event that there exists some $A\in\mathcal{A}(m)$ such that $A$ is good.
Then

$\displaystyle\mathbb{P}[|C_{o}|\geq
n]\leq\mathbb{P}[F_{m}]+\mathbb{P}[|C_{o}|\geq n,F_{m}^{c}].$ (38)

First we bound $\mathbb{P}[|C_{o}|\geq n,F_{m}^{c}]$. We define
$C^{*}_{o}:=\\{x\in\kappa\mathbb{Z}^{d}\ |\ \Lambda_{\kappa}(x)\cap
C_{o}\neq\varnothing\\}$, to be the minimal animal that contains $C_{o}$. Then
$C_{o}^{*}$ is connected, $o\in C_{o}^{*}$ and if $|C_{o}^{*}|>D+1$, then all
sites in $C_{o}^{*}$ are good. It follows that if $m\geq D+1$ and
$|C_{o}^{*}|\geq m$ then $F_{m}$ occurs. Also if $|C_{o}^{*}|<m$ then there
exists at least one animal $A\in{\cal A}(m)$ with $C_{o}^{*}\subset A$. Hence,
if $n$ is large enough so that $m\geq D+1$, then

$\displaystyle\mathbb{P}[|C_{o}|\geq n,F_{m}^{c}]$
$\displaystyle\leq\mathbb{P}[|C_{o}|\geq n,|C_{o}^{*}|<m]$
$\displaystyle\leq\mathbb{P}\bigg{[}\bigcup_{A\in\mathcal{A}(m)}\\{\eta\big{(}\cup_{x\in
A}\Lambda_{\kappa}(x)\big{)}\geq n\\}\bigg{]}$
$\displaystyle\leq\sum_{A\in\mathcal{A}(m)}\mathbb{P}[\text{Pois}(\lambda\kappa^{d}|A|)\geq
n].$

By [14, Lemma 9.3] we know that $|\mathcal{A}(m)|\leq 2^{Dm}$. Hence,

$\displaystyle\mathbb{P}[|C_{o}|\geq n,F_{m(n)}^{c}]$ $\displaystyle\leq
2^{mD}\mathbb{P}[\text{Pois}(\lambda m\kappa^{d})\geq n]$ $\displaystyle\leq
2^{mD}\mathbb{P}[\text{Pois}(c^{\prime}n)\geq n].$

Assume $c^{\prime}\leq e^{-4}$. Then by [14, Lemma 1.2, eq. (1.12)] we have
that $\mathbb{P}[\text{Pois}(c^{\prime}n)\geq n]\leq e^{-2n}$. Assume also
that $c^{\prime}\leq\lambda\kappa^{d}/(D\log 2)$; then

$2^{mD}\leq\exp((D\log 2)c^{\prime}n/(\lambda\kappa^{d}))\leq e^{n}.$

Combining these estimates yields

$\mathbb{P}[|C_{o}|\geq n,F_{m}^{c}]\leq\exp(-n).$ (39)

Next we bound $\mathbb{P}[F_{m}]$. We use the fact that the goodness of sites
in $T(A)$ are independent, since the boxes that define them do not intersect.
It is always possible to find a set $T(A)$ of cardinality at least $(m-1)/D$.
Such a set can be constructed iteratively by using a ‘greedy’ approach;
subsequently adding vertices adjacent to neighbors of already added vertices
in such a way there they are not a direct neighbor of any already chosen
vertex. Then by the union bound,

$\displaystyle\mathbb{P}[F_{m}]$
$\displaystyle\leq\sum_{A\in\mathcal{A}(m)}\mathbb{P}[A\text{ is good}]$
$\displaystyle\leq\sum_{A\in\mathcal{A}(m)}\mathbb{P}[\forall x\in
T(A):x\text{ is good}]$
$\displaystyle=\sum_{A\in\mathcal{A}(m)}\mathbb{P}[\Lambda_{\kappa}\leftrightarrow\Lambda_{3\kappa/2}^{c}]^{|T(A)|}$
$\displaystyle\leq|\mathcal{A}(m)|\mathbb{P}[\Lambda_{\kappa}\leftrightarrow\Lambda_{3\kappa/2}^{c}]^{m/D}.$

We can now apply Lemma 5.1 and pick $\kappa>2$ sufficiently large such that

$\mathbb{P}[\Lambda_{\kappa}\leftrightarrow\Lambda_{3\kappa/2}^{c}]\leq\lambda\kappa^{d}\exp(-c\kappa/2)\leq\frac{1}{e}\cdot
2^{-D^{2}}.$

Using $|A(m)|\leq 2^{Dm}$ again, we thus find that

$\displaystyle\mathbb{P}[F_{m}]$ $\displaystyle\leq
2^{mD}(e^{-m/D}2^{-mD})\leq\exp(-c^{\prime}n/(2\lambda\kappa^{d}D)).$ (40)

Putting (38), (39) and (40) together gives us (37). ∎

###  5.2 Properties of the correlation length

To prove Theorem 2.7, the only missing ingredient is a more detailed
understanding of the correlation length $\zeta$. We recall the Definition 2.6.

