1. Introduction
Determinantal point processes (DPPs) were introduced in quantum mechanics to study configurations of fermions [Reference Macchi20]. Due to their repulsive nature, they play a fundamental role in applied sciences, for example, as a model for base stations in a wireless network [Reference Miyoshi and Shirai21]. In mathematics, DPPs arise naturally in different fields, such as eigenvalues of random matrices [Reference Diaconis11] and random spanning trees [Reference Burton and Pemantle5]. 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. [Reference Decreusefond, Flint, Privault, Torrisi, Peccati and Reitzner9]). 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 a Gaussian kernel given in Section 2.
We study the following model. Let
$\eta$
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 some measurable
$W \subset \mathbb{R}^d$
, let
\begin{equation*} \Xi [\eta ]\,:\!=\,\sum _{x \in \eta \cap W} g(x,\eta ) \delta _x, \end{equation*}
where
$\delta _x$
denotes the Dirac measure in
$x$
. Here, the function
$g$
has the effect of a thinning of
$\eta$
, in the sense that
$\Xi$
is the point process of all points
$x \in \eta$
in the set
$W$
, which satisfy
$g(x,\eta )=1$
. The random measure
$\Xi$
is a flexible model, which appears in the study of random spatial graphs, stochastic topology, and geometric extreme value theory.
In this article, we study the distance (in an appropriate metric of point processes) of
$\Xi$
and a Poisson point process. To the best of our knowledge, this is the first paper that systematically studies Poisson approximation for determinantal processes. This continues the studies for stabilising functionals of Poisson point processes [Reference Bobrowski, Schulte and Yogeshwaran4,Reference Decreusefond, Schulte and Thäle10,Reference Otto24], Poisson hyperplane processes [Reference Otto23], and Gibbs point processes [Reference Last and Otto16]. However, the repulsive behaviour of a DPP requires additional tools that are not needed if Poisson input is considered in
$\Xi$
. As main contributions, this article shows:
-
(i) If the correlation kernel
$K$
is fast decaying and if the thinning function
$g$
is stabilising and satisfies some natural assumptions, the bound of the distance of
$\Xi$
and a Poisson process is comparable to the bounds obtained for thinned Poisson point processes (see [Reference Bobrowski, Schulte and Yogeshwaran4] and [Reference Otto23]). -
(ii) If
$\eta$
is the Ginibre process and if
$\Xi$
is the point process of elements in
$\eta \cap W$
with a large distance to its nearest neighbour, we prove, in an asymptotic scenario where the volume of
$W$
tends to infinity, that an appropriate scaling of
$\Xi$
converges to a Poisson process.
Our paper is organised as follows. In Section 2, we introduce DPPs and state Theorems2.1 and 2.2 as our two main results. In Section 3, we introduce important notions such as Palm theory, negative association, and correlation decay. The proof of Theorem2.1 is given in Section 4. In Section 5, we provide the proof of Theorem2.2.
2. Model and main results
We work on the Euclidean space
$\mathbb{R}^d$
(
$d \geqslant 1$
) equipped with its Borel
$\sigma$
-field
$\mathcal B^d$
, Lebesgue measure
$\lambda$
, 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
$\eta$
of
$\mathbf N$
, defined over some fixed probability space
$(\Omega, \mathcal A,\mathbb{P})$
. The intensity measure of
$\eta$
is the measure
$\mathbb{E}[\eta ]$
defined by
$\mathbb{E}[\eta ](B)\,:\!=\,\mathbb{E}[\eta (B)]$
,
$B\in \mathcal B^d$
. For
$z \in \mathbb{R}^d$
and
$r\gt 0$
, let
$B_r(z)$
be the closed Euclidean ball with radius
$r$
around
$z$
. We denote
$|B|\,:\!=\,\lambda (B)$
and write
$A \oplus B$
for the Minkowski sum of
$A, B \subset \mathbb{R}^d$
.
Let
$K\,:\,(\mathbb{R}^d)^2 \to \mathbb{C}$
be a complex function. We say that
$\eta$
is a determinantal point process with correlation kernel
$K$
, if for every
$n \in \mathbb{N}$
and pairwise disjoint
$A_1,\ldots, A_n \in \mathcal B^d$
, we have that
\begin{align*} \mathbb{E} [\eta (A_1)\cdots \eta (A_n)]=\int _{A_1 \times \cdots \times A_n} \det\!(K(x_i,x_j))_{i,j=1}^n d(x_1,\ldots x_n), \end{align*}
where
$d\ldots$
denotes integration with respect to
$\lambda$
,
$(K(x_i,x_j))_{i,j=1}^m$
is the
$m\times m$
-matrix with entry
$K(x_i,x_j)$
at position
$(i,j)$
, and
$\det M$
is the determinant of the complex-valued
$m\times m$
-matrix
$M$
. This says that
$\eta$
has correlation functions of all orders, that the
$m$
th order correlation function
$\rho ^{(m)}$
is given by
\begin{align*} \rho ^{(m)}(x_1,\ldots, x_m)= \det \!(K(x_i,x_j))_{i,j=1}^m,\quad x_1,\ldots, x_m \in \mathbb{R}^d,\,\quad m \in \mathbb{N}, \end{align*}
and that it is locally integrable. In this article, we assume that
$K$
satisfies the following assumptions (i)–(iv):
-
(i)
$K$
is Hermitian, that is,
$K(x,y)=\overline {K(y,x)}$
,
$x,y \in \mathbb{R}^d$
. -
(ii)
$K$
is locally square integrable, that is, for every compact
$B \in \mathcal B^d$
, the integralis finite.
\begin{equation*}\int _B \int _B |K(x,y)|^2 dy dx\end{equation*}
-
(iii)
$K$
is locally of trace class, that is, for every compact
$B \in \mathcal B^d$
, the integral
$\int _B K(x,x) dx$
is finite.
Under the assumptions (i)–(iii), it follows from Mercer’s theorem that for every compact
$B \subset \mathbb{R}^d$
, the restriction
$\eta _B$
of
$\eta$
to
$B$
is a determinantal point process whose kernel
$K_B$
is for almost all
$(x,y)\in B \times B$
given by
\begin{align*} K_B(x,y)=\sum _{k=1}^\infty \lambda _k^B \phi _k^B(x) \overline {\phi _k^B(y)}, \end{align*}
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
-
(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 exists a unique (in distribution) determinantal point process with correlation kernel
$K$
(see [[Reference Soshnikov28], Theorem 3]).
For
$x \in \mathbb{R}^d$
, we call
$\eta ^x$
a Palm version of the point process
$\eta$
at
$x$
, if for all measurable
$\,f:\,\mathbb{R}^d \times \mathbf N \to \mathbb{R}_+$
,
\begin{align} \mathbb{E}\Big [\int f(x,\eta ) \eta (dx)\Big ]=\int \mathbb{E} [f(x,\eta ^x)] \mathbb{E}[\eta ](dx). \end{align}
Later, we will generalise this definition and define Palm processes of
$\eta$
with respect to
$\Xi$
.
