thm prop defi lma cor re

#  ON THE RATE OF NORMAL APPROXIMATION FOR POISSON CONTINUUM PERCOLATION

Tiffany Y. Y. Lo email<EMAIL_ADDRESS>Work supported by the
Australian Research Council Grant No DP190100613, Knut and Alice Wallenberg
Foundation, Ragnar Söderberg Foundation and Swedish Research Council.
Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37
Uppsala, Sweden  Aihua Xia email<EMAIL_ADDRESS>Work supported by
the Australian Research Council Grant No DP190100613. School of Mathematics
and Statistics, the University of Melbourne, Parkville VIC 3010, Australia

(August 28, 2024)

###### Abstract

It is known that the number of points in the largest cluster of a percolating
Poisson process restricted to a large finite box is asymptotically normal. In
this note, we establish a rate of convergence for the statement. As each point
in the largest cluster is determined by points as far as the diameter of the
box, known results in the literature of normal approximation for Poisson
functionals cannot be directly applied. To disentangle the long-range
dependence of the largest cluster, we use the fact that the second largest
cluster has comparatively shorter range of dependence to restrict the range of
dependence, apply a recently established result in Stein’s method to obtain a
Berry-Esseen type bound for the normal approximation of the number of points
belonging to clusters that have a restricted range of dependence, and then
estimate the gap between this quantity and the number of points in the largest
cluster.

Key words and phrases: Berry-Esseen bound; Poisson percolation; Stein’s
method.

AMS 2020 Subject Classification: primary 60K35, 60F05; secondary 60D05, 60G57,
82B43, 62E20.

##  1 Introduction and the main result

Let m be the $m$-dimensional Euclidean space equipped with the Euclidean norm
$\|\cdot\|$. For each $A,B\subset^{m}$, we define $d(A,B)=\inf\\{\|x-y\|:\
x\in A,y\in B\\}$, where $\inf\emptyset:=\infty$. For simplicity, we write
$d(\\{x\\},B)=:d(x,B)$. For $r>0$, we define $B(A,r)=\\{y\in^{m}:\
d(y,A)<r\\}$ and write $B(\\{x\\},r)=B(x,r)$, so that $B(x,r)$ is simply a
ball of radius $r$ with its centre at $x$. We use ${\rm card}(A)$ or $|A|$ to
denote the cardinality of the set $A$. For convenience, we use the term
cardinality and size interchangeably.

######  Definition 1.1.

For a Borel subset ${\mathscr{X}}\subset^{m}$, a subset
$A\subset{\mathscr{X}}$ is called a cluster or connected component of
${\mathscr{X}}$ with radius $r$ if $B(A,r)\cap{\mathscr{X}}=A$, and either
${\rm card}(A)=1$ or ${\rm card}(A)>1$ with $d(x,A\backslash\\{x\\})<r$ for
all $x\in A$. Two clusters $A_{1},A_{2}$ are said to be disconnected if
$d(A_{1},A_{2})\geq r$.

For fixed $r>0$, we say that a point process ${\mathscr{X}}$ percolates with
radius $r$ if ${\mathscr{X}}$ almost surely contains a unique infinite cluster
with radius $r$. Let ${\mathscr{P}}^{\lambda}$ be the homogeneous Poisson
point process with rate $\lambda>0$ on m. For any $r>0$ and $m\geq 2$, it is
well known that there exists $0<\lambda_{c}(r,m)<\infty$ such that
${\mathscr{P}}^{\lambda}$ percolates if and only if
$\lambda>\lambda_{c}(r,m)$; see [Zuev and Sidorenko (1985a), Zuev and
Sidorenko (1985b)]. Since ${\mathscr{P}}^{\lambda}$ is scale invariant, it is
enough to consider only $r=1$ and write
$\lambda_{c}:=\lambda_{c}(m):=\lambda_{c}(1,m)$. Proving the exact values of
$\lambda_{c}(m)$ remains an open question, although for $m=2$, a sharp
estimate was given in [Balister, Bollobás and Walter (2005)]. There is also a
vast literature considering more general continuum percolation since [Gilbert
(1961)] initiated the study, where the point processes are not necessarily
homogeneous Poisson and each ball $B(x,r)$ can be replaced by a random shape
centred at $x$; we refer the reader to [Meester and Roy (1996)] for a
comprehensive overview.

In practice, any physical system is finite and the percolation phenomenon is
examined through growing observation windows, hence it is of practical
interest to study the asymptotic behaviour of the cardinality of the largest
cluster inside a growing window in $\mathbb{R}^{m}$. The statistical behaviour
of the largest cluster in a large finite observation window under both the
regimes $\lambda<\lambda_{c}$ and $\lambda>\lambda_{c}$ have been thoroughly
investigated in [Penrose (2003), Penrose and Pisztora (1996)], and here we
briefly summarise some results for the case $\lambda>\lambda_{c}$. Let
$\Gamma_{n}:=[-n/2,n/2]^{m}$ be such a window and $N_{n}$ be the number of
points in the largest cluster in ${\mathscr{P}}^{\lambda}\cap\Gamma_{n}$. It
was shown in [Penrose (2003), Chapter 10] that when $m\geq 2$,
$n^{-m}N_{n}\to\lambda p(\lambda)$ in probability as $n\to\infty$, where
$0<p(\lambda)<1$ is the probability that the infinite cluster contains the
origin. Furthermore, with probability tending to one as $n\to\infty$, the size
of the second largest cluster in $\Gamma_{n}$ is of the exact order $(\ln
n)^{m/(m-1)}$, thus establishing the uniqueness of the largest cluster. Large
deviation estimates for the size, volume, and diameter of the largest cluster
were provided in [Penrose and Pisztora (1996)]. The result that is most
pertinent to our work here, is the central limit theorem for $N_{n}$
established in [Penrose (2003), Theorem 10.22] that holds for $m\geq 2$ and
$\lambda>\lambda_{c}$. Let $B_{n}^{2}=\mathrm{Var}(N_{n})$. [Penrose (2003),
Theorem 10.22] and the errata in [Penrose (2010)] showed that, for $m\geq 2$,
there exists a constant $0<\sigma^{2}:=\sigma^{2}(\lambda,m)<\infty$ such that

$n^{-m}B_{n}^{2}\to\sigma^{2}$ (1.1)

and

$\mathscr{L}(n^{-m/2}(N_{n}-\mathbb{E}N_{n}))\to{\cal
N}(0,\sigma^{2})\quad\text{as $n\to\infty$.}$

Our main result below compliments this central limit theorem by providing a
convergence rate in the Kolmogorov distance.