$\displaystyle\zeta(\lambda):=\zeta(\lambda,\psi):$
$\displaystyle=\lim_{n\to\infty}-\frac{1}{n}\log\mathbb{P}[|C_{o}|=n\text{ in
}\xi_{\lambda}^{o}]$ (6)
$\displaystyle=\lim_{n\to\infty}-\frac{1}{n}\log\mathbb{P}[n\leq|C_{o}|<\infty\text{
in }\xi_{\lambda}^{o}].$ (7)

We start with showing that the limit exists using the following lemma, a proof
of which can be found in [7].

######  Lemma 5.3 (Fekete’s Subadditivity Lemma).

Let $(u_{n})_{n}\subset\mathbb{R}$ be a sequence of numbers. If for all $n$
and $m$ we have $u_{n+m}\leq u_{n}+u_{m}$, then

$\lim_{n\to\infty}\frac{u_{n}}{n}=\inf_{n}\frac{u_{n}}{n}\in[-\infty,\infty).$

######  Theorem 5.4.

Let $\lambda\in(0,\lambda_{c})$. The correlation length $\zeta(\lambda)$ is
well defined. It is continuous and non-increasing in $\lambda$. As $\lambda$
goes to zero $\zeta(\lambda)\to\infty$.

The following proof is based on [14], where the same fact is shown for the
case where $\psi(x)=1\\{\|x\|\leq 1\\}$.

###### Proof of Theorem 5.4.

First we show convergence of (6). We then show equality of (6) and (7). Then
we show continuity and the limiting behavior as $\lambda$ goes to zero.

######  Lemma 5.5 ($\zeta$ is well-defined).

The limit (6) exists, and $\zeta(\lambda)\in[0,\infty)$.

###### Proof.

To prove convergence we use  (Fekete’s Subadditivity Lemma).. We define
$p_{n}:=\mathbb{P}[|C_{o}(\xi_{\lambda}^{o})|=n]$. Showing subadditivity for
$-\log p_{n}$ is equivalent to showing supermultiplicativity for $p_{n}$. We
will show $p_{n+1}p_{m+1}\leq p_{n+m+1}$ which implies Lemma 5.5 by a simple
reindexing via $q_{n}=p_{n+1}$.

We know by [12] (see also [11, Proposition 6.2]) that

$p_{n+1}={\lambda^{n}}\int_{(\mathbb{R}^{d})^{n}}g_{2}(o,x_{1},\dots,x_{n})\times\exp\big{(}-\lambda\int_{\mathbb{R}^{d}}g_{1}(y;o,x_{1},\dots,x_{n})\mathrm{d}y\big{)}\mathrm{d}\vec{x},$

where $g_{1}(y;z_{1},\dots,z_{k})$ is the probability that $y$ is not isolated
from the points $z_{1},\dots,z_{k}$ in $\xi[\\{y,z_{1},\dots,z_{n}\\}]$ and
$g_{2}(z_{1},\dots,z_{k})$ is the probability that $z_{1},\dots,z_{k}$ are in
one connected component and ordered $z_{2}<\dots<z_{n}$ for some order (e.g.
lexicographic) in $\xi[\\{z_{1},\dots,z_{k}\\}]$. We use $\vec{z}$ to
represent $(z_{1},\dots,z_{k})$. Combining we find

