# Gravitational Wave luminosity distance in viscous cosmological models

Giuseppe Fanizza  Eliseo Pavone  and Luigi Tedesco

###### Abstract

We study the so-called Gravitational Wave luminosity distance-redshift
relation $d_{L}^{\,GW}(z)$ during cosmological eras driven by non-perfect
fluids. In particular, we show that the presence of a shear viscosity in the
energy momentum tensor turns out to be the most relevant effect. Within this
scenario, a constant shear viscosity imprints the gravitational wave
propagation through a friction term $\delta(z)$ with a uniquely given redshift
dependence. This peculiar evolution predicts a specific shape for the ratio
$d_{L}^{GW}/d_{L}^{EM}$ which tends to a constant value when the sources are
at $z\gtrsim 1$, whereas scales linearly with the shear viscosity at lower
redshifts, regardless of the value of $\Omega_{m0}$. According to our final
discussion, the predicted redshift dependence $\delta(z)$ provided by a shear
viscosity could be tested by upcoming surveys of multi-messenger sources
against analogous scenarios provided by some widely studied theories of
modified gravity.

##  1 Introduction

The detection of Gravitational Waves (GWs) from black holes coalescence [1, 2,
3, 4, 5] has opened a new window for exploring new phenomena of interest for
astrophysics and cosmology. Moreover, the subsequent detection of GWs emitted
by the binary neutron stars [6], associated to the analysis of the
electromagnetic (EM) counterpart [7] (in particular $\gamma$-ray burst [8, 9,
10]) led us to the era of the GWs astrophysics and multimessenger cosmology.
In this regards, GWs have opened a new window to understand the nature of
gravity [11, 12, 13, 14, 15]. Among all these new kinds of observations, the
possibility to have an absolute measurement of the luminosity distance of
coalescing compact binaries has gained an increasing interest [16]. Indeed,
the signal of coalescing binaries combines intrinsic and apparent luminosity
in order to have a determination of the luminosity distance (from which the
name standard sirens) [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30].

On the other hand, detection of signals like GW170817, observed at $z\simeq
0.009$, can be used to constrain deviations from $\Lambda$CDM model for what
concerns either the nature of gravity or the features of the sources. One of
the most remarkable cases consists of the simultaneous study of
electromagnetic and gravitational signals emitted by the same source: this has
provided stringent constraints about differences in their speed of propagation
[31, 32, 33, 34, 35] which, for instance, ruled out entire classes of models
such as scalar-tensor and vector-tensor theories of gravity [31, 33, 34, 35].
Another idea to be mentioned has been proposed in [30], where it has been
shown that GWs could be used as an independent tool to study the Hubble-
Lemaître diagram. In fact, the localization of the host galaxy for GW170817
allows to determine $H_{0}$ by the gravitational luminosity distance whereas
the electromagnetic counterpart provides the redshift. Another interesting but
futuristic idea regards the possibility to study different messengers from the
same source through the so-called time-of-flight distance [36, 37].

The different nature of the gravitational signals can shed light on the
structure and the evolution of the universe up to early epochs (from the
inflationary period to the recombination) that we are unable to probe directly
using EM waves. However, for “near” sources, in the next future we could be
able to compare both EM and GW signals. It is known that distances in
astronomy are calibrated using different techniques, the so-called distance
ladder. For near astronomical objects ($z\approx 1$), measurements are usually
done using the so called Standard Candles (Supernova IA), from which the
accelerated expansion of the universe was firstly observed in 1998 [38]. It is
possible to use also binary mergers as standard sirens, which could emit GW
and also EM when probably one of the objects is a neutron star. These events
are known as Ultranovas and Kilonovas and in [39] it is argued how the latter
can be standardized (like Supernova IA) and how the analysis of the distance
given by the GW measurements (which does not rely on any distance ladder nor
cosmological model), with the redshift, given by the EM counterpart, might
provide new probes for measuring the Hubble constant $H_{0}$.

In this work we discuss how these new observational probes can be investigated
to detect modifications on the gravitational luminosity distance as provided
by a cosmological viscous fluid. The machinery that we will adopt to this end
is in line with to the one developed for the study of the GW’s distance-
redshift relation concerning theories of modified gravity [40, 41, 42]. The
key aspect that we will show is that a viscous fluid can provide a non-trivial
ratio between the GW and the EM luminosity distance already within the
$\Lambda$CDM model. This interesting aspect emerges when a pure shear
viscosity is considered in the cosmic fluid. Indeed, such a modification of
the perfect fluid description leaves unchanged the Friedmann equations but can
affect the linear part of the perturbations related to the Weyl components of
the Riemann tensor, as one would expect by a traceless perturbations of the
energy-momentum tensor like a shear viscosity.

After having discussed the general framework for the GW’s luminosity distance
in the above-mentioned scenario, we will discuss possible observational tests
and forecasts for the detection/exclusion of the shear viscosity, according to
the specifics of futuristic surveys, such as LISA and Einstein Telescope (ET).

This paper is organized as follows: in Sect. 2, we recall the evolution
equation for linear gravitational waves in presence of viscous fluids and
provide the general expression for $d_{L}^{\,GW}(z)$. In Sect. 3, we first
study our results with a generic power-law solution for the scale factor.
Hence we provide an analytical expression for $d_{L}^{\,GW}(z)$ in presence of
a shear viscosity in a $\Lambda$CDM-like cosmological model. Possible
observational tests and forecasts regarding the presence of shear viscosity
are discussed in Sect. 4. Conclusions and discussions are finally provided in
Sect. 5.

##  2 The Gravitational Wave luminosity distance in viscous scenarios

Let us suppose to describe the contents of the Universe as a dissipative
fluid. From a physical point of view the energy momentum tensor of such fluid
must depend on the shear viscosity coefficient $\eta_{V}$ and the bulk
viscosity coefficient $\zeta$. By introducing the effective pressure

$\widetilde{p}=p-\left(\zeta-{2\over
3}\eta_{V}\right)\nabla_{\lambda}u^{\lambda},$ (2.1)

we have the full energy-momentum tensor of the viscous fluid [43, 44, 45] in
the following convenient form (see e.g. [46, 47] for recent discussions)

$\widetilde{T}_{\alpha\beta}=\left(\rho+\widetilde{p}\right)u_{\alpha}u_{\beta}-\widetilde{p}\,g_{\alpha\beta}-\eta_{V}\left[u_{\alpha}u^{\rho}\nabla_{\rho}u_{\beta}+u_{\beta}u^{\rho}\nabla_{\rho}u_{\alpha}-\nabla_{\alpha}u_{\beta}-\nabla_{\beta}u_{\alpha}\right].$
(2.2)

For the purposes of this work, we will only consider the presence of shear
viscosity $\eta_{V}$. This is a simple but interesting case study since only a
bulk viscosity can modify the background equations (see [48]). Hence,
neglecting its presence will allow us to study the GWs propagation by using
the standard solutions for the background expansion.

Having this in mind, in this section we will analyze what physical
consequences arise from the presence of a shear viscous component in the
stress-energy tensor of the cosmological matter fluid: more precisely, we will
study how the gravitational luminosity distance is affected and how the
gravitational waves propagates in this background. The presence of a shear
viscous source introduces an additional friction term in the standard
gravitational wave equation in an FLRW universe, so we expect that the overall
differences will be in an increased damping effect of the GW amplitude. We
start from the results of [48], where it has been shown that the perturbed
Einstein-Hilbert action and the energy momentum tensor of a viscous fluid to
the second order in the metric, using an unperturbed FLRW metric for the
background and applying the TT gauge, leads to the propagation equation for
the tensor perturbations which reads as

$\displaystyle{h^{\prime\prime}_{i}}^{\,j}+2\left(\mathcal{H}+\lambda_{p}^{2}\,\eta_{V}\,a\right)\,{h^{\prime}_{i}}^{\,j}-\nabla^{2}{h_{i}}^{j}=0\,,$
(2.3)

where $\lambda^{2}_{p}=8\pi G$, $\mathcal{H}=a^{\prime}/a$, the prime is
referred to derivative with respect to conformal time $\eta$,
$\nabla^{2}=\delta^{ij}\partial_{i}\partial_{j}$ and $a(\eta)$ is the FLRW
scale factor.

To lighten the notation, let us define $\alpha=\lambda_{p}^{2}\,\eta_{V}$ as
the viscosity parameter. We remark that $\alpha$ is a dimensional quantity,
scaling as an inverse length. It is worth then to compare its value to
$H_{0}$, in order to discuss the regimes when shear viscosity could play a
role. For instance, interesting physical values for the shear viscosity have
been given in [49], where upper-bounds for
$\alpha<10^{-2}\,\text{Mpc}^{-1}\approx 50\,H_{0}$ have been given by the
GW150914 data analyzed supposing a viscous background and by viscous dark
matter in galaxies,. To be mentioned is also the upper bound
$\alpha<10^{-6}\,\text{Mpc}^{-1}\approx 5\times 10^{-3}\,H_{0}$, obtained by
estimation of the cross section per unit mass mass $\sigma/m$ from Abell 3827
[50] in a self-interaction model of dark matter. For the sake of simplicity,
let us then omit the spatial indices in Eq. (2.3) and

$h^{\prime\prime}+2\mathcal{H\,}\left[1-\delta(\eta)\right]\,h^{\prime}-\nabla^{2}h=0.$
(2.4)

where, following [40], we have defined
$\delta(\eta)\equiv-\alpha\,a\,\mathcal{H}^{-1}$. It is useful to express
$h\left(\eta,\mathbf{x}\right)$ in terms of its Fourier transform defined as
follows111From the technical point of view, Eq. (2.5) should contain also the
polarization tensor $\epsilon_{ij}({\bf k})$ accounting for the two
independent polazation modes of $h_{i}^{j}$. With any loss of generality, we
have omitted it, since Eq. (2.4) provides the same equation for the two
polarization states.

$h\left(\eta,\mathbf{\mathbf{x}}\right)=\frac{1}{(2\pi)^{3}}\int
d^{3}k\,e^{i\mathbf{k}\cdot\mathbf{x}\,}h_{\mathbf{k}}\left(\eta\right).$
(2.5)

From Eq. (2.4) we obtain the time evolution equation for each mode $k$

$h_{k}^{\prime\prime}+2\mathcal{H\,}h_{k}^{\prime}+2\alpha\,a\,h_{k}^{\prime}+k^{2}\,h_{k}=0,$
(2.6)

where we denote $k=|\mathbf{k}|$ and assume
$h_{\mathbf{k}}\left(\eta\right)=h_{k}\left(\eta\right)$.