Let
$\eta$
be a stationary determinantal process satisfying (i)–(iv) with intensity
$\rho \gt 0$
. Let
$g\,:\,\mathbb{R}^d \times \mathbf N \to \{0,1\}$
be a measurable function (called score function), and let
$W \in \mathcal B^d$
. Recall from the Introduction that
\begin{align} \Xi [\omega ]=\sum _{x \in \omega \cap W} g(x,\omega ) \delta _{x}, \end{align}
and set
$\Xi \,:\!=\,\Xi [\eta ]$
. Note that by (1), the intensity measure
$\mathbf L$
of
$\Xi$
is given by
\begin{equation*} \mathbf L(A)=\rho \int _{W \cap A} \mathbb{E}[g(x,\eta ^x)]\, dx,\quad A \in \mathcal B^d. \end{equation*}
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 [Reference Decreusefond, Schulte and Thäle10]. For finite point processes
$\zeta$
and
$\eta$
on
$\mathbb{R}^d$
, let
\begin{align*} \mathbf {d}_{\mathbf{KR}}(\zeta, \eta ) \,:\!=\, \sup _{h \in \text {Lip}}|\mathbb{E} h(\zeta )-\mathbb{E} h(\eta )|, \end{align*}
where
$\text {Lip}$
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
\begin{align*} d_{\text {TV}}(\omega _1,\omega _2)\,:\!=\,\sup |\omega _1(A)-\omega _2(A)|, \end{align*}
where the supremum is taken over all
$A \in \mathcal B^d$
with
$\omega _1(A), \omega _2(A)\lt \infty$
. Under appropriate conditions on
$\eta$
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
$\omega \in \mathbf N$
,
\begin{align} \sum _{x \in \omega \cap A} g(x,\omega ) \lt \alpha |A|, \end{align}
and assume that
$g$
is monotonic in the sense that for all
$x \in W$
, we have
\begin{align} 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. \end{align}
We further assume that
$g$
is stabilising with respect to a Borel set
$S\subset \mathbb{R}^d$
, by which we mean that
\begin{equation*} g(x, \omega ) = g(x, \omega \cap S_x) \end{equation*}
holds for any
$\omega \in \mathbf {N}$
and any
$x \in \mathbb{R}^d$
, where
$ S_x\,:\!=\,x+S$
.
Moreover, we assume that the kernel
$K\,:\,(\mathbb{R}^d)^2 \to \mathbb{C}$
satisfies
\begin{align} |K(x,y)|\leqslant \phi (\|x-y\|),\quad x,y \in \mathbb{R}^d, \end{align}
for some decreasing function
$\phi \,:\,\mathbb{R}_+ \to \mathbb{R}_+$
with
$\lim _{r \to \infty } \phi (r) =0$
.
Theorem 2.1.
Let
$\Xi$
be the point process defined at (
2
) with compact
$W \subset \mathbb{R}^d$
. Let
$S, T \subset \mathbb{R}^d$
be closed with
$o\in S$
and
$(S \oplus T^c) \cap S=\varnothing$
. Suppose that
$g$
is stabilising with respect to
$S$
and satisfies (
3
) and (
4
) with
$\alpha \gt 0$
and that (
5
) holds. Let
$\zeta$
be a finite Poisson process on
$\mathbb{R}^d$
with intensity measure
$\mathbf M$
. Then,
\begin{align*} &\mathbf {d}_{\mathbf{KR}}(\Xi, \zeta ) \leqslant d_{TV}(\mathbf L, \mathbf M) + 2(E_1+E_2 +F) \end{align*}
with
\begin{align*} E_1&\,:\!=\,\rho ^2 \int _W \int _{W \cap T_x} \mathbb{E}[ g(x,\eta ^x)] \mathbb{E}[ g(y,\eta ^y)]\,dy dx,\\ E_2&\,:\!=\,\int _W \int _{W \cap T_x} \mathbb{E}[ g(x,\eta ^{x,y}) g(y,\eta ^{x,y})]\rho ^{(2)}(x,y) dy dx,\\ F&\,:\!=\,\|K\| (1+\rho +3^{5/2} \|K\|) (\alpha |S|+1) \max \!(|S|,1)|W\oplus S|^2 \phi (d(S,T^c)), \end{align*}
where
$T_x\,:\!=\,x+T$
,
$\|K\|=\sup _{x,y \in \mathbb{R}^d} |K(x,y)|$
, and
$d(S,T^c)$
is the Hausdorff distance of
$S$
and
$T^c$
.
In the second part of this paper, we give an application of Theorem2.1 for a concrete choice of
$\eta$
and of
$g$
. Let
$\eta$
be the (infinite) Ginibre process
$\xi$
, which is a stationary determinantal point process on
$\mathbb C$
with correlation kernel given by
\begin{equation*}K(z,w)= \pi ^{-1}e^{-(|z|^2+|w|^2)/2} e^{z\overline {w}},\quad z,w \in \mathbb C.\end{equation*}
Hence,
$\xi$
has intensity
$\rho =\pi ^{-1}$
, and it holds that
$|K(z,w)|\leqslant \phi (\|z-w\|)$
with
$\phi (r)\,:\!=\,\pi ^{-1} \exp \!(-r^2/2)$
for
$r\gt 0$
and
$z,w \in \mathbb C$
.
In Theorem2.2 below, we choose
$g$
depending on
$n \in \mathbb{N}$
. Let
$g_n$
be 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 neighbour 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 [Reference Otto and Thäle25]).
We consider
$\xi$
as a random set in
$\mathbb{R}^2$
. Let
$B_n\,:\!=\,B_n(o)$
the closed ball with radius
$n\gt 0$
in
$\mathbb{R}^2$
centred at the origin
$o$
. We consider the process
\begin{equation} \Xi _n\,:\!=\,\sum _{x \in \xi \cap B_n} \unicode {x1D7D9}\{\xi (B_{v_n}(x)\setminus \{x\})=0\} \ \delta _{x}. \end{equation}
as well as the scaled process
\begin{equation} \Psi _n\,:\!=\,\sum _{y \in \Xi _n}\delta _{y/n}=\sum _{x \in \xi \cap B_n} \unicode {x1D7D9}\{\xi (B_{v_n}(x)\setminus \{x\})=0\} \ \delta _{x/n}. \end{equation}
In the following theorem, we compare
$\Psi _n$
with a Poisson process on the unit ball
$B_1$
in
$\mathbb{R}^2$
.
Theorem 2.2.
Let
$\nu$
be a stationary Poisson process on
$\mathbb{R}^2$
with intensity
$\tau \gt 0$
. There exists a sequence
$(v_n)_{n \in \mathbb{N}}$
with
$v_n^4 \sim 8\log n$
as
$n \to \infty$
and a constant
$C\gt 0$
such that for all
$n \in \mathbb{N}$
and any
$\varepsilon \gt 0$
,
\begin{equation*} \mathbf {d}_{\mathbf{KR}}(\Psi _n,\nu \cap B_1) \leqslant Cn ^{\varepsilon -1/16}. \end{equation*}
As an application of the above theorem, we consider largest distances to the nearest neighbour.
Corollary 2.3.
We have as
$n \to \infty$
,
\begin{equation*} \frac {1}{8 \log n} \max _{x \in \xi \cap B_n} \min _{y\in \xi \setminus \{x\}} \|x-y\|^4 \stackrel {\mathbb{P}}{\longrightarrow } 1. \end{equation*}
The proof of Corollary 2.3 is quite standard (see, e.g. [[Reference Otto24], Corollary 4.2] or [[Reference Chenavier and Hirsch7], Corollary 1]) and therefore omitted.
Remark 2.4.
(i) One should compare Theorem
2.1
with [[Reference Bobrowski, Schulte and Yogeshwaran4], Theorem 4.1] (or the refined version [[Reference Otto23], Theorem 4.1]) that discusses Poisson process approximation for score sums built on a Poisson process. The terms
$E_1$
,
$E_2$
, and
$E_3$
in Theorem
2.1
are the analogues to the terms
$E_1$
,
$E_2$
, and
$E_3$
in [[Reference Bobrowski, Schulte and Yogeshwaran4], 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
$\eta$
) in [Reference Bobrowski, Schulte and Yogeshwaran4] and [Reference Otto23]. Note also that for a wide class of DPPs (including the Ginibre process), the
$\beta$
-mixing coefficient does not decay exponentially fast (see [[Reference Poinas26], Proposition 4.2]). Therefore, the exponential decay dependence property from [Reference Chenavier and Otto8] is violated, and general results for Poisson approximation of strongly mixing processes do not apply.