$\displaystyle p_{n+1}p_{m+1}$
$\displaystyle=\lambda^{n+m}\int_{(\mathbb{R}^{d})^{n}}\int_{(\mathbb{R}^{d})^{m}}g_{2}(o,x_{1},\dots,x_{n})g_{2}(o,y_{1},\dots,y_{m})$
$\displaystyle\quad\times\exp\bigg{(}-\lambda\Big{[}\int_{\mathbb{R}^{d}}g_{1}(z;o,x_{1},\dots,x_{n})\mathrm{d}z+\int_{\mathbb{R}^{d}}g_{1}(z;o,y_{1},\dots,y_{m})\mathrm{d}z\Big{]}\bigg{)}\mathrm{d}\vec{y}\mathrm{d}\vec{x}$
$\displaystyle=\lambda^{n+m}\int_{(\mathbb{R}^{d})^{n}}\int_{(\mathbb{R}^{d})^{m}}g_{2}(o,x_{1},\dots,x_{n})g_{2}(x_{n},y_{1},\dots,y_{m})$
$\displaystyle\quad\times\exp\bigg{(}-\lambda\Big{[}\int_{\mathbb{R}^{d}}g_{1}(z;o,x_{1},\dots,x_{n})+g_{1}(z;x_{n},y_{1},\dots,y_{m})\mathrm{d}z\Big{]}\bigg{)}\mathrm{d}\vec{y}\mathrm{d}\vec{x}$

The second equality follows from translation invariance and substitution
$y_{i}\to y_{i}+x_{n}$. Connecting $m$ different points to $o$ has the same
probability as connecting $m$ different points to $x_{n}$ when they are all
shifted by $x_{n}$. We rename $y_{1},\dots,y_{m}$ to $x_{n+1},\dots,x_{n+m}$.
We now reduced the problem to showing

$g_{2}(o,x_{1},\dots,x_{n})g_{2}(x_{n},x_{n+1},\dots,x_{n+m})\leq
g_{2}(o,x_{1},\dots,x_{n+m})$

and

$g_{1}(z;o,x_{1},\dots,x_{n})+g_{1}(z;x_{n},\dots,x_{n+m})\geq
g_{1}(z;o,x_{1},\dots,x_{n+m})$

The first equation follows, since the union of two connected graphs with a
vertex in common is again a connected graph and the ordering is maintained.
The second equation holds by the union bound.

We now get existence by choosing $u_{n}:=-\log\mathbb{P}_{\lambda}[|C_{o}|=n]$
and applying Lemma 5.3. Furthermore, since the $u_{n}$’s are lower bounded by
$0$ we get the stronger bound that $\zeta(\lambda)\in[0,\infty)$. We also find
immediately by Lemma 5.2 that if $\lambda<\lambda_{c}$ we have
$\zeta(\lambda)>0$:

$-\frac{1}{n}\log(\tilde{C}\exp(-\tilde{c}n))\geq\tilde{c}-\frac{\log\tilde{C}}{n}\xrightarrow{n\to\infty}\tilde{c}>0.$
(41)

∎

##### Equivalence of definitions

Next, we show that the two different definitions (6) and (7), of the
correlation length are indeed equivalent. It suffices to show that

$q_{n}:=q_{n}(\lambda):=\Big{(}\frac{\mathbb{P}[|C_{o}|=n]}{\mathbb{P}[n\leq|C_{o}|<\infty]}\Big{)}^{\frac{1}{n}}\xrightarrow{n\to\infty}1$

We immediately know that $q_{n}\leq 1$, so we only need to bound it from
below. Let $0<\zeta^{-}<\zeta(\lambda)<\zeta^{+}$, to be chosen later. For now
$\zeta(\lambda)$ will refer to the definition using $\mathbb{P}[|C_{o}|=n]$,
i.e. (39). Then, by existence of the limit (Lemma 5.5), we know that for $n$
large enough

$e^{-n\zeta^{+}}<\mathbb{P}[|C_{o}|=n]<e^{-n\zeta^{-}}.$ (42)

This gives us the following bounds on
$\mathbb{P}[n\leq|C_{o}|<\infty]=\sum_{k\geq n}\mathbb{P}[|C_{o}|=n]$,

$\frac{e^{-\zeta^{+}n}}{1-e^{-\zeta^{+}}}<\mathbb{P}[n\leq|C_{o}|<\infty]<\frac{e^{-\zeta^{-}n}}{1-e^{-\zeta^{-}}}.$
(43)

For the above bound we used the fact that $\zeta^{-}>0$ to obtain convergence
for the geometric series. This ceases to be possible if $\zeta(\lambda)=0$.
This, however, only happens in the critical and supercritical cases, which we
disregard here.

