††thanks: ORCiD: 0000-0002-6286-0722††thanks: ORCiD: 0000-0001-6841-9850

# Compact equations for the envelope theory

Lorenzo Cimino<EMAIL_ADDRESS>Claude Semay
<EMAIL_ADDRESS>Service de Physique Nucléaire et Subnucléaire,
Université de Mons, UMONS Research Institute for Complex Systems, Place du
Parc 20, 7000 Mons, Belgium

(August 27, 2024)

###### Abstract

Abstract The envelope theory is a method to easily obtain approximate, but
reliable, solutions for some quantum many-body problems. Quite general
Hamiltonians can be considered for systems composed of an arbitrary number of
different particles in $D$ dimensions. In the case of identical particles, a
compact set of 3 equations can be written to find the eigensolutions. This set
provides also a nice interpretation and a starting point to improve the
method. It is shown here that a similar set of 7 equations can be determined
for a system containing an arbitrary number of two different particles.

Envelope theory; Many-body quantum systems; Approximation methods

##  I introduction

The envelope theory (ET) hall80 ; hall83 ; hall04  is a technique to compute
approximate eigenvalues and eigenvectors of $N$-body systems. This method,
first developed for systems with identical particles, has been extended to
treat non-standard kinematics in $D$ dimensions in sema13 ; sema19 , and it
has been recently generalized for systems with different particles sema20 .
The big advantage of this method is that the computation cost is independent
of the number of particles. Quite general Hamiltonians can be considered, and
the approximate eigenvalues are lower or upper bounds in favorable cases. The
method relies on the existence of an exact solution for the $N$-body harmonic
oscillator Hamiltonian hall79 ; cint01 . The accuracy of the method has been
checked for various three-dimensional systems sema15a  and one-dimensional
systems containing up to 100 bosons sema19 .

It is worth noting that the ET method has been rediscovered in 2008 under the
name of the auxiliary field method, following an approach different from the
one used by Hall hall80 ; hall83 ; hall04 . It has been recognized later that
both methods are actually completely equivalent. This story is described in
silv12 , where a lot of information is given about this approximation method.

The ET has been used to obtain physical results about hadronic systems as in
sema09 , and is especially useful when the number of particles can be
arbitrary large as in the large-$N$ formulation of QCD buis11 ; buis12 . The
method has allowed the study of a possible quasi Kepler’s third law for
quantum many-body systems sema21 . It can also be simply used to test accurate
numerical calculations as in char15 .

Let us consider the $N$-body Hamiltonian

$H=\sum_{i=1}^{N}T_{i}(p_{i})+\sum_{i<j=2}^{N}V_{ij}(r_{ij}),$ (1)

where $T_{i}$ is an arbitrary kinetic energy with some constraints sema18a
and $V_{ij}$ is a two-body central potential. We also define
$p_{i}=\left\lvert\bm{p}_{i}\right\rvert$ and
$r_{ij}=\left\lvert\bm{r}_{i}-\bm{r}_{j}\right\rvert$, where $\bm{r}_{i}$ and
$\bm{p}_{i}$ are respectively the position and the momentum of the $i$th
particle. It is assumed in the following that we are always working in the
centre of mass (CM) frame, $\bm{P}=\sum_{i=1}^{N}\bm{p}_{i}=\bm{0}$, and with
natural units ($\hbar=c=1$).

As explained in silv10 ; sema20 , in the framework of the ET, Hamiltonian (1)
is replaced by an auxiliary Hamiltonian (it is the origin of the other name of
the method)

$\tilde{H}(\\{\alpha\\})=\sum_{i=1}^{N}\left[\frac{\bm{p}_{i}^{2}}{2\mu_{i}}+T_{i}(G_{i}(\mu_{i}))-\dfrac{G_{i}^{2}(\mu_{i})}{2\mu_{i}}\right]+\sum_{i<j=2}^{N}\left[\rho_{ij}\bm{r}_{ij}^{2}+V_{ij}(J_{ij}(\rho_{ij}))-\rho_{ij}J_{ij}^{2}(\rho_{ij})\right],$
(2)

where $\\{\alpha\\}=\\{\\{\mu_{i}\\},\\{\rho_{ij}\\}\\}$ is a set of auxiliary
parameters to determine later, and where the auxiliary functions $G_{i}$ and
$J_{ij}$ are such that

$\begin{array}[]{cc}T^{\prime}_{i}(G_{i}(x))-\dfrac{G_{i}(x)}{x}=0,\\\\[14.22636pt]
V^{\prime}_{ij}(J_{ij}(x))-2xJ_{ij}(x)=0,\end{array}$ (3)

where $U^{\prime}(x)=dU(x)/dx$. It is useful to write Hamiltonian (2) in the
form

$\tilde{H}(\\{\alpha\\})=H_{\text{ho}}(\\{\alpha\\})+B(\\{\alpha\\}),$ (4)

where $H_{\text{ho}}$ is the harmonic oscillator part and $B$ is a function
obtained by subtracting the harmonic oscillator contributions from (2). An
eigenvalue of (2) is given by

$\tilde{E}(\\{\alpha\\})=E_{\text{ho}}(\\{\alpha\\})+B(\\{\alpha\\}),$ (5)

where $E_{\text{ho}}$ is an eigenvalue of $H_{\text{ho}}$. A procedure in
silv10 ; sema20  explains how to compute $E_{\text{ho}}$ but an example will
be given below. An eigenvalue $\tilde{E}$ also depends on the set of
parameters $\\{\alpha\\}=\\{\\{\mu_{i}\\},\\{\rho_{ij}\\}\\}$. The principle
of the method is to search for the set of parameters
$\\{\alpha_{0}\\}=\\{\\{\mu_{i0}\\},\\{\rho_{ij0}\\}\\}$ such that

$\frac{\partial\tilde{E}}{\partial\mu_{i}}\biggr{\rvert}_{\\{\alpha_{0}\\}}=\frac{\partial\tilde{E}}{\partial\rho_{ij}}\biggr{\rvert}_{\\{\alpha_{0}\\}}=0\hskip
14.22636pt\forall\ i,j.$ (6)

Equations (6) can be easily implemented and solutions $\\{\alpha_{0}\\}$ are
easily found since we only need to find an extremum sema20 . After solving
(6), we obtain the desired approximate energy by substituting the set
$\\{\alpha_{0}\\}$ back to (5), $\tilde{E}(\\{\alpha_{0}\\})=\tilde{E}_{0}$.

In the case of identical particles, it has been showed sema13 ; sema19  that
we can equivalently find the eigenvalue $\tilde{E}_{0}$ by using a set of
three compact equations

$\tilde{E}_{0}=N\,T(p_{0})+C^{2}_{N}\,V(\rho_{0}),$ (7a)
$N\,T^{\prime}(p_{0})\,p_{0}=C^{2}_{N}\,V^{\prime}(\rho_{0})\,\rho_{0},$ (7b)
$Q(N)=\sqrt{C^{2}_{N}}\,p_{0}\,\rho_{0},$ (7c)

where $C^{2}_{N}=N(N-1)/2$ is the number of pairs, and where
$p_{0}^{2}=\left\langle\bm{p}_{i}^{2}\right\rangle$ and
$\rho_{0}^{2}=\left\langle\bm{r}_{ij}^{2}\right\rangle\hskip 5.69054pt\forall\
i,j$. The mean values are taken with an eigenstate of the auxiliary
Hamiltonian corresponding to the global quantum number $Q(N)$ for the set
$\\{\alpha_{0}\\}$ insuring the constraints (6). The eigenstate is also
completely (anti)symmetric for the exchange between particles.

$Q(N)=\begin{cases}\sum\limits_{i=1}^{N-1}\left(2n_{i}+l_{i}+\frac{D}{2}\right)&\text{
if }D\geq 2\\\\[15.0pt]
\sum\limits_{i=1}^{N-1}\left(n_{i}+\frac{1}{2}\right)&\text{ if
}D=1\end{cases},$ (8)

where the quantum numbers $\\{n_{i},l_{i}\\}$ are associated with the internal
Jacobi variables. Some values of $Q(N)$ for the bosonic and fermionic ground
states are given in sema20 ; sema19 . In previous papers sema13 ; sema19 , the
variable
$r_{0}^{2}=N^{2}\left\langle\left(\bm{r}_{i}-\bm{R}\right)^{2}\right\rangle$,
where $\bm{R}$ is the CM position, was used instead of $\rho_{0}$ because one-
body and two-body potentials are treated together.

These equations are called compact because all the relevant variables appear
in 3 equations giving the definition of the energy (7a), the equation of
motion (7b) and the rule for the quantization (7c). Moreover, the
uninteresting auxiliary parameters and functions are not present. Equations
(7) can also be easily implemented and solved. There are good reasons to
prefer the compact equations (7) over the “extremization” equations (6).
First, the quantities $p_{0}$ and $\rho_{0}$ give direct access to more
interesting expectation values than $\\{\alpha_{0}\\}$. Secondly, these
equations have a nice semiclassical interpretation as explained in sema13 .
Thirdly, it is possible to improve the ET with the dominantly orbital state
method starting from these equations sema15b , which is the main motivation to
write these equations. As the improvement obtained can be significant in some
cases, it is worth generalizing it beyond systems of identical particles. In
the following section, we will present the compact equations for a system
composed of two different sets of $N_{a}$ and $N_{b}$ identical particles.

##  II $\bm{N_{a}+N_{b}}$ systems

Let us specify the auxiliary Hamiltonian (2) for this system. The harmonic
oscillator Hamiltonian for a system of $N_{a}$ particles of type $a$ and
$N_{b}$ particles of type $b$ is given by

$H_{\text{ho}}=\sum_{i=1}^{N_{a}}\frac{\bm{p}_{i}^{2}}{2\mu_{a}}+\sum_{j=1}^{N_{b}}\frac{\bm{p}_{j}^{2}}{2\mu_{b}}+\sum_{i<i^{\prime}=2}^{N_{a}}\rho_{aa}\bm{r}_{ii^{\prime}}^{2}+\sum_{j<j^{\prime}=2}^{N_{b}}\rho_{bb}\bm{r}_{jj^{\prime}}^{2}+\sum_{i=1}^{N_{a}}\sum_{j=1}^{N_{b}}\rho_{ab}\bm{r}_{ij}^{2}.$
(9)

In the following, letters $i\text{ }(j)$ are reserved for particles of type
$a\text{ }(b)$. As explained in sema20 ; hall79 , it is useful to write (9) in
the form

$H_{\text{ho}}=H_{a}+H_{b}+H_{\text{CM}}\hskip 28.45274pt\text{with}$ (10a)
$H_{a}=\sum_{i=1}^{N_{a}}{\frac{\bm{p}_{i}^{2}}{2\mu_{a}}}-\frac{\bm{P}_{a}^{2}}{2M_{a}}+\sum_{i<i^{\prime}=2}^{N_{a}}\left(\rho_{aa}+\frac{N_{b}}{N_{a}}\rho_{ab}\right)\bm{r}_{ii^{\prime}}^{2},$
(10b)
$H_{b}=\sum_{j=1}^{N_{b}}{\frac{\bm{p}_{j}^{2}}{2\mu_{b}}}-\frac{\bm{P}_{b}^{2}}{2M_{b}}+\sum_{j<j^{\prime}=2}^{N_{b}}\left(\rho_{bb}+\frac{N_{a}}{N_{b}}\rho_{ab}\right)\bm{r}_{jj^{\prime}}^{2},$
(10c) $H_{\text{CM}}=\frac{\bm{p}^{2}}{2\mu}+N_{a}N_{b}\rho_{ab}\bm{r}^{2},$
(10d)

where $\bm{P}_{\alpha}$ and $M_{\alpha}=N_{\alpha}\,\mu_{\alpha}$ are the
total momentum and mass for the set $\alpha=\\{a,b\\}$,
$\mu=\frac{M_{a}M_{b}}{M_{a}+M_{b}}$ is a reduced mass, and
$\bm{p}=\frac{M_{b}\bm{P}_{a}-M_{a}\bm{P}_{b}}{M_{a}+M_{b}}$ and
$\bm{r}=\bm{R}_{a}-\bm{R}_{b}$ are the relative momentum and position between
the CM of the two sets, respectively. The three parts of (10a) are entirely
decoupled since (10b) and (10c) depends on the internal coordinates of their
respective set, and (10d) on the relative coordinates between the two CM.

