2.1. Particle processes

Let $\mathbb{R}^d$ be Euclidean d-dimensional space with Borel $\sigma$ -field $\mathcal{B}^d$ , and let $\mathcal{B}^d_{b}$ denote the system of bounded Borel sets. Let $\mathcal{C}^d$ be the space of compact subsets (particles) of $\mathbb{R}^d$ . Let ${\mathcal{C}^{(d)}}\coloneqq \mathcal{C}^{d}\setminus\{\emptyset\}$ be equipped with the Hausdorff metric $d_H$ (see [Reference Last and Penrose18] and [Reference Schneider and Weil25]) and the associated Borel $\sigma$ -field $\mathcal{B}({\mathcal{C}^{(d)}})$ . As usual for metric spaces, for non-empty sets $\Psi, \Gamma\subset\mathcal{C}^{(d)}$ we define

(2.1) \begin{equation} d(\Psi,\Gamma)\coloneqq \inf_{A\in\Psi,\,B\in\Gamma}d_H(A,B).\end{equation}

To avoid confusion, our notation $d(\Psi,\Gamma)$ does not reflect the underlying metric $d_H$ . Let ${\textbf{N}}$ denote the space of all measures $\xi$ on ${\mathcal{C}^{(d)}}$ with values in $\mathbb{N}_0\cup\{\infty\}$ such that $\xi(B(K,r))< \infty$ , for each $K\in {\mathcal{C}^{(d)}}$ and each $r\ge 0$ , where $B(K,r)\coloneqq \{L\in{\mathcal{C}^{(d)}}\mid{}d_H(K,L)\le r\}$ is the ball with radius r centred at K. As usual (see e.g. [Reference Last and Penrose18]), we equip this space with the smallest $\sigma$ -field $\mathcal{N}$ such that the mappings $\xi\mapsto \xi(\Psi)$ are measurable for each $\Psi \in \mathcal{B}({\mathcal{C}^{(d)}})$ .

Let z(K) denote the centre of the circumscribed ball of $K \in {\mathcal{C}^{(d)}}$ and note that $z(K+x)=z(K)+x$ for all $(K,x)\in {\mathcal{C}^{(d)}}\times\mathbb{R}^d$ . We say that a particle process is simple, if $\Xi(z^{-1}(x))\in\{0,1\}$ , $x\in \mathbb{R}^{d}$ , holds almost surely. In the following, we consider only simple stationary particle processes. We also assume that $\mathbb{P}(\Xi({\mathcal{C}^{(d)}} )\ne 0)=1$ . The intensity $\gamma$ of such a particle process $\Xi$ is defined by

\begin{equation*}\gamma\coloneqq \mathbb{E}\biggl[\int {\bf{1}}\{z(K)\in[0,1]^d\}\,\Xi(\textrm{d} K)\biggr].\end{equation*}

The intensity measure $\mathbb{E}[\Xi]$ of $\Xi$ is the measure $A\mapsto\mathbb{E}[\Xi(A)],\;A\in \mathcal{B}({\mathcal{C}^{(d)}})$ .

An important example of a particle process is a Poisson process $\Pi_{\mu}$ on ${\mathcal{C}^{(d)}}$ , whose intensity measure $\mu$ is defined by

(2.2) \begin{equation} \mu\coloneqq \iint {\bf{1}}\{K+x\in \cdot\}\,\mathbb{Q}(\textrm{d} K)\,\textrm{d} x,\end{equation}

where $\textrm{d} x$ refers to integration with respect to the Lebesgue measure $\mathcal{L}_d$ on $\mathbb{R}^d$ and $\mathbb{Q}$ is some fixed probability measure on ${\mathcal{C}^{(d)}}$ . We refer to [Reference Last and Penrose18, Chapter 3] for the definition and fundamental properties of general Poisson processes. More generally, we consider the Poisson processes $\Pi_{\lambda\mu}$ with intensity measure $\lambda\mu$ , where $\lambda >0$ . Under some integrability assumptions on $\mathbb{Q}$ , the Poisson process $\Pi_{\lambda\mu}$ exists as a stationary particle process [Reference Schneider and Weil25]. The number $\lambda$ is the intensity of $\Pi_{\lambda\mu}$ while $\mathbb{Q}$ is called the particle distribution of $\Pi_{\lambda\mu}$ . It is no restriction of generality to assume that

(2.3) \begin{equation} \mathbb{Q}(\mathcal{C}_{\textbf{0}}^{(d)})=1,\end{equation}

where $\mathcal{C}_{\textbf{0}}^{(d)}\coloneqq \{K\in {\mathcal{C}^{(d)}} \mid{}z(K) = \textbf{0}\}$ , and $\textbf{0}$ denotes the origin in $\mathbb{R}^d$ . However, we make the crucial assumption that there exists $R>0$ such that

(2.4) \begin{equation} \mathbb{Q}(\{K\in {\mathcal{C}^{(d)}} \mid{} K \subseteq B(\textbf{0},R)\}) = 1,\end{equation}

where B(x, R) is the closed Euclidean ball with radius R centred at $x\in\mathbb{R}^d$ . This is the deterministic bound on the particle size.