######  Theorem 1.2.

Suppose that $m\geq 2$ and $\lambda>\lambda_{c}$, and let
$W_{n}:=(N_{n}-\mathbb{E}N_{n})/B_{n}$. Then

$d_{\mathrm{K}}(\mathscr{L}(W_{n}),{\cal
N}(0,1)):=\sup_{x\in}|\mathbb{P}(W_{n}\leq x)-\mathbb{P}(Z\leq x)|\leq
O\left(n^{-m/2}(\ln n)^{\frac{2m}{m-1}}\right),$ (1.2)

where $Z\sim{\cal N}(0,1)$.

######  Remark 1.3.

The logarithmic factor in (1.2) seems unavoidable because the percolation is a
long-range dependent structure and it differs significantly from the geometric
structures studied in [Schreiber and Yukich (2021)].

The proof of the central limit theorem in [Penrose (2003)] hinges on a
martingale argument, while here we rely on Stein’s method [Chen, Goldstein and
Shao (2011)] to deduce the convergence and the rate in Theorem 1.2. Besides
Stein method, one may also consider other tools such as the stabilisation tool
[Penrose and Yukich (2001), Penrose and Yukich (2005)], the Malliavin-Stein
technique via the Wiener-Itô expansion [Peccati et al. (2010)] and the second
order Poincaré inequalities [Last, Peccati and Schulte (2016)]. In fact, using
these tools, a variety of central limit theorems have been developed for
random quantities of the form $\sum_{x\in{\mathscr{P}}^{\lambda}\cap
A}\xi(x,{\mathscr{P}}^{\lambda})$, where $A\subset\mathbb{R}^{m}$ is a bounded
Borel set, and $\xi(x,{\mathscr{P}}^{\lambda})$ is a score function that
measures the contribution of $x$ with respect to ${\mathscr{P}}^{\lambda}$;
see for examples [Barbour and Xia (2001), Penrose and Yukich (2001), Penrose
and Yukich (2003), Penrose and Yukich (2005), Peccati et al. (2010), Schulte
(2012), Schulte (2016), Last, Peccati and Schulte (2016), Lachièze-Rey,
Schulte and Yukich (2019), Cong and Xia (2020), Schreiber and Yukich (2021),
Bhattacharjee and Molchanov (2022)]. In our setting, $A$ is replaced by
$\Gamma_{n}$, and $\xi(x,{\mathscr{P}}^{\lambda})$ can be taken as the
indicator function that takes value one if the Poisson point $x$ belongs to
the largest cluster in $\Gamma_{n}$. To obtain rates of convergence, the
existing literature on normal approximation generally requires the score
functions to have short-range dependence, which loosely speaking, is the
condition that the score function $\xi(x,{\mathscr{P}}^{\lambda})$ depends
only on points that are not too far away from $x$. For instance,
[Bhattacharjee and Molchanov (2022), Penrose (2007), Schulte (2016), Xia and
Yukich (2015), Schreiber and Yukich (2021)] require the score function
$\xi(x,{\mathscr{P}}^{\lambda})$ to be determined by the points of
${\mathscr{P}}^{\lambda}$ in a ball $B(x,R)$ such that $\mathbb{P}(R>t)$
decreases as the reciprocal of a polynomial or an exponential function of $t$
as $t\to\infty$. In our case, with the long-range dependence of the cells in
the percolation, $\mathbb{P}(R=\Theta(n))\approx 1$, so the score function in
consideration here does not fit into the framework of such literature.

Strategy of the proof. To disentangle the long-range dependence, we use the
characteristic of the second largest cluster to construct a suitable score
function $\xi^{\prime}(x,{\mathscr{P}}^{\lambda})$ that takes value one if $x$
belongs to a ‘local’ cluster that is typically larger than the second largest
cluster, apply [Chen, Röllin and Xia (2021), Corollary 3.2] to obtain a Berry-
Esseen type bound for the normal approximation of the sum
$N_{\theta,n}^{\prime}$ of these score functions, and then bound the gap
between $N_{n}$ and $N_{\theta,n}^{\prime}$.

##  2 The proof of Theorem 1.2

Recall that $N_{n}$ is the number of Poisson points contained in the largest
cluster in ${\mathscr{P}}^{\lambda}\cap\Gamma_{n}$. To represent $N_{n}$ as
the sum of appropriate score function, for any ${\mathscr{X}}\subset^{m}$, we
write ${\cal C}({\mathscr{X}})$ as the set of all clusters of ${\mathscr{X}}$,
and for $x\in{\mathscr{X}}$, let ${\cal C}(x,{\mathscr{X}})=\cup_{x\in
A\in{\cal C}({\mathscr{X}})}A$ be the largest cluster of ${\mathscr{X}}$
containing $x$, and write ${\cal C}_{0}({\mathscr{X}})$ as the largest cluster
of ${\mathscr{X}}$ if it is unique. Furthermore, let $\Gamma=\Gamma_{n}$,
${\mathscr{P}}^{\lambda}_{n}={\mathscr{P}}^{\lambda}\cap\Gamma_{n}$, and
define the score function of the point configuration
${\mathscr{P}}^{\lambda}_{n}$ at $x$ as
$\xi(x,{\mathscr{P}}^{\lambda}_{n}):={\bf 1}_{[{\cal
C}(x,{\mathscr{P}}^{\lambda}_{n})={\cal
C}_{0}({\mathscr{P}}^{\lambda}_{n})]}$. The score function collects the points
in the largest cluster in ${\mathscr{P}}^{\lambda}_{n}$ and
$N_{n}=\sum_{x\in{\mathscr{P}}^{\lambda}_{n}}\xi(x,{\mathscr{P}}^{\lambda}_{n})$.