Now let $\varepsilon>0$. We want to show the existence of an $n_{0}$ such that
for all $n\geq n_{0}$ we have $q_{n}\in(1-\varepsilon,1]$. We now choose
$\zeta^{\pm}=\zeta(\lambda)\pm\varepsilon/4$. Let $n$ be sufficiently large,
such that (42) holds and

$e^{-\varepsilon/2}(1-e^{-\zeta^{-}})^{\frac{1}{n}}>(1-\varepsilon).$ (44)

Now we bound $q_{n}$ using (42) and (43):

$\displaystyle
q_{n}>\frac{e^{-\zeta^{+}}(1-e^{-\zeta^{-}})^{\frac{1}{n}}}{e^{-\zeta^{-}}}.$
(45)

Substituting for the defintion of $\zeta^{\pm}$ together with (44) we find
that

$\displaystyle q_{n}>1-\varepsilon.$

Thus the two definitions of $\zeta$ are equivalent.

##### Continuity and monotonicity

For continuity and monotonicity of $\zeta$ we follow [14, Theorem 10.1]. First
we show that $\zeta$ is non increasing and continuous. Consider the quantities
$q_{n}(\lambda)=\mathbb{P}[|C_{o}|=n]$ and
$q^{+}_{n}:=\mathbb{P}[n\leq|C_{o}|<\infty]$. It is easy to see that
$q^{+}_{n}$ is increasing in $\lambda$ for every $n$ in the subcritical
regime. We now define

$\rho(\lambda):=\lim_{n\to\infty}q^{+}_{n}(\lambda)^{1/n}=e^{-\zeta(\lambda)}.$

We see that $\rho$ is non-decreasing, which in turn shows that $\zeta$ is non-
increasing in $\lambda$.

We move on to continuity in $\lambda$. We couple the RCM at different
intensities $\lambda$ by giving every point $X_{i}$ a marking $\lambda_{i}$.
For a given intensity $\lambda$ we simply retain all points where
$\lambda_{i}\leq\lambda$. This will help us reason about thinnings.

Consider $0<\lambda<\mu<\lambda_{c}$. One way for $|C_{o}|=n$ in
$\xi^{o}_{\lambda}$ to hold is to require $|C_{o}|=n$ in $\xi^{o}_{\mu}$ and
all $(X_{i},\lambda_{i})\in C_{o}$ have the property that
$\lambda_{i}<\lambda$, which has probability $\lambda/\mu$, per vertex. This
gives

$\mathbb{P}[|C_{o}|=n\text{ in
}\xi^{o}_{\lambda}]\geq(\frac{\lambda}{\mu})^{n-1}\mathbb{P}[|C_{o}|=n\text{
in }\xi^{o}_{\mu}].$ (46)

Then

$\rho(\lambda)=\lim_{n\to\infty}q_{n}(\lambda)^{1/n}\geq\lim_{n\to\infty}((\frac{\lambda}{\mu})^{n-1}q_{n}(\mu))^{1/n}=\frac{\lambda}{\mu}\rho(\mu).$

This together with the fact that $\rho$ is non-decreasing gives the continuity
for $\rho$. Fix some $\lambda\in(0,\lambda_{c})$. Let $\varepsilon>0$,
$\delta=\lambda\varepsilon/{\rho(\lambda)}$ and
$\lambda^{\prime}\in(\lambda-\delta,\lambda+\delta)$. Assume
$\lambda^{\prime}>\lambda$. Then

$\rho(\lambda^{\prime})-\rho(\lambda)\leq\frac{\lambda^{\prime}}{\lambda}\rho(\lambda)-\rho(\lambda)\leq\rho(\lambda)\frac{\delta}{\lambda}\leq\varepsilon.$

The case where $\lambda^{\prime}<\lambda$ is analogous.

Continuity of $\rho$ implies that $\zeta$ is also continuous in $\lambda$ in
the range $(0,\lambda_{c})$.

##### Near zero behavior

We show as $\lambda\to 0$ that $\zeta(\lambda,\psi)\to\infty$. We show this by
bounding the size of the component of the origin by the total progeny (total
number of vertices) of a Galton-Watson tree. For more details see [13] where a
similar strategy was used. We use the following theorem by [6].