Then, an eigenvalue $E_{\text{ho}}$ is easily obtained since (10) is composed
of three decoupled parts sema20

$E_{\text{ho}}=Q(N_{a})\sqrt{\frac{2}{\mu_{a}}(N_{a}\rho_{aa}+N_{b}\rho_{ab})}+Q(N_{b})\sqrt{\frac{2}{\mu_{b}}(N_{b}\rho_{bb}+N_{a}\rho_{ab})}+Q(2)\sqrt{\frac{2}{\mu}N_{a}N_{b}\rho_{ab}}.$
(11)

To be complete, the expression of the function $B(\\{\alpha\\})$ is given by

$\displaystyle B$
$\displaystyle=N_{a}\left[T_{a}(G_{a}(\mu_{a}))-\frac{G^{2}_{a}(\mu_{a})}{2\mu_{a}}\right]+C^{2}_{N_{a}}\left[V_{aa}(J_{aa}(\rho_{aa}))-\rho_{aa}J^{2}_{aa}(\rho_{aa})\right]$
(12)
$\displaystyle+N_{b}\left[T_{b}(G_{b}(\mu_{b}))-\frac{G^{2}_{b}(\mu_{b})}{2\mu_{b}}\right]+C^{2}_{N_{b}}\left[V_{bb}(J_{bb}(\rho_{bb}))-\rho_{bb}J^{2}_{bb}(\rho_{bb})\right]$
$\displaystyle+N_{a}N_{b}\left[V_{ab}(J_{ab}(\rho_{ab}))-\rho_{ab}J^{2}_{ab}(\rho_{ab})\right].$

When combining (10) and (12), we can see that our auxiliary Hamiltonian (2) is
also composed of three distinct parts: one for the particles of type $a$,
another for the particles of type $b$ and a last one for the relative motion
between the two sets.

The compact equations can then be established in a similar way as done for
identical particles silv12 . First, we apply the Hellmann-Feynman theorem hell
on Hamiltonian (2) to evaluate extremization conditions (6). By using
definitions (3) we get the following results

$\begin{array}[]{lllll}&G_{a}^{2}(\mu_{a0})=p_{a}^{2}+\frac{P_{0}^{2}}{N_{a}^{2}}={p^{\prime}_{a}}^{2},\\\\[8.5359pt]
&G_{b}^{2}(\mu_{b0})=p_{b}^{2}+\frac{P_{0}^{2}}{N_{b}^{2}}={p^{\prime}_{b}}^{2},\\\\[8.5359pt]
&J^{2}_{aa}(\rho_{aa0})=r_{aa}^{2},\\\\[8.5359pt]
&J^{2}_{bb}(\rho_{bb0})=r_{bb}^{2},\\\\[8.5359pt]
&J^{2}_{ab}(\rho_{ab0})=\frac{N_{a}-1}{2N_{a}}r_{aa}^{2}+\frac{N_{b}-1}{2N_{b}}r_{bb}^{2}+R_{0}^{2}={r^{\prime}_{0}}^{2},\end{array}$
(13)

where we have defined the six physical parameters

$\begin{array}[]{lll}&p_{a}^{2}=\left\langle\bm{p}_{i}^{2}-\frac{\bm{P}_{a}^{2}}{N_{a}^{2}}\right\rangle\text{
and
}p_{b}^{2}=\left\langle\bm{p}_{j}^{2}-\frac{\bm{P}_{b}^{2}}{N_{b}^{2}}\right\rangle,\\\\[8.5359pt]
&r_{aa}^{2}=\left\langle\bm{r}_{ii^{\prime}}^{2}\right\rangle\text{ and
}r_{bb}^{2}=\left\langle\bm{r}_{jj^{\prime}}^{2}\right\rangle,\\\\[8.5359pt]
&P_{0}^{2}=\left\langle\bm{p}^{2}\right\rangle\text{ and
}R_{0}^{2}=\left\langle\bm{r}^{2}\right\rangle.\end{array}$ (14)

The mean values are taken with an eigenstate of the auxiliary Hamiltonian
corresponding to the quantum numbers $Q(N_{a})$, $Q(N_{b})$ and $Q(2)$ for the
set $\\{\alpha_{0}\\}$ insuring the constraints (6). The eigenstate is also
completely (anti)symmetric for the exchange between the $N_{a}$ or the $N_{b}$
particles.

Then, by evaluating
$\tilde{E}_{0}=\left\langle\tilde{H}(\\{\alpha_{0}\\})\right\rangle$ and using
results (13), we obtain the following equation for the energy

$\tilde{E}_{0}=N_{a}T_{a}\left(p^{\prime}_{a}\right)+N_{b}T_{b}\left(p^{\prime}_{b}\right)+C^{2}_{N_{a}}V_{aa}\left(r_{aa}\right)+C^{2}_{N_{b}}V_{bb}\left(r_{bb}\right)+N_{a}N_{b}V_{ab}\left(r_{0}^{\prime}\right)$
(15)

It is interesting to look at the meaning of the linear combinations
$p^{\prime}_{a}$, $p^{\prime}_{b}$ and $r^{\prime}_{0}$ since they appear in
(15). As shown in sema20 , we can derive a similar equation for the energy by
using the form (9), instead of (10), of the harmonic oscillator. By comparing,
one can identify
${p^{\prime}_{a}}^{2}=\left\langle\bm{p}_{i}^{2}\right\rangle$,
${p^{\prime}_{b}}^{2}=\left\langle\bm{p}_{j}^{2}\right\rangle$ and
${r^{\prime}_{0}}^{2}=\left\langle\bm{r}_{ij}^{2}\right\rangle$.

In order to find these parameters, we need 6 additional equations. We can find
three of them by applying the virial theorem separately on each of the three
parts of the auxiliary Hamiltonian sema20 . One gets

$N_{a}T^{\prime}_{a}(p^{\prime}_{a})\frac{p_{a}^{2}}{p^{\prime}_{a}}=C^{2}_{N_{a}}V^{\prime}_{aa}(r_{aa})r_{aa}+\frac{N_{b}}{N_{a}}C^{2}_{N_{a}}V^{\prime}_{ab}(r^{\prime}_{0})\frac{r_{aa}^{2}}{r^{\prime}_{0}},$
(16a)
$N_{b}T^{\prime}_{b}(p^{\prime}_{b})\frac{p_{b}^{2}}{p^{\prime}_{b}}=C^{2}_{N_{b}}V^{\prime}_{bb}(r_{bb})r_{bb}+\frac{N_{a}}{N_{b}}C^{2}_{N_{b}}V^{\prime}_{ab}(r^{\prime}_{0})\frac{r_{bb}^{2}}{r^{\prime}_{0}},$
(16b)
$\frac{1}{N_{a}}T^{\prime}_{a}(p^{\prime}_{a})\frac{P_{0}^{2}}{p^{\prime}_{a}}+\frac{1}{N_{b}}T^{\prime}_{b}(p^{\prime}_{b})\frac{P_{0}^{2}}{p^{\prime}_{b}}=N_{a}N_{b}V^{\prime}_{ab}(r^{\prime}_{0})\frac{R_{0}^{2}}{r^{\prime}_{0}}.$
(16c)

Finally, we obtain three last equations by using the exact eigenvalue (11) of
the harmonic oscillator and comparing it to $\left\langle
H_{\text{ho}}\\{\alpha_{0}\\}\right\rangle$. Thanks to (10), the comparison is
done in a similar way as in silv12  and one gets

$Q(N_{a})=\sqrt{C^{2}_{N_{a}}}p_{a}r_{aa},$ (17a)
$Q(N_{b})=\sqrt{C^{2}_{N_{b}}}p_{b}r_{bb},$ (17b) $Q(2)=P_{0}R_{0}.$ (17c)

Equations (16) and (17) form a set of six equations which, combined with (15),
form the ET compact equations for a system of $N_{a}+N_{b}$ particles, and
allow us to compute the approximate eigenvalue $\tilde{E}_{0}$. We have
verified on several systems that these equations give the same results than
those found with the extremization equations in sema20 . We can note that the
three equations (16) can be derived by minimizing (15) with respect to
$r_{aa}$, $r_{bb}$ and $R_{0}$, and using (17).

Equations (15)-(17) are more complicated than equations (7). But, when
comparing the two sets, it is possible to find an interpretation for equations
(15)-(17). Equation (15) is obviously the energy computed in terms of the mean
momenta and relative distances. Equations (16) are the equations of motion
determining these mean quantities, and equations (17) are the semiclassical
quantifications of the various orbital and radial motions. These equations
make clear what are the relevant quantities appearing in a quantum system
containing two different sets of identical particles. It is worth recalling
that solutions obtained by the ET are full quantum ones with eigenfunctions
associated silv10 ; sema20  and that observables can be computed sema15a .

As a first check for these equations, we need to recover the three equations
(7) when considering all particles identical. In this case $T_{a}=T_{b}$ and
$V_{aa}=V_{bb}=V_{ab}$, and we must impose the following symmetries

$\begin{array}[]{cc}&\left\langle\bm{p}_{i}^{2}\right\rangle=\left\langle\bm{p}_{j}^{2}\right\rangle,\hskip
14.22636pt\forall\ i,j,\\\\[14.22636pt]
&\left\langle\bm{r}_{ii^{\prime}}^{2}\right\rangle=\left\langle\bm{r}_{jj^{\prime}}^{2}\right\rangle=\left\langle\bm{r}_{ij}^{2}\right\rangle,\hskip
14.22636pt\forall\ i,i^{\prime},j,j^{\prime}.\end{array}$ (18)

From the definitions of our 6 parameters, we conclude

$\begin{array}[]{cc}&p^{\prime}_{a}=p^{\prime}_{b}=p_{0},\\\\[14.22636pt]
&r_{aa}=r_{bb}=r^{\prime}_{0}=\rho_{0},\end{array}$ (19)

where $p_{0}$ and $\rho_{0}$ are defined as before in (7). Then, we easily see
that equation (15) reduces to (7a) with $N=N_{a}+N_{b}$. It is a matter of
algebra to show that the sum of the three equations (16)

$N_{a}T^{\prime}_{a}(p^{\prime}_{a})p^{\prime}_{a}+N_{b}T^{\prime}_{b}(p^{\prime}_{b})p^{\prime}_{b}=C^{2}_{N_{a}}V^{\prime}_{aa}(r_{aa})r_{aa}+C^{2}_{N_{b}}V^{\prime}_{bb}(r_{bb})r_{bb}+N_{a}N_{b}V^{\prime}_{ab}(r^{\prime}_{0})r^{\prime}_{0},$
(20)

reduces to (7b). When all the particles are identical, it is not relevant to
separate the energy on several subsets. We notice that
$Q(N_{a})+Q(N_{b})+Q(2)=Q(N_{a}+N_{b})$. This is a hint that the sum of the
three equations (17) must reduce to (7c), but the proof is more subtle. Thanks
to the symmetries (18) and (19), one can express $R_{0}$ in terms of
$\rho_{0}$, and $p_{a}$, $p_{b}$ and $P_{0}$ in terms $p_{0}$. Then, simple
calculations show that (17) reduces to (7c). Finally, all equations (7) are
recovered. Note that (19) also implies symmetries on the auxiliary parameters,
$\mu_{a}=\mu_{b}$ and $\rho_{aa}=\rho_{bb}=\rho_{ab}$, which is also expected
as explained in sema20 .