To tackle the long-range dependence, we first observe that the typical size of
the second largest cluster in $\Gamma$ is no more than $c(\ln n)^{m/(m-1)}$
for a constant $c>0$ not depending $n$. Next, for each $x\in^{m}$, we take the
cube $A_{x,\theta,1}$ with the centre $x$ and edge length $2(\theta\ln
n)^{\frac{1}{m-1}}$, $A_{x,\theta,2}=A_{x,\theta,1}\cap\Gamma$, and show that
the point $x$ is in the largest cluster in $\Gamma$ is essentially the same as
that ${\cal C}(x,{\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})$ is the largest
cluster in ${\mathscr{P}}^{\lambda}\cap A_{x,\theta,2}$. However, the latter
characterisation ensures that its corresponding score function has short-range
dependence, so the tools of normal approximation to the sum of locally
dependent score functions can be applied. For the size of the second largest
cluster, the following lemma is a direct consequence of [Penrose (2003),
(10.58)] and [Penrose and Pisztora (1996), Theorem 2].

######  Lemma 2.1.

Let ${\cal C}^{(2)}_{0}({\mathscr{P}}^{\lambda}_{n})$ be the second largest
cluster in $\Gamma$. Then there exists $k_{0}>0$ such that for any $k_{1}\geq
k_{0}$, there exist constants $k_{2}(k_{1})=:k_{2}>0$ and
$n_{0}(k_{1})=:n_{0}>0$ not depending $n$ such that

$\mathbb{P}\left(|{\cal C}^{(2)}_{0}({\mathscr{P}}^{\lambda}_{n})|\geq
k_{1}(\ln n)^{m/(m-1)}\right)\leq\exp\\{-k_{2}\ln n\\}$ (2.1)

for $n\geq n_{0}$.

Write ${\cal C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})$ as the
largest cluster in ${\mathscr{P}}^{\lambda}\cap A_{x,\theta,2}$ if it is
unique, and define another score function
$\xi^{\prime}(x,\theta,{\mathscr{P}}^{\lambda}_{n}):={\bf 1}_{[{\cal
C}(x,{\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})={\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})]}$, so that
$\xi^{\prime}(x,\theta,{\mathscr{P}}^{\lambda}_{n})=1$ if $x$ belongs to the
largest cluster of ${\mathscr{P}}^{\lambda}\cap A_{x,\theta,2}$. We now assess
the difference between $N_{n}$ and
$N_{\theta,n}^{\prime}:=\sum_{x\in{\mathscr{P}}^{\lambda}_{n}}\xi^{\prime}(x,\theta,{\mathscr{P}}^{\lambda}_{n})$.

######  Lemma 2.2.

There exists a constant $\theta>0$ such that

$\displaystyle\mathbb{P}\left(N_{\theta,n}^{\prime}-N_{n}\neq 0\right)$
$\displaystyle=O\left(n^{-10m}\right).$

Proof. Let $\tilde{\theta}:=\tilde{\theta}(\lambda)$ denote the probability
that there is an unbounded cluster $D$ such that $\cup_{x\in D}B(x,1)$
intersects the ball of unit volume centred at the origin ${\bf 0}\in^{m}$.
[Penrose and Pisztora (1996), Theorem 1] states that there exist constants
$k_{3}>0$ and $n_{1}>0$ such that

$\displaystyle\mathbb{P}\left(|{\cal
C}_{0}({\mathscr{P}}^{\lambda}_{n})|<0.5\lambda\tilde{\theta}n^{m}\right)$
$\displaystyle\leq\exp(-k_{3}n^{m-1}),\ \ \ n\geq n_{1},$ (2.2)
$\displaystyle\mathbb{P}\left(\mbox{the largest cluster in
}{\mathscr{P}}^{\lambda}_{n}\mbox{ is not unique}\right)$
$\displaystyle\leq\exp(-k_{3}n^{m-1}),\ \ \ n\geq n_{1}.$ (2.3)

In fact, the argument for (2.3) can be adapted to show that, for any
$x\in\Gamma$,

$\mathbb{P}\left(\mbox{the largest cluster in
}{\mathscr{P}}^{\lambda}_{n}\cup\\{x\\}\mbox{ is not
unique}\right)\leq\exp(-k_{3}n^{m-1}),\ \ \ n\geq n_{1}.$ (2.4)

With $k_{1},k_{2}$ in (2.1) and $k_{3}$ in (2.2), we define
$\theta:=11m\max\left(\frac{1}{k_{3}},\frac{2(2k_{1})^{(m-1)/m}}{k_{2}(\lambda\tilde{\theta})^{(m-1)/m}}\right)$,
$\theta_{1}:=\theta(\lambda\tilde{\theta}/(2k_{1}))^{(m-1)/m}$,
$a_{n}:=k_{1}(\theta_{1}\ln
n)^{\frac{m}{m-1}}=\frac{1}{2}\lambda\tilde{\theta}(\theta\ln
n)^{\frac{m}{m-1}}$.