It is convenient to rewrite Eq. (2.6) by introducing a new auxiliary variable
$v_{k}(\eta)$ as

$v_{k}\left(\eta\right)\equiv\widetilde{a}(\eta)\,h_{k}(\eta),$ (2.7)

where we required that the friction term proportional to
$v^{\prime}_{k}(\eta)$ in the evolution equation for $v_{k}(\eta)$ is null.
Substituting Eq. (2.7) in Eq. (2.6), we are left with two coupled evolution
equations for $v_{k}$ and $\widetilde{a}$

$\begin{cases}v_{k}^{\prime\prime}+\left(k^{2}-\frac{\widetilde{a}^{\prime\prime}}{\widetilde{a}}\right)\,v_{k}=0\\\
\frac{\widetilde{a}^{\prime}}{\widetilde{a}}=\mathcal{H}+\alpha\,a\,.\end{cases}$
(2.8)

Interestingly, first of Eq. (2.8) shows that $v_{k}(\eta)$ behaves as the
canonical Mukhanov variable where $\tilde{a}(\eta)$ plays the role of the so-
called pump field. Eqs. (2.8) clearly show that GWs in presence of shear
viscosity couple to an effective scale factor which fully encodes the viscous
content. First of Eqs. (2.8) also shows that on small scales, when
$k^{2}\gg\tilde{a}^{\prime\prime}/\tilde{a}$, $v_{k}$ propagates as free-wave.
From Eq. (2.7), we then conclude that $h_{k}$ oscillates as well but its
amplitude scales as $\tilde{a}^{-1}$.

On the other hand, general relativistic light-like geodesics in a FLRW
geometry are not affected by the presence of shear viscosity. Hence, the
photon wave-vector scales as $a^{-1}$ [40, 41, 42, 14]. It follows that we can
directly apply the same results for the GW luminosity-distance relation
already discussed for modified gravity theories. We then have

$\frac{d_{L}^{GW}}{d_{L}^{EM}}\left(z\right)=\frac{a\left(\eta\right)}{\widetilde{a}\left(\eta\right)},$
(2.9)

where the equality is obtained by imposing
$\widetilde{a}\left(\eta_{0}\right)=a\left(\eta_{0}\right)=\widetilde{a}_{0}=a_{0}$
as initial condition in Eq. (2.8).

Strictly speaking, Eq. (2.9) is fully meaningful when
$k^{2}\gg\tilde{a}^{\prime\prime}/\tilde{a}$, where we can write

$\frac{\tilde{a}^{\prime\prime}}{\tilde{a}}=\frac{a^{\prime\prime}}{a}+3\,\alpha\,a\mathcal{H}+\alpha^{2}a^{2}\,.$
(2.10)

Moreover, the GW’s that we can actually probe propagates on sub-horizon
scales. This condition gives $k\eta\gg 1$ for a class of cosmological models
such as power-law scale factor or $\Lambda$CDM, where it also holds
$a^{\prime\prime}/a\sim\eta^{-2}$ and $\mathcal{H}\sim\eta^{-1}$. This means
that the requirement $k^{2}\gg\tilde{a}^{\prime\prime}/\tilde{a}$ translates
to

$\left(\frac{k}{\mathcal{H}}\right)^{2}\gg
1+3\,a\,\frac{\alpha}{\mathcal{H}}+a^{2}\,\left(\frac{\alpha}{\mathcal{H}}\right)^{2}\,.$
(2.11)

Since $a<1$, for the redshift of interest for upcoming observations such as
LISA [51] and ET [52], i.e. $z\lesssim 2$, Eq. (2.11) tells us that sub-
horizon modes can be safely considered for our models as long as
$\alpha\,H^{-1}_{0}\lesssim 1$. Instead, for higher values of $\alpha$ our
results are meaningful for $k\gg\alpha$.

In the following sections we will discuss the explicit cases for a power-law
scale factor and for the $\Lambda$CDM model.

##  3 Viscous models

In this section, we study the GW distance-redshift relation for some
particular cases of physical interest. Firstly, we analyze the case of a
generic power law behaviour for the scale factor. An interesting application
for this case could provided by models where also the bulk viscosity is not
null. Indeed, in this case the Friedmann equations are modified by the bulk
viscosity and some interesting deviation from the Cosmic Concordance model
might occur. Secondly, we will discuss the case of a viscous extension of
$\Lambda$CDM model, where only shear viscosity is allowed.

###  3.1 Power-law solution

We first present the illustrative case of a power-law solution for the
cosmological background. Under these assumptions, the scale factor in term of
the conformal time $\eta$ reads

$a\left(\eta\right)=a_{0}\left(\frac{\eta}{\eta_{0}}\right)^{\beta},$ (3.1)

and this yields to $\mathcal{H}=\frac{\beta}{\eta}$, where $\eta_{0}$ is the
conformal time today. Inserting the last two equalities in the second line of
Eq. (2.8) we then have

$\frac{\widetilde{a}^{\prime}}{\widetilde{a}}=\frac{\beta}{\eta}+\alpha\,a_{0}\left(\frac{\eta}{\eta_{0}}\right)^{\beta}$
(3.2)

with the initial condition given by
$\widetilde{a}\left(\eta_{0}\right)=a\left(\eta_{0}\right)$. We have done this
choice in order to recover $d_{L}^{\,GW}(0)=d_{L}^{\,EM}(0)$ (see Eq. (2.9).
For $\beta\neq-1$, we then have

$\widetilde{a}\left(\eta\right)=a(\eta)\exp\left[\frac{\alpha\,a_{0}\left(\eta^{\beta+1}-\eta_{0}^{\beta+1}\right)}{(\beta+1)\eta_{0}^{\beta}}\right].$
(3.3)

A direct comparison between Eqs. (3.1) and (3.3) shows that the damping
effects introduced by the shear viscosity is an exponential suppression in the
past of the effective scale factor $\tilde{a}$, which governs the GWs
propagation.

Finally, we substitute Eq. (3.3) in Eq. (2.9) to obtain

$\frac{d_{L}^{GW}\left(\eta\right)}{d_{L}^{EM}\left(\eta\right)}=\exp\left\\{\frac{\alpha\,a_{0}\eta_{0}}{(\beta+1)}\left[1-\left(\frac{a}{a_{0}}\right)^{(\beta+1)/\beta}\right]\right\\}\,.$
(3.4)

Since our observations relate distances to redshift, the last equation (3.4)
can be further manipulated in order to make explicit the dependence on $z$. To
this purpose, we first recall that $1+z=\frac{a_{0}}{a}$. Hence by expressing
the conformal time today in terms of the Hubble constant $H_{0}$ we obtain

$\frac{d_{L}^{GW}}{d_{L}^{EM}}\left(z\right)=\text{exp}\left[\frac{\beta}{\beta+1}\frac{\alpha}{H_{0}}\left(1-\frac{1}{\left(1+z\right)^{\frac{\beta+1}{\beta}}}\right)\right]\,.$
(3.5)

This is the most general equation that relates the gravitational luminosity
distance to the redshift in a power law Universe with constant shear
viscosity. Setting $\beta=2$, we apply the general result given in Eq. (3.5)
to the physical scenario of a matter dominated Universe (Einstein-de Sitter
Universe). In Fig. 1 we show the relative correction between the gravitational
and the electromagnetic luminosity distance-redshift relation as given by Eq.
(3.5) for different values of viscosities.

Figure 1: Luminosity distance ratio for different values of $\alpha$ in unit
of $H_{0}$ for CDM model.

Regardless of the value of $\alpha$, we notice that all the solutions tend to
a constant value of the ratio beyond a certain redshift $z^{*}$. For a generic
value of $\beta$, this overall offset is given by

$\frac{d_{L}^{GW}}{d_{L}^{EM}}\approx
e^{\frac{\alpha}{H_{0}}\frac{\beta}{(\beta+1)}}\,,$ (3.6)

which clearly shows that is controlled by the ratio $\alpha/H_{0}$. Although
an Einstein-de Sitter universe is a very good approximation of the evolution
of the Universe after the CMB, it fails to reproduce the recent Dark Energy
epoch and a better model of the Universe is required. In the next subsection
we will discuss the effects of viscosity within a $\Lambda$CDM-like model.

###  3.2 A viscous $\Lambda$CDM model

After recombination, the radiation contribution to the expansion of the
Universe has become negligible and since we are interested in the recent
epoch, we can safely neglect it. The first difficulty we have to deal with is
the absence of an analytic expression for $a(\eta)$ in the Cosmic Concordance
model. In order to circumvent this technical aspect, it is convenient to
rewrite the second line of Eq. (2.8) in terms of the redshift. With this in
mind, we first provide the following well-known relations ($a_{0}=1$)

$\frac{d}{d\eta}=-H(z)\frac{d}{dz}\qquad\text{and}\quad\mathcal{H}(z)=\frac{H(z)}{1+z}\,,$
(3.7)

where

$H(z)=H_{0}\sqrt{\Omega_{m0}(1+z)^{3}+\Omega_{\Lambda 0}}\,,$ (3.8)

and clearly $\Omega_{\Lambda 0}+\Omega_{m0}=1$. Therefore, first of Eqs. (2.8)
becomes222Here a possible time dependence in $\alpha$ is considered for a sake
of generality. This case will not be considered along this work but might be
of interest for other scenarios, where it has been claimed that a running
value $\alpha$ could emerge from the coarse graining of the small-scales
inhomogeneities [53, 54].

$\frac{d\ln\widetilde{a}}{dz}=-\frac{1}{1+z}-\frac{\alpha(z)}{H(z)(1+z)}\,.$
(3.9)

By integrating the lhs between $\widetilde{a}_{0}=a_{0}=1$ and $\widetilde{a}$
and the rhs between 0 and $z$ we obtain

$\widetilde{a}(z)=\frac{1}{1+z}\exp\left(-\int_{0}^{z}\frac{\alpha(z^{\prime})}{(1+z^{\prime})H(z^{\prime})}dz^{\prime}\right)\,,$
(3.10)

Once inserted in Eq. (2.9), Eq. (3.10) gives

$\frac{d_{L}^{GW}}{d_{L}^{EM}}\left(z\right)=\text{exp}\left[\int_{0}^{z}\frac{\alpha(z^{\prime})}{(1+z^{\prime})H(z^{\prime})}dz^{\prime}\right].$
(3.11)

Last equation is a general expression that holds true in any FLRW cosmology
with a viscous fluid component in the stress-energy tensor, where the shear
viscosity parameter is not constant. All the information about the background
gravitational sources is encoded in $H(z)$.

We remark that Eq. (3.11) holds true also for values of $H(z)$ which are more
general of Eq. (3.8). Indeed, as long as the Hubble function is evaluated
accordingly to the total stress-energy tensor, it follows that the equation is
valid even if the fluids do not have a barotropic equation of state (e.g.
polytropic fluids) and it remains valid also in the presence of bulk viscous
contributions. However, since now on, we will only consider background
dynamics driven by dark matter and $\Lambda$, in agreement with the value of
$H(z)$ provided in Eq. (3.8).