######  Lemma 5.6 (Dwass’ formula).

Let $\tau$ be a Galton-Watson tree with offspring distribution $\nu$. Let
$|\tau|$ be its total progeny. We let $N_{k}:=X_{1}+\dots+X_{k}$ where
$X_{j}\sim\nu$ iid. Then

$\mathbb{P}[|\tau|=k]=\frac{1}{k}\mathbb{P}[N_{k}=k-1].$ (47)

We know that the origin has $\operatorname{Pois}(\lambda\int\psi\mathrm{d}x)$
many offspring. For simplicity we write $\alpha=\lambda\int\psi\mathrm{d}x$.
We call $\tau$ the Galton-Watson tree with offspring distribution
$\operatorname{Pois}(\alpha)$. We know that $|\tau|$ dominates $|C_{o}|$ in
the sense that (see [13])

$\mathbb{P}[|\tau|\geq k]\geq\mathbb{P}[|C_{o}|\geq k].$ (48)

We know that for $\alpha<1$ the Galton-Watson tree is subcritical and so the
total progeny $|\tau|$ of the tree is almost-surely finite. By using  (Dwass’
formula). we find that

$\displaystyle\mathbb{P}[|\tau|\geq k]$
$\displaystyle=\sum_{j=k}^{\infty}\frac{1}{j}\mathbb{P}[\operatorname{Pois}(j\alpha)=j-1]$
(49)
$\displaystyle=\sum_{j=k}^{\infty}\frac{e^{-j\alpha}(j\alpha)^{j-1}}{j!}.$

By the fact that for all $j\in\mathbb{N}$ we have $\frac{j^{j}}{j!}\leq e^{j}$
we find that

$\displaystyle\mathbb{P}[|\tau|\geq k]$
$\displaystyle\leq\frac{1}{\alpha}\sum_{j=k}^{\infty}(e^{(1-\alpha)}\alpha)^{j}$
(50)
$\displaystyle=\frac{(e^{1-\alpha}\alpha)^{k}}{\alpha(1-e^{1-\alpha}\alpha)}.$

By substituting $\lambda\int\psi\mathrm{d}x$ back in for $\alpha$ we find that

$\displaystyle\limsup_{n\to\infty}\frac{1}{n}\log\mathbb{P}[|C_{o}|\geq n]$
$\displaystyle\leq\limsup_{n\to\infty}\frac{1}{n}\log\mathbb{P}[|\tau|\geq n]$
(51)
$\displaystyle\leq\limsup_{n\to\infty}\Big{(}(1-\alpha)+\log\alpha-\frac{1}{n}\log\alpha-\frac{1}{n}\log(1-e^{1-\alpha}\alpha)\Big{)}$
$\displaystyle\leq
1-\lambda\int\psi\mathrm{d}x+\log\big{(}\lambda\int\psi\mathrm{d}x\big{)}.$

And so the upper bound tends to $-\infty$ as $\lambda$ goes to zero. Hence, by
applying Definition 2.6, we have that $\zeta(\lambda,\psi)\to\infty$,
completing the proof of Theorem 5.4. ∎

###  5.3 Proof of Theorem 2.7

The following proof closely follows the proof on the log bound of the Poisson
Boolean model by Penrose in [14, Theorem 10.3].

###### Proof of Theorem 2.7.

We start by showing that the largest component in a box with side lengths $s$
is no larger than $\frac{d}{\zeta(\lambda)}\log s$. Let
$\alpha>\frac{d}{\zeta(\lambda)}$. By applying the Markov bound and then the
Mecke formula we find

$\displaystyle\mathbb{P}[|L_{1}(\mathcal{\xi}_{\lambda}\cap\Lambda_{s})|\geq\alpha\log
s]$
$\displaystyle\leq\mathbb{E}[\sum_{x\in\mathcal{P}_{\lambda}}1\\{|C(x,\xi^{x}\cap\Lambda_{s})|\geq\alpha\log
s\\}]$
$\displaystyle=\lambda\int_{\Lambda_{s}}\mathbb{P}[|C(x,\xi^{x}\cap\Lambda_{s})|\geq\alpha\log
s\text{ in }\xi^{x}_{\lambda}]dx$

Now consider some $\zeta^{\prime}\in(\frac{d}{\alpha},\zeta(\lambda))$. By
definition (7) of $\zeta(\lambda)$ as a limit, we know that for $n$ and $s$
large enough

$\displaystyle\mathbb{P}[|C(x,\xi^{x}\cap\Lambda_{s})|\geq\alpha\log s\text{
in }\xi^{x}_{\lambda}]$ $\displaystyle\leq\mathbb{P}[|C_{o}|\geq\alpha\log
s\text{ in }\xi^{o}_{\lambda}]$
$\displaystyle\leq\exp(-\zeta^{\prime}\alpha\log
s)=s^{-\zeta^{\prime}\alpha}.$