The event $\\{N_{\theta,n}^{\prime}\neq N_{n}\\}$ can be analysed through the
sizes of the clusters ${\cal C}_{0}({\mathscr{P}}^{\lambda}\cap
A_{x,\theta,2})$, $x\in{\mathscr{P}}^{\lambda}_{n}$. More precisely, let
$E_{0,x}$ be the event that the largest cluster in
${\mathscr{P}}^{\lambda}\cap A_{x,\theta,2}$ is not unique, $E_{1,x}:=\\{{\cal
C}(x,{\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})\neq{\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2}),{\cal
C}(x,{\mathscr{P}}^{\lambda}_{n})={\cal
C}_{0}({\mathscr{P}}^{\lambda}_{n})\\}$, $E_{2,x}:=\\{{\cal
C}(x,{\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})={\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2}),{\cal
C}(x,{\mathscr{P}}^{\lambda}_{n})\neq{\cal
C}_{0}({\mathscr{P}}^{\lambda}_{n})\\}$, then
$\\{\xi(x,{\mathscr{P}}^{\lambda}_{n})\neq\xi^{\prime}(x,\theta,{\mathscr{P}}^{\lambda}_{n})\\}\subset
E_{0,x}\cup E_{1,x}\cup E_{2,x}$. Let $E_{3,x}$ be the event that ${\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})$ and ${\cal
C}_{0}({\mathscr{P}}^{\lambda}_{n})$ are disconnected, we now show that
$E_{1,x}\cup E_{2,x}\subset E_{3,x}$ by contradiction. Suppose that ${\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})$ and ${\cal
C}_{0}({\mathscr{P}}^{\lambda}_{n})$ are connected, then ${\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})\subset{\cal
C}_{0}({\mathscr{P}}^{\lambda}_{n})$: on $E_{1,x}$, we have ${\cal
C}(x,{\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})={\cal
C}(x,{\mathscr{P}}^{\lambda})\cap A_{x,\theta,2}={\cal
C}_{0}({\mathscr{P}}^{\lambda}_{n})\cap A_{x,\theta,2}\supset{\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})$, so ${\cal
C}(x,{\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})={\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})$, which contradicts ${\cal
C}(x,{\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})\neq{\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})$; on $E_{2,x}$, we obtain

$\left({\cal C}(x,{\mathscr{P}}^{\lambda}_{n})\cap{\cal
C}_{0}({\mathscr{P}}^{\lambda}_{n})\right)\cap A_{x,\theta,2}={\cal
C}(x,{\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})\cap{\cal
C}_{0}({\mathscr{P}}^{\lambda}_{n})={\cal C}_{0}({\mathscr{P}}^{\lambda}\cap
A_{x,\theta,2}),$

which contradicts ${\cal C}(x,{\mathscr{P}}^{\lambda}_{n})\neq{\cal
C}_{0}({\mathscr{P}}^{\lambda}_{n})$.

Let $E_{0}=\cup_{x\in{\mathscr{P}}^{\lambda}_{n}}E_{0,x}$. Below, using
$E_{1,x}\cup E_{2,x}\subset E_{3,x}$ in the second inequality, the size of the
second largest cluster ${\cal C}^{(2)}_{0}({\mathscr{P}}^{\lambda}_{n})$ is
not smaller than the size of ${\cal C}_{0}({\mathscr{P}}^{\lambda}\cap
A_{x,\theta,2})$ on $E_{3,x}$ in the third inequality, the Palm theorem in the
equality, the Cauchy-Schwarz inequality in the fifth inequality, (2.1) and
(2.2) in the second last inequality, we have

$\displaystyle\mathbb{P}\left(\\{N_{\theta,n}^{\prime}\neq N_{n}\\}\cap
E_{0}^{c}\right)$ $\displaystyle\leq\mathbb{E}\int_{\Gamma}{\bf
1}_{[E_{1,x}\cup
E_{2,x}]}{\mathscr{P}}^{\lambda}(dx)\leq\mathbb{E}\int_{\Gamma}{\bf
1}_{[E_{3,x}]}{\mathscr{P}}^{\lambda}(dx)$
$\displaystyle\leq\mathbb{E}\int_{\Gamma}{\bf 1}_{[|{\cal
C}^{(2)}_{0}({\mathscr{P}}^{\lambda}_{n})|\geq|{\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap
A_{x,\theta,2})|]}{\mathscr{P}}^{\lambda}(dx)$
$\displaystyle\leq\mathbb{E}\int_{\Gamma}{\bf 1}_{[|{\cal
C}^{(2)}_{0}({\mathscr{P}}^{\lambda}_{n})|\geq
a_{n}]}{\mathscr{P}}^{\lambda}(dx)+\mathbb{E}\int_{\Gamma}{\bf 1}_{[|{\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap
A_{x,\theta,2})|<a_{n}]}{\mathscr{P}}^{\lambda}(dx)$
$\displaystyle=\mathbb{E}\left\\{{\bf 1}_{[|{\cal
C}^{(2)}_{0}({\mathscr{P}}^{\lambda}_{n})|\geq
a_{n}]}|{\mathscr{P}}^{\lambda}_{n}|\right\\}+\lambda\int_{\Gamma}\mathbb{P}(|{\cal
C}_{0}(({\mathscr{P}}^{\lambda}\cup\\{x\\})\cap A_{x,\theta,2})|<a_{n})dx$
$\displaystyle\leq\sqrt{\mathbb{E}\left(|{\mathscr{P}}^{\lambda}_{n}|^{2}\right)\mathbb{P}(|{\cal
C}^{(2)}_{0}({\mathscr{P}}^{\lambda}_{n})|\geq
a_{n})}+\lambda\int_{\Gamma}\mathbb{P}(|{\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})|<a_{n})dx$
$\displaystyle\leq O\left(n^{m}e^{-\frac{1}{2}k_{2}\theta_{1}\ln
n}\right)+O\left(n^{m}e^{-k_{3}\theta\ln n}\right)\leq
O\left(n^{-10m}\right).$

Moreover, a similar argument together with (2.4) gives

$\displaystyle\mathbb{P}\left(\\{N_{\theta,n}^{\prime}\neq N_{n}\\}\cap
E_{0}\right)$
$\displaystyle\leq\mathbb{P}(E_{0})\leq\mathbb{E}\int_{\Gamma}{\bf
1}_{[E_{0,x}]}{\mathscr{P}}^{\lambda}(dx)$ $\displaystyle\leq
O\left(n^{m}e^{-k_{3}\theta\ln n}\right)\leq O\left(n^{-10m}\right),$

completing the proof.