(ii) Note that the scaling of the maximum nearest neighbour ball in Corollary 2.3 is different from its analogue for independent points, where the second power is proportional to
$\log n$
as
$n \to \infty$
(see [Reference Chenavier, Henze and Otto6]). It seems interesting to investigate whether Theorem
2.2
can be extended to
$k$
-nearest neighbour distances (
$k \geqslant 2$
). However, this extension would require delicate 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 associations
Following [[Reference Kallenberg14], Chapter 6], we next introduce Palm measures and thereby generalise the definition given in (1). Let
$\eta$
be a stationary determinantal process as introduced in Section 2, and let
$\Xi$
be its thinned process with intensity measure
$\mathbf L$
. Then there are point processes
$\eta ^{x,\Xi },\,x \in \mathbb{R}^d$
, such that for all measurable mappings
$f$
from
$\mathbb{R}^d \times \mathbf {N}$
to
$[0,\infty )$
,
\begin{align} \mathbb{E} \Big [\int f(x,\eta )\,\Xi (\mathrm {d}x)\Big ]=\int \mathbb{E} [f(x,\eta ^{x,\Xi })]\mathbf L(dx). \end{align}
The processes
$\eta ^{x,\Xi }, x \in \mathbb{R}^d,$
are called Palm processes of
$\eta$
with respect to
$\Xi$
at
$x$
, and the distribution
$P^{x,\Xi }$
is called the Palm measure of
$\eta$
with respect to
$\Xi$
. Since
$\Xi$
is simple,
$\eta ^{x,\Xi }$
can be interpreted as the process
$\eta$
seen from
$x$
and conditioned on
$\Xi$
having a point in
$x$
. Since
$\Xi \subset \eta$
, it follows from [[Reference Kallenberg14], Lemma 6.2 (ii)] that
$\delta _x \in \eta ^{x,\Xi }$
a.s. This allows us to define the reduced Palm process
$\eta ^{x!,\Xi }\,:\!=\,\eta ^{x,\Xi }-\delta _x$
with distribution
$P^{x!,\Xi }$
. If
$\eta =\Xi$
(i.e.
$g \equiv 1$
and
$W =\mathbb{R}^d$
), we write
$\eta ^x$
for a Palm process of
$\eta$
(with respect to itself) at
$x$
(c.f. [1]) and
$\eta ^{x!}$
for a reduced Palm process.
Recall that
$K$
is the correlation kernel of
$\eta$
and write
$P$
for its distribution. For
$x \in \mathbb{R}^d$
, let
$\eta ^{x!}$
be reduced Palm processes of
$\eta$
at
$x$
and denote their distribution by
$P^{x!}$
. Then
$\eta ^{x!},\, x \in \mathbb{R}^d,$
are determinantal processes with correlation kernel
$K^x,\, x \in \mathbb{R}^d,$
given by
\begin{align} K^x(z,w)=K(z,w)-\frac {K(z,x)K(x,w)}{\rho },\quad z,w \in \mathbb{R}^d, \end{align}
(see [[Reference Shirai and Takahashi27], Theorem 1.7]). By [[Reference Goldman12], Theorem 3] (see also [Reference Møller and O’Reilly22]), the process
$\eta ^{x!}$
is stochastically dominated by
$\eta$
(denoted by
$P^{x!}\leqslant P$
) which means that
\begin{align} \mathbb{E} [F(\eta ^{x!})] \leqslant \mathbb{E} [F(\eta )] \end{align}
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,
$\eta ^x$
be a Palm process of
$\eta$
at
$x$
and
$\eta ^{x,\Xi }$
a Palm process of
$\eta$
with respect to
$\Xi$
at
$x$
. Then we have
\begin{align} \mathbb{E} \Big [ \int f(x,\eta ) \,\Xi (dx) \Big ] = \mathbb{E}\Big [ \int _W f(x,\eta ) g(x,\eta ) \eta (\mathrm {d}x)\Big ]&= \int _W \mathbb{E} [f(x,\eta ^x) g(x,\eta ^x)] \rho (x) \lambda (\mathrm {d}x)\nonumber \\ &=\int _W \mathbb{E} [f(x,\eta ^{x,\Xi })] \mathbb{E} [g(x,\eta ^x)] \rho (x) \lambda (\mathrm {d}x). \end{align}
An important property of DPPs is that they have negative associations (see [[Reference Lyons19], Theorem 3.7] or [[Reference Last and Szekli17], Theorem 3.2]), by which we mean that
\begin{align} { \mathbb E[F(\eta )G(\eta )] \leqslant \mathbb E[F(\eta )] \mathbb{E}[G(\eta )],} \end{align}
for any real, bounded, and increasing functions
$F, G\,:\, \mathbf N \to \mathbb R$
that are measurable with respect to complementary subsets (see [Reference Last and Szekli17]).
Let
$F\,:\,\mathbf {N} \to \mathbb{R}$
be measurable, bounded, and increasing and assume that
$g$
is measurable with respect to
$S$
and increasing in the second argument. Then we find for almost all
$x \in W$
from (11) and (12) (applied to the determinantal point process
$\eta ^{x!}$
) that
\begin{align} \mathbb{E} [F(\eta ^{x,\Xi } \cap S_x^c)] \mathbb{E} [g(x,\eta ^x)] = \mathbb{E} [F(\eta ^{x}\cap S_x^c) g(x,\eta ^{x})] \leqslant \mathbb{E} [F(\eta ^{x}\cap S_x^c)] \mathbb{E} [g(x,\eta ^{x})], \end{align}
implying that
$P^{x!,\Xi }|_{S_x^c} \leqslant P^{x!}|_{S_x^c}$
for
$\lambda$
-a.a.
$x \in W$
. On the other hand, if
$g$
is measurable with respect to
$S$
and decreasing in the second argument, then we find by taking
$-g$
in (12) that
\begin{align} \mathbb{E} [F(\eta ^{x,\Xi }\cap S_x^c)] \mathbb{E} [g(x,\eta ^x)] = \mathbb{E} [F(\eta ^{x}\cap S_x^c) g(x,\eta ^{x})] \geqslant \mathbb{E} [F(\eta ^{x}\cap S_x^c)] \mathbb{E} [g(x,\eta ^{x})], \end{align}
implying that
$P^{x!}|_{S_x^c} \leqslant P^{x!,\Xi }|_{S_x^c}$
for
$\lambda$
-a.a.
$x \in W$
.
3.2 Fast decay of correlation
Let
$\eta$
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 [[Reference Blaszczyszyn, Yogeshwaran and Yukich3], Lemma 1.3] that the correlation functions
$\rho ^{(m)}$
,
$m \in \mathbb{N}$
, of
$\eta$
satisfy
\begin{align} |\rho ^{(p+q)}(x_1,\ldots, x_{p+q})- \rho ^{(p)}(x_1,\ldots, x_p)\rho ^{(q)}(x_{p+1},\ldots, x_{p+q})|\leqslant m^{1+\frac m2} \phi (s) \|K\|^{m-1}, \end{align}
where
$m\,:\!=\,p+q$
,
$s\,:\!=\,d(\{x_1,\ldots, x_p\},\{x_{p+1},\ldots, x_{p+q}\})\,:\!=\,\inf _{i \in \{1,\ldots, p\}, j \in \{p+1,\ldots, p+q\}} |x_i-x_j|$
, and
$\|K\|\,:\!=\,\sup _{x,y\in \mathbb{R}^d} |K(x,y)|$
.