With this in mind, we then combine Eq. (3.11) and with Eq. (3.8) and consider
a constant $\alpha(z)=\alpha$. The integral in Eq. (3.11) hence becomes

$\int_{0}^{z}\frac{1}{(1+z^{\prime})H(z^{\prime})}dz^{\prime}=\frac{1}{3H_{0}\sqrt{\Omega_{\Lambda
0}}}\ln\left(\frac{H(z)/H_{0}-\sqrt{\Omega_{\Lambda
0}}}{H(z)/H_{0}+\sqrt{\Omega_{\Lambda 0}}}\,\frac{1+\sqrt{\Omega_{\Lambda
0}}}{1-\sqrt{\Omega_{\Lambda 0}}}\right)$ (3.12)

bringing then to

$\frac{d_{L}^{GW}}{d_{L}^{EM}}\left(z\right)=\left(\frac{\sqrt{\Omega_{m0}(1+z)^{3}+\Omega_{\Lambda
0}}-\sqrt{\Omega_{\Lambda 0}}}{\sqrt{\Omega_{m0}(1+z)^{3}+\Omega_{\Lambda
0}}+\sqrt{\Omega_{\Lambda 0}}}\,\frac{1+\sqrt{\Omega_{\Lambda
0}}}{1-\sqrt{\Omega_{\Lambda
0}}}\right)^{\frac{\alpha}{3H_{0}\sqrt{\Omega_{\Lambda 0}}}}\,.$ (3.13)

Just as discussed in the previous section, we have again a critical redshift
$z^{*}$ such that, when $z>z^{*}$, we have:

$\Xi_{0}\equiv\frac{d_{L}^{GW}}{d_{L}^{EM}}(z\gg
z^{*})=\left(\frac{1+\sqrt{\Omega_{\Lambda 0}}}{1-\sqrt{\Omega_{\Lambda
0}}}\right)^{\frac{\alpha}{3H_{0}\sqrt{\Omega_{\Lambda 0}}}}\,.$ (3.14)

Here we have called this limit value $\Xi_{0}$ since we have in mind the
widely used parametrization of $d_{L}^{\,GW}/d_{L}^{\,EM}$ proposed in [41]

$\frac{d_{L}^{\,GW}}{d_{L}^{\,EM}}(z)=\Xi_{0}+\frac{1-\Xi_{0}}{(1+z)^{n}}\,.$
(3.15)

According to Eq. (3.13), the parametrization in Eq. (3.15) is not expected to
work perfectly for any value of $\alpha$.

Figure 2: Luminosity distance ratio for different values of $\alpha$ in unit
of $H_{0}$ for $\Lambda$CDM where $\Omega_{m0}=0.307$ has been taken in
accordance to the local estimation from the Pantheon sample [55].

Indeed, as we can see from Fig. 2, for high value of $\alpha/H_{0}$, the
emergence of an intermediate slope at redshifts $\sim 1$ renders Eq. (3.15)
hard to be used. However, when $\alpha$ is perturbatively smaller than
$H_{0}$, Eq. (3.15) works quite well.

Figure 3: Relative error between the exact expression in Eq. (3.13) and the
parametric expression in Eq. (3.15). We fix $\Omega_{m0}=0.307$ in accordance
to the local estimation of the Pantheon sample [55]. This plot shows that the
widely used parametrization in Eq. (3.15) is reliable only for small values of
$\alpha\,H^{-1}_{0}$.

This is more evident in Fig. 3, where the slope $n$ in Eq. (3.15) has been
modeled as (see [56] for a detailed discussion about different parametrization
of $n$)

$n\equiv\frac{\delta(0)}{1-\Xi_{0}}\,.$ (3.16)

The dependences of $\Xi_{0}$ and $n$ on the ratio $\alpha/H_{0}$ are shown in
Fig. 4, where $\Omega_{m0}=0.307$ has been taken in accordance to [55].

Figure 4: Dependence of $\Xi_{0}$ (left) and $n$ (right) in terms of $\alpha
H^{-1}_{0}$, according to the widely used parametrization given in Eq. (3.15).
The value of $\Omega_{m0}=0.307$ has been taken in accordance to the local
estimation from the Pantheon sample [55].

From both these panels, we notice that the discrepancy between the
gravitational and electromagnetic luminosity distance deviates exponentially
when the shear viscosity becomes of order of $H_{0}$.

What emerges from Figs. 3 and 4 is that the parametrization of Eq. (3.15) can
be safely adopted also for the viscous models when its value is at maximum of
few percent of the $H_{0}$. Accordingly, also the constraints on the value of
$\alpha$ from Eq. (3.15) are not trustworthy beyond this regime. The reason
for this discrepancy stands in the fact that Eq. (3.15) holds when
$d_{L}^{\,GW}/d_{L}^{\,EM}$ exhibits only two regimes: top curve of Fig. 2
clearly shows that this is not the case when $\alpha\gg H_{0}$. We remark that
in this case the estimation of $n$ becomes meaningless whereas $\Xi_{0}$ can
still be interpreted as the limit value of $d_{L}^{\,GW}/d_{L}^{\,EM}$.

As a final remark we show that in the limit $\Omega_{\Lambda 0}\rightarrow 0$
we retrieve the Einstein-de Sitter solution given in Eq. (3.5) by setting
$\beta=2$. To this end, it is more straightforward to expand in terms of
$\Omega_{\Lambda 0}$ Eq. (3.12), having in mind the constraint
$\Omega_{m0}=1-\Omega_{\Lambda 0}$. Since we are interested in the leading
term and the logarithm is divided $\sqrt{\Omega_{\Lambda 0}}$, we neglect
terms of order $\mathcal{O}(\Omega_{\Lambda 0})$ or higher. First fraction in
the argument of the logarithm of Eq. (3.12) then becomes

$\displaystyle\frac{\sqrt{\Omega_{m0}(1+z)^{3}+\Omega_{\Lambda
0}}-\sqrt{\Omega_{\Lambda 0}}}{\sqrt{\Omega_{m0}(1+z)^{3}+\Omega_{\Lambda
0}}+\sqrt{\Omega_{\Lambda 0}}}=$ $\displaystyle\frac{\sqrt{1-\Omega_{\Lambda
0}+\Omega_{\Lambda 0}(1+z)^{-3}}-\sqrt{\Omega_{\Lambda
0}}(1+z)^{-3/2}}{\sqrt{1-\Omega_{\Lambda 0}+\Omega_{\Lambda
0}(1+z)^{-3}}+\sqrt{\Omega_{\Lambda 0}}(1+z)^{-3/2}}$ $\displaystyle=$
$\displaystyle\frac{1-\sqrt{\Omega_{\Lambda
0}}(1+z)^{-3/2}}{1+\sqrt{\Omega_{\Lambda
0}}(1+z)^{-3/2}}+\mathcal{O}\left(\Omega_{\Lambda 0}\right)$ $\displaystyle=$
$\displaystyle 1-2\,\sqrt{\Omega_{\Lambda
0}}\,(1+z)^{-3/2}+\mathcal{O}\left(\Omega_{\Lambda 0}\right)\,.$ (3.17)

In the same way, for the second fraction in Eq. (3.12) we have

$\frac{1+\sqrt{\Omega_{\Lambda 0}}}{1-\sqrt{\Omega_{\Lambda
0}}}=1+2\sqrt{\Omega_{\Lambda 0}}+\mathcal{O}(\Omega_{\Lambda 0})\,.$ (3.18)

Hence, thanks to Eqs. (3.17) and (3.18), the expansion of the logarithm in Eq.
(3.12) is

$\ln\left(\frac{H(z)/H_{0}-\sqrt{\Omega_{\Lambda
0}}}{H(z)/H_{0}+\sqrt{\Omega_{\Lambda 0}}}\,\frac{1+\sqrt{\Omega_{\Lambda
0}}}{1-\sqrt{\Omega_{\Lambda 0}}}\right)=2\sqrt{\Omega_{\Lambda
0}}\left[1-(1+z)^{-3/2}\right]+\mathcal{O}\left(\Omega^{3/2}_{\Lambda
0}\right)\,.$ (3.19)

By inserting this last expansion in Eq. (3.12), we then arrive at

$\int_{0}^{z}\frac{1}{(1+z^{\prime})H(z^{\prime})}dz^{\prime}=\frac{2}{3H_{0}}\left[1-(1+z)^{-3/2}\right]+\mathcal{O}\left(\Omega_{\Lambda
0}\right)\,.$ (3.20)

Finally, by using Eq. (3.11) with a constant $\alpha$, we recover the result
of Eq. (3.5).

##  4 A smoking gun for the shear viscosity?

An interesting feature emerging from the discussion in the previous section is
the neat distinction between two regimes of the ratio
$d_{L}^{\,GW}/d_{L}^{\,EM}$ for small values of $\alpha/H_{0}$. The crossing
from one regime to the other occurs at a given redshift, where a knee between
the linear and constant behaviour is evident. To quantify this transition, we
first expand Eq. (3.13) for low redshift

$\frac{d_{L}^{\,GW}}{d_{L}^{\,EM}}=1-\delta(0)\,z+\mathcal{O}\left(z^{2}\right)=1+\frac{\alpha}{H_{0}}z+\mathcal{O}\left(z^{2}\right)\,,$
(4.1)

where, following [40], we have used $\delta(0)=-\alpha\,H^{-1}_{0}$. It is
already interesting to notice from Eq. (4.1) that the linear regime appearing
at low redshift of Fig. 2 is independent of $\Omega_{m0}$: a detection of a
linear increasing behaviour for close sources is entirely addressed to a non-
null shear viscosity in this scenario, regardless of the value of $\alpha$.
However, the amplitude still look prohibitive according to the forecasted
precision at those redshifts. Indeed, following [41], we can use the the
stringent limit on $\delta(0)$ coming from combined observations of the
standard siren GW170817 [6] and electromagnetic counterpart of its host galaxy
NGC4993 [57, 58]. This provides the bounds $\delta(0)=-7.8^{+9.7}_{-18.4}$.
Despite this is a test to directly constrain $\alpha H^{-1}_{0}$, the provided
bound is unfortunately not so stringent.

Stronger constraints can be put by considering the forecasts for LISA [56] and
ET [41]. To this end, we make use of the parametrization in Eq. (3.15) and
consider the forecasted error $\Delta\Xi_{0}$ respectively given by $0.044$
for LISA [56] and $0.008$ for ET [41]. Indeed, for small values of $\alpha
H^{-1}_{0}$, Eq. (3.14) gives

$\Xi_{0}\approx
1+\frac{1}{3}\frac{\alpha}{H_{0}}\ln\left(\frac{1+\sqrt{\Omega_{\Lambda
0}}}{1-\sqrt{\Omega_{\Lambda 0}}}\right)\,.$ (4.2)

Around the fiducial values of the standard cosmic concordance model
($\Xi_{0}=1$), we then have that $\alpha/H_{0}$ could be constrained by future
missions with an error given by

$\Delta\left(\frac{\alpha}{H_{0}}\right)\approx\frac{3\,\Delta\Xi_{0}}{\ln\left(\frac{1+\sqrt{\Omega_{\Lambda
0}}}{1-\sqrt{\Omega_{\Lambda 0}}}\right)}\approx\Delta\Xi_{0}\,,$ (4.3)

where we have used the fiducial value $\Omega=0.307$ as given by the
independent estimation from Pantheon sample [55]. Interestingly, the
forecasted error $\alpha/H_{0}$ is almost of the same order of magnitude as
the one for $\Xi_{0}$. This opens an interesting window to investigate whether
a shear viscosity of order of few percents of $H_{0}$ is present in the
cosmological fluids. We also remark that these constraints are more stringent
than the ones provided by the same analysis for particular classes of modified
gravity theories, where the forecasted errors for the free parameters are
worsen by almost one order of magnitude wrt to $\Delta\Xi_{0}$ [59].