As a second test, we have substituted the harmonic oscillator Hamiltonian (9)
into our 7 equations. Then, it is a matter of algebra to find the exact
solution (11). A third check is given in the following section.

##  III $\bm{N_{a}=1}$ or/and $\bm{N_{b}=1}$

The 7 equations (15), (16) and (17) were computed for a system with
$N_{a}+N_{b}$ particles. It is interesting to look at what happens when only
one particle is present in a set. For example, let’s look at the case
$N_{b}=1$. Then, all the terms in $C^{2}_{N_{b}}$ and $Q(N_{b})$ vanish.
Equation (17b) becomes trivial and (16b) leads to $p_{b}=0$. As $p_{b}=0$, we
also have $p^{\prime}_{b}=P_{0}$. At the end, we are left with a system of 5
equations

$\tilde{E}_{0}=N_{a}T_{a}\left(p^{\prime}_{a}\right)+T_{b}\left(P_{0}\right)+C^{2}_{N_{a}}V_{aa}\left(r_{aa}\right)+N_{a}V_{ab}\left(r_{0}^{\prime}\right),$
(21a)
$N_{a}T^{\prime}_{a}(p^{\prime}_{a})\frac{p_{a}^{2}}{p^{\prime}_{a}}=C^{2}_{N_{a}}V^{\prime}_{aa}(r_{aa})r_{aa}+\frac{N_{a}-1}{2}V^{\prime}_{ab}(r^{\prime}_{0})\frac{r_{aa}^{2}}{r^{\prime}_{0}},$
(21b)
$\frac{1}{N_{a}}T^{\prime}_{a}(p^{\prime}_{a})\frac{P_{0}^{2}}{p^{\prime}_{a}}+T^{\prime}_{b}(P_{0})P_{0}=N_{a}V^{\prime}_{ab}(r^{\prime}_{0})\frac{R_{0}^{2}}{r^{\prime}_{0}},$
(21c) $Q(N_{a})=\sqrt{C^{2}_{N_{a}}}p_{a}r_{aa},$ (21d) $Q(2)=P_{0}R_{0},$
(21e)

where our four parameters are now defined as
$p_{a}^{2}=\left\langle\bm{p}_{i}^{2}-\frac{\bm{P}_{a}^{2}}{N_{a}^{2}}\right\rangle$,
$P_{0}^{2}=\left\langle\left(\frac{\mu_{b}\bm{P}_{a}-M_{a}\bm{p}_{b}}{M_{a}+\mu_{b}}\right)^{2}\right\rangle$,
$\bm{r}_{aa}^{2}=\left\langle\bm{r}_{ii^{\prime}}^{2}\right\rangle$ and
$R_{0}^{2}=\left\langle\left(\bm{R}_{a}-\bm{r}_{b}\right)^{2}\right\rangle$.
We also have ${p^{\prime}_{a}}^{2}=p_{a}^{2}+\frac{P_{0}^{2}}{N_{a}^{2}}$ and
${r^{\prime}_{0}}^{2}=\frac{N_{a}-1}{2N_{a}}r_{aa}^{2}+R_{0}^{2}$. The five
equations (21) can also be found from scratch with the above explained
procedure.

Another special case is when $N_{a}=N_{b}=1$, that is we have a two-body
system. We then have similar simplifications as in the previous case and we
obtain the equations of the envelope theory at $N=2$, which are a
generalization of the results obtained in sema13 ; sema12

$\tilde{E}_{0}=T_{a}(P_{0})+T_{b}(P_{0})+V_{ab}(R_{0}),$ (22a)
$T^{\prime}_{a}(P_{0})P_{0}+T^{\prime}_{b}(P_{0})P_{0}=V^{\prime}_{ab}(R_{0})R_{0},$
(22b) $Q(2)=P_{0}R_{0},$ (22c)

where
$P_{0}^{2}=\left\langle\left(\frac{\mu_{b}\bm{p}_{a}-\mu_{a}\bm{p}_{b}}{\mu_{a}+\mu_{b}}\right)^{2}\right\rangle$
and
$R_{0}^{2}=\left\langle\left(\bm{r}_{a}-\bm{r}_{b}\right)^{2}\right\rangle$.
The fact that the correct limits are obtained for $N_{a}=1$ or $N_{b}=1$ is
also a test of coherence for the set (15)-(16)-(17).

##  IV concluding remarks

We were able to build the 7 compact equations of the envelope theory for a
system of $N_{a}+N_{b}$ particles. These equations reduce to the 3 usual ones
when considering identical particles. We also presented the special cases when
$N_{b}=1$ or/and $N_{a}=1$. Starting from these equations, it is possible to
improve the envelope theory using a similar procedure than the one used in
sema15b . This is performed for a system of $N_{a}+1$ particles in chev21 .
With these 7 equations, it is possible to open new domains of applicability of
the envelope theory, especially in hadronic physics where the method is proven
to be useful as mentioned in the introduction. But the method can also be used
to estimate the binding energies of other systems such as nuclei or clusters
of cold atoms for which ab-initio calculations are already available, as for
instance in gatt13 . In particular, accurate calculations have been performed
for large helium clusters gatt11 ; kiev20 . For such systems, it is necessary
to take into account a three-body forces that can be handled by the envelope
theory sema18b .

###### Acknowledgements.

L.C. would thank the Fonds de la Recherche Scientifique - FNRS for the
financial support. This work was also supported under Grant Number 4.45.10.08.

## References

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# Couplings and Poisson approximation for stabilizing functionals of
determinantal point processes

Moritz Otto  Department of Mathematics, Aarhus University, Denmark
<EMAIL_ADDRESS>
###### Abstract.

We prove a Poisson process approximation result for stabilizing functionals of
a determinantal point process. Our results use concrete couplings of
determinantal processes with different Palm measures and exploit their
association properties. Second, we focus on the Ginibre process and show in
the asymptotic scenario of an increasing window size that the process of
points with a large nearest neighbor distance converges after a suitable
scaling to a Poisson point process. As a corollary, we obtain the scaling of
the maximum nearest neighbor distance in the Ginibre process, which turns out
to be different from its analogue for independent points.

###### Key words and phrases:

coupling method, determinantal process, Ginibre process, Kantorovich-
Rubinstein distance, Palm calculus, Poisson approximation, scaling limit

###### 2010 Mathematics Subject Classification:

Primary 60K35. Secondary 60G55, 60D05.

##  1\. Introduction

Determinantal point processes (DPPs) were introduced in quantum mechanics to
study configurations of fermions [21]. Due to their repulsive nature, the play
a fundamental role in applied sciences, e.g. as a model for base stations in a
wireless network [22]. In mathematics, DPPs arise naturally in different
fields, such as eigenvalues of random matrices [13] and random spanning trees
[6]. DPPs have notable probabilistic properties. Amongst others, the (reduced)
Palm process of a DPP is again a determinantal process and important
quantities such as the Laplace transform and Janossy densities admit closed-
form expressions (see e.g. [11]). A DPP on $\mathbb{R}^{d}$ is determined by
its correlation kernel $K$, which is a Hermitian function from
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ to $\mathbb{C}$. An important DPP is the
Ginibre process on $\mathbb{R}^{2}$ with Gaussian kernel given in Section 2.

Let $\xi$ be a stationary DPP on $\mathbb{R}^{d}$ and let $g$ be a measurable
function from $\mathbb{R}^{d}\times\mathbf{N}$ to $\\{0,1\\}$, where we write
$\mathbf{N}$ for the set of $\sigma$ -finite point configurations on
$\mathbb{R}^{d}$. For $x\in\mathbb{R}^{d}$, let $\delta_{x}$ denote the Dirac
measure in $x$. For some measurable $W\subset\mathbb{R}^{d}$, let

$\Xi[\xi]:=\sum_{x\in\xi\cap W}g(x,\xi)\delta_{x}.$

Here, the function $g$ has the effect of a thinning of $\xi$, in the sense
that $\Xi$ is the point process of all points $x\in\xi$ in the set $W$ which
satisfy $g(x,\xi)=1$. Random measures of this type are flexible models that
appear in the study of random spatial graphs, stochastic topology and
geometric extreme value theory.

In this article, we study the distance (in an appropriate distance of point
processes) of $\Xi$ and a Poisson point process. To the best of our knowledge,
this is the first paper which systematically studies Poisson approximation for
determinantal processes. This continues the studies for stabilizing
functionals of Poisson point processes [12, 5, 25], Poisson hyperplanes
processes [24] and Gibbs point processes [18]. However, the tools used in our
paper are different then in above mentioned. As main contributions, this
article shows:

(i) If the correlation kernel $K$ is fast decaying and if the thinning
function $g$ is stabilizing and satisfies some natural assumptions, the bound
of the distance of $\Xi$ and a Poisson process is analogous to the bounds
obtained for thinned Poisson point processes (see [5] and [24]).

(ii) If $\xi$ is the Ginibre process and $g(x,\xi)$ is indicator which is one
if $\xi\setminus\\{x\\}$ does not have points in a ball with a given radius
$v$ around $x$, we prove in as asymptotic scenario where the size of $W$ and
$v$ tend to infinity, that an appropriate scaling of $\Xi$ converges to a
Poisson process.

Our paper is organized as follows. In Section 2 we introduce determinantal
point processes state our two main results. In Section 3, we provide important
notions such as Palm theory, negative association and correlation decay. The
proof of Theorem 1 is given in Section 4. In Section 5 we provide the proof of
Theorem 2.

##  2\. Model and main results

We work on the Euclidean space $\mathbb{R}^{d}$ ($d\geqslant 1$) equipped with
its Borel $\sigma$-field $\mathcal{B}^{d}$ and Euclidean norm $\|\cdot\|$. We
denote by $\mathbf{N}$ the space of all $\sigma$-finite counting measures on
$\mathbb{R}^{d}$, and by $\widehat{\mathbf{N}}$ the space of all finite
counting measures on $\mathbb{R}^{d}$ and equip $\mathbf{N}$ and
$\widehat{\mathbf{N}}$ with their corresponding $\sigma$-fields $\mathcal{N}$
and $\widehat{\mathcal{N}}$, which are induced by the maps
$\omega\mapsto\omega(B)$ for all $B\in\mathcal{B}^{d}$. A point process is a
random element $\xi$ of $\mathbf{N}$, defined over some fixed probability
space $(\Omega,\mathcal{F},\mathbb{P})$. The intensity measure of $\xi$ is the
measure $\mathbb{E}[\xi]$ defined by $\mathbb{E}[\xi](B):=\mathbb{E}[\xi(B)]$,
$B\in\mathcal{B}^{d}$. For $z\in\mathbb{R}^{d}$ and $r>0$ let $B_{r}(z)$ be
the closed Euclidean ball with radius $r$ around $z$. For
$B\in\mathcal{B}^{d}$ we write $|B|$ for the Lebesgue measure of $B$.