Hence, by our choices of $\alpha$ and $\zeta^{\prime}$, we find

$\displaystyle\mathbb{P}[|L_{1}(\mathcal{\xi}_{\lambda}\cap\Lambda_{s})|\geq\alpha\log
s]\leq\lambda s^{d-\zeta^{\prime}\alpha}\xrightarrow{s\to\infty}0.$

For the other direction we choose $\beta<{d}/{\zeta(\lambda)}$ and
$\zeta^{\prime\prime}\in(\zeta(\lambda),d/\beta)$. We will tile the box
$\Lambda_{s}$ with smaller boxes. Let $m(s)=\lfloor\frac{s}{2\beta\log
s}\rfloor^{d}$ denote the number of boxes. Let
$\\{B_{1,s},\dots,B_{m(s),s}\\}$ be the maximal collection of disjoint boxes
with side-length $s/\sqrt[d]{m(s)}\geq 2\beta\log s$. Thus, the boxes
$B_{i,s}$ fill $\Lambda_{s}$ exactly. Let $x_{i,s}$ denote the center of the
box $B_{i,s}$. Now take $\lambda^{\prime}\in(0,\lambda)$ such that
$\zeta(\lambda^{\prime})<d/\beta$. This is possible by the continuity of
$\zeta$. We can separate $\mathcal{P}_{\lambda}$ into a union of
$\mathcal{P}_{\lambda^{\prime}}$ and

$\mathcal{P}_{\lambda^{\prime},\lambda}=\big{\\{}(X_{i},\lambda_{i})\mid\lambda^{\prime}\leq\lambda_{i}<\lambda\big{\\}}.$

Now take $\zeta(\lambda^{\prime})<\zeta^{\prime\prime}<d/\beta$.

Let $x\in\mathbb{R}^{d}$ and $r>0$. Denote by $D(x,r)$ the closed ball with
radius $r$ centered at $x$. If $\mathcal{P}_{\lambda^{\prime},\lambda}\cap
D(x_{i,s},1)$ consists of a single point we denote that point by $X_{i,s}$ and
let

$V_{i,s}:=|C(X_{i,s},\mathcal{\xi}^{X_{i,s}}_{\lambda^{\prime}}\cap
B_{i,s})|,$

where $X_{i,s}$ inherits the connections from the original sampling of
$\xi_{\lambda}$. If $|\mathcal{P}_{\lambda^{\prime},\lambda}\cap B_{i,s}|\neq
1$ then let $V_{i,s}=0$. Let $\mu$ be the volume of a $d$-dimensional unit
ball. By our choice of box size we know that $\\{0<V_{i,s}<\beta\log
s\\}\subset\\{C_{X_{i,s}}(\xi_{\lambda^{\prime}})\subset B_{i,s}\\}$ and so
$V_{i,s}$ is has the distribution of the size of the component of the origin.
Then, by independence of $\mathcal{P}_{\lambda^{\prime}}$ and
$\mathcal{P}_{\lambda^{\prime},\lambda}$, we find for large $s$ that

$\displaystyle\mathbb{P}[V_{i,s}\geq\beta\log s]$
$\displaystyle\geq\mu(\lambda-\lambda^{\prime})e^{-\mu(\lambda-\lambda^{\prime})}\mathbb{P}[|C_{o}|\geq\beta\log
s\text{ in }\xi_{\lambda^{\prime}}^{o}]$ $\displaystyle\geq
c^{\prime}\exp(-\zeta^{\prime\prime}\beta\log
s)=c^{\prime}s^{-\zeta^{\prime\prime}\beta},$

where the inequality follows from the definition of $\zeta$ and
$c^{\prime}=\mu(\lambda-\lambda^{\prime})e^{-\mu(\lambda-\lambda^{\prime})}$.
The random variables $V_{i,s}$ are independent, since they are dependent on
configurations in disjoint boxes. It follows that

$\displaystyle\mathbb{P}\big{[}\bigcap_{i=1}^{m(s)}\\{V_{i,s}<\beta\log
s\\}\big{]}$
$\displaystyle\leq\big{(}1-c^{\prime}s^{-\zeta^{\prime\prime}\beta}\big{)}^{m(s)}$
$\displaystyle\leq\exp(-c^{\prime}s^{-\zeta^{\prime\prime}\beta}m(s))$

which tends to zero by the definition of $m(s)$ and the fact that
$\zeta^{\prime\prime}\beta<d$. On the other hand, if for some $i$ we have
$V_{i,s}\geq\beta\log s$, then
$L_{1}(\xi_{\lambda}\cap\Lambda_{s})\geq\beta\log s$. This gives us the
desired result. ∎

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