3.3 Poisson process approximation
The following Poisson approximation result is inspired by [[Reference Bobrowski, Schulte and Yogeshwaran4], Theorem 3.1].
Proposition 3.1.
Let the assumptions of Theorem
2.1
prevail. For
$x,y \in W$
, let
$\eta ^x$
be a Palm version of
$\eta$
at
$x$
, let
$\eta ^{x,y}$
be a Palm version of
$\eta ^x$
at
$y$
, and let
$\eta ^{x,\Xi } \sim P^{x,\Xi }|_{S_x^c}$
be a Palm version of
$\eta$
with respect to
$\Xi$
at
$x$
, restricted to
$S_x^c$
. Let
$\zeta$
be a finite Poisson process with intensity measure
$\mathbf M$
. Then we have
\begin{align*} \mathbf {d}_{\mathbf{KR}}(\Xi, \zeta ) \leqslant d_{TV}(\mathbf L,\mathbf M) +2(R_1+R_2+R_3+R_4) \end{align*}
with
\begin{align*} R_1\,:\!=\,& \int _W \int _{T_x} \mathbb{E}[g(x,\eta ^{x})] \mathbb{E}[g(y,\eta ^y)] \rho ^2 dy dx,\\ R_2\,:\!=\,&\int _W \int _{T_x} \mathbb{E}[g(x,\eta ^{x,y})g(y,\eta ^{x,y})] \rho ^{(2)}(x,y)dy dx,\\ R_3\,:\!=\,&\rho |W|\textrm {ess sup}_{x \in W} \mathbb{E}[g(x,\eta ^{x})] \mathbb{E}[(\eta \Delta \eta ^{x,\Xi }) (W \setminus T_x)],\\ R_4\,:\!=\,&\rho |W| \textrm {ess sup}_{x \in W} \mathbb{E}[g(x,\eta ^{x})] \mathbb{E}\Big [ \sum _{y \in \eta \cap \eta ^{x,\Xi }\cap W\setminus T_x} |g(y,\eta )-g(y,\eta ^{x,\Xi })|\Big ], \end{align*}
where
$\Delta$
stands for the symmetric difference.
Proof.
Without loss of generality, we can assume that
$\mathbf L=\mathbf M$
(otherwise apply [[Reference Bobrowski, Schulte and Yogeshwaran4], (2.6)]). We adapt the proof of [[Reference Bobrowski, Schulte and Yogeshwaran4], Theorem 3.1] to our setting. Let
$\mathcal L$
be the generator of the Glauber dynamics from [[Reference Bobrowski, Schulte and Yogeshwaran4], (2.7)] with associated Markov semigroup
$P_s$
, and let
$D_xP_sh(\omega )\,:\!=\,P_sh(\omega +\delta _x)-P_sh(\omega )$
for
$\omega \in \widehat {\mathbf {N}}$
and
$h \in \text {Lip}$
. By definition of KR distance and [[Reference Bobrowski, Schulte and Yogeshwaran4], (3.1)], we have
\begin{align*} \mathbf {d}_{\mathbf{KR}}(\Xi, \zeta ) &= \sup _{h \in \text {Lip}} \Big |\int _0^{\infty } \mathbb{E} [\mathcal L P_s h(\Xi )]ds\Big |\\ &= \sup _{h \in \text {Lip}} \Big |\int _0^{\infty } \int _W \Big (\mathbb{E}[D_xP_sh(\Xi )]-\mathbb{E}[D_xP_sh(\Xi ^{x!})] \Big )\mathbf L(dx) ds\Big |. \end{align*}
We can bound the absolute value of the last integrand by
\begin{align} &\mathbb{E} [|D_x P_sh(\Xi ) - D_xP_sh(\Xi ^{x!})|] =\mathbb{E}[|P_sh(\Xi +\delta _x)-P_sh(\Xi )-P_sh(\Xi ^x)+P_sh(\Xi ^{x!}|]\nonumber \\ &\quad \leqslant \mathbb{E}[|P_sh(\Xi +\delta _x)-P_sh(\Xi \cap {T_x^c}+\delta _x)| + |P_sh(\Xi )-P_sh(\Xi \cap {T_x^c})|]\nonumber \\ &\qquad + \mathbb{E}[|P_sh(\Xi ^{x!}+\delta _x)-P_sh( \Xi ^{x!} \cap {T_x^c}+\delta _x)| + |P_sh( \Xi ^{x!})-P_sh( \Xi ^{x!} \cap T_x^c)|]\nonumber \\ &\qquad + \mathbb{E}[| P_sh(\Xi \cap {T_x^c}+\delta _x)-P_sh( \Xi ^{x!} \cap {T_x^c}+\delta _x)| +|P_sh(\Xi \cap {T_x^c})-P_sh( \Xi ^{x!} \cap {T_x^c}) |]. \end{align}
By [[Reference Bobrowski, Schulte and Yogeshwaran4], (2.9)] we have that
$|P_sh(\omega _1)-P_sh(\omega _1)|\leqslant e^{-s} (\omega _1\Delta \omega _2)(W)$
for
$\omega _1, \omega _2 \in \widehat {\mathbf {N}}$
. Hence,
\begin{align*} \mathbb{E}[|P_sh(\Xi +\delta _x)-P_sh(\Xi \cap {T_x^c}+\delta _x)| + |P_sh(\Xi )-P_sh(\Xi \cap {T_x^c})|]&\leqslant 2e^{-s} \mathbb{E}[\Xi (T_x)],\\ \mathbb{E}[|P_sh(\Xi ^{x!}+\delta _x)-P_sh( \Xi ^{x!} \cap {T_x^c}+\delta _x)| + |P_sh( \Xi ^{x!})-P_sh( \Xi ^{x!} \cap T_x^c)|]&\leqslant 2e^{-s} \mathbb{E}[\Xi ^{x!}(T_x)]. \end{align*}
We now specify a coupling of
$\Xi$
and
$ \Xi ^{x!}$
that we use to bound the third term in the right-hand side of (16). Let
$\eta \sim P$
and let
$\eta ^{x,\Xi }\sim P^{x,\Xi }|_{S_x^c}$
be a Palm version of
$\eta$
with respect to
$\Xi$
at
$x$
, restricted to
$S_x^c$
. Since
$(S \oplus T^c) \cap S=\varnothing$
, we have
$S_y\cap S_x=\varnothing$
for all
$y \in T_x^c$
. Hence, using that
$g$
stabilises with respect to
$S$
,
$\Xi [ \eta ^{x,\Xi }] \cap T_x^c$
and
$\Xi [\hat \eta ^{x,\Xi }] \cap T_x^c$
agree in distribution, where
$\hat \eta ^{x,\Xi } \sim P^{x,\Xi }$
. Therefore, it follows from the definition of
$P^{x,\Xi }$
that
$\Xi [ \eta ^{x,\Xi }] \cap T_x^c$
and
$\Xi ^{x!} \cap T_x^c$
agree in distribution. This gives
\begin{align*} \mathbb{E}[|P_sh(\Xi \cap {T_x^c}+\delta _x)-P_sh( \Xi ^{x!} \cap {T_x^c}+\delta _x)|] &\leqslant e^{-s} \mathbb{E} [(\Xi [\eta ] \Delta \Xi [\eta ^{x,\Xi }])(T_x^c)],\\ \mathbb{E}[|P_sh(\Xi \cap {T_x^c})-P_sh( \Xi ^{x!} \cap {T_x^c})|] &\leqslant e^{-s} \mathbb{E} [(\Xi [\eta ] \Delta \Xi [\eta ^{x,\Xi }])(T_x^c)]. \end{align*}
From the particular form of the score functional