######  Lemma 2.3.

With $\theta$ in Lemma 2.2, we have

$\displaystyle\mathbb{E}|N^{\prime}_{\theta,n}-N_{n}|$ $\displaystyle\leq
O\left(n^{-4m}\right),$ $\displaystyle|{\rm Var}(N_{\theta,n}^{\prime})-{\rm
Var}(N_{n})|$ $\displaystyle=O\left(n^{-m}\right),$ $\displaystyle{\rm
Var}(N_{\theta,n}^{\prime})$ $\displaystyle=\Theta(n^{m}).$

Proof. For the first claim, we apply the Cauchy-Schwarz inequality in the
second inequality and Lemma 2.2 in the last inequality below to get

$\displaystyle\mathbb{E}|N^{\prime}_{\theta,n}-N_{n}|$
$\displaystyle\leq\mathbb{E}\left(|{\mathscr{P}}^{\lambda}_{n}|{\bf
1}_{[N_{\theta,n}^{\prime}\neq N_{n}]}\right)$
$\displaystyle\leq\sqrt{\mathbb{E}(|{\mathscr{P}}^{\lambda}_{n}|^{2})\mathbb{P}(N_{\theta,n}^{\prime}\neq
N_{n})}\leq O\left(n^{m}\right)O\left(n^{-5m}\right)=O\left(n^{-4m}\right).$

Likewise, we have

$\displaystyle{\rm Var}(N^{\prime}_{\theta,n}-N_{n})$
$\displaystyle\leq\mathbb{E}[(N^{\prime}_{\theta,n}-N_{n})^{2}]\leq\mathbb{E}\left(|{\mathscr{P}}^{\lambda}_{n}|^{2}{\bf
1}_{[N_{\theta,n}^{\prime}\neq N_{n}]}\right)$
$\displaystyle\leq\sqrt{\mathbb{E}(|{\mathscr{P}}^{\lambda}_{n}|^{4})\mathbb{P}(N_{\theta,n}^{\prime}\neq
N_{n})}\leq O\left(n^{2m}\right)O\left(n^{-5m}\right)=O\left(n^{-3m}\right),$

therefore,

$\displaystyle|{\rm Var}(N^{\prime}_{\theta,n})-{\rm Var}(N_{n})|$
$\displaystyle=|{\rm Var}(N^{\prime}_{\theta,n}-N_{n})+2{\rm
Cov}(N_{n},N_{\theta,n}^{\prime}-N_{n})|$ $\displaystyle\leq
O\left(n^{-3m}\right)+2\sqrt{{\rm Var}(N_{n}){\rm
Var}(N_{\theta,n}^{\prime}-N_{n})}=O\left(n^{-m}\right).$

The third claim follows from (1.1) and the second claim. It can also be
directly obtained from [Xia and Yukich (2015), Lemma 4.6] and the fact that
the score function $\xi^{\prime}$ is locally dependent.

We now establish the error bound of the normal approximation to
$N^{\prime}_{\theta,n}$. Let $B^{\prime}_{\theta,n}$ be the standard deviation
of $N^{\prime}_{\theta,n}$.

######  Lemma 2.4.

For any constant $\theta>0$, let
$W_{\theta,n}^{\prime}=\frac{N^{\prime}_{\theta,n}-\mathbb{E}(N^{\prime}_{\theta,n})}{B^{\prime}_{\theta,n}}$,
then

$d_{\mathrm{K}}(\mathscr{L}(W_{\theta,n}^{\prime}),{\cal N}(0,1))=O\left((\ln
n)^{\frac{2m}{m-1}}n^{-m/2}\right),$

where $Z\sim{\cal N}(0,1)$.

Proof. Recall that $\xi^{\prime}(x,\theta,{\mathscr{P}}^{\lambda}_{n})={\bf
1}_{[{\cal C}(x,{\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})={\cal
C}_{0}({\mathscr{P}}^{\lambda}\cap A_{x,\theta,2})]}$, define the point
process
$\Xi^{\prime}(dx)=\xi^{\prime}(x,\theta,{\mathscr{P}}^{\lambda}_{n}){\mathscr{P}}^{\lambda}_{n}(dx)$,
$\Lambda^{\prime}(dx)=\mathbb{E}\Xi^{\prime}(dx)$, let $\Xi^{\prime}_{x}$ be
its Palm process at $x$ [Kallenberg (1983), Chapter 10]. Let $A_{x,\theta,3}$
be the cube with centre $x$ and edge length $4(\theta\ln n)^{\frac{1}{m-1}}$
and $A_{x,\theta,4}:=A_{x,\theta,3}\cap\Gamma$. Because the score function
$\xi^{\prime}(x,\theta,{\mathscr{P}}^{\lambda}_{n})$ is completely determined
by the point configuration ${\mathscr{P}}^{\lambda}\cap A_{x,\theta,2}$, we
can construct $\Xi^{\prime}$ and $\Xi^{\prime}_{x}$ together such that
$\Xi^{\prime}_{x}$ and $\Xi^{\prime}$ are identical outside $A_{x,\theta,4}$.
Let
$Y^{\prime}_{x}=\Xi^{\prime}_{x}(\Gamma)-\Xi(\Gamma)=\Xi^{\prime}_{x}(A_{x,\theta,4})-\Xi(A_{x,\theta,4})$,
$\Delta^{\prime}_{x}=Y^{\prime}_{x}/B^{\prime}_{\theta,n}$,
$D=\\{(x,y)\in\Gamma^{2}:\ d(x,y)\leq 4\sqrt{m}(\theta\ln
n)^{\frac{1}{m-1}}\\}$, then $\Delta^{\prime}_{x}$ and $\Delta^{\prime}_{y}$
are independent when $(x,y)\in\Gamma^{2}\setminus D$. [Chen, Röllin and Xia
(2021), Corollary 3.2] states that