As a further remark, an interesting idea would be to use Eq. (4.3) as a
forecast for $\alpha$ itself, rather than just $\alpha\,H^{-1}_{0}$. This
could be achieved by combining Eq. (4.3) with the analysis of the expected
Hubble-Lemaître diagram from Superluminous Supernovae [60] for forthcoming
Large-Scale Structure surveys at higher redshift (up to $z=4$). Indeed, it has
been shown in [61] that this analysis could provide a (model dependent)
estimation of $H_{0}$ at a precision level of $\sim 0.1\%$.

Finally, we discuss another test that could be done by looking at the
transition between the linear evolution and the constant offset given by Eq.
(3.14). This transition would occur at the redshift $z^{*}$ when the linear
growth in Eq. (4.1) reaches the offset value in Eq. (3.14). This simple
reasoning returns an expression for $z^{*}$ given by

$z^{*}=\frac{H_{0}}{\alpha}\left[\left(\frac{1+\sqrt{1-\Omega_{m0}}}{1-\sqrt{1-\Omega_{m0}}}\right)^{\frac{\alpha}{3H_{0}\sqrt{1-\Omega_{m0}}}}-1\right]\,.$
(4.4)

We remark that Eq. (4.4) is meaningful only for small values of the ratio
$\alpha/H_{0}$, since the transition between the linear and the constant
regime occurs directly only in this case. Indeed, Fig. 2 for values
$\alpha/H_{0}>1$, a third intermediate regime appears and this breaks the
validity of the discussed case study. Fortunately, the values of $\alpha$ when
this limitation occurs are too high to be seriously considered. It is worth to
notice that when $\alpha\ll H_{0}$, within the uncertainties of
$\Omega_{m0}=0.307\pm 0.012$ given by [55], the value of $z^{*}$ is quite
stable around the value of $z^{*}=1$.

##  5 Conclusions and outlook

In this paper we have studied the influence of shear viscosity on the so-
called gravitational luminosity distance. The main property of the shear
viscosity is that it leaves unchanged the Friedmann equations and modifies the
linear tensor perturbations by an additional friction terms. As a consequence,
the most stringent constraints on the speed of GW’s propagation are satisfied
but a lot of room for other effects can be investigated.

In particular, the friction term $\delta(\eta)$ arising from the viscosity
impacts the GW luminosity distance-redshift relation in two specific ways:

  * •

the shear viscosity introduces a new length scale $\alpha$;

  * •

the friction term depends on the Hubble function through the ratio
$\alpha/\mathcal{H}$.

From these two properties we infer that the relevant parameter to study and/or
discard the presence of the shear viscosity in the cosmological fluid is the
relative amplitude between $\alpha$ and $H_{0}$. In this regard, we have shown
that when $\alpha H^{-1}_{0}\ll 1$, most of the well known techniques for the
study of GW luminosity distance in modified gravity theories can be directly
applied.

More specifically, for a $\Lambda$CDM-like viscous model, the gravitational
distance given in Eq. (3.13) exhibits a constant asymptotic value at redshifts
$z>1$ as given by the parameter $\Xi_{0}$. For the case of perturbative
amplitude of $\alpha/H_{0}$, $\Xi_{0}$ can actually be interpreted as usually
done [41] for the parametrization of Eq. (3.15) and consequently forecast
whether it will be possible to discriminate the presence shear viscosity from
futuristic surveys, such as LISA and ET. Our analysis indicates that non-
perfect fluids with a shear viscosity could be studied in details up to values
of the $\alpha$ of few percents of $H_{0}$.

On the contrary, non-perturbative values of the shear viscosity
($\alpha\gtrsim H_{0}$) could be easily distinguished from all the class of
models compatible with Eq. (3.15) and there are two reasons for that:

  * •

first of all, the offset value $\Xi_{0}$ exponentially grows with
$\alpha/H_{0}$. This means that a large deviation of $d^{\,GW}_{L}$ from the
EM luminosity distance could be easily detectable, if present.

  * •

Secondly, the redshift dependence of $d_{L}^{\,GW}/d_{L}^{\,EM}$ largely
differs from the parametric fitting functions in Eq. (3.15), also at
intermediate redshifts around $z\sim 1$ (see the curve for $\alpha=10\,H_{0}$
in Fig. 2).

As a final remark, we stress that the behaviour of $d_{L}^{\,GW}/d_{L}^{\,EM}$
for small redshifts ($z\ll 1$), is insensitive to $\Omega_{m0}$ and depends
only linearly on the ratio $\alpha/H_{0}$, regardless of its amplitude. For
this kind of close sources, the independent estimation of $z$, $d_{L}^{\,EM}$
and $d_{L}^{\,GW}$ may allow to have independent measurements of $H_{0}$ and
$\alpha$ rather than just their ratio. Such a scenario could provide a highly
promising test to estimate the shear viscosity if the precision concerning the
measurement of $d_{L}^{\,GW}$ would dramatically increase in the next years.

## Acknowledgements

The authors wish to warmly thank Maurizio Gasperini for his comments and
suggestions. GF is thankful to José Fonseca and Noemi Frusciante for valuable
comments on the results presented in this work. GF acknowledges support by
Fundação para a Ciência e a Tecnologia (FCT) under the program “Stimulus" with
the grant no. CEECIND/04399/2017/CP1387/CT0026 and through the research
project with ref. number PTDC/FIS-AST/0054/2021. GF is also member of the
Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di
Alta Matematica (INdAM). LT is supported in part by INFN under the program
TAsP (“Theoretical Astroparticle Physics"), and by the research grant number
2017W4HA7S “NAT-NET: Neutrino and Astroparticle Theory Network", under the
program PRIN 2017 funded by the Italian Ministero dell’Università e della
Ricerca (MUR).

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11footnotetext:  Department of Mathematical Sciences, University of Bath, Bath
BA2 7AY,
United Kingdom<EMAIL_ADDRESS>
# Giant component of the soft random geometric graph

Mathew D. Penrose1
University of Bath

###### Abstract

Consider a 2-dimensional soft random geometric graph $G(\lambda,s,\phi)$,
obtained by placing a Poisson($\lambda s^{2}$) number of vertices uniformly at
random in a square of side $s$, with edges placed between each pair $x,y$ of
vertices with probability $\phi(\|x-y\|)$, where $\phi:\mathbb{R}_{+}\to[0,1]$
is a finite-range connection function. This paper is concerned with the
asymptotic behaviour of the graph $G(\lambda,s,\phi)$ in the large-$s$ limit
with $(\lambda,\phi)$ fixed. We prove that the proportion of vertices in the
largest component converges in probability to the percolation probability for
the corresponding random connection model, which is a random graph defined
similarly for a Poisson process on the whole plane. We do not cover the case
where $\lambda$ equals the critical value $\lambda_{c}(\phi)$.

AMS classifications: 60C05, 60D05, 60K35.

##  1 Introduction and statement of results

Let $\phi:[0,\infty)\to[0,1]$ be a nonincreasing function. Given a locally
finite point set ${\cal X}\subset\mathbb{R}^{2}$, let $G({\cal X},\phi)$
denote the graph with vertex set ${\cal X}$, where for each
$\\{x,y\\}\subset{\cal X}$, the edge $xy$ is included with probability
$\phi(\|y-x\|)$, independently of the other pairs. Here $\|\cdot\|$ is the
Euclidean norm.

Given $\lambda>0$, let ${\cal H}_{\lambda}$ denote a homogeneous Poisson
process of intensity $\lambda$ in $\mathbb{R}^{2}$. Given also $s>0$, set
$B(s):=[-s/2,s/2]^{2}$ and let ${\cal H}_{\lambda,s}$ denote the restriction
of ${\cal H}_{\lambda}$ to $B(s)$, which is a homogeneous Poisson process of
intensity $\lambda$ in $B(s)$. We are interested in the graphs $G({\cal
H}_{\lambda},\phi)$ and $G({\cal H}_{\lambda,s},\phi)$, which are known as the
random connection model [6] and soft random geometric graph [9] respectively,
with connection function $\phi$.

Let ${\cal H}^{{\bf o}}_{\lambda}$ denote the point process ${\cal
H}_{\lambda}\cup\\{{\bf o}\\}$, where ${\bf o}$ is the origin in
$\mathbb{R}^{2}$. For $k\in\mathbb{N}$, let $\pi_{k}(\phi,\lambda)$ denote the
probability that the component of $G({\cal H}^{{\bf o}}_{\lambda},\phi)$
containing the origin is of order $k$. The percolation probability
$\theta(\phi,\lambda)$ is the probability that ${\bf o}$ lies in an infinite
component of the graph $G({\cal H}^{{\bf o}}_{\lambda},\phi)$, that is,

$\theta(\phi,\lambda):=1-\sum_{k=1}^{\infty}\pi_{k}(\phi,\lambda).$

A standard coupling argument shows that $\theta(\phi,\lambda)$ is
nondecreasing in $\lambda$. The critical value (continuum percolation
threshold) $\lambda_{\rm c}(\phi)$ is defined by

$\displaystyle\lambda_{\rm
c}(\phi):=\inf\\{\lambda>0:\theta(\phi,\lambda)>0\\}.$ (1.1)

It is known (see [6]) that $0<\lambda_{\rm c}<\infty$, provided
$0<\int_{\mathbb{R}^{2}}\phi(\|x\|)dx<\infty$.

For any finite graph $G$, let $L_{j}(G)$ denote the order of its $j$th-largest
component, that is, the $j$th-largest of the orders of its components, or zero
if it has fewer than $j$ components. In this paper we prove the following
results about convergence in probability of $s^{-2}L_{j}(G({\cal
H}_{\lambda,s},\phi))$ for $j=1,2$, as $s\to\infty$ with $\phi,\lambda$ fixed.