$\Xi$
, we obtain for the last term on the right-hand side above,
\begin{equation*} (\Xi [\eta ] \Delta \Xi [\eta ^{x,\Xi }]) (T_x^c)\leqslant (\eta \Delta \eta ^{x,\Xi }) (W \setminus T_x) + \sum _{y \in \eta \cap \eta ^{x,\Xi }\cap W\setminus T_x} |g(y,\eta )-g(y,\eta ^{x!,\Xi })|\quad \text {a.s.} \end{equation*}
Hence, we have shown that
\begin{align*} \mathbf {d}_{\mathbf{KR}}(\Xi, \zeta ) &\leqslant 2\Big \{\int _W \mathbb{E}[\Xi (T_x)] +\mathbb{E}[\Xi ^{x!}(T_x)] \,\mathbf L(dx) \\ &\quad + \rho |W|\textrm {ess sup}_{x \in W} \mathbb{E}[g(x,\eta ^{x})] \mathbb{E}[(\eta \Delta \eta ^{x,\Xi }) (W \setminus T_x)]\\ &\quad + \rho |W| \textrm {ess sup}_{x \in W} \mathbb{E}[g(x,\eta ^{x})] \mathbb{E}\Big [ \sum _{y \in \eta \cap \eta ^{x,\Xi }\cap W\setminus T_x} |g(y,\eta )-g(y,\eta ^{x,\Xi })|\Big ]\Big \}. \end{align*}
From here, the asserted bound follows from the observations that
$\mathbf L(dx)=\unicode {x1D7D9}\{x \in W\}\mathbb{E}[g(x,\eta ^x)] \rho dx$
and that
\begin{align*} \int _W \mathbb{E}[\Xi ^{x!}(T_x)] \,\mathbf L(dx)&= \int _W \mathbb{E}[(\Xi [\eta ^{x,\Xi }]-\delta _x)(T_x)] \,\mathbf L(dx)\\ &=\mathbb{E}\Big [\int _W (\Xi [\eta ]-\delta _x)(T_x) \, \Xi (dx)\Big ]\\ &=\int _W \int _{T_x} \mathbb{E}[g(x,\eta ^{x,y})g(y,\eta ^{x,y})] \rho ^{(2)}(x,y)dy dx. \end{align*}
4. Proof of Theorem 2.1
Proof of Theorem
2.1. For
$x \in W$
, let
$S_x\,:\!=\,x+S$
and
$T_x\,:\!=\,x+T$
. Let
$\mathbf L$
be the intensity measure of
$\Xi$
. We apply Proposition 3.1. Since the terms
$R_1$
and
$R_2$
directly translate into
$E_1$
and
$E_2$
, it remains to bound
$R_3$
and
$R_4$
.
Bounding
$R_3$
.
Recall that
$P$
is the distribution of a determinantal process with correlation kernel
$K$
, that
$P^x$
is its Palm measure and that
$P^{x,\Xi }$
is the Palm measure with respect to
$\Xi$
. The idea is to show the existence of a construct
$(\eta, \eta ^{x,\Xi })$
of
$P$
and
$P^{x,\Xi }$
for each
$x \in W$
with a ’small’ symmetric difference
$(\eta \Delta \eta ^{x,\Xi }) (W\setminus T_x)$
. We discuss the coupling 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
$P^{x!} \leqslant P$
and
$P^{x!,\Xi }|_{S_x^c} \leqslant P^{x!}$
for
$\mathbf L$
-a.a.
$x \in W$
, implying that
$P^{x!,\Xi }|_{S_x^c} \leqslant P$
. By Strassen’s theorem (see [Reference Lindvall18]), this implies that there are processes
$\eta \sim P$
and
$\eta ^{x!,\Xi } \sim P^{x!,\Xi }|_{S_x^c}$
such that
$\eta ^{x!,\Xi } \subset \eta$
. Thus, we have
\begin{align} \mathbb{E} [(\eta \Delta \eta ^{x!,\Xi }) (W\setminus T_x)]&=\mathbb{E}[\eta (W\setminus T_x)] -\mathbb{E}[\eta ^{x!,\Xi }(W\setminus T_x)]\nonumber \\ &=\big \{\mathbb{E} [\eta (W\setminus T_x)]- \mathbb{E}[ \eta ^{x!}(W\setminus T_x)]\big \} +\mathbb{E}[\eta ^{x!}(W\setminus T_x)]-\mathbb{E}[ \eta ^{x!,\Xi }(W\setminus T_x)]. \end{align}
The term in
$\{\cdots \}$
on the right-hand side above is by (9), (5) and since
$\phi$
is decreasing, given by
\begin{align} \rho ^{-1}\int _{W \setminus T_x} |K(x,y)|^2 dy \leqslant \rho ^{-1} \|K\|\int _{W\setminus T_x} \phi (\|x-y\|) dy \leqslant \rho ^{-1} \|K\| |W| \sup _{y \in T^c}\phi (\|y\|),\quad x \in W. \end{align}
Next we consider the second term on the right-hand side in (17). By definition of the reduced Palm process
$\eta ^{x!,\Xi }$
, we have for
$\lambda$
-almost all
$x\in W$
,
\begin{align} &\mathbb{E}[g(x,\eta ^{x})]\big \{\mathbb{E}[\eta ^{x!}(W\setminus T_x)]-\mathbb{E}[ \eta ^{x!,\Xi }(W\setminus T_x)]\big \}\nonumber \\ &\quad =\mathbb{E} [\eta ^{x} (W\setminus T_x)]\mathbb{E} [g(x,\eta ^x)]- \mathbb{E}[\eta ^{x}(W \setminus T_x) g(x,\eta ^x)]\nonumber \\ &\quad =-\texttt {Cov}(\eta ^{x} (W\setminus T_x), g(x,\eta ^{x})). \end{align}
Now we use that the reduced Palm process
$\eta ^{x!}$
is a determinantal process itself and therefore has negative associations (see (12)). For
$k \in \mathbb{N}$
, we consider the auxiliary functions
\begin{align*} f^{(k)}(\omega )\,:\!=\,\min \{k,\omega (S_x)- g(x,\omega )\},\quad f(\omega )\,:\!=\,\omega (S_x)- g(x,\omega ),\quad \omega \in \mathbf {N}. \end{align*}
It is easy to see that
$f^{(k)},\, k \in \mathbb{N},$
and
$f$
are bounded and increasing. Since
$\eta ^{x!}$
has negative associations, we have that
\begin{align*} \texttt {Cov}(\min \{k,\eta ^{x!}(W\setminus T_x)\}, f^{(k)}(\eta ^{x!}))\leqslant 0. \end{align*}
Hence, by monotone convergence,
\begin{align*} &\texttt {Cov}(\eta ^{x!}(W\setminus T_x),\eta ^{x!}(S_x))-\texttt {Cov}(\eta ^{x!}(W\setminus T_x), g(x,\eta ^{x}))\\ &=\texttt {Cov}(\eta ^{x!}(W\setminus T_x), f(\eta ^{x!}))=\lim _{k \to \infty } \texttt {Cov}(\min \{k,\eta ^{x!}(W\backslash T_x)\}, f^{(k)}(\eta ^{x!}))\leqslant 0. \end{align*}