Let $K:(\mathbb{R}^{d})^{2}\to\mathbb{C}$ be a complex function. We say that
$\xi$ is a determinantal point process with correlation kernel $K$, if for
every $n\in\mathbb{N}$ and pairwise disjoint
$A_{1},\dots,A_{n}\in\mathcal{B}^{d}$ we have that

$\displaystyle\mathbb{E}[\xi(A_{1})\cdots\xi(A_{n})]=\int_{A_{1}\times\cdots\times
A_{n}}\det(K(x_{i},x_{j}))_{i,j=1}^{n}\mathrm{d}(x_{1},\dots x_{n}),$

where $\mathrm{d}\dots$ denotes integration with respect to the Lebesgue
measure on $\mathbb{R}^{d}$, $(K(x_{i},x_{j}))_{i,j=1}^{m}$ is the $m\times
m$-matrix with entry $K(x_{i},x_{j})$ at position $(i,j)$, and $\det M$ is the
determinant of a complex-valued $m\times m$-matrix $M$. This says that $\xi$
has correlation functions of all orders and the $m$th order correlation
function $\rho^{(m)}$ is given by

$\displaystyle\rho^{(m)}(x_{1},\dots,x_{m})=\det(K(x_{i},x_{j}))_{i,j=1}^{m},\quad
x_{1},\dots,x_{m}\in\mathbb{R}^{d},\,\quad m\in\mathbb{N},$

and that it is locally integrable. In this article we assume that $K$
satisfies the following assumptions (i)-(iv).

  1. (i)

$K$ is Hermitian, i.e. $K(x,y)=\overline{K(y,x)}$, $x,y\in\mathbb{R}^{d}$,

  2. (ii)

$K$ is locally square integrable, i.e. for every compact $B\in\mathcal{B}^{d}$
the integral

$\int_{B}\int_{B}|K(x,y)|^{2}\mathrm{d}y\mathrm{d}x$

is finite,

  3. (iii)

$K$ is locally of trace class, i.e. for every compact $B\in\mathcal{B}^{d}$
the integral $\int_{B}K(x,x)\mathrm{d}x$ is finite,

Under the assumptions (i)–(iii), it follows from Mercer’s theorem that for
every compact $B\subset\mathbb{R}^{d}$, the restriction $\xi_{B}$ of $\xi$ to
$B$ is a determinantal point process whose kernel $K_{B}$ is for almost all
$(x,y)\in B\times B$ given by

$\displaystyle
K_{B}(x,y)=\sum_{k=1}^{\infty}\lambda_{k}^{B}\phi_{k}^{B}(x)\overline{\phi_{k}^{B}(y)},$

where $\lambda_{k}^{B}\in\mathbb{R}$, $k\in\mathbb{N}$, and the functions
$\phi_{k}^{B}$, $k\in\mathbb{N}$, form on orthonormal base of $L^{2}(B)$.
Finally, we assume that

  1. (iv)

$0\leqslant\lambda_{k}^{B}\leqslant 1$ for all $k\in\mathbb{N}$ and all
compact $B\in\mathcal{B}^{d}$.

Under the assumptions (i)–(iv) there is a unique (in distribution)
determinantal point process with correlation kernel $K$ (see [29, Theorem 3]).

For $x\in\mathbb{R}^{d}$ we call $\xi^{x}$ a Palm version of the point process
$\xi$ at $x$, if for all measurable
$f:\mathbb{R}^{d}\times\mathbf{N}\to\mathbb{R}_{+}$,

(1) $\displaystyle\mathbb{E}\Big{[}\int f(x,\xi)\xi(\mathrm{d}x)\Big{]}=\int
f(x,\xi^{x})\mathbb{E}[\xi](\mathrm{d}x).$

Later, we will generalize this definition and define a Palm process with
respect to another point process.

Let $\xi$ be a stationary determinantal process satisfying (i)–(iv) with
intensity $\rho>0$. Let $g:\mathbb{R}^{d}\times\mathbf{N}\to\\{0,1\\}$ be a
measurable function (called score function) and let $W\in\mathcal{B}^{d}$. We
define

(2) $\displaystyle\Xi[\omega]:=\sum_{x\in\omega\cap W}g(x,\omega)\delta_{x},$

and set $\Xi:=\Xi[\xi]$. Note that by (1), the intensity measure $\mathbf{L}$
of $\Xi$ is given by

$\mathbf{L}(A)=\rho\int_{W\cap A}\mathbb{E}[g(x,\xi^{x})]\,\mathrm{d}x,\quad
A\in\mathcal{B}^{d}.$

In this article we study the Kantorovich-Rubinstein (KR) distance of $\Xi$ and
a finite Poisson process. We recall the definition of the KR distance from
[12]. For finite point processes $\zeta$ and $\xi$ on $\mathbb{R}^{d}$ the KR
distance is given by

$\displaystyle\mathbf{d_{KR}}(\zeta,\xi):=\sup_{h\in\text{Lip}}|\mathbb{E}h(\zeta)-\mathbb{E}h(\xi)|,$

where Lip is the class of all measurable 1-Lipschitz functions
$h:\widehat{\mathbf{N}}\to\mathbb{R}$ with respect to the total variation
between measures $\omega_{1},\omega_{2}$ on $\mathbb{R}^{d}$ given by

$\displaystyle
d_{\text{TV}}(\omega_{1},\omega_{2}):=\sup|\omega_{1}(A)-\omega_{2}(A)|,$

where the supremum is taken over all $A\in\mathcal{B}^{d}$ with
$\omega_{1}(A),\omega_{2}(A)<\infty$. Under appropriate conditions on $\xi$
and $g$, we prove that $\Xi$ can be approximated by a Poisson process.

We suppose that there exists $\alpha\in(0,\infty)$ such that for all
$A\in\mathcal{B}^{d}$ and all $\omega\in\mathbf{N}$,

(3) $\displaystyle\sum_{x\in\omega\cap A}g(x,\omega)<\alpha|A|,$

and assume that $g$ is monotonic in the sense that for all $x\in W$, it holds
that

(4) $\displaystyle g(x,\omega_{1})\leqslant g(x,\omega_{2})\quad\text{or}\quad
g(x,\omega_{1})\geqslant g(x,\omega_{2}),\qquad\omega_{1}\subset\omega_{2}.$

We further assume that $g$ is stabilizing, by which we mean that there es a
measurable function $\mathcal{S}:\mathbb{R}^{d}\times\mathbf{N}\to\mathcal{F}$
such that

$g(x,\omega)=g(x,\omega\cap\mathcal{S}(x,\omega))$

holds for any $\omega\in\mathbf{N}$ and any $x\in\mathbb{R}^{d}$, where
$\mathcal{F}$ is the set of closed subsets in $\mathbb{R}^{d}$. Further
suppose that $\mathcal{S}$ is a stopping set, which says that

$\\{\omega\in\mathbf{N}:\,\mathcal{S}(x,\omega)\subset
S\\}=\\{\omega\in\mathbf{N}:\,\mathcal{S}(x,\omega\cap S)\subset S\\}.$

Moreover, we assume that the kernel $K:(\mathbb{R}^{d})^{2}\to\mathbb{C}$
satisfies

(5) $\displaystyle|K(x,y)|\leqslant\phi(\|x-y\|),\quad x,y\in\mathbb{R}^{d},$

for some decreasing function $\phi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with
$\lim_{r\to\infty}\phi(r)=0$.

######  Theorem 1.

Let $\xi$ be a stationary determinantal process with kernel $K$ satisfying (5)
and intensity $\rho\in(0,\infty)$. Let $\Xi$ be the score sum defined in (2)
with intensity measure $\mathbf{L}$ and suppose that $g$ satisfies (3) and (4)
and is stabilizing with respect to the stopping set $\mathcal{S}$. For Borel
sets $S,T\subset\mathbb{R}^{d}$ with $o\in S\subset T$ let $S_{x}:=x+S$ and
$T_{x}:=x+T,\,x\in W,$ and define

$\tilde{g}(x,\omega):=g(x,\omega)\mathds{1}\\{\mathcal{S}(x,\omega)\subseteq
S_{x}\\},\quad x\in\mathbb{R}^{d},\,\omega\in\mathbf{N}.$

Let $\zeta$ be a finite Poisson process on $\mathbb{R}^{d}$ with intensity
measure $\mathbf{M}$. Then,

$\displaystyle\mathbf{d_{KR}}(\Xi,\zeta)\leqslant
d_{TV}(\mathbf{L},\mathbf{M})+2(E_{1}+E_{2}+E_{3})+F,$

where

$\displaystyle E_{1}$
$\displaystyle:=\rho\int_{W}\mathbb{P}(\mathcal{S}(x,\eta^{x})\not\subset
S)\,\mathrm{d}x,$ $\displaystyle E_{2}$
$\displaystyle:=\rho^{2}\int_{W}\int_{W\cap
T_{x}}\mathbb{E}[\tilde{g}(x,\xi^{x})]\mathbb{E}[\tilde{g}(y,\xi^{y})]\,\mathrm{d}y\mathrm{d}x,$
$\displaystyle E_{3}$ $\displaystyle:=\rho^{2}\int_{W}\int_{W\cap
T_{x}}\mathbb{E}[\tilde{g}(x,\xi^{x,y})\tilde{g}(y,\xi^{x,y})]\,\mathrm{d}y\mathrm{d}x,$
$\displaystyle F$ $\displaystyle:=c\|K\|\max(|S|,1)|W\oplus
S|^{2}\phi(d(T,S)),$

where $\|K\|=\sup_{x,y\in\mathbb{R}^{d}}|K(x,y)|$, $W\oplus S$ is the
Minkowski sum of $W$ and $S$, $d(T,S)$ is the Hausdorff distance of $T$ and
$S$ and the constant $c>0$ does not depend on $K$, $g$, $W$, $S$ and $T$.

In the second part of this paper, we give an application of Theorem 1 for a
concrete choice of $\xi$ and of $g$. Let $\xi$ be the (infinite) Ginibre
process, which is a stationary determinantal point process on $\mathbb{C}$
with correlation kernel given by

$K(z,w)=\pi^{-1}e^{-(|z|^{2}+|w|^{2})/2}e^{z\overline{w}},\quad
z,w\in\mathbb{C}.$

Hence, $\xi$ has intensity $\rho=\pi^{-1}$ and it holds that
$|K(z,w)|\leqslant\phi(\|z-w\|)$ with $\phi(r):=\pi^{-1}\exp(-r^{2}/2)$,
$r>0$, for all $z,w\in\mathbb{C}$.

In Theorem 2 below, we choose $g$ depending on $n\in\mathbb{N}$. Let $g_{n}$
is the indicator function which is one if and only if the process
$\xi\setminus\\{x\\}$ is empty in a ball with a certain radius $v_{n}$ (chosen
such that $v_{n}\to\infty$ as $n\to\infty$) around $x$. This choice leads to
the study of large nearest neighbor balls. It is also an important prototype
for more sophisticated models in stochastic geometry and has been studied
extensively for different point processes in various spaces (see [26]).

We consider $\xi$ as a random set in $\mathbb{R}^{2}$. Let $B_{n}:=B_{n}(o)$
the closed ball with radius $n>0$ in $\mathbb{R}^{2}$ centered at the origin
$o$. We consider the process

(6) $\Xi_{n}:=\sum_{x\in\xi\cap
B_{n}}\mathds{1}\\{\xi(B_{v_{n}}(x)\setminus\\{x\\})=0\\}\ \delta_{x}.$

as well as the scaled process

(7) $\Psi_{n}:=\sum_{y\in\Xi}\delta_{y/n}=\sum_{x\in\xi\cap
B_{n}}\mathds{1}\\{\xi(B_{v_{n}}(x)\setminus\\{x\\})=0\\}\ \delta_{x/n}.$

In the following theorem we compare $\Psi_{n}$ with a Poisson process on the
unit ball $B_{1}$ in $\mathbb{R}^{2}$.