######  Theorem 1.1.

Suppose $\lambda>0$ with $\theta(\phi,\lambda)=0$. Then

$\displaystyle s^{-2}L_{1}(G({\cal
H}_{\lambda,s},\phi))\stackrel{{\scriptstyle
P}}{{\longrightarrow}}0{\rm~{}~{}~{}~{}~{}~{}as~{}~{}}s\to\infty.$ (1.2)

######  Theorem 1.2.

Suppose $\sup\\{r>0:\phi(r)>0\\}\in(0,\infty)$ and
$\lambda>\lambda_{c}(\phi)$. Then as $s\to\infty$ we have that
$s^{-2}L_{1}(G({\cal H}_{\lambda,s},\phi))\stackrel{{\scriptstyle
P}}{{\longrightarrow}}\lambda\theta(\phi,\lambda)$ and $s^{-2}L_{2}(G({\cal
H}_{\lambda,s},\phi))\stackrel{{\scriptstyle P}}{{\longrightarrow}}0.$

These theorems do not address the case with $\lambda=\lambda_{c}(\phi)$,
unless we know $\theta(\phi,\lambda_{c}(\phi))=0$ (if we had this, we could
apply Theorem 1.1 when $\lambda=\lambda_{c}(\phi)$). Theorem 1.2 also does not
address the case of $\phi$ having unbounded range (recall we are assuming
$\phi$ is nonincreasing).

In the case with $\phi={\bf 1}_{[0,1]}$, these results were already proved in
[7], but the method here provides an alternative and possibly shorter proof
(the proof in [7] relies on a lengthy RSW argument from [6], as well as not
working for general $\phi$). When $\phi={\bf 1}_{[0,1]}$ it is known [6] that
$\theta(\phi,\lambda_{c}(\phi))=0$ since this case is equivalent to a Boolean
model.

##  2 Proof of theorems

###  2.1 Preliminaries

We introduce some further notation that will be used in proving Theorems 1.1
and 1.2.

For ${\cal X}\subset\mathbb{R}^{2}$ and $x,y\in\mathbb{R}^{2}$ we sometimes
write ${\cal X}^{x}$ for ${\cal X}\cup\\{x\\}$ and ${\cal X}^{x,y}$ for ${\cal
X}\cup\\{x,y\\}$.

Let $\phi:[0,\infty)\to[0,1]$ be nonincreasing. We shall view $\phi$ as fixed
from now on, and for any locally finite ${\cal X}\subset\mathbb{R}^{2}$ write
simply $G({\cal X})$ instead of $G({\cal X},\phi)$. Also for $x\in{\cal X}$,
let ${\cal C}_{x}({\cal X})$ denote the vertex set of the component of
$G({\cal X})$ containing $x$.

Given $x\in\mathbb{R}^{2}$ and $r>0$, we write $D_{r}(x)$ for the disk
$\\{y\in\mathbb{R}^{2}:\|y-x\|\leq r\\}$ and $D_{r}$ for $D_{r}({\bf o})$.
Also let $S_{r}:=B(2r)=[-r,r]^{2}$, and let ${\bf e}:=(1,0)$.

Given $A,B\subset\mathbb{R}^{2}$, and locally finite ${\cal
X}\subset\mathbb{R}^{2}$, we write $\\{A\leftrightarrow B$ in $G({\cal X})\\}$
for the event that there exist $x\in A\cap{\cal X}$ and $y\in B\cap{\cal X}$
such that there is a path in $G({\cal X})$ from $x$ to $y$. We write
$\\{A\leftrightarrow\infty$ in $G({\cal X})\\}$ for the intersection over all
$n\geq 1$ of events $\\{A\leftrightarrow\mathbb{R}^{2}\setminus D_{n}$ in
$G({\cal X})\\}$

Next, we assemble some known facts which will be used later.

We say a real-valued function $f$, defined on graphs $({\cal V},E)$, with
${\cal V}\subset\mathbb{R}^{2}$ locally finite, is increasing if $f({\cal
V},E)\leq f({\cal V}^{\prime},F)$ whenever ${\cal V}\subset{\cal V}^{\prime}$
and $E\subset E^{\prime}$. We say $f$ is decreasing if $-f$ is increasing.
Given $\lambda>0$ and given $\phi$, we say $E$ is an increasing (resp.
decreasing) event on $G({\cal H}_{\lambda})$ if ${\bf 1}_{E}$ is an increasing
(resp. decreasing) function of $G({\cal H}_{\lambda})$.

######  Lemma 2.1 (Harris-FKG inequality).

Suppose $f,g$ are measurable bounded increasing real-valued functions defined
on graphs $({\cal V},E)$ with ${\cal V}\subset\mathbb{R}^{2}$ locally finite,
Then

${\mathbb{E}}[f(G({\cal H}_{\lambda}))g(G({\cal
H}_{\lambda}))]\geq{\mathbb{E}}[f(g({\cal
H}_{\lambda}))]{\mathbb{E}}[g(G({\cal H}_{\lambda}))].$

The same inequality holds if $f$ and $g$ are both decreasing.

###### Proof.

See [2], where measurability issues are also dealt with. ∎

######  Corollary 2.2 (Square Root trick).

Let $\lambda>0$, $k\in\mathbb{N}$, ${\varepsilon}\in(0,1)$. Suppose for
$i=1,\ldots,k$ we have increasing events $A_{i}$ defined on $G({\cal
H}_{\lambda})$, such that ${\mathbb{P}}[\cup_{i=1}^{k}A_{i}]>1-{\varepsilon}$.

Then $\max_{1\leq i\leq k}{\mathbb{P}}[A_{i}]>1-{\varepsilon}^{1/k}$.

###### Proof.

Set $M=\max_{1\leq i\leq k}{\mathbb{P}}[A_{i}]$. The events $A_{i}^{c}$ are
all decreasing, so by Lemma 2.1,

$\displaystyle{\varepsilon}>{\mathbb{P}}[\cap_{i=1}^{k}A_{i}^{c}]\geq\prod_{i=1}^{k}{\mathbb{P}}[A_{i}^{c}]\geq(1-M)^{k},$

so that $1-M<{\varepsilon}^{1/k}$ and $M>1-{\varepsilon}^{1/k}$. ∎

Given $\lambda>0$, let $N_{\infty}(\phi,\lambda)$ be the number of infinite
components of the graph $G({\cal H}_{\lambda})$. It is not hard to show that
if $\theta(\phi,\lambda)=0$, then
${\mathbb{P}}[N_{\infty}(\phi,\lambda)=0]=1$. The next preliminary result
concerns uniqueness of the infinite cluster, in the other case, where
$\theta(\phi,\lambda)>0$.

######  Lemma 2.3.

Suppose $\theta(\phi,\lambda)>0$. Then
${\mathbb{P}}[N_{\infty}(\phi,\lambda)=1]=1$.

###### Proof.

See [6]. ∎

Another useful fact is the Mecke formula for the random connection model. Let
$s,\lambda>0$ and suppose $f(x,G)\in\mathbb{R}_{+}$ is defined for all pairs
$(x,G)$ where $G$ is a finite graph with vertex set ${\cal V}(G)\subset B(s)$
and $x\in{\cal V}(G)$. Then whenever the following expectations are defined we
have

$\displaystyle{\mathbb{E}}\sum_{x\in{\cal H}_{\lambda,s}}f(x,G({\cal
H}_{\lambda,s}))=\lambda\int_{B(s)}{\mathbb{E}}[f(x,G({\cal
H}_{\lambda,s}^{x}))]dx$ (2.1)

and moreover if $g(x,y,G)\in\mathbb{R}_{+}$ is defined whenever additionally
$y\in{\cal V}(G)$ then

$\displaystyle{\mathbb{E}}\sum\sum_{x,y\in{\cal H}_{\lambda,s},x\neq
y}g(x,y,G({\cal
H}_{\lambda,s}))=\lambda^{2}\int_{B(s)}\int_{B(s)}{\mathbb{E}}[g(x,y,G({\cal
H}_{\lambda,s}^{x,y}))]dydx.$ (2.2)

The Mecke formulae (2.1) and (2.2) can be derived by conditioning on ${\cal
H}_{\lambda,s}$ and using the usual Mecke formulae from e.g. [3].

Also of use to us is the following sequential construction of clusters in
$G({\cal H}_{\lambda,s})$. Let $\lambda,s>0$ and $A\subset B(s)$ (typically a
disk). The set $\cup_{x\in{\cal H}_{\lambda}\cap A}{\cal C}_{x}({\cal
H}_{\lambda,s})$ can be created as follows:

First generate ${\cal H}_{\lambda}\cap A$. Denote the points so created as
active points and let the initial intensity function of unexplored points,
i.e. Poisson points that are not yet generated, is $g=\lambda{\bf
1}_{B(s)\setminus A}(\cdot)$.

Next, choose an active point $x$ and generate a Poisson process of intensity
$g(\cdot)\phi(\cdot-x)$, representing the previously unexplored points of
${\cal H}_{\lambda,s}$ that are connected directly to $x$. Label all the new
points as ‘active’, and change the status of $x$ from ‘active’ to ‘finished’.
Also change the intensity of unexplored points from $g(\cdot)$ to
$g(\cdot)(1-\phi(\cdot-x))$.

Then pick a new active point and repeat the above, using the new intensity of
unexplored points. Keep repeating until we run out of active points, then
stop.

We shall refer to the above procedure as growing the cluster sequentially.
This method is described in detail for the case $\phi={\bf 1}_{[0,1]}$ in [8],
and for the general random connection model in [5].

###  2.2 The subcritical case

###### Proof of Theorem 1.1.

Suppose $\lambda>0$ with $\theta(\phi,\lambda)=0$.

Let ${\varepsilon}>0$. Let $N_{s}:=\sum_{x\in{\cal H}_{\lambda,s}}{\bf
1}\\{|{\cal C}_{x}({\cal H}_{\lambda,s})|\geq{\varepsilon}s^{2}\\}$. If
$L_{1}(G({\cal H}_{\lambda,s}))\geq{\varepsilon}s^{2}$, then
$N_{s}\geq{\varepsilon}s^{2}$. Hence by Markov’s inequality and the Mecke
formula,

$\displaystyle{\mathbb{P}}[L_{1}(G({\cal
H}_{\lambda,s}))\geq{\varepsilon}s^{2}]\leq({\varepsilon}s^{2})^{-1}{\mathbb{E}}[N_{s}]=({\varepsilon}s^{2})^{-1}\int_{B(s)}{\mathbb{P}}[|{\cal
C}_{x}({\cal H}_{\lambda,s}^{x})|\geq{\varepsilon}s^{2}]\lambda dx$
$\displaystyle\leq({\varepsilon}s^{2})^{-1}\int_{B(s)}{\mathbb{P}}[|{\cal
C}_{x}({\cal H}_{\lambda}^{x})|\geq{\varepsilon}s^{2}]\lambda dx$
$\displaystyle=\lambda{\varepsilon}^{-1}\sum_{k\geq{\varepsilon}s^{2}}\pi_{k}(\phi,\lambda)$

which tends to zero as $s\to\infty$. Therefore $s^{-2}L_{1}(G({\cal
H}_{\lambda,s}))\stackrel{{\scriptstyle P}}{{\longrightarrow}}0$. ∎

###  2.3 Renormalization

From now on we assume $\phi$ is nonincreasing with
$\inf\\{r>0:\phi(x)=0\\}=1$. We shall prove Theorem 1.2 only for this case,
since simple scaling arguments then yield the general finite-range case in the
statement of the theorem.