$d_{\mathrm{K}}(\mathscr{L}(W^{\prime}),{\cal N}(0,1))\leq
7s_{1}+5.5s_{2}+10s_{3},$

where, with $B:=B^{\prime}_{\theta,n}$,

$\displaystyle s_{1}$ $\displaystyle=\frac{1}{B^{2}}\left(\int_{(x,y)\in
D}\mathbb{E}\\{(Y^{\prime}_{x})^{2}{\bf 1}_{[|Y^{\prime}_{x}|\leq
B]}\\}\Lambda^{\prime}(dx)\Lambda^{\prime}(dy)\right)^{\frac{1}{2}};$
$\displaystyle s_{2}$
$\displaystyle=\frac{1}{B^{3}}\int_{\Gamma}\mathbb{E}\\{(Y^{\prime}_{x})^{2}\\}\Lambda^{\prime}(dx);$
$\displaystyle s_{3}$ $\displaystyle=\frac{1}{B^{3}}\int_{(x,y)\in
D}\mathbb{E}\\{|Y^{\prime}_{x}|{\bf 1}_{[|Y^{\prime}_{x}|\leq
B]}\\}\Lambda^{\prime}(dx)\Lambda^{\prime}(dy).$

Since $|Y^{\prime}_{x}|\leq{\mathscr{P}}^{\lambda}(A_{x,\theta,4})$, we have
$\mathbb{E}|Y^{\prime}_{x}|\leq 4^{m}\lambda(\theta\ln n)^{\frac{m}{m-1}}$ and

$\mathbb{E}\\{(Y^{\prime}_{x})^{2}\\}\leq 2\cdot 4^{2m}\lambda^{2}(\theta\ln
n)^{\frac{2m}{m-1}}$

for large $n$. Hence,

$\displaystyle s_{1}$ $\displaystyle\leq\frac{1}{B^{2}}\left(\int_{(x,y)\in
D}2\lambda^{2}(4(\theta\ln
n)^{\frac{1}{m-1}})^{2m}\Lambda^{\prime}(dx)\Lambda^{\prime}(dy)\right)^{\frac{1}{2}}\leq
O\left(\frac{n^{m/2}(\ln n)^{\frac{1.5m}{m-1}}}{B^{2}}\right);$ $\displaystyle
s_{2}$ $\displaystyle\leq\frac{1}{B^{3}}\int_{\Gamma}2\lambda^{2}(4(\theta\ln
n)^{\frac{1}{m-1}})^{2m}\Lambda^{\prime}(dx)=O\left(\frac{n^{m}(\ln
n)^{\frac{2m}{m-1}}}{B^{3}}\right);$ $\displaystyle s_{3}$
$\displaystyle\leq\frac{1}{B^{3}}\int_{(x,y)\in D}4^{m}\lambda(\theta\ln
n)^{\frac{m}{m-1}}\Lambda^{\prime}(dx)\Lambda^{\prime}(dy)=O\left(\frac{n^{m}(\ln
n)^{\frac{2m}{m-1}}}{B^{3}}\right).$

By Lemma 2.3, we have $B^{2}=\Theta(n^{m})$, the proof is complete.

With these preparations, we are now ready to prove Theorem 1.2.

Proof of Theorem 1.2. Using the triangle inequality, we have

$d_{\mathrm{K}}\left(\mathscr{L}(W_{n}),{\cal N}(0,1)\right)\leq
d_{\mathrm{K}}\left(\mathscr{L}(W_{n}),\mathscr{L}(W_{\theta,n}^{\prime})\right)+d_{\mathrm{K}}\left(\mathscr{L}(W_{\theta,n}^{\prime}),{\cal
N}(0,1)\right),$

hence, by Lemma 2.4, it suffices to show that

$d_{\mathrm{K}}\left(\mathscr{L}(W_{n}),\mathscr{L}(W_{\theta,n}^{\prime})\right)\leq
2d_{\mathrm{K}}\left(\mathscr{L}(W_{\theta,n}^{\prime}),{\cal
N}(0,1)\right)+O\left(n^{-2m}\right).$ (2.5)

To this end, let
$V_{\theta,n}:=\frac{N^{\prime}_{\theta,n}-\mathbb{E}(N_{n})}{B_{n}}$,
$v_{\theta,n}=\mathbb{E}(V_{\theta,n})$, $r_{\theta,n}^{2}:={\rm
Var}(V_{\theta,n})=(B_{\theta,n}/B_{n})^{2}$, applying the triangle inequality
in the first inequality and [Xia and Yukich (2015), (5.9)] in the second
inequality below, we have

$\displaystyle
d_{\mathrm{K}}(\mathscr{L}(W_{n}),\mathscr{L}(W_{\theta,n}^{\prime}))$
$\displaystyle\leq
d_{\mathrm{K}}\left(\mathscr{L}(W_{n}),\mathscr{L}(V_{\theta,n})\right)+d_{\mathrm{K}}\left(\mathscr{L}(V_{\theta,n}),{\cal
N}\left(v_{\theta,n},r_{\theta,n}^{2}\right)\right)$ $\displaystyle\ \ \
+d_{\mathrm{K}}\left({\cal N}\left(v_{\theta,n},r_{\theta,n}^{2}\right),{\cal
N}(0,1)\right)+d_{\mathrm{K}}\left(\mathscr{L}(W_{\theta,n}^{\prime}),{\cal
N}(0,1)\right)$ $\displaystyle\leq\mathbb{P}(N_{n}\neq
N^{\prime}_{\theta,n})+2d_{\mathrm{K}}\left(\mathscr{L}(W_{\theta,n}^{\prime}),{\cal
N}(0,1)\right)+\frac{|v_{\theta,n}|}{\sqrt{2\pi}}+\frac{|r_{\theta,n}^{2}-1|}{\sqrt{2e\pi}}.$
(2.6)

Lemma 2.3 gives

$|v_{\theta,n}|\leq O\left(n^{-4.5m}\right),\ \ \ |r_{\theta,n}^{2}-1|\leq
O\left(n^{-2m}\right),$ (2.7)

hence (2.5) follows from combining Lemma 2.2 and the estimates (2.6) and
(2.7).