This shows that (19) is bounded by
\begin{align} &-\texttt {Cov}(\eta ^{x!}(W\setminus T_x),\eta ^{x!}(S_x))\nonumber \\ &\quad =\mathbb{E} [\eta ^{x!} (W\setminus T_x)] \mathbb{E}[\eta ^{x!}(S_x)] -\mathbb{E}[\eta ^{x!}(W\setminus T_x)\eta ^{x!}(S_x)] \nonumber \\ &\quad \leqslant \big |\mathbb{E} [\eta ^{x!} (W\setminus T_x)] - \mathbb{E}[\eta (W \setminus T_x)]\big | \mathbb{E}[\eta ^{x!}(S_x)] \end{align}
\begin{align} &\qquad +\big |\mathbb{E}[\eta ^{x!}(W\setminus T_x)\eta ^{x!}(S_x)] - \mathbb{E}[\eta (W\setminus T_x)]\mathbb{E}[\eta ^{x!}(S_x)]\big |. \end{align}
Here, we use (18) and
$\mathbb{E}[\eta ^{x!}(S_x)]\leqslant \mathbb{E}[\eta (S_x)]= \rho |S|$
to obtain for (20)
\begin{equation*} \big |\mathbb{E} [\eta ^{x!} (W\setminus T_x)] - \mathbb{E}[\eta (W \setminus T_x)]\big | \mathbb{E}[\eta ^{x!}(S_x)] \leqslant \|K\| |W| |S| \sup _{y \in T^c}\phi (\|y\|). \end{equation*}
Next we consider (21). We write
$\rho _x^{(m)}$
for the
$m$
-th correlation function of
$\eta ^{x!}$
and find by [[Reference Shirai and Takahashi27], Lemma 6.4], by the definition of
$\eta ^{x!}$
and by (15) that
\begin{align*} &\rho \Big (\mathbb{E}[\eta ^{x!}(W\setminus T_x)\eta ^{x!}(S_x)] - \mathbb{E}[\eta (W\setminus T_x)]\mathbb{E}[\eta ^{x!}(S_x)]\Big )\\ &\quad =\rho \int _{S_x} \int _{W\setminus T_x} (\rho _x^{(2)}(y,z) - \rho _{x}(y) \rho ) \,dz dy \\ &\quad = \int _{S_x} \int _{W\setminus T_x} (\rho ^{(3)}(x,y,z) - \rho ^{(2)}(x,y) \rho ) \,dz dy \\ &\quad \leqslant 3^{5/2} \|K\|^2 \int _{S_x} \int _{W\setminus T_x} \!\!\!\!\! \phi (d(\{x,y\},\{z\}))\, dz dy\\ &\quad \leqslant 3^{5/2} \|K\|^2 |W| |S| \phi (d(S,T^c)). \end{align*}
Thus, since
$\mathbf L(W) \leqslant \rho |W|$
, we can conclude that
\begin{align} R_3&=\rho |W|\textrm {ess sup}_{x \in W} \mathbb{E}[g(x,\eta ^{x})] \mathbb{E} [(\eta ^x \Delta \eta ^{x!,\Xi }) (W\setminus T_x)]\nonumber \\ &\leqslant (1+\rho |S|) \|K\| |W|^2\sup _{y \in T^c}\phi (\|y\|)+3^{5/2} \|K\|^2 |W|^2 |S| \phi (d(S,T^c))\nonumber \\ &\leqslant \|K\| (1+\rho +3^{5/2} \|K\|) \max (|S|,1)|W|^2 \phi (d(S,T^c)). \end{align}
(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
$P^{x!} \leqslant P$
and
$P^{x!}|_{S_x^c} \leqslant P^{x!,\Xi }|_{S_x^c}$
for
$\lambda$
-a.a.
$x \in W$
. Let
$\eta ^{x} \sim P^x$
. By Strassen’s theorem and [[Reference Kallenberg13], Theorem 2.15], there exist point processes
$\eta \sim P$
and
$\eta ^{x,\Xi }\sim P^{x,\Xi }|_{S_x^c}$
such that
$\eta ^{x!} \subset \eta$
and
$\eta ^{x!} \cap S_x^c \subset \eta ^{x!,\Xi }$
. This gives
\begin{align} \mathbb{E} [(\eta \Delta \eta ^{x,\Xi }) (W\setminus T_x)]&\leqslant \mathbb{E}[(\eta \setminus \eta ^{x!})(W\setminus T_x)] +\mathbb{E}[(\eta ^{x!,\Xi } \setminus \eta ^{x!})(W\setminus T_x)] \nonumber \\ &= \big \{\mathbb{E} [\eta (W\setminus T_x)]- \mathbb{E}[ \eta ^{x!}(W\setminus T_x)]\big \} +\mathbb{E}[ \eta ^{x!,\Xi }(W\setminus T_x)]-\mathbb{E}[\eta ^{x!}(W\setminus T_x)]. \end{align}
Here we bound the term in
$\{\cdots \}$
as in (18). Moreover, we obtain
\begin{align} &\mathbb{E}[ g(x,\eta ^{x})]\big \{\mathbb{E}[ \eta ^{x!,\Xi }(W\setminus T_x)]-\mathbb{E}[\eta ^{x!}(W\setminus T_x)]\big \}\nonumber \\ &\quad =\mathbb{E}[\eta ^x(W \setminus T_x) g(x,\eta ^x)]-\mathbb{E} [\eta ^{x} (W\setminus T_x)]\mathbb{E} [ g(x,\eta ^x)]\nonumber \\ &\quad =\texttt {Cov}(\eta ^{x} (W\setminus T_x), g(x,\eta ^{x})). \end{align}
Now we use that the reduced Palm process
$\eta ^{x!}$
is a determinantal process itself and therefore has negative associations (see [12]). For
$k \in \mathbb{N}$
, we consider the auxiliary functions
\begin{align*} f^{(k)}(\omega )\,:\!=\,\min \{k,\omega (S_x)+ g(x,\omega )\},\quad f(\omega )\,:\!=\,\omega (S_x)+ g(x,\omega ),\quad \omega \in \mathbf {N}. \end{align*}
It is easy to see that
$f^{(k)},\, k \in \mathbb{N},$
and
$f$
are bounded and increasing. Since
$\eta ^{x!}$
has negative associations, we have that
\begin{align*} \texttt {Cov}(\min \{k,\eta ^{x!}(W\setminus T_x)\}, f^{(k)}(\eta ^{x!}))\leqslant 0. \end{align*}
Hence, by monotone convergence,
\begin{align*} &\texttt {Cov}(\eta ^{x!}(W\setminus T_x),\eta ^{x!}(S_x))+\texttt {Cov}(\eta ^{x!}(W\setminus T_x), g(x,\eta ^{x!}))\\ &\quad =\texttt {Cov}(\eta ^{x!}(W\setminus T_x), f(\eta ^{x!}))\\ &\quad =\lim _{k \to \infty } \texttt {Cov}(\min \{k,\eta ^{x!}(W\backslash T_x)\}, f^{(k)}(\eta ^{x!})) \leqslant 0. \end{align*}
This shows that (24) is bounded by
$-\texttt {Cov}(\eta ^{x!}(W\setminus T_x),\eta ^{x!}(S_x))$
. Therefore, we can proceed as for increasing scores and obtain the same bound for
$R_3$
as in (22).
Bounding
$R_4$
.