Given $\lambda,K,L,M\in(0,\infty)$ with $L>K,M>2K$, define the following
events:

  * •

$U_{K,L,\lambda}$ is the event that there is a unique component of $G({\cal
H}_{\lambda}\cap D_{L+1})$ that meets both $D_{K}$ and
$\mathbb{R}^{2}\setminus D_{L}$.

  * •

$F_{K,M,\lambda}=\\{D_{K}\leftrightarrow D_{K}(M{\bf e})$ in $G({\cal
H}_{\lambda}\cap D_{3M})\\}$.

######  Proposition 2.4.

Suppose $\lambda>\lambda_{c}(\phi)$ and let ${\varepsilon}\in(0,1)$. There
exist finite constants $K>0$ and $M>3K$ such that (i)
${\mathbb{P}}[D_{K}\leftrightarrow\infty$ in $G({\cal
H}_{\lambda})]>1-{\varepsilon}$, and (ii)
${\mathbb{P}}[U_{K,M/3,\lambda}]>1-{\varepsilon}$, and (iii)
${\mathbb{P}}[F_{K,M,\lambda}]>1-{\varepsilon}$.

We shall use this to establish the limiting behaviour of $s^{-2}L_{1}(G({\cal
H}_{\lambda,s}))$ for $\lambda>\lambda_{c}(\phi)$. The point is that we can
use it to compare $G({\cal H}_{\lambda})$ with a finite-range dependent
percolation process on the lattice $\mathbb{Z}^{2}$.

To prepare for the proof of Proposition 2.4, fix $\lambda>\lambda_{c}(\phi)$
and ${\varepsilon}\in(0,1)$. Set $\mu=(\lambda_{c}(\phi)+\lambda)/2$, so
$\mu\in(\lambda_{c}(\phi),\lambda)$. It is not hard to see that we can (and
do) choose $\eta>0$ such that for any two distinct points $x,y\in[0,3]^{2}$,

$\displaystyle{\mathbb{P}}[\\{x\\}\leftrightarrow\\{y\\}~{}{\rm in}~{}G({\cal
H}_{\lambda-\mu}^{x,y}\cap[0,3]^{2})]\geq\eta.$ (2.3)

We then choose $\nu\in\mathbb{N}$ such that
$(1-\eta)^{\nu/9}<{\varepsilon}/3$. Then take

$\displaystyle\delta:=({\varepsilon}/3)e^{-25\lambda\nu}.$ (2.4)

###### Proof of Proposition 2.4.

We adapt an argument in [1]. Let $\mu,\eta,\nu$ and $\delta$ be as given above
and let ${\varepsilon}_{1}=(\delta/3)^{32}$. Choose $K$ such that
${\mathbb{P}}[D_{K}\leftrightarrow\infty{\rm~{}in~{}}G({\cal
H}_{\mu})]>1-{\varepsilon}_{1}$. Since ${\varepsilon}_{1}<{\varepsilon}$ and
$\mu<\lambda$ this yields (i) at once. Since
${\varepsilon}_{1}<{\varepsilon}$, we claim that we can (and do) choose
$n_{1}\in\mathbb{N}$ with $n_{1}>K$ such that

$\displaystyle{\mathbb{P}}[U_{K,n,\lambda}]>1-{\varepsilon},~{}~{}~{}\forall
n\in[n_{1},\infty).$ (2.5)

We leave the proof of this claim, using Lemma 2.3, as an exercise.

Now for integer $n\geq n_{1}$, and for $0\leq\alpha\leq\beta\leq n+1$, define
the event

$E_{n}(\alpha,\beta):=\\{D_{K}\leftrightarrow[n,n+1]\times[\alpha,\beta]~{}{\rm
in}~{}G({\cal H}_{\mu}\cap S_{n+1})\\}.$

Since ${\mathbb{P}}[D_{K}\leftrightarrow S_{n}^{c}~{}{\rm in}~{}G({\cal
H}_{\mu}\cap S_{n+1})]\geq{\mathbb{P}}[D_{K}\leftrightarrow\infty~{}{\rm
in}~{}G({\cal H}_{\mu})]>1-{\varepsilon}_{1}$, using the Square Root trick we
can deduce that

$\displaystyle{\mathbb{P}}[E_{n}(0,n+1)]>1-{\varepsilon}_{1}^{1/8}.$ (2.6)

Next, observe that for fixed $n$, we have that as a function of $\alpha$,
${\mathbb{P}}[E_{n}(0,\alpha)]-{\mathbb{P}}[E_{n}(\alpha,n+1)]$ increases
continuously from a value of $-{\mathbb{P}}[E_{n}(0,n+1)]$ at $\alpha=0$ to a
value of $+{\mathbb{P}}[E_{n}(0,n+1)]$ at $\alpha=n+1$. Therefore we can and
do choose $\alpha_{n}\in(0,n+1)$ such that
${\mathbb{P}}[E_{n}(0,\alpha_{n})]={\mathbb{P}}[E_{n}(\alpha_{n},n+1)].$ Since
$E_{n}(0,n+1)=E_{n}(0,\alpha_{n})\cup E_{n}(\alpha_{n},n+1)$, by (2.6) and a
further application of the Square Root trick we obtain that

$\displaystyle{\mathbb{P}}[E_{n}(\alpha_{n},n+1)]={\mathbb{P}}[E_{n}(0,\alpha_{n})]>1-{\varepsilon}_{1}^{1/16}.$
(2.7)

By yet another application of the Square Root trick we obtain that

$\max({\mathbb{P}}[E_{n}(0,\alpha_{n}/2)],{\mathbb{P}}[E_{n}(\alpha_{n}/2,\alpha_{n})])>1-{\varepsilon}_{1}^{1/32}$

so we can and do choose $y_{n}$, with either $y_{n}=\alpha_{n}/4$ or
$y_{n}=3\alpha_{n}/4$, such that

$\displaystyle{\mathbb{P}}[E_{n}(y_{n}-\alpha_{n}/4,y_{n}+\alpha_{n}/4)]>1-{\varepsilon}_{1}^{1/32}.$
(2.8)

Set $n_{2}=3n_{1}$. We claim that there exists integer $N\geq n_{2}$ such that
$\alpha_{3N}<4\alpha_{N}$. Indeed, if this were not true then we would have
for all $k\geq 1$ that $\alpha_{3^{k}n_{2}}\geq 4^{k}\alpha_{n_{2}}$, but
since $\alpha_{n}\leq n+1$ for all $n$, this would imply $3^{k}(n_{2}+1)\geq
4^{k}\alpha_{n_{2}}$ so that $(4/3)^{k}\leq(n_{2}+1)/\alpha_{n_{2}}$ for all
$k$, which is not true (since $\alpha_{n_{2}}>0$), justifying the claim.

Choose (deterministic) integer $N\geq n_{2}$ such that
$\alpha_{3N}<4\alpha_{N}$. Then by (2.7) and (2.8), setting
${\varepsilon}_{2}:={\varepsilon}_{1}^{1/32}$ we have

$\displaystyle\min({\mathbb{P}}[E_{N}(\alpha_{N},N+1)],{\mathbb{P}}[E_{3N}(y_{3N}-\alpha_{3N}/4,y_{3N}+\alpha_{3N}/4)])>1-{\varepsilon}_{2}.$
(2.9)

Now set ${\bf x}=(2N,y_{3N})$ (we use bold face to indicate certain fixed
2-vectors such as ${\bf o}$ and ${\bf e}$). Let $S_{N+1}({\bf
x}):=S_{N+1}+{\bf x}=[N-1,3N+1]\times[y_{3N}-N-1,y_{3N}+N+1]$. Define the
vertical blocks (see Figure 1)

$I:=[3N,3N+1]\times[y_{3N}-\alpha_{3N}/4,y_{3N}+\alpha_{3N}/4],$
$J^{+}:=[3N,3N+1]\times[y_{3N}+\alpha_{N},y_{3N}+N+1],$
$J^{-}:=[3N,3N+1]\times[y_{3N}-N-1,y_{3N}-\alpha_{N}].$

Let $A^{+}$ be the event $\\{D_{K}({\bf x})\leftrightarrow J^{+}$ in $G({\cal
H}_{\mu}\cap S_{N+1}({\bf x}))\\}$, and let $A^{-}$ be the event
$\\{D_{K}({\bf x})\leftrightarrow J^{-}$ in $G({\cal H}_{\mu}\cap S_{N+1}({\bf
x}))\\}$. Then
${\mathbb{P}}[A^{+}]={\mathbb{P}}[A^{-}]={\mathbb{P}}[E_{N}(\alpha_{N},N+1)]$.
Also
$E_{3N}(y_{3N}-\alpha_{3N}/4,y_{3N}+\alpha_{3N}/4)=\\{D_{K}\leftrightarrow I$
in $G({\cal H}_{\mu}\cap S_{3N+1})\\}$. By (2.9) and the union bound,

${\mathbb{P}}[A^{+}\cap A^{-}\cap\\{D_{K}\leftrightarrow I{\rm~{}in~{}}G({\cal
H}_{\mu}\cap S_{3N+1})\\}]>1-3{\varepsilon}_{2}=1-\delta.$

Set $M=\|{\bf x}\|$. Then $M\geq 2N\geq 3n_{1}$. By (2.5),
${\mathbb{P}}[U_{K,M/3,\lambda}]>1-{\varepsilon}$ so we have (ii). The proof
is then completed by the following ‘gluing lemma’. ∎

######  Lemma 2.5.

If ${\mathbb{P}}[A^{+}\cap A^{-}\cap\\{D_{K}\leftrightarrow
I{\rm~{}in~{}}G({\cal H}_{\mu}\cap S_{3N+1})\\}>1-\delta$, then
${\mathbb{P}}[F_{K,M,\lambda}]>1-{\varepsilon}$, where we take $M=\|{\bf
x}\|$.

#### Remark.

The proof below is not needed for the special case $\phi={\bf 1}_{[0,1]}$,
since in this case the lemma is immediate because $A^{+}\cap
A^{-}\cap\\{D_{K}\leftrightarrow I{\rm~{}in~{}}G({\cal H}_{\mu}\cap
S_{3N+1})\\}$ implies $F_{K,M,\mu}$.

###### Proof of Lemma 2.5.

Divide $\mathbb{R}^{2}$ into half-open rectilinear squares $Q_{i}$ of side 1
and for each $i$ let $Q_{i}^{+}$ (respectively $Q_{i}^{++}$ be the half-open
square of side 3 (resp. 5) with the same centre. We shall define a random
variable $Z$ taking values in $\mathbb{Z}_{+}\cup\\{+\infty\\}$, as follows.

Grow the cluster ${\cal C}:=\cup_{x\in{\cal H}_{\mu}\cap D_{K}}{\cal
C}_{x}({\cal H}_{\mu}\cap S_{3N+1})$ sequentially. Let ${\cal P}$ be the point
process of unexplored points of ${\cal H}_{\mu}\cap S_{N+1}({\bf x})$ at the
end of this procedure, i.e. the points of ${\cal H}_{\mu}\cap S_{N+1}({\bf
x})$ that either lie outside $S_{3N+1}$ (since $S_{N+1}({\bf x})$ is
conceivably not entirely contained in $S_{3N+1}$) or are not connected by an
edge to any point of ${\cal C}$. If ${\cal C}\cap D_{K}({\bf
x})\neq\varnothing$ then set $Z=+\infty$.