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11institutetext:  Astronomy Instrumentation Group, University of Cardiff, The
Parade, CF24 3AA, United Kindgom 22institutetext: Núcleo de Astronomía,
Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejército
441, Santiago, Chile 33institutetext: Univ. Grenoble Alpes, CNRS, Grenoble
INP, Institut Néel, 38000 Grenoble, France 44institutetext: Groupement
d’Interet Scientifique KID, 38000 Grenoble and 38400 Saint Martin d’Héres,
France 55institutetext: Aix Marseille Université, CNRS, LAM (Laboratoire
d’Astrophysique de Marseille), F-13388 Marseille, France 66institutetext:
Univ. Grenoble Alpes, CNRS, LPSC/IN2P3, 38000 Grenoble, France
77institutetext: European Southern Observatory, Karl Schwarzschild StraÃe 2,
85748 Garching, Germany 88institutetext: Univ. Grenoble Alpes, CNRS, IPAG,
38400 Saint Martin d’Héres, France 99institutetext: European Southern
Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile
1010institutetext: School of Earth and Space Exploration and Department of
Physics, Arizona State University, Tempe, AZ 85287, USA

# CONCERTO at APEX: Installation and first phase of on-sky commissioning

A. Catalano 4466<EMAIL_ADDRESS>P. Ade 11 M. Aravena 22 E. Barria 3344
A. Beelen 55 A. Benoit 3344 M. Béthermin 55 J. Bounmy 6644 O. Bourrion 6644 G.
Bres 3344 C. De Breuck 77 M. Calvo 3344 F.-X. Désert 8844 C.A Durán 99 G.
Duvauchelle 55 L. Eraud 66 A. Fasano 3344 T. Fenouillet 55 J. Garcia 55 G.
Garde 3344 J. Goupy 3344 C. Groppi 1010 C. Hoarau 6644 W. Hu 55 G. Lagache 55
J.-C. Lambert 55 J.-P. Leggeri 3344 F. Levy-Bertrand 3344 J. Macías-Pérez 6644
H. Mani 1010 J. Marpaud 6644 M. Marton 66 P. Mauskopf 1010 A. Monfardini 3344
G. Pisano 11 N. Ponthieu 8844 L. Prieur 55 G. Raffin 66 S. Roni 66 S. Roudier
66 D. Tourres 6644 C. Tucker 11 L. Vivargent 66

###### Abstract

CONCERTO (CarbON CII line in post-rEionisation and ReionisaTiOn) is a large
field-of-view (FoV) spectro-imager that has been installed on the Cassegrain
Cabin of Atacama Pathfinder EXperiment (APEX) telescope in April 2021.
CONCERTO hosts 2 focal planes and a total number of 4000 Kinetic Inductance
Detectors (KID), with an instantaneous FoV of 18.6 arc-minutes in the range of
130-310 GHz. The spectral resolution can be easily tuned down to 1 GHz
depending on the scientific target. The scientific program of CONCERTO has
many objectives, with two main programs focused on mapping the fluctuations of
the [CII] line intensity in the reionisation and post-reionisation epoch
(4.5<z<8.5), and on studying galaxy clusters via the thermal and kinetic
Sunyaev-Zel’dovich (SZ) effect. CONCERTO will also measure the dust and
molecular gas contents of local and intermediate-redshift galaxies, it will
study the Galactic star-forming clouds and finally it will observe the CO
intensity fluctuations arising from 0.3<z<2 galaxies.

The design of the instrument, installation at APEX and current status of the
commissioning phase and science verification will be presented. Also we
describe the deployment and first on-sky tests performed between April and
June 2021.

##  1 Introduction

Starting from January 2019, the Grenoble collaboration (LPSC111 _Laboratoire
de Physique Subatomique et Cosmologie_ , Institut Néel and IPAG 222 _Institut
de planétologie et d’astrophysique de Grenoble_) together with the LAM333
_Laboratoire d’Astrophysique de Marseille_ , funded by an Advanced Grant ERC
has worked to design, fabricate, install, commission, and observe with the
CONCERTO Monfardini & Lagache  (2021) instrument. CONCERTO is installed at the
12-metre APEX telescope and now it is open to the APEX community, the
scientific goals will be manifold. The CONCERTO collaboration will focus on
two cosmological goals:

  * •

The measurement of variations of the [CII] line emission at redshift z>5.2.
The atomic [CII] line is one of the most valuable tracers of star formation.
At high redshifts it is observed in the sub-millimetre and millimetre
atmospheric windows Kovetz et al. (2017); Concerto Collab.  (2020). We will
use the [CII] line emission as a tracer of cosmic density structure, and make
the first constraints on the power spectrum of dusty star-forming matter. Our
experiment will also observe the CO intensity fluctuations arising from
galaxies at redshifts between 0.2 and 2, mapping the spatial distribution and
abundance of molecular gas over a broad range of cosmic time.

  * •

Precise measurements of the shape of the SZ electromagnetic spectrum for
cluster of galaxies for redshifts between 0.2 and 0.8 to disentangle the
kinetic and relativistic corrections from the main thermal contribution
Concerto Collab.  (2020).

Since April 2021 CONCERTO is installed at the 12 metre APEX telescope, located
at a 5105 m altitude on the Llano de Chajnantor in Northern Chile apex (4).
The commissioning phase has been performed until the end of June. Starting
from July 2021 the instrument has started its regular scientific program. In
the next sections we will present a description of the instrument, its
installation at the telescope and some technical and scientific commissioning
results.

Figure 1: Top: CONCERTO optical design (using Zemax software with details of
the cold optics inside the cryostat and the 100 mK stage cold pupil, polariser
and the two arrays). Bottom left: picture of the fabricated CONCERTO cryostat
and warm electronics in the laboratory at Neel in Grenoble. Bottom right:
baseline sweeps for one array (6 feedlines) taken with Vector Network Analyser
(VNA).