For each
$x \in W$
, we have
\begin{align*} &\mathbb{E} \Big [\sum _{y \in \eta ^{x,\Xi } \cap \eta \cap W\setminus T_x} |g(y,\eta ^{x,\Xi })- g(y, \eta )|\Big ]\\ &\quad = \mathbb{E} \Big [\sum _{y \in \eta ^{x,\Xi } \cap \eta \cap W\setminus T_x} \unicode {x1D7D9}\{(\eta ^{x,\Xi } \Delta \eta )\cap S_x \neq \emptyset \} |g(y,\eta ^{x,\Xi })-g(y,\eta )|\Big ]\\ &\quad \leqslant \mathbb{E} \Big [\sum _{z \in (\eta ^{x,\Xi } \Delta \eta ) \cap (W\oplus S_x) \setminus T_x} \sum _{y \in \eta ^{x,\Xi } \cap \eta \cap S_z} | g(y,\eta ^{x,\Xi })- g(y,\eta )|\Big ]\\ &\quad \leqslant \mathbb{E} \Big [\sum _{z \in (\eta ^{x,\Xi } \Delta \eta ) \cap (W\oplus S_x) \setminus T_x} \max _{\omega \in \{\eta ^{x,\Xi },\eta \}}\sum _{y \in \omega \cap S_z} g(y,\omega )\Big ]. \end{align*}
Here we obtain from Condition (3) that the above is bounded by
\begin{equation*} \alpha |S| \mathbb{E} \Big [(\eta ^{x,\Xi } \Delta \eta ) ((W\oplus S) \setminus T_x)\Big ]. \end{equation*}
Hence, we obtain from the estimate in (22) (with
$W$
replaced by
$W \oplus S$
) that
\begin{align} R_4&=\rho |W|\textrm {ess sup}_{x \in W} \mathbb{E}[g(x,\eta ^{x})] \mathbb{E} \Big [\sum _{x \in \eta ^{x,\Xi } \cap \eta \cap W\setminus T_x} |g(x,\eta ^{x,\Xi })- g(x,\eta )|\Big ] \nonumber \\ & \leqslant \alpha |S|\|K\| (1+\rho +3^{5/2} \|K\|) \max \!(|S|,1)|W \oplus S|^2 \phi (d(S,T^c)). \end{align}
5. Proof of Theorem 2.2
In the proof of Theorem2.2, 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 [Reference Kostlan15] or [[Reference Hough, Krishnapur, Peres and Virág1], 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
$\tau \gt 0$
, there exists an unbounded increasing sequence
$(v_n)_{n \in \mathbb{N}}$
such that
\begin{equation} \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|}{n^2},\quad n \in \mathbb N,\, A \in \mathcal B^2. \end{equation}
To determine the asymptotic behaviour of
$v_n$
as
$n \to \infty$
, we use that by [[Reference Hough, Krishnapur, Peres and Virág1], Theorem 26],
\begin{equation} \mathbb{P}(\xi ^{o!}(B_{v_n})=0)=e^{v_n^2}\mathbb{P}(\xi (B_{v_n})=0). \end{equation}
Moreover, by [[Reference Hough, Krishnapur, Peres and Virág2], Proposition 7.2.1],
\begin{equation*} \lim _{r \to \infty } \frac {1}{r^4} \log \mathbb{P}(\xi (B_{r})=0)=-\frac 14. \end{equation*}
Therefore, re-writing
$\log \mathbb{P}(\xi (B_{v_n})=0)$
as
\begin{equation*} \log\! (n^2 e^{v_n^2}\mathbb{P}(\xi (B_{v_n})=0))-2\log n -v_n^2, \end{equation*}
we find from (26) and (27) that
$\frac {v_n^4}{\log n}\to 8$
as
$n \to \infty$
.
Proof of Theorem
2.2. Given
$n \in \mathbb{N}$
, we choose
$\xi$
as the Ginibre process, let
$g(x,\omega )\,:\!=\,\unicode {x1D7D9}\{\omega (B_{v_n}(x)\backslash \{x\})=0\}$
,
$S\,:\!=\,B_{v_n}\,:\!=\,B_{v_n}(o)$
and
$T\,:\!=\,B_{\log n}\,:\!=\,B_{\log n}(o)$
. Note that
$g$
is stabilising with respect to
$S$
. The idea is to apply Theorem2.1 to the process
$\Xi _n$
defined at (6), where we choose
$\zeta$
as a stationary Poisson process with intensity
$\frac {\tau }{n^2}$
. Then,
$\zeta \cap B_n$
has intensity measure
$\mathbf L_n$
given at (26). First, we check the Conditions (3) and (4). Note that for all
$\omega \in \mathbf N$
and
$n \in \mathbb{N}$
,
\begin{align*} \bigcap _{x \in \Xi _n[\omega ]} B_{v_n/2}(x)=\emptyset . \end{align*}
Therefore, (3) holds with
$\alpha \,:\!=\,\frac {1}{\pi (v_n/2)^2}=\frac {4}{\pi v_n^2}\leqslant \frac {4}{\pi v_1^2}$
. Moreover, it clearly holds that
$g(x,\omega _1)\geqslant g(x,\omega _2)$
for
$\omega _1 \subset \omega _2$
, verifying (4).
Thus, by invariance of KR distance under scalings and Theorem2.1, we have
\begin{align} \mathbf {d}_{\mathbf{KR}}(\Psi _n,\nu \cap B_1) &= \mathbf {d}_{\mathbf{KR}}(\Xi _n,\zeta \cap B_n) \leqslant 2(E_{1,n}+E_{2,n}+F_n), \end{align}
where the error terms
$E_{1,n}$
,
$E_{2,n}$
, and
$F_n$
depend on
$n$
.