Next, assuming $Z<\infty$, grow the cluster $\cup_{x\in{\cal P}\cap D_{K}({\bf
x})}{\cal C}_{x}({\cal P})$ sequentially but do not continue the exploration
from any points created that lie in $\cup_{\\{i:{\cal C}\cap
Q_{i}\neq\varnothing\\}}Q_{i}^{+}$; leave these points as ‘active’. We denote
this second cluster by ${\cal C}^{\prime}$. Let ${\cal I}:=\\{i:{\cal C}\cap
Q_{i}\neq\varnothing,{\cal C}^{\prime}\cap Q_{i}^{+}\neq\varnothing\\}$ and
let $Z=|{\cal I}|$, as illustrated in Figure 1.

Define the event $E:=\\{D_{K}\leftrightarrow I{\rm~{}in~{}}G({\cal
H}_{\mu}\cap S_{3N+1})\\}=\\{{\cal C}\cap I\neq\varnothing\\}$.

Recall that $\nu$ was defined just after (2.3). Suppose $\nu<Z<\infty$. Then
we can find $i_{1},\ldots i_{\lceil\nu/9\rceil}\in{\cal I}$ such that the
squares $Q_{i_{1}}^{+},\ldots,Q^{+}_{i_{\lceil\nu/9\rceil}}$ are disjoint.
Then sprinkling an independent Poisson process ${\cal H}_{\lambda-\mu}$ on top
of ${\cal H}_{\mu}$ in these squares, for each square we have a chance at
least $\eta$ to join ${\cal C}$ to ${\cal C}^{\prime}$ via the sprinkled
points in that square. Since we may assume ${\cal H}_{\lambda}={\cal
H}_{\mu}\cup{\cal H}_{\lambda-\mu}$, setting
$F^{\prime}=\\{D_{K}\leftrightarrow D_{K}({\bf x})~{}{\rm in}~{}G({\cal
H}_{\lambda}\cap(S_{3N+1}\cup S_{N+1}({\bf x})))\\}$ we obtain that

$\displaystyle{\mathbb{P}}[(F^{\prime})^{c}|E\cap\\{Z>\nu\\}]\leq(1-\eta)^{\nu/9}<{\varepsilon}/3.$
(2.10)

Note that if $Z=+\infty$ then $F^{\prime}$ must occur so this case is included
in (2.10).

Now suppose $Z\leq\nu$. If also event $E$ occurs then it is not possible to
find paths in $S_{N+1}({\bf x})$ both from $D_{K}({\bf x})$ to $J^{+}$, and
from $D_{K}({\bf x})$ to $J^{-}$, with neither path passing through
$\cup_{\\{i:{\cal C}\cap Q_{i}\neq\varnothing\\}}Q_{i}^{+}$ (see Figure 1, and
also [1, Figure 2.2]; note $I\cap J^{+}=I\cap J^{-}=\varnothing$ since
$\alpha_{3N}/4<\alpha_{N}$). Therefore we have not yet achieved event
$A^{+}\cap A^{-}$ at this stage.

Figure 1: The top cluster is the part of ${\cal C}$ within $S_{N+1}({\bf x})$.
The other clusters are ${\cal C}^{\prime}$, which is grown within the square
$S_{N+1}({\bf x})$ but without further exploration from the points created in
the shaded region, which is $\cup_{i:{\cal C}\cap
Q_{i}\neq\varnothing}Q_{i}^{+}$. In this case $Z=5$; the left-most branch of
${\cal C}^{\prime}$ reaching the shaded region contributes 2 to $Z$. It is not
possible for ${\cal C}^{\prime}$ to reach both $J^{+}$ and $J^{-}$ by this
stage.

At the next stage sample all of the new Poisson points (not part of ${\cal C}$
or ${\cal C}^{\prime}$) in the union of squares $Q_{i}^{++}$, $i\in{\cal I}$.
With probability at least $e^{-25\mu\nu}$ no new Poisson points are generated
at this stage, and if this is the case then $A^{+}\cap A^{-}$ does not occur
because the cluster ${\cal C}^{\prime}$ dies out without having reached
$J^{+}$ and $J^{-}$. Thus we have

${\mathbb{P}}[(A^{+}\cap
A^{-})^{c}|E\cap\\{Z\leq\nu\\}]\geq\exp(-25\mu\nu)\geq\exp(-25\lambda\nu).$

Therefore if ${\mathbb{P}}[A^{+}\cap A^{-}\cap E]>1-\delta$ we have

$\delta>{\mathbb{P}}[(A^{+}\cap A^{-})^{c}\cap
E\cap\\{Z\leq\nu\\}]>\exp(-25\lambda\nu){\mathbb{P}}[E\cap\\{Z\leq t\\}],$

so that ${\mathbb{P}}[E\cap\\{Z\leq\nu\\}]<{\varepsilon}/3$ by (2.4). Combined
with (2.10), this shows that

$\displaystyle{\mathbb{P}}[(F^{\prime})^{c}]\leq{\mathbb{P}}[E^{c}]+{\mathbb{P}}[(F^{\prime})^{c}|E\cap\\{Z>\nu\\}]+{\mathbb{P}}[E\cap\\{Z\leq\nu\\}]<{\varepsilon}.$
(2.11)

Since $(S_{3N+1}\cup S_{N+1}({\bf x}))\subset D_{6N}\subset D_{3\|{\bf x}\|}$,
we have $F^{\prime}\subset\\{D_{K}\leftrightarrow D_{K}({\bf x})$ in $G({\cal
H}_{\lambda}\cap D_{3\|{\bf x}\|})\\}$. By rotation invariance, with
$M:=\|{\bf x}\|$, we thus have
${\mathbb{P}}[F_{K,M,\lambda}]>1-{\varepsilon},$ as required. ∎

###  2.4 Connection probability

Given $s>0$, let $V_{s},W_{s}$ be independent uniformly distributed points in
$B(s)$, independent of ${\cal H}_{\lambda}$. We shall characteraize the giant
component of $G({\cal H}_{\lambda,s})$ in terms of those vertices which are
path-connected to a fixed disk centred at ${\bf o}$. For this, the following
is useful.

######  Proposition 2.6.

Suppose $\lambda>\lambda_{c}(\phi)$. Let ${\varepsilon}>0$. Then there exists
$K>0$ such that ${\mathbb{P}}[D_{K}\leftrightarrow\infty{\rm~{}in~{}}G({\cal
H}_{\lambda})]>1-{\varepsilon}$, and

$\displaystyle\liminf_{s\to\infty}{\mathbb{P}}[\\{V_{s}\\}\leftrightarrow
D_{K}~{}{\rm in}~{}G({\cal
H}_{\lambda,s}^{V_{s}})]\geq\theta(\phi,\lambda)-{\varepsilon}.$ (2.12)

###### Proof.

Assume $\lambda>\lambda_{c}(\phi)$. Let
${\varepsilon}_{1}\in(0,{\varepsilon}/3)$ be chosen such that if
$(X_{x})_{x\in\mathbb{Z}^{2}}$ is a $7$-dependent Bernoulli random field on
$\mathbb{Z}^{2}$ with ${\mathbb{P}}[X_{x}=1]>1-5{\varepsilon}_{1}$ for all
$x\in\mathbb{Z}^{d}$, than for all $s>0$ and all $x\in\mathbb{Z}^{2}\cap
B(s)$, there is a lattice path of 1’s from $x$ to ${\bf o}$ in
$\mathbb{Z}^{2}\cap B(s)$ with probability greater than $1-{\varepsilon}/3$.
The proof that such an ${\varepsilon}_{1}$ exists is standard, using e.g. [4]
and a Peierls argument (e.g. [7, Theorem 9.8]).

Using Proposition 2.4, choose $K,M$ such that $0<K<M/3$, and

$\min({\mathbb{P}}[D_{K}\leftrightarrow\infty{\rm~{}in~{}}G({\cal
H}_{\lambda})],{\mathbb{P}}[U_{K,M/3,\lambda}],{\mathbb{P}}[F_{K,M,\lambda}])>1-{\varepsilon}_{1}.$

For each $x,y\in\mathbb{Z}^{2}$ with $\|x-y\|=1$, let $U_{x}$ denote the event
that there is a unique component of $G({\cal H}_{\lambda}\cap
D_{(M/3)+1}(Mx))$ that meets both $D_{K}(Mx)$ and $\mathbb{R}^{2}\setminus
D_{M/3}(Mx)$. Let $F_{xy}:=\\{D_{K}(Mx)\leftrightarrow D_{K}(My)$ in $G({\cal
H}_{\lambda}\cap D_{3M}(Mx))\\}$.

By translation and rotation invariance of ${\cal H}_{\lambda}$,
${\mathbb{P}}[U_{x}]>1-{\varepsilon}_{1}$ for each $x$, and
${\mathbb{P}}[F_{xy}]>1-{\varepsilon}_{1}$ for each $(x,y)$.

For each $x\in\mathbb{Z}^{2}$, let us set $X_{x}=1$ if event $U_{x}$ occurs,
and also $F_{xy}$ occurs for each of the four $y\in\mathbb{Z}^{2}$ with
$\|y-x\|=1$; otherwise set $X_{x}=0$. Then by the union bound
${\mathbb{P}}[X_{x}=1]\geq 1-5{\varepsilon}_{1}$. Also $X_{x}$ is determined
by ${\cal H}_{\lambda}|_{D_{3M}(Mx)}$, so $(X_{x},x\in\mathbb{Z}^{2})$ is a
$7$-dependent Bernoulli random field. By the choice of ${\varepsilon}_{1}$,
for any $x\in B(s/M)\cap\mathbb{Z}^{2}$ there is a lattice path in
$\mathbb{Z}^{2}\cap B(s/M)$ from $x$ to ${\bf o}$ of sites $z$ with $X_{z}=1$,
with probability at least $1-{\varepsilon}/3$.

Since ${\mathbb{P}}[D_{M+K}\leftrightarrow\infty$ in $G({\cal
H}_{\lambda})]\geq{\mathbb{P}}[D_{K}\leftrightarrow\infty$ in $G({\cal
H}_{\lambda})]>1-{\varepsilon}_{1}$, by a similar argument to (2.5) we can
(and do) take $M_{1}>M+K$ such that
${\mathbb{P}}[U_{M+K,M_{1},\lambda}]>1-{\varepsilon}_{1}$.

Let $x_{V}$ be the closest point in $\mathbb{Z}^{2}\cap B(s/M)$ to
$M^{-1}V_{s}$. Consider the following events:

  * •

$A_{1}:=\\{V_{s}\in B(s-4M_{1})\setminus D_{3M_{1}}\\}$. Provided $s$ is large
enough ${\mathbb{P}}[A_{1}]>1-{\varepsilon}/3$.