######  Theorem 2.

Let $\tau>0$ and let $\nu$ be a stationary Poisson process on $\mathbb{R}^{2}$
with intensity $\tau>0$. There exists a sequence $(v_{n})_{n\in\mathbb{N}}$
with $v_{n}^{4}\sim 4\log n$ as $n\to\infty$ such that for all
$n\in\mathbb{N}$ and any $\varepsilon>0$,

$\mathbf{d_{KR}}(\Psi_{n},\nu\cap B_{1})\leqslant Cn^{\varepsilon-1/16}.$

As an application of the above theorem, we consider largest distances to the
nearest neighbor. That is, for any $x\in\mathbb{R}^{2}$, we denote by
$\mathsf{nn}(x,\xi):=\arg\min_{y\in\xi\setminus\\{x\\}}|x-y|$ the nearest
neighbor of $x$ in $\xi\setminus\\{x\\}$.

######  Corollary 3.

We have as $n\to\infty$,

$\frac{1}{2\pi\sqrt{\log n}}\max_{x\in\xi\cap
B_{n}}|B_{\mathsf{nn}(x,\xi)}|\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}1.$

The proof of Corollary 3 is quite standard (see e.g. [25, Corollary 4.2] or
[9, Corollary 1]) and therefore omitted.

######  Remark 4.

(i) One should compare Theorem 1 with [5, Theorem 4.1] (or the refined version
[24, Theorem 7]) that discusses Poisson process approximation for score sums
built on a Poisson process. The terms $E_{1}$, $E_{2}$, $E_{3}$ in Theorem 1
are the analogues to the terms $E_{1}$, $E_{2}$, $E_{3}$ in [5, Theorem 4.1].
Due to the spatial independence property of the Poisson process, there is no
analogue of the term $F$ (which reflects the correlation decay of the
determinantal process $\xi$) in [5] and [24]. Note also that for a wide class
of determinantal point processes (including the Ginibre process), the
$\beta$-mixing coefficient does not decay exponentially fast (see [27,
Proposition 4.2]). Therefore, the exponential decay dependence (EDD) property
from [10] is violated and general results for Poisson approximation of
strongly mixing processes do not apply.

(ii) Note that the scaling of the maximum nearest neighbor ball in Corollary 3
is different from its analogue for independent points, which is proportional
to $\log n$ as $n\to\infty$ (see [8]). It seems interesting to investigate
whether Theorem 2 can be extended to $k$-nearest neighbor distances
($k\geqslant 2$). However, this extension would require fine probabilistic
estimates on empty-space probabilities of the Ginibre process, which are
beyond the scope of this article.

##  3\. Preliminaries

###  3.1. Palm calculus and negative association

Following [16, Chapter 6] we next introduce Palm measures and thereby
generalize the definition given in (1). Let $\xi,\Xi$ be point processes on
$(\mathbb{R}^{d},\mathcal{B}^{d})$ and assume that $\Xi$ has $\sigma$-finite
intensity measure $\mathbf{L}$. Then there are point processes
$\xi^{x,\Xi},\,x\in\mathbb{R}^{d}$, such that

(8) $\displaystyle\mathbb{E}\Big{[}\int
f(x,\xi)\,\Xi(\mathrm{d}x)\Big{]}=\int\mathbb{E}[f(x,\xi^{x,\Xi})]\mathbf{L}(\mathrm{d}x),$

where $f:\mathbb{R}^{d}\times\mathbf{N}\to[0,\infty)$ is assumed to be
measurable. The processes $\xi^{x,\Xi},x\in\mathbb{R}^{d},$ are called Palm
processes of $\xi$ with respect to $\Xi$ at $x$. If $\Xi$ is simple,
$\xi^{x,\Xi}$ can be interpreted as the process $\xi$ seen from $x$ and
conditioned on $\Xi$ having a point in $x$. From [16, Lemma 6.2(ii)] it
follows that $\delta_{x}\in\xi^{x,\Xi}$ a.s. This allows us to define the
reduced Palm process $\xi^{x!,\Xi}:=\xi^{x,\Xi}-\delta_{x}$. If $\xi=\Xi$
a.s., we write $\xi^{x}$ for a Palm process of $\xi$ (with respect to itself)
at $x$ (c.f. (1)) and $\xi^{x!}$ for a reduced Palm process.

Let $\xi$ be a determinantal point process satisfying the conditions (i)–(iv)
from Section 2 with correlation kernel $K$. Then the reduced Palm processes
$\xi^{x!},\,x\in\mathbb{R}^{d},$ are determinantal processes with correlation
kernel $K^{x},\,x\in\mathbb{R}^{d},$ given by

(9) $\displaystyle K^{x}(z,w)=K(z,w)-\frac{K(z,x)K(x,w)}{K(x,x)},\quad
z,w\in\mathbb{R}^{d},$

whenever $K(x,x)>0$ (see [28, Theorem 1.7]). By [14, Theorem 3] (see also
[23]), the process $\xi^{x!}$ is stochastically dominated by $\xi$ (denoted by
$\xi^{x!}\leqslant\xi$) which means that

(10) $\displaystyle\mathbb{E}[F(\xi^{x!})]\leqslant\mathbb{E}[F(\xi)]$

for each measurable $F:\mathbf{N}\to\mathbb{R}$ which is bounded and
increasing, by which we mean that $F(\omega_{1})\leqslant F(\omega_{2})$ if
$\omega_{1}\subset\omega_{2}$.

For $x\in\mathbb{R}^{d}$ let $\xi^{x}$ be a Palm process of $\xi$ at $x$ and
$\xi^{x,\Xi}$ a Palm process of $\xi$ with respect to $\Xi$ at $x$. Then we
have

$\displaystyle\mathbb{E}\Big{[}\int
f(x,\xi)\,\Xi(\mathrm{d}x)\Big{]}=\mathbb{E}\Big{[}\int
f(x,\xi)g(x,\xi)\xi(\mathrm{d}x)\Big{]}$
$\displaystyle=\int\mathbb{E}[f(x,\xi^{x})g(x,\xi^{x})]\rho(x)\lambda(\mathrm{d}x)$
(11)
$\displaystyle=\int\mathbb{E}[f(x,\xi^{x,\Xi})]\mathbb{E}[g(x,\xi^{x})]\rho(x)\lambda(\mathrm{d}x).$

An important property of determinantal point processes is that they are
negatively associated (NA) (see [20, Theorem 3.7]). We say that a point
process $\xi$ is NA if for each collection of disjoint sets
$B_{1},\dots,B_{m}\in\mathcal{B}^{d}$ and each subset
$I\subset\\{1,\dots,m\\}$ we have that

(12) $\displaystyle\mathsf{Cov}(F(\xi(B_{i}),\,i\in I),G(\xi(B_{i}),\,i\in
I^{c}))\leqslant 0,$

where $F,G$ are real bounded and increasing functions (see [7]).

Let $F:\mathbb{R}\to\mathbb{R}$ be bounded and increasing and assume that $g$
is increasing in the second argument. Then we find for almost all
$x\in\mathbb{R}^{d}$ from (11) and (12) (applied to the determinantal point
process $\xi^{x!}$) that

(13)
$\displaystyle\mathbb{E}[F(\xi^{x,\Xi})]\mathbb{E}[g(x,\xi^{x})]=\mathbb{E}[F(\xi^{x})g(x,\xi^{x})]\leqslant\mathbb{E}[F(\xi^{x})]\mathbb{E}[g(x,\xi^{x})],$

implying that $\xi^{x,\Xi}\leqslant\xi^{x}$ for $\mathbb{E}[\Xi]$-a.a.
$x\in\mathbb{R}^{d}$. On the other hand, if $g$ is decreasing in the second
argument, then we find by taking $-g$ in (12) that

(14)
$\displaystyle\mathbb{E}[F(\xi^{x,\Xi})]\mathbb{E}[g(x,\xi^{x})]=\mathbb{E}[F(\xi^{x})g(x,\xi^{x})]\geqslant\mathbb{E}[F(\xi^{x})]\mathbb{E}[g(x,\xi^{x})],$

implying that $\xi^{x,\Xi}\geqslant\xi^{x}$ for $\mathbb{E}[\Xi]$-a.a.
$x\in\mathbb{R}^{d}$.

###  3.2. Fast decay of correlation

Let $\xi$ be a stationary determinantal process on $\mathbb{R}^{d}$ with
covariance kernel $K$ that satisfies the conditions (i)–(iv) and
$|K(x,y)|\leqslant\phi(\|x-y\|)$ for some exponentially decreasing function
$\phi$ (see (5)). Then we have from [4, Lemma 1.3] that the correlation
functions $\rho^{(m)}$, $m\in\mathbb{N}$, of $\xi$ satisfy

(15)
$\displaystyle|\rho^{(p+q)}(x_{1},\dots,x_{p+q})-\rho^{(p)}(x_{1},\dots,x_{p})\rho^{(q)}(x_{p+1},\dots,x_{p+q})|\leqslant
m^{1+\frac{m}{2}}\phi(s)\|K\|^{m-1},$

where $m:=p+q$,
$s:=d(\\{x_{1},\dots,x_{p}\\},\\{x_{p+1},\dots,x_{p+q}\\}):=\inf_{i\in\\{1,\dots,p\\},j\in\\{p+1,\dots,p+q\\}}|x_{i}-x_{j}|$,
and $\|K\|:=\sup_{x,y\in\mathbb{R}^{d}}|K(x,y)|$.

###  3.3. Poisson process approximation

Let $\mathbf{L}$ be the (finite) intensity measure of the point process $\Xi$
defined at (2). Suppose for $x\in W$ that $\Xi^{x}$ is a Palm version of $\Xi$
at $x$ and that $\tilde{\Xi}\stackrel{{\scriptstyle d}}{{=}}\Xi$. Then we
obtain from [5, Theorem 3.1] that the Kantorovich-Rubinstein distance of $\Xi$
defined at (2) and a finite Poisson process $\zeta$ with intensity measure
$\mathbf{M}$ is bounded by

(16) $\displaystyle\mathbf{d_{KR}}(\Xi\cap W,\zeta)\leqslant
d_{TV}(\mathbf{L}(\cdot\cap
W),\mathbf{M})+2\int_{W}\mathbb{E}[(\Xi^{x}\Delta\tilde{\Xi}^{x})(W)]\,\mathbf{L}(\mathrm{d}x).$

The overall idea of the proofs of Theorem 1 is to apply this inequality with a
particular choice of $\Xi_{n}^{x}$ and $\tilde{\Xi}_{n}^{x}$. Note that if
$\xi^{x,\Xi}$ is a Palm version of $\xi$ at $x$ with respect to $\Xi$, then
$\Xi[\xi^{x,\Xi}]-\delta_{x}$ is a reduced Palm version of $\Xi$. This allows
us to reduce the problem of constructing a coupling of $\Xi$ and its reduced
Palm measure to constructing a coupling of $\xi$ and its Palm measure with
respect to $\Xi$.