##  2 The CONCERTO Instrument

The instrument is based on several sub-systems: an optical box which selects
the optical inputs to be injected in a second box that contains a Martin-
Puplett interferometer. The signal is then focused on two arrays of about 4000
KID detectors, cooled at 100 mK through a closed-cycle dilution cryostat (see
Fig. 1).

The design and the control of the systematic errors of such a configuration
drives the major requirement for the whole instrument. This is because the MPI
must be able to perform continuously up to 9 cm path interferograms, which
gives a spectral resolution of about 1 GHz within the band, at a frequency of
about 4 Hz in order to maintain a low sky noise level. The fast KID time
response permits us to use this method without any loss of information. The
readout sampling frequency is constrained by this requirement; data are
sampled at 4 kHz producing roughly 128 MByte per second.

The optical system has been arranged to have a FoV of 18.6 arcmin. Since an
MPI measures the difference between two optical inputs D. H. Martin and E.
Puplett  (1970), we split the beam after passing through the telescope into
two fully polarised beams before reaching the MPI. This can be done with the
use an initial wire-grid polariser. After interfering in the MPI, the first
half of the beam is focused onto the detectors, creating an image of 18.6
arcmin on the sky. The second half of the beam is first de-focused from a
focused image to a pupil (corresponding to the image of the primary mirror of
the telescope) and then collected to the detectors. This makes an initial
hardware subtraction of the constant common mode from the atmosphere to
simplify the off-line analysis. In addition, instead of the sky itself, the
second input can be taken from a cold reference, which is a cold black-body at
a given tunable temperature.

The 18.6 arcmin focal plane is fully covered by two arrays of single
polarisation aluminium LEKID containing 2152 pixels each and separated by a 45
degrees polariser. Each array has a different optical band 195-310 GHz for the
High frequency array (HF) and 130-270 GHz for low frequency array (LF).

We recall in Table 1 the main characteristics of CONCERTO.

Telescope primary mirror diameter [m] | 12
---|---
Field-of-view diameter [arcmin] | 18.6
Beam Widths [arcsec] | 30 (HF) 35 (LF)
Absolute spectral resolution [GHz] |  $\geq$ 1
Relative spectral resolution R [#] | 1–300
Frequency range HF $\mid$ LF [GHz] | 195–310 $\mid$ 130–270
Pixels on Sky HF $\mid$ LF [#] | 2,152 $\mid$ 2,152
Instrument geometrical throughput [sr m2] | 2.5$\times$10-3
Single Pixel geometrical throughput [sr m2] | 1.16$\times$10-6
Data rate [MBytes/sec] | 128
Table 1: Main characteristics of CONCERTO at APEX. Figure 2: LF (left) and HF
(right) focal plane geometries. Around 90% of the designed pixels exhibit a
beam and are healthy.

##  3  Installation and first phase of on-sky commissioning

After installation of the whole instrument, the cryostat has been first cooled
down. After reaching a base temperature of 70 mK CONCERTO has been switch on
for the first time the 12th of April 2021. The first phase of the technical
commissioning started right after. During this phase, the sub-systems
operations and typical electrical characteristic of the electronics and
detectors (feedlines connections, quality factors of the detectors, feedline
cosmetics, etc..) were checked. The optical alignment has been achieved using
an optical laser and a very first light on Jupiter was obtained.

After these first checks we started to point the telescope in order to start
the scientific commissioning. We list, among others, the following
achievements:

  * •

About 90% of the detectors are valid and able to detect bright sources
individually on the sky.

  * •

A fraction of about 70% of the designed KIDs exhibits a beam with an
eccentricity lower than 0.7.

  * •

A preliminary Noise Equivalent Temperature (NET) has been evaluated and equal
to 2.2 mK/$\sqrt{Hz}$, which is well in agreement with the expected value.
Note that we expect to have a better sensitivity per beam at low frequency
where we have more than one KID per beam, and at intermediate frequencies
where the two arrays overlap in frequency.

The presented results must be consolidated with detailed data analysis, which
is still in progress. Before the end of the scientific commissioning few
selected astronomical targets were observed to demonstrate the capabilities of
the instrument and optimize the observing strategies. In particular in this
paper we present in Figure 3 photometric maps (integrated emission on the full
CONCERTO band) of the star forming region NGC6334 (the Cat Paw Nebula) and of
the Crab nebula supernova remnant.

Figure 3: First results of the CONCERTO scientific commissioning. Two
observations of continuum emission are shown as observed by the two bands of
CONCERTO (1). Left plot: star forming region NGC6334 (Cat Paw Nebula) mapped
in a field of 37$\times$25 arcmin2 for 16 minutes of integration. Right plot:
Crab nebula observed with 2.5 minutes integration time mapped in a field of
10$\times$10 arcmin2. In both cases we represent composite images where the LF
array is in blue, and the HF array in red (ESO/CONCERTO collaboration:
https://www.eso.org/public/unitedkingdom/announcements/ann21010/?lang).

##  4 Conclusions, Future work and perspectives

The CONCERTO instrument has been successfully installed on the APEX telescope
in April 2021. Operations were started few days after the installation and
scientific commissioning was performed in less than two months.

Here we presented preliminary results of the scientific commissioning showing
estimates of the instrument performance well in agreement with expectations.

The regular scientific observations have already started in July 2021 thanks
to the fact that the instrument has a complete remote operations design.
Indeed, after installation it became impossible to travel to Chile, therefore
operations of the instrument are all conducted remotely from France.

Spectra and further results of the commissioning will be presented in a more
extensive paper in the coming months. Scientific commissioning data are
available in the ESO data base at
https://www.eso.org/sci/publications/announcements/sciann17437.html.

_Acknowledgements._ This project has received funding from the European
Research Council (ERC) under the European Unionâs Horizon 2020 research and
innovation programme (grant agreement No 788212) also from the Excellence
Initiative of Aix-Marseille University-AMidex, a French "Investissements
dâAvenir" programme and the labex FOCUS. We thank the electronics, mechanics,
cryogenics and administrative groups at Institut Néel, LPSC and LAM. For the
outstanding technological and human support we thank the whole APEX staff.

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