  * •

$A_{2}$ is the event that there is a lattice path from $x_{V}$ to ${\bf o}$
within $\mathbb{Z}^{2}\cap B(s/M)$ with $X_{x}=1$ for all sites $x$ in the
path. By the previous discussion, ${\mathbb{P}}[A_{2}]>1-{\varepsilon}/3$.

  * •

$A_{3}$ is the event that there is a unique component in $G({\cal
H}_{\lambda}^{V_{s}}\cap D_{M_{1}+1}(V_{s}))$ that meets both $D_{M+K}(V_{s})$
and $\mathbb{R}^{2}\setminus D_{M_{1}}(V_{s})$. Then
${\mathbb{P}}[A_{3}]>1-{\varepsilon}/3$.

  * •

$A_{4}$ is the event that ${\cal C}_{V_{s}}({\cal H}_{\lambda}^{V_{s}})\cap
D_{M_{1}}(V_{s})^{c}\neq\varnothing$. Then
${\mathbb{P}}[A_{4}]\geq\theta(\phi,\lambda)$.

By the union bound
${\mathbb{P}}[\cap_{i=1}^{4}A_{i}]\geq\theta(\phi,\lambda)-{\varepsilon},$ for
all large enough $s$. Therefore it suffices to prove that if
$\cap_{i=1}^{4}A_{i}$ occurs, then $\\{V_{s}\\}\leftrightarrow D_{K}$ in
$G({\cal H}_{\lambda,s}^{V_{s}})$.

To see this, suppose $\cap_{i=1}^{4}A_{i}$ occurs. Then we have
$\|Mx_{V}-V_{s}\|\leq M$ so using $A_{1}$ we have ${\rm
dist}(Mx_{V},B(s)^{c})\geq 2M_{1}-M>(M/3)+1$. Since $X_{x_{V}}=1$ by $A_{2}$,
there is a unique component of $G({\cal H}_{\lambda}\cap D_{(M/3)+1}(Mx_{V}))$
that meets both $D_{K}(Mx_{V})$ and $D_{(M/3)}(Mx_{V})^{c}$, and since
$D_{(M/3)+1}(Mx_{V})\subset B(s)$, this extends to a unique component of
$G({\cal H}_{\lambda,s})$ that meets both $D_{K}(Mx_{V})$ and
$D_{M/3}(Mx_{V})^{c}$. We denote this component by ${\cal C}$.

By $A_{2}$, the component ${\cal C}$ includes a vertex in $D_{K}$. Choose such
a vertex and denote it by $z$.

Next, using $A_{1}$ observe that $\|V_{s}-z\|\geq\|V_{s}\|-\|z\|\geq
3M_{1}-K\geq 2M_{1}$. Also $D_{K}(Mx_{V})\subset D_{M+K}(V_{s})$, so ${\cal
C}$ meets both $D_{M+K}(V_{s})$ and $D_{M_{1}}(V_{s})^{c}$. Therefore using
$A_{3}$ and $A_{4}$ we have that $V_{s}$ is connected to ${\cal C}$, and thus
$\\{V_{s}\\}\leftrightarrow D_{K}$ in $G({\cal H}_{\lambda,s}^{V_{s}})$ as
required. ∎

###  2.5 Proof of the giant component phenomenon

We now write $L_{i,s}$ for $L_{i}(G({\cal H}_{\lambda,s}))$ (we are thinking
of $\lambda$ and $\phi$ as fixed with $\sup\\{r:\phi(r)>0\\}=1$). For
convenience, we re-state Theorem 1.2, which we are now ready to prove.

######  Theorem 2.7.

If $\lambda>\lambda_{c}(\phi)$, then $s^{-2}L_{1,s}\stackrel{{\scriptstyle
P}}{{\longrightarrow}}\lambda\theta(\phi,\lambda)$ and
$s^{-2}L_{2,s}\stackrel{{\scriptstyle P}}{{\longrightarrow}}0$ as
$s\to\infty$.

###### Proof.

Assume $\lambda>\lambda_{c}(\phi)$. Let ${\varepsilon}>0$ and using
Proposition 2.6, choose $K>0$ such that
${\mathbb{P}}[D_{K}\leftrightarrow\infty{\rm~{}in~{}}G({\cal
H}_{\lambda})]>1-{\varepsilon}$, and (2.12) holds. Consider the sum

$N_{s}:=\sum_{x\in{\cal H}_{\lambda,s}}{\bf 1}\\{\\{x\\}\leftrightarrow
D_{K}{\rm~{}in~{}}G({\cal H}_{\lambda,s})\\}.$

Let $V_{s},W_{s}$ be as in Section 2.4. By the Mecke formula,
${\mathbb{E}}N_{s}=\lambda s^{2}{\mathbb{P}}[\\{V_{s}\\}\leftrightarrow
D_{K}{\rm~{}in~{}}G({\cal H}_{\lambda,s}^{V_{s}})].$ Then using (2.12), and
writing just $\theta$ for $\theta(\phi,\lambda)$, we deduce that

$\displaystyle\liminf_{s\to\infty}s^{-2}{\mathbb{E}}N_{s}\geq\lambda(\theta-{\varepsilon}).$
(2.13)

Next, let $N^{\prime}_{s}=\sum_{x\in{\cal H}_{\lambda,s}}{\bf 1}\\{|{\cal
C}_{x}({\cal H}_{\lambda,s})|\geq s^{1/2}\\}$ (here $|\cdot|$ represents
number of elements). Using the Mecke formula (2.1) we have that
${\mathbb{E}}[N^{\prime}_{s}]=\lambda s^{2}{\mathbb{P}}[|{\cal
C}_{V_{s}}({\cal H}_{\lambda,s}^{V_{s}})|\geq s^{1/2}]$, and hence

$\displaystyle\lim_{s\to\infty}s^{-2}{\mathbb{E}}N^{\prime}_{s}=\lambda\theta.$
(2.14)

Also
${\mathbb{E}}[N^{\prime}_{s}(N^{\prime}_{s}-1)]=\lambda^{2}s^{4}{\mathbb{P}}[|{\cal
C}_{V_{s}}({\cal H}_{\lambda,s}^{V_{s},W_{s}})|\geq s^{1/2},|{\cal
C}_{W_{s}}({\cal H}_{\lambda,s}^{V_{s},W_{s}})|\geq s^{1/2}|]$ by (2.2), so
that

$\lim_{s\to\infty}s^{-4}{\mathbb{E}}[N^{\prime}_{s}(N^{\prime}_{s}-1)]=\lambda^{2}\theta^{2}.$

Thus $s^{-2}N^{\prime}_{s}\to\lambda\theta$ in $L^{2}$ and hence in
probability, as $s\to\infty$.

Since $(N_{s}-N^{\prime}_{s})^{+}\leq{\cal H}_{\lambda}(D_{K+s^{1/2}})$ we
have that $s^{-2}{\mathbb{E}}[(N_{s}-N^{\prime}_{s})^{+}]\to 0$ as
$s\to\infty$. Hence by (2.13) and (2.14),

$\limsup{\mathbb{E}}[s^{-2}(N^{\prime}_{s}-N_{s})^{+}]=\limsup{\mathbb{E}}[s^{-2}(N^{\prime}_{s}-N_{s})]\leq\lambda{\varepsilon}.$

Hence by Markov’s inequality
$\limsup_{s\to\infty}{\mathbb{P}}[s^{-2}(N_{s}^{\prime}-N_{s})\geq{\varepsilon}^{1/2}]\leq\lambda{\varepsilon}^{1/2},$
and hence

$\displaystyle\limsup_{s\to\infty}{\mathbb{P}}[s^{-2}N_{s}\leq\lambda\theta-2{\varepsilon}^{1/2}]\leq\limsup_{s\to\infty}$
$\displaystyle\left({\mathbb{P}}[s^{-2}N^{\prime}_{s}\leq\lambda\theta-{\varepsilon}^{1/2}]\right.$
$\displaystyle\left.+{\mathbb{P}}[s^{-2}(N_{s}-N^{\prime}_{s})\leq-{\varepsilon}^{1/2}]\right)\leq\lambda{\varepsilon}^{1/2}.$

As at (2.5), we can and do choose $M_{2}>K$ so
${\mathbb{P}}[U_{K,M_{2}}]>1-{\varepsilon}$. If $(s/2)>M_{2}+1$ and
$U_{K,M_{2}}$ occurs then $L_{1,s}\geq N_{s}-{\cal H}_{\lambda}(D_{M_{2}})$,
since all $x\in{\cal H}_{\lambda,s}\setminus D_{M_{2}}$ that are path-
connected to $D_{K}$ lie in the same component of $G({\cal H}_{\lambda,s})$.
Therefore

${\mathbb{P}}[s^{-2}L_{1,s}\leq\lambda\theta-3{\varepsilon}^{1/2}]\leq{\mathbb{P}}[s^{-2}N_{s}\leq\lambda\theta-2{\varepsilon}^{1/2}]+{\mathbb{P}}[s^{-2}{\cal
H}_{\lambda}(D_{M_{2}})\geq{\varepsilon}^{1/2}]+{\mathbb{P}}[U_{K,M_{2}}^{c}]$

so that

$\displaystyle\limsup_{s\to\infty}{\mathbb{P}}[s^{-2}L_{1,s}\leq\lambda\theta-3{\varepsilon}^{1/2}]\leq\lambda{\varepsilon}^{1/2}+{\varepsilon}.$
(2.15)

Conversely, note that if $s^{2}\lambda(\theta+{\varepsilon})>s^{1/2}$ then

${\mathbb{P}}[s^{-2}L_{1,s}\geq\lambda(\theta+{\varepsilon})]\leq{\mathbb{P}}[s^{-2}N^{\prime}_{s}\geq\lambda(\theta+{\varepsilon})]$

which tends to zero. Combined with (2.15) this shows that
$s^{-2}L_{1,s}\stackrel{{\scriptstyle P}}{{\longrightarrow}}\lambda\theta$.

If $s^{2}\lambda(\theta+{\varepsilon})>s^{1/2}$ and $L_{1,s}+L_{2,s}\geq
s^{2}\lambda(\theta+{\varepsilon})$ then either $N^{\prime}_{s}\geq
s^{2}\lambda(\theta+{\varepsilon})$ or $L_{1,s}+s^{1/2}\geq
s^{2}\lambda(\theta+{\varepsilon})$. Hence
${\mathbb{P}}[s^{-2}(L_{1,s}+L_{2,s})>\lambda(\theta+{\varepsilon})]\to 0.$
Combined with (2.15) this shows that
$s^{-2}(L_{1,s}+L_{2,s})\stackrel{{\scriptstyle
P}}{{\longrightarrow}}\lambda\theta$ and hence by Slutsky’s theorem,
$s^{-2}L_{2,s}\stackrel{{\scriptstyle P}}{{\longrightarrow}}0$. ∎

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