Moreover, for each $x\in W$ we will split the expected symmetric difference
$\mathbb{E}[(\Xi_{n}^{x}\Delta\tilde{\Xi}_{n}^{x})(W)]$ into a part that
represents local contributions around $x$ and the rest. This is done as
follows. For $x\in W$ let $T_{x}\in\mathcal{B}^{d}$ with $x\in T_{x}$ and
$T_{x}\subset W$. Then we have for $\Xi_{n}^{x}:=\Xi[\xi^{x}]$ and
$\tilde{\Xi}^{x}:=\Xi[\hat{\xi}^{x}]$ that almost surely,

$\displaystyle(\Xi^{x}\Delta\tilde{\Xi}^{x})(W)$
$\displaystyle\leqslant\Xi[\xi^{x}](T_{x})+\Xi[\xi^{x,\Xi}](T_{x})+(\Xi^{x}\Delta\tilde{\Xi}^{x})(W\setminus
T_{x})$
$\displaystyle\leqslant\xi^{x}(T_{x})+\xi^{x,\Xi}(T_{x})+(\Xi^{x}\Delta\tilde{\Xi}^{x})(W\setminus
T_{x}).$

From the particular form of the score functional $\Xi$ we obtain that the last
term is almost surely bounded by

$(\Xi^{x}\Delta\tilde{\Xi}^{x})(W\setminus
T_{x})\leqslant(\xi^{x}\Delta\xi^{x,\Xi})(W\setminus
T_{x})+\sum_{y\in\xi^{x}\cap\xi^{x,\Xi}\cap W\setminus
T_{x}}|g(y,\xi^{x})-g(y,\xi^{x,\Xi})|.$

Since we will rely on these observations in the sequel, we summarize them in
the following proposition.

######  Proposition 5.

For $x,y\in W$ let $\xi^{x}$ be a Palm version of $\xi$ at $x$, let
$\xi^{x,y}$ be a Palm version of $\xi^{x}$ at $y$ and let $\xi^{x,\Xi}$ be a
Palm version of $\xi$ with respect to $\Xi$ at $x$. Then we have

$\displaystyle\mathbf{d_{KR}}(\Xi\cap W,\zeta)\leqslant
d_{TV}(\mathbf{L}(\cdot\cap W),\mathbf{M})+T_{1}+T_{2}+T_{3}+T_{4},$

where

$\displaystyle T_{1}:=$
$\displaystyle\rho^{2}\int_{W}\int_{T_{x}}\mathbb{E}[g(x,\xi^{x})]\mathbb{E}[g(y,\xi^{y})]\mathrm{d}y\mathrm{d}x,$
$\displaystyle T_{2}:=$
$\displaystyle\rho^{2}\int_{W}\int_{T_{x}}\mathbb{E}[g(x,\xi^{x,y})g(y,\xi^{x,y})]\mathrm{d}y\mathrm{d}x,$
$\displaystyle T_{3}:=$
$\displaystyle\rho\int_{W}\mathbb{E}[(\xi^{x}\Delta\xi^{x,\Xi})(W\setminus
T_{x})]\mathrm{d}x,$ $\displaystyle T_{4}:=$
$\displaystyle\rho\int_{W}\mathbb{E}\Big{[}\sum_{y\in\xi^{x}\cap\xi^{x,\Xi}\cap
W\setminus T_{x}}|g(y,\xi^{x})-g(y,\xi^{x,\Xi})|\Big{]}\mathrm{d}x.$

##  4\. Proof of Theorem 1

###### Proof of Theorem 1.

For $x\in W$ let $S_{x}:=x+S$ and $T_{x}:=x+T$. We let
$\tilde{g}(x,\omega):=g(x,\omega)\mathds{1}\\{\mathcal{S}(x,\omega)\subset
S_{x}\\}$, $\omega\in\mathbf{N}$, and consider the truncated functional

(17) $\displaystyle\Xi_{\mathsf{tr}}:=\sum_{x\in\xi\cap
W}\tilde{g}(x,\xi)\delta_{x}.$

Note that, due to the stopping set property of $\mathcal{S}(x,\xi)$,
$\tilde{g}(x,\xi)$ is measurable with respect to $\xi\cap S_{x}$.

Step 1: In a first step we assume that $g=\tilde{g}$ and, therefore,
$\Xi=\Xi_{\mathsf{tr}}$. Let $\mathbf{L}$ be the intensity measure of $\Xi$.
We apply Proposition 5 and bound the terms $T_{3}$ and $T_{4}$.

Bounding $T_{3}$. For each $x\in W$, we construct a coupling
$(\xi^{\prime},\xi^{\prime\prime})$ of $P^{x!}$ and $P^{x!,\Xi}$ such that the
symmetric difference $(\xi^{\prime}\Delta\xi^{\prime\prime})(W\setminus
T_{x})$ becomes small. We discuss this for increasing and decreasing scores
separately.

(i) Increasing scores. If $g(x,\omega_{1})\leqslant g(x,\omega_{2})$ for
$\omega_{1}\subset\omega_{2}$, we have by (10) and (13) that
$\xi^{x!}\leqslant\xi$ and $\xi^{x!,\Xi}\leqslant\xi^{x!}$, implying that
$\xi^{x!,\Xi}\leqslant\xi$. By Strassen’s theorem, there exists a Palm version
$\xi^{x,\Xi}$ of $\xi$ with respect to $\Xi$ and a process
$\tilde{\xi}^{x}\stackrel{{\scriptstyle d}}{{=}}\xi$ such that
$\xi^{x,\Xi}\subset\tilde{\xi}^{x}$ almost surely. Thus, we have

$\displaystyle\mathbb{E}[(\xi^{x!,\Xi}\Delta\tilde{\xi}^{x})(W\setminus
T_{x})]$ $\displaystyle=\mathbb{E}[\xi(W\setminus
T_{x})]-\mathbb{E}[\xi^{x!,\Xi}(W\setminus T_{x})]$ (18)
$\displaystyle=\big{\\{}\mathbb{E}[\xi(W\setminus
T_{x})]-\mathbb{E}[\xi^{x!}(W\setminus
T_{x})]\big{\\}}+\mathbb{E}[\xi^{x!}(W\setminus
T_{x})]-\mathbb{E}[\xi^{x!,\Xi}(W\setminus T_{x})].$

The term in $\\{\cdots\\}$ on the right-hand side above is by (9), the fact
that $K(x,x)=\rho$ for all $x\in\mathbb{R}^{d}$, (5) and since $\phi$ is
decreasing, given by

(19) $\displaystyle\rho^{-1}\int_{W\setminus
T_{x}}|K(x,y)|^{2}\mathrm{d}y\leqslant\rho^{-1}\int_{W\setminus
T_{x}}\phi(\|x-y\|)^{2}\mathrm{d}y\leqslant\rho^{-1}|W|\sup_{y\in
T}\phi(\|y\|)^{2},\quad x\in W.$

Next we consider the second term on the right-hand side in (18). By definition
of the reduced Palm process $\xi^{x!,\Xi}$, we have for almost all $x\in W$,

$\displaystyle\mathbb{E}[g(x,\xi^{x})]\big{\\{}\mathbb{E}[\xi^{x!}(W\setminus
T_{x})]-\mathbb{E}[\xi^{x!,\Xi}(W\setminus T_{x})]\big{\\}}$
$\displaystyle=\mathbb{E}[\xi^{x}(W\setminus
T_{x})]\mathbb{E}[\tilde{g}(x,\xi^{x})]-\mathbb{E}[\xi^{x}(W\setminus
T_{x})\tilde{g}(x,\xi^{x})]$ (20)
$\displaystyle=-\mathsf{Cov}(\xi^{x}(W\setminus T_{x}),\tilde{g}(x,\xi^{x})).$

Now we use that the reduced Palm process $\xi^{x!}$ is a determinantal process
itself and therefore negatively associated (see (12)). For $k\in\mathbb{N}$ we
consider the auxiliary functions

$\displaystyle
f^{(k)}(\omega):=\min\\{k,\omega(S_{x})-\tilde{g}(x,\omega)\\},\quad
f(\omega):=\omega(S_{x})-\tilde{g}(x,\omega),\quad\omega\in\mathbf{N}.$

It is easy to see that $f^{(k)},\,k\in\mathbb{N},$ and $f$ are bounded and
increasing. Since $\xi^{x!}$ is negatively associated, we have that

$\displaystyle\mathsf{Cov}(\min\\{k,\xi^{x!}(W\setminus
T_{x})\\},f^{(k)}(\xi^{x!}))\leqslant 0.$

Hence, by monotone convergence,

$\displaystyle\mathsf{Cov}(\xi^{x!}(W\setminus
T_{x}),\xi^{x!}(S_{x}))-\mathsf{Cov}(\xi^{x!}(W\setminus
T_{x}),\tilde{g}(x,\xi^{x!}))$ $\displaystyle=\mathsf{Cov}(\xi^{x!}(W\setminus
T_{x}),f(\xi^{x!}))=\lim_{k\to\infty}\mathsf{Cov}(\min\\{k,\xi^{x!}(W\setminus
T_{x})\\},f^{(k)}(\xi^{x!}))\leqslant 0.$

This shows that (20) is bounded by

$\displaystyle-\mathsf{Cov}(\xi^{x!}(W\setminus T_{x}),\xi^{x!}(S_{x}))$
$\displaystyle\quad=\big{(}\mathbb{E}[\xi^{x!}(W\setminus
T_{x})]-\mathbb{E}[\xi(W\setminus
T_{x})]\big{)}\mathbb{E}[\xi^{x!}(S_{x})]-\big{(}\mathbb{E}[\xi^{x!}(W\setminus
T_{x})\xi^{x!}(S_{x})]-\mathbb{E}[\xi(W\setminus
T_{x})]\mathbb{E}[\xi^{x!}(S_{x})]\big{)}$ (21)
$\displaystyle\quad\leqslant\big{|}\mathbb{E}[\xi^{x!}(W\setminus
T_{x})]-\mathbb{E}[\xi(W\setminus
T_{x})]\big{|}\mathbb{E}[\xi^{x!}(S_{x})]+\big{|}\mathbb{E}[\xi^{x!}(W\setminus
T_{x})\xi^{x!}(S_{x})]-\mathbb{E}[\xi(W\setminus
T_{x})]\mathbb{E}[\xi^{x!}(S_{x})]\big{|}.$

Here, we use (19) and
$\mathbb{E}[\xi^{x!}(S_{x})]\leqslant\mathbb{E}[\xi(S_{x})]=\rho|S|$ to bound
the first term on the right-hand by

$\big{|}\mathbb{E}[\xi^{x!}(W\setminus T_{x})]-\mathbb{E}[\xi(W\setminus
T_{x})]\big{|}\mathbb{E}[\xi^{x!}(S_{x})]\leqslant|W||S|\sup_{y\in
T}\phi(\|y\|)^{2}.$

Next we consider the second term on the right-hand side of (21). We write
$\rho_{m}^{x}$ for the $m$-th correlation function of $\xi^{x!}$, we obtain by
[28, Lemma 6.4], by the definition of $\xi^{x!}$ and by (15) that

$\displaystyle\rho\int_{W}\Big{(}\mathbb{E}[\xi^{x!}(W\setminus
T_{x})\xi^{x!}(S_{x})]-\mathbb{E}[\xi(W\setminus
T_{x})]\mathbb{E}[\xi^{x!}(S_{x})]\Big{)}\mathrm{d}x$
$\displaystyle=\rho\int_{W}\int_{S_{x}}\int_{W\setminus
T_{x}}(\rho_{2}^{x}(y,z)-\rho^{x}(y)\rho)\,\mathrm{d}z\mathrm{d}y\mathrm{d}x$
$\displaystyle=\int_{W}\int_{S_{x}}\int_{W\setminus
T_{x}}(\rho_{3}(x,y,z)-\rho_{2}(x,y)\rho)\,\mathrm{d}z\mathrm{d}y\mathrm{d}x$
$\displaystyle\leqslant 3^{5/2}\|K\|^{2}\int_{W}\int_{S_{x}}\int_{W\setminus
T_{x}}\phi(d(\\{x,y\\},\\{z\\}))\,\mathrm{d}z\mathrm{d}y\mathrm{d}x$
$\displaystyle\leqslant 3^{5/2}\|K\||W|^{2}|S|\phi(d(T,S)).$