Next we bound
$E_{1,n}$
,
$E_{2,n}$
, and
$F_n$
. Since
$\|K\|=\pi ^{-1}$
and
$\phi (r)=\pi ^{-1} \exp \!(-r^2/2)$
for
$r\gt 0$
, we obtain
\begin{align*} F_n = \frac {1}{\pi ^{2}} \Big (1+\frac {1+3^{5/2}}{\pi }\Big ) \Big (\frac {4}{\pi v_1^2}|B_{v_n}|+1\Big ) \max \!(|B_{v_n}|,1) |B_{n+v_n}|^2 \exp \!\Big (-\frac {|\log n - v_n|^2}{2}\Big ). \end{align*}
Using here that
$v_n^4 \sim 8 \log n$
, we obtain that
$F_n\leqslant 1/n$
for
$n$
large enough. Next we bound
$E_{1,n}$
, where we recall that
$\rho =\pi ^{-1}$
. From (26), we obtain
\begin{align*} E_{1,n}&= \frac {1}{\pi ^2} \int _{B_n} \int _{B_n \cap B_{\log n}(x)} \mathbb{P}(\xi ^{x!} (B_{v_n}(x))=0) \mathbb{P}(\xi ^{y!} (B_{v_n}(y))=0) dy dx\\ &\leqslant \frac {\tau ^2}{n^4} |B_n| \sup _{x \in B_n} |B_{\log n}(x) \cap B_n| \leqslant \frac {\tau ^2 \pi ^2(\log n)^2}{n^2}. \end{align*}
Thus, it remains to bound
\begin{align} E_{2,n}&=\int _{B_n}\int _{B_n \cap B_{\log n}} \mathbb{E} [\unicode {x1D7D9}\{\xi ^{x,y}(B_{v_n}(x)\setminus \{x\})=\emptyset \} \unicode {x1D7D9}\{\xi ^{x,y}(B_{v_n}(y)\setminus \{y\})=\emptyset \}] \rho ^{(2)}(x,y) dy dx\nonumber \\ &=\int _{B_n}\int _{B_n \cap B_{\log n}} \mathbb{E} [\unicode {x1D7D9}\{\xi ^{x!,y!}(B_{v_n}(x)\cup B_{v_n}(y)) =\emptyset \}] \unicode {x1D7D9}\{\|x-y\|\gt v_n\} \rho ^{(2)}(x,y) dy dx, \end{align}
where
$\xi ^{x!,y!}\,:\!=\,\xi ^{x,y} \setminus \{x,y\}$
. By [[Reference Shirai and Takahashi27], Theorem 6.5], the reduced Palm process
$\xi ^{x!,y!}$
is a determinantal process itself. Hence, we can conclude from [[Reference Lyons19], Theorem 3.7] that
$\xi ^{x!,y!}$
has negative associations, that is, that
$\mathbb{E}[F(\xi ^{x!,y!})G(\xi ^{x!,y!})]\leqslant \mathbb{E}[F(\xi ^{x!,y!})] \mathbb{E}[G(\xi ^{x!,y!})]$
for every pair
$F,G$
of real bounded increasing (or decreasing) functions that are measurable with respect to complementary subsets of
$\mathbb{R}^d$
. We apply this with the decreasing functions
$F(\mu )=\unicode {x1D7D9}\{\mu (B_{v_n}(x))=0\}$
and
$G(\mu )=\unicode {x1D7D9}\{\mu (B_{v_n}(y) \setminus B_{v_n}(x))=0\}$
. This gives
\begin{equation} \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). \end{equation}
To bound the first probability, we note that by [[Reference Goldman12], Theorem 1], there is a reduced Palm process
$\xi ^{x!,y!}$
of
$\xi ^{x!}$
such that
$\xi ^{x!,y!}\subset \xi ^{x!}$
and
$|\xi ^{x!} \setminus \xi ^{x!,y!}| \leqslant 1$
a.s. This gives
\begin{equation*} \mathbb{P}(\xi ^{x!,y!}(B_{v_n}(x))=0) \leqslant \mathbb{P}(\xi ^{x!}(B_{v_n}(x)) \leqslant 1). \end{equation*}
Now we apply the same argument to the determinantal process
$\xi ^{x!}$
and obtain the bound
\begin{equation*} \mathbb{P}(\xi ^{x!}(B_{v_n}(x))\leqslant 1)\leqslant \mathbb{P}(\xi (B_{v_n}(x))\leqslant 2)=\mathbb{P}(\xi (B_{v_n})\leqslant 2), \end{equation*}
where the last equality holds due to the stationarity of
$\xi$
. As mentioned at the beginning of this section, the set of absolute values of the points of the Ginibre process
$\xi$
has the same distribution as a sequence
$(X_i)_{i \in N}$
of independent random variables with
$X_i^2 \sim \text {Gamma}(i,1)$
. Similarly to [[Reference Hough, Krishnapur, Peres and Virág2], Section 7.2], this gives
\begin{align*} \mathbb{P}(\xi (B_{v_n}) \leqslant 2) &=\mathbb{P}(\#\{j \in \mathbb{N}\,:\,\,X_j \leqslant v_n\} \leqslant 2)\\ & \leqslant \mathbb{P}(\#\{j \in \{1,\ldots, v_n^2\}\,:\,\,X_j \leqslant v_n\} \leqslant 2)\\ &=\mathbb{P}\Big (\bigcup _{i=1}^{v_n^2} \bigcup _{\substack {j=1\\ j \neq i}}^{v_n^2} \{ \forall k \in \{1,\ldots, v_n^2\}\setminus \{i,j\}\,:\,\,X_k \gt v_n \}\Big ). \end{align*}
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
\begin{align*} \sum _{i=1}^{v_n^2} \sum _{\substack {j=1\\ j \neq i}}^{v_n^2} \mathbb{P}(\forall k \in \{1,\ldots, v_n^2\}\setminus \{i,j\}\,:\,\,X_k \gt v_n) =\sum _{i=1}^{v_n^2} \sum _{\substack {j=1\\ j \neq i}}^{v_n^2} \prod _{\substack {k =1\\ k\neq i,j}}^{v_n^2} \mathbb P(X_k^2 \gt v_n^2). \end{align*}
Let
$t\lt 1$
. 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
\begin{equation*} \mathbb{P}(X_k^2\gt r^2) \leqslant e^{-t r^2} \mathbb{E} [e^{t X_k^2}]=e^{-t r^2} (1-t)^{-k}. \end{equation*}
For
$k\lt r^2$
, this bound is maximised for
$t=1- \frac {k}{r^2}$
, which gives
\begin{equation*} \mathbb{P}(\xi (B_{v_n}) \leqslant 2)\leqslant \sum _{i=1}^{v_n^2} \sum _{\substack {j=1\\ j \neq i}}^{v_n^2} \prod _{\substack {k =1\\ k\neq i,j}}^{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 _{\substack {j=1\\ j \neq i}}^{v_n^2} \prod _{\substack {k =1\\ k\neq i,j}}^{v_n^2} e^{-v_n^2+k-k \log \big (\frac {k}{v_n^2}\big )}. \end{equation*}
Using here that
$u \mapsto u-u\log \!(u/r^2)$
is increasing for
$u\leqslant r^2$
, we find that
\begin{align*} \mathbb{P}(\xi (B_{v_n}) \leqslant 2) & \leqslant \sum _{i=1}^{v_n^2} \sum _{\substack {j=1\\ j \neq i}}^{v_n^2}\prod _{k=3}^{v_n^2} e^{-v_n^2+k-k \log \big (\frac {k}{v_n^2}\big )}\\ &\leqslant v_n^4 \prod _{k=3}^{v_n^2} e^{-v_n^2+k-k \log \big (\frac {k}{v_n^2}\big )}\\ &=v_n^2 e^{-\frac 12 (v_n^2-3)(v_n^2-2)-v_n^4\int _{3/v_n^2}^{1}u \log \!(u) \mathrm {d}x+O(v_n^2 \log v_n)}\\ &=e^{-\frac 14 v_n^4(1+o(1))} \end{align*}
as
$n \to \infty$
, where we have used that
$\int _0^1 u \log \!(u) dx=-\frac 14$
.
Next we bound the second probability in (30). By the same coupling argument as above, we find that
\begin{align} \mathbb{P}(\xi ^{x!,y!}(B_{v_n}(y)\backslash B_{v_n}(x))=0)\leqslant \mathbb{P}(\xi (B_{v_n}(y)\backslash B_{v_n}(x))\leqslant 2). \end{align}
Next we note that
$B_{v_n/2}\left (y+\frac {v_n(y-x)}{2|y-x|}\right ) \subset B_{v_n}(y)\setminus B_{v_n}(x)$
if
$\|x-y\|\geqslant v_n$
. (This is sufficient, since the integrand of (29) vanishes if
$\|x-y\|\lt v_n$
.) Hence, (31) is for
$\|x-y\|\geqslant v_n$
bounded by
\begin{equation*} \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 14 (v_n/2)^4 (1+o(1))} \end{equation*}
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
$\varepsilon \gt 0$
at the bound
\begin{align*} E_2&=\int _{B_n}\int _{B_n \cap B_{\log n}} \mathbb{E} [\unicode {x1D7D9}\{\xi ^{x!,y!}(B_{v_n}(x)\cup B_{v_n}(y)) =\emptyset \}] \unicode {x1D7D9}\{\|x-y\|\gt v_n\} \rho ^{(2)}(x,y) dy dx\\ &\leqslant n^2 (\log n)^2 e^{-\frac 14 v_n^4 (1+o(1))} e^{-\frac 1{64} v_n^4(1+o(1))} \leqslant n^{\varepsilon -1/16}, \end{align*}
where we have used (26) and that
$\frac {v_n^4}{\log n}\to 8$
as
$n \to \infty$
. Hence, the assertion follows from (28).
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