Thus, we can conclude that

(22) $\displaystyle
T_{3}=\rho\int_{W}\mathbb{E}[(\xi^{x!,\Xi}\Delta\tilde{\xi}^{x})(W\setminus
T_{x})]\mathrm{d}x$ $\displaystyle\leqslant(1+\rho|S|)|W|^{2}\sup_{y\in
T}\phi(\|y\|)^{2}+3^{5/2}\|K\||W|^{2}|S|\phi(d(T,S)).$

(ii) Decreasing scores. If $g(x,\omega_{1})\geqslant g(x,\omega_{2})$ for
$\omega_{1}\subset\omega_{2}$, we have by (10) and (14) that
$\xi^{x!}\leqslant\xi$ and $\xi^{x!}\leqslant\xi^{x!,\Xi}$. Let $\xi^{x}$ be a
Palm version of $\xi$ at $x$. By Strassen’s theorem and [15, Theorem 2.15],
there exists a point process $\tilde{\xi}^{x}\stackrel{{\scriptstyle
d}}{{=}}\xi$ and a Palm process $\xi^{x,\Xi}$ of $\xi$ with respect to $\Xi$
such that $\xi^{x!}\subset\tilde{\xi}^{x}$ and $\xi^{x!}\subset\xi^{x!,\Xi}$.
This gives

$\displaystyle\mathbb{E}[(\xi^{x,\Xi}\Delta\tilde{\xi}^{x})(W\setminus
T_{x})]$
$\displaystyle\leqslant\mathbb{E}[(\tilde{\xi}^{x}\setminus\xi^{x!})(W\setminus
T_{x})]+\mathbb{E}[(\xi^{x,\Xi}\setminus\xi^{x!})(W\setminus T_{x})]$ (23)
$\displaystyle=\big{\\{}\mathbb{E}[\xi(W\setminus
T_{x})]-\mathbb{E}[\xi^{x!}(W\setminus
T_{x})]\big{\\}}+\big{\\{}\mathbb{E}[\xi^{x!,\Xi}(W\setminus
T_{x})]-\mathbb{E}[\xi^{x!}(W\setminus T_{x})]\big{\\}}.$

Here we bound the term in $\\{\cdots\\}$ as in (19). For the second term in
(23) we obtain

$\displaystyle\mathbb{E}[g(x,\xi^{x})]\big{\\{}\mathbb{E}[\xi^{x!,\Xi}(W\setminus
T_{x})]-\mathbb{E}[\xi^{x!}(W\setminus T_{x})]\big{\\}}$
$\displaystyle=\mathbb{E}[\xi^{x}(W\setminus
T_{x})\tilde{g}(x,\xi^{x})]-\mathbb{E}[\xi^{x}(W\setminus
T_{x})]\mathbb{E}[\tilde{g}(x,\xi^{x})]$ (24)
$\displaystyle=\mathsf{Cov}(\xi^{x}(W\setminus T_{x}),\tilde{g}(x,\xi^{x})).$

Now we use that the reduced Palm process $\xi^{x!}$ is a determinantal process
itself and therefore negatively associated (see (12)). For $k\in\mathbb{N}$ we
consider the auxiliary functions

$\displaystyle
f^{(k)}(\omega):=\min\\{k,\omega(S_{x})+\tilde{g}(x,\omega)\\},\quad
f(\omega):=\omega(S_{x})+\tilde{g}(x,\omega),\quad\omega\in\mathbf{N}.$

It is easy to see that $f^{(k)},\,k\in\mathbb{N},$ and $f$ are bounded and
increasing. Since $\xi^{x!}$ is negatively associated, we have that

$\displaystyle\mathsf{Cov}(\min\\{k,\xi^{x!}(W\setminus
T_{x})\\},f^{(k)}(\xi^{x!}))\leqslant 0.$

Hence, by monotone convergence,

$\displaystyle\mathsf{Cov}(\xi^{x!}(W\setminus
T_{x}),\xi^{x!}(S_{x}))+\mathsf{Cov}(\xi^{x!}(W\setminus
T_{x}),\tilde{g}(x,\xi^{x!}))$ $\displaystyle=\mathsf{Cov}(\xi^{x!}(W\setminus
T_{x}),f(\xi^{x!}))$
$\displaystyle=\lim_{k\to\infty}\mathsf{Cov}(\min\\{k,\xi^{x!}(W\setminus
T_{x})\\},f^{(k)}(\xi^{x!}))\leqslant 0.$

This shows that (24) is bounded by $-\mathsf{Cov}(\xi^{x!}(W\setminus
T_{x}),\xi^{x!}(S_{x}))$. Therefore, we can process as for increasing scores
and obtain the same bound for $T_{3}$ as in (22).

Bounding $T_{4}$. For each $x\in W$ we have

$\displaystyle\mathbb{E}\Big{[}\sum_{y\in\xi^{x!,\Xi}\cap\tilde{\xi}^{x}\cap
W\setminus
T_{x}}|\tilde{g}(y,\xi^{x!,\Xi})-\tilde{g}(y,\tilde{\xi}^{x})|\Big{]}$
$\displaystyle\quad=\mathbb{E}\Big{[}\sum_{y\in\xi^{x!,\Xi}\cap\tilde{\xi}^{x}\cap
W\setminus T_{x}}\mathds{1}\\{(\xi^{x!,\Xi}\Delta\tilde{\xi}^{x})\cap
S_{x}\neq\emptyset\\}|\tilde{g}(y,\xi^{x!,\Xi})-\tilde{g}(y,\tilde{\xi}^{x})|\Big{]}$
$\displaystyle\quad\leqslant\mathbb{E}\Big{[}\sum_{z\in(\xi^{x!,\Xi}\Delta\tilde{\xi}^{x})\cap(W\oplus
S_{x})\setminus T_{x}}\sum_{y\in\xi^{x!,\Xi}\cap\tilde{\xi}^{x}\cap
S_{z}}|\tilde{g}(y,\xi^{x!,\Xi})-\tilde{g}(y,\tilde{\xi}^{x})|\Big{]}$
$\displaystyle\quad\leqslant\mathbb{E}\Big{[}\sum_{z\in(\xi^{x!,\Xi}\Delta\tilde{\xi}^{x})\cap(W\oplus
S_{x})\setminus
T_{x}}\max_{\omega\in\\{\xi^{x!,\Xi},\tilde{\xi}^{x}\\}}\sum_{y\in\omega\cap
S_{z}}\tilde{g}(y,\omega)\Big{]}.$

Here we obtain from Condition (3) that the above is bounded by

$\alpha|S|\mathbb{E}\Big{[}(\xi^{x!,\Xi}\Delta\tilde{\xi}^{x})((W\oplus
S_{x})\setminus T_{x})\Big{]}.$

Hence, we obtain from the estimate in (22) (with $W$ replaced by $W\oplus
S_{x}$) that

$\displaystyle T_{4}$
$\displaystyle=\rho\int_{W}\mathbb{E}\Big{[}\sum_{x\in\xi^{x!,\Xi}\cap\tilde{\xi}^{x}\cap
W\setminus
T_{x}}|\tilde{g}(x,\xi^{x!,\Xi})-\tilde{g}(x,\tilde{\xi}^{x})|\Big{]}\mathrm{d}x$
(25) $\displaystyle\leqslant(1+\rho\sup_{x\in W}|S|)|W\oplus S|^{2}\sup_{y\in
T}\phi(\|y\|)^{2}+3^{5/2}\|K\||W\oplus S|^{2}|S|\phi(d(T,S)).$

Step 2: Finally, we shall complete the proof in a second step. Let
$\mathbf{L}$ and $\mathbf{L}_{\mathsf{tr}}$ denote the intensity measure of
the truncated process $\Xi_{\mathsf{tr}}$ from (17) and of $\Xi$,
respectively. By definition of the Palm process $\xi^{x}$, we have

$\displaystyle\mathbf{L}_{\mathsf{tr}}(A)=\rho\int_{A\cap
W}\mathbb{E}[g(x,\xi^{x})\mathds{1}\\{\mathcal{S}(x,\xi^{x})\subset
S_{x}\\}]\,\mathrm{d}x,\quad A\in\mathcal{B}^{d}.$

Let $\zeta_{\mathsf{tr}}$ be a Poisson process with intensity measure
$\mathbf{L}_{\mathsf{tr}}$. From the triangle inequality for the KR distance
we find that

$\displaystyle\mathbf{d_{KR}}(\Xi,\zeta)\leqslant\mathbf{d_{KR}}(\Xi,\Xi_{\mathsf{tr}})+\mathbf{d_{KR}}(\Xi_{\mathsf{tr}},\zeta_{\mathsf{tr}})+\mathbf{d_{KR}}(\zeta_{\mathsf{tr}},\zeta).$

From Section 3 we know that
$\mathbf{d_{KR}}(\zeta_{\mathsf{tr}},\zeta)\leqslant
d_{TV}(\mathbf{L}_{\mathsf{tr}},\mathbf{L})$ and
$\mathbf{d_{KR}}(\Xi,\Xi_{\mathsf{tr}})\leqslant\mathbf{d_{TV}}(\Xi,\Xi_{\mathsf{tr}})$,
where

$\displaystyle\mathbf{d_{TV}}(\Xi,\Xi_{\mathsf{tr}})=\mathbb{E}\Xi(W)-\mathbb{E}\Xi_{\mathsf{tr}}(W)=d_{TV}(\mathbf{L},\mathbf{L}_{\mathsf{tr}})$

and

$\displaystyle
d_{TV}(\mathbf{L},\mathbf{L}_{\mathsf{tr}})=\rho\int_{W}\mathbb{E}[g(x,\xi^{x})\mathds{1}\\{\mathcal{S}(x,\xi^{x})\not\subset
S_{x}\\}]\,\mathrm{d}x.$

This shows that

(26) $\displaystyle\mathbf{d_{KR}}(\Xi,\zeta)$
$\displaystyle\leqslant\mathbf{d_{KR}}(\Xi_{\mathsf{tr}},\zeta_{\mathsf{tr}})+2\int_{W}\mathbb{E}[g(x,\xi^{x})\mathds{1}\\{\mathcal{S}(x,\xi^{x})\not\subset
S_{x}\\}]\,\mathrm{d}x.$

Combining $\eqref{step2}$ with (22) and (25) from Step 1 gives the assertion.
∎

##  5\. Proof of Theorem 2

In the proof of Theorem 2, we repeatedly use that the set of absolute values
of the points of the (infinite) Ginibre process $\xi$ has the same
distribution as a sequence $(X_{i})_{i\in\mathbb{N}}$ of independent random
variables with $X_{i}^{2}\sim\text{Gamma}(i,1)$ (see [17] or [1, Theorem 26]).
This implies that the mapping $r\mapsto\mathbb{P}(\xi(B_{r})=0)$ is continuous
and that $\mathbb{P}(\xi(B_{r})=0)\downarrow 0$ as $r\to\infty$. Hence, for
all $\tau>0$ there exists an increasing sequence $(v_{n})_{n\in\mathbb{N}}$
such that

(27) $\mathbf{L}_{n}(A):=\mathbb{E}[\Xi_{n}(A)]=\frac{|A\cap
B_{n}|}{\pi}\mathbb{P}(\xi^{o!}(B_{v_{n}})=0)=\frac{\tau|A\cap B_{n}|}{\pi
n^{2}},\quad n\in\mathbb{N},\,A\in\mathcal{B}^{2}.$

To determine the asymptotic behavior of $v_{n}$ as $n\to\infty$, we use that
by [1, Theorem 26],

(28)
$\mathbb{P}(\xi^{o!}(B_{v_{n}})=0)=e^{v_{n}^{2}}\mathbb{P}(\xi(B_{v_{n}})=0).$

Moreover, by [2, Proposition 7.2.1],

$\lim_{r\to\infty}\frac{1}{r^{4}}\log\mathbb{P}(\xi(B_{r})=0)=-\frac{1}{4}.$

Therefore, re-writing $\log\mathbb{P}(\xi(B_{r})=0)$ as

$\log(ne^{v_{n}^{2}}\mathbb{P}(\xi(B_{r}=0))-\log(n)v_{n}^{2},$

we find from (27) and (28) that $\frac{v_{n}^{4}}{\log n}\to 4$ as
$n\to\infty$.

###### Proof of Theorem 2.

Given $n\in\mathbb{N}$, we choose $\xi$ as the Ginibre process, let
$g(x,\omega):=\mathds{1}\\{\omega(B_{v_{n}}(x)\setminus\\{x\\})=0\\}$,
$S_{x}:=B_{v_{n}}(x)$ and $T_{x}:=B_{\log n}(x)$, $x\in\mathbb{R}^{2}$. Note
that $g$ is stabilizing with respect to the (deterministic) stopping set
$\mathcal{S}(x,\omega):=S_{x},\,\omega\in\mathbf{N}$. The idea is now to apply
Theorem 1 to the process $\Xi_{n}$ defined at (6), where we choose $\zeta$ is
a stationary Poisson process with intensity $\frac{\tau}{\pi n^{2}}$. Then,
$\zeta\cap B_{n}$ has intensity measure $\mathbf{L}_{n}$ given at (27). Since
$g$ is deterministically stabilizing, we have that $E_{1}=0$. Note moreover
that $E_{2}$ and $F$ are bounded by constant multiples of $\frac{(\log
n)^{2}}{n}$. Thus, by the contractivity property of the KR distance and by
Theorem 1, we have

$\displaystyle\mathbf{d_{KR}}(\Psi_{n},\nu\cap B_{1})$
$\displaystyle\leqslant\mathbf{d_{KR}}(\Xi_{n},\zeta\cap B_{n})$ (29)
$\displaystyle\leqslant\int_{B_{n}}\int_{T_{x}}\mathbb{P}(\xi^{x!,y!}(B_{v_{n}}(x)\cup
B_{v_{n}}(y))=0)\rho_{2}(x,y)\mathrm{d}y\mathrm{d}x+\beta\frac{(\log
n)^{2}}{n}$

for some $\beta>0$. Thus, it remains to bound the integral on the right-hand
side above. We first note that by [28, Theorem 6.5], the reduced Palm process
$\xi^{x!,y!}$ is a determinantal process itself. Hence, we can conclude from
[20, Theorem 3.7] (see also [19, Theorem 3.2]) that $\xi^{x!,y!}$ has negative
associations. Recall that a point process $\zeta$ has negative associations if
$\mathbb{E}[f(\zeta)g(\zeta)]\leqslant\mathbb{E}[f(\zeta)]\mathbb{E}[g(\zeta)]$
for every pair $f,g$ of real bounded increasing (or decreasing) functions that
are measurable with respect to complementary subset $A,A^{c}$ of
$\mathbb{R}^{d}$, meaning that a function is measurable with respect to $A$ if
it is measurable with respect to $\mathcal{N}_{A}$. We apply this with the
decreasing functions $f(\mu)=\mathds{1}\\{\mu(B_{v_{n}}(x))=0\\}$ and
$g(\mu)=\mathds{1}\\{\mu(B_{v_{n}}(y)\setminus B_{v_{n}}(x))=0\\}$. This gives

(30) $\mathbb{P}(\xi^{x!,y!}(B_{v_{n}}(x)\cup
B_{v_{n}}(y))=0)\leqslant\mathbb{P}(\xi^{x!,y!}(B_{v_{n}}(x))=0)\,\mathbb{P}(\xi^{x!,y!}(B_{v_{n}}(y)\setminus
B_{v_{n}}(x))=0).$

To bound the first probability, we note that by [14, Theorem 1] there is a
coupling $(\xi^{\prime},\xi^{\prime\prime})$ of $(\xi^{x!},\xi^{x!,y!})$ such
that $\xi^{\prime\prime}\subset\xi^{\prime}$ and
$|\xi^{\prime}\setminus\xi^{\prime\prime}|\leqslant 1$ a.s. This gives

$\mathbb{P}(\xi^{x!,y!}(B_{v_{n}}(x))=0)\leqslant\mathbb{P}(\xi^{x!}(B_{v_{n}}(x))\leqslant
1).$

Now we apply the same argument with a coupling of $(\xi,\xi^{x!})$ and obtain
the bound

$\mathbb{P}(\xi(B_{v_{n}}(x))\leqslant 2)=\mathbb{P}(\xi(B_{v_{n}})\leqslant
2),$

where the last equality holds due to the stationarity of $\xi$. As mentioned
at beginning of this section, the set of absolute values of the points of the
(infinite) Ginibre process $\xi$ has the same distribution as a sequence
$(X_{i})_{i\in N}$ of independent random variables with
$X_{i}^{2}\sim\text{Gamma}(i,1)$. This gives

$\displaystyle\mathbb{P}(\xi(B_{v_{n}})\leqslant 2)$
$\displaystyle=\mathbb{P}(\\#\\{j\in\mathbb{N}:\,X_{j}\leqslant
v_{n}\\}\leqslant 2)$
$\displaystyle\leqslant\mathbb{P}(\\#\\{j\in\\{1,\dots,v_{n}^{2}\\}:\,X_{j}\leqslant
v_{n}\\}\leqslant 2)$
$\displaystyle=\mathbb{P}\Big{(}\bigcup_{i=1}^{v_{n}^{2}}\bigcup_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{v_{n}^{2}}\\{\forall
k\in\\{1,\dots,v_{n}^{2}\\}\setminus\\{i,j\\}:\,X_{k}>v_{n}\\}\Big{)}.$

In the above equation, with a slight abuse of notation, we have written
$v_{n}^{2}$ instead of $\lfloor v_{n}^{2}\rfloor$. The union bound yields that
the above is bounded by

$\displaystyle\sum_{i=1}^{v_{n}^{2}}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{v_{n}^{2}}\mathbb{P}(\forall
k\in\\{1,\dots,v_{n}^{2}\\}\setminus\\{i,j\\}:\,X_{k}>v_{n})=\sum_{i=1}^{v_{n}^{2}}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{v_{n}^{2}}\prod_{\begin{subarray}{c}k=1\\\ k\neq
i,j\end{subarray}}^{v_{n}^{2}}\mathbb{P}(X_{k}^{2}>v_{n}^{2}).$

Let $t<1$. The moment generating function
$M_{X_{k}^{2}}(t)=\mathbb{E}[e^{tX_{k}^{2}}]=(1-t)^{-k}$ of $X_{k}^{2}$ exists
and we obtain from the Chernoff bound that

$\mathbb{P}(X_{k}^{2}>r^{2})\leqslant
e^{-tr^{2}}\mathbb{E}[e^{tX_{k}^{2}}]=e^{-tr^{2}}(1-t)^{-k}.$

For $k<r^{2}$, this bound is maximized for $t=1-\frac{k}{r^{2}}$, which gives

$\mathbb{P}(\xi(B_{v_{n}})\leqslant
2)\leqslant\sum_{i=1}^{v_{n}^{2}}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{v_{n}^{2}}\prod_{\begin{subarray}{c}k=1\\\ k\neq
i,j\end{subarray}}^{v_{n}^{2}}e^{-(1-\frac{k}{v_{n}^{2}})v_{n}^{2}-k\log\big{(}\frac{k}{v_{n}^{2}}\big{)}}=\sum_{i=1}^{v_{n}^{2}}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{v_{n}^{2}}\prod_{\begin{subarray}{c}k=1\\\ k\neq
i,j\end{subarray}}^{v_{n}^{2}}e^{-v_{n}^{2}+k-k\log\big{(}\frac{k}{v_{n}^{2}}\big{)}}.$

Using here that $u\mapsto u-u\log(u/r^{2})$ is increasing for $u\leqslant
r^{2}$, we find that

$\displaystyle\mathbb{P}(X_{k}^{2}>r^{2})$
$\displaystyle\leqslant\sum_{i=1}^{v_{n}^{2}}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{v_{n}^{2}}\prod_{k=3}^{v_{n}^{2}}e^{-v_{n}^{2}+k-k\log\big{(}\frac{k}{v_{n}^{2}}\big{)}}$
$\displaystyle\leqslant
v_{n}^{4}\prod_{k=3}^{v_{n}^{2}}e^{-v_{n}^{2}+k-k\log\big{(}\frac{k}{v_{n}^{2}}\big{)}}$
$\displaystyle=v_{n}^{2}e^{-\frac{1}{2}(v_{n}^{2}-3)(v_{n}^{2}-2)-v_{n}^{4}\int_{3/v_{n}^{2}}^{1}x\log(x)\mathrm{d}x+O(v_{n}^{2}\log
v_{n})}$ $\displaystyle=e^{-\frac{1}{4}v_{n}^{4}(1+o(1))}$

as $n\to\infty$, where we have used that
$\int_{0}^{1}x\log(x)\mathrm{d}x=-\frac{1}{4}$.

Next we bound the second probability in (30). By the same coupling argument as
above we find that

$\mathbb{P}(\xi^{x!,y!}(B_{v_{n}}(y)\setminus
B_{v_{n}}(x))=0)\leqslant\mathbb{P}(\xi(B_{v_{n}}(y)\setminus
B_{v_{n}}(x))\leqslant 2).$

Next we note that $B_{v_{n}/2}\left(y+\frac{v_{n}(y-x)}{2|y-x|}\right)\subset
B_{v_{n}}(y)\setminus B_{v_{n}}(x)$. Hence, the last probability is bounded by

$\mathbb{P}\Big{(}\xi\big{(}B_{v_{n}/2}\big{(}y+\frac{v_{n}(y-x)}{2|y-x|}\big{)}\big{)}\leqslant
2\Big{)}=\mathbb{P}(\xi(B_{v_{n}/2})\leqslant 2)\leqslant
e^{-\frac{1}{4}(v_{n}/2)^{4}(1+o(1))}$

by the same estimates as above (with $v_{n}/2$ instead of $v_{n}$). Since
$\rho_{2}(x,y)\leqslant 1/\pi^{2}$ for all $x,y\in\mathbb{R}^{2}$, we arrive
for all $\varepsilon>0$ at the bound

$\displaystyle\int_{B_{n}}\int_{T_{x}}\mathbb{P}(\xi^{x!,y!}(B_{v_{n}}(x)\cup
B_{v_{n}}(y))=0)\rho_{2}(x,y)\mathrm{d}(x,y)\leqslant\frac{n(\log
n)^{2}}{\pi}e^{-\frac{1}{4}v_{n}^{4}}e^{-\frac{1}{64}v_{n}^{4}}\leqslant\frac{(\log
n)^{2}}{\pi}n^{\varepsilon-1/16},$

where we have used (27) and that $\frac{v_{n}^{4}}{\log n}\to 4$ as
$n\to\infty$. Hence, the assertion follows from (29). ∎

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