Given $m\in\mathbb{N}$ and $\xi\in{\textbf{N}}$ , the mth factorial measure $\xi^{(m)}$ of $\xi$ is the measure on $({\mathcal{C}^{(d)}})^m$ defined by

\begin{equation*} \xi^{(m)}(\!\cdot\!)\coloneqq \int {\bf{1}}\{(K_1,\ldots,K_m)\in\cdot\}\text{${\bf{1}}\{K_{i}\ne K_{j}$ for $i\ne j$} \} \,\xi^{m}(\textrm{d}(K_{1},\ldots,K_{m})).\end{equation*}

For us this is only of relevance if $\xi(\{K\})\le 1$ , for each $K\in\mathcal{C}^{(d)}$ . Then $\xi$ is called simple. In this case $\xi^{(m)}$ coincides with the standard definition of the factorial measure [Reference Last and Penrose18, Chapter 4]. The mth factorial moment measure $\alpha^{(m)}$ of a simple particle process $\Xi$ is defined by $\alpha^{(m)}\coloneqq \mathbb{E} [\Xi^{(m)}]$ .

Definition 2. Let $p\in\mathbb{N}$ . The pth Palm distributions of a particle process $\Xi$ is a family $\mathbb{P}_{K_1, \ldots, K_p}$ , $K_1, \ldots, K_p \in {\mathcal{C}^{(d)}}$ , of probability measures on ${\textbf{N}}$ satisfying

(2.5) \begin{align}& \mathbb{E}\bigg[\int f(K_1,\ldots,K_p,\Xi)\,\Xi^{(p)}(\textrm{d}(K_1,\ldots,K_p))\bigg] \notag \\*&\quad =\iint f(K_1,\ldots,K_p,\xi ) \,\mathbb{P}_{K_1,\ldots,K_p}(\textrm{d}\xi) \,\alpha^{(p)}(\textrm{d}(K_1,\ldots,K_p)),\end{align}

for each non-negative measurable function f on $({\mathcal{C}^{(d)}} )^p\times{\textbf{N}}$ .

Palm distributions are well-defined whenever the pth factorial moment measure $\alpha^{(p)}$ of $\Xi$ is $\sigma$ -finite. They can be chosen such that $(K_1,\ldots,K_p)\mapsto \mathbb{P}_{K_1,\ldots,K_p}(A)$ is a measurable function on $({\mathcal{C}^{(d)}} )^p$ , for each $A\in\mathcal{N}$ . The reduced Palm distribution $\mathbb{P}^{!}_{K_{1},\ldots,K_{p}}$ of $\Xi$ is defined by means of the equality

(2.6) \begin{equation} \int f(K_1,\ldots,K_p,\xi)\,\mathbb{P}^!_{K_1,\ldots,K_p}(\textrm{d}\xi) =\int f(K_1,\ldots,K_p,\xi-\delta_{K_1}-\cdots-\delta_{K_p}) \,\mathbb{P}_{K_1,\ldots,K_p}(\textrm{d}\xi),\end{equation}

valid for every non-negative measurable function f on $(\mathcal{C}^{(d)})^p\times{\textbf{N}}$ . We abuse our notation by writing, for each measurable $g\colon{\textbf{N}}\to\mathbb{R}$ ,

\begin{equation*}\mathbb{E}_{K_1,\ldots,K_p}[g(\Xi)]\coloneqq \int g(\xi)\,\mathbb{P}_{K_1,\ldots,K_p}(\textrm{d}\xi)\quad\text{and}\quad\mathbb{E}^!_{K_1,\ldots,K_p}[g(\Xi)]\coloneqq \int g(\xi)\,\mathbb{P}^!_{K_1,\ldots,K_p}(\textrm{d}\xi).\end{equation*}

Given $\Psi\in \mathcal{B}({\mathcal{C}^{(d)}})$ , we define ${\textbf{N}}_{\Psi} \coloneqq \{\xi \in {\textbf{N}} \mid{} \xi(\Psi^{c}) = 0 \}$ and let $\mathcal{N}_{\Psi}$ denote the $\sigma$ -field on this set of measures. Given $\xi\in{\textbf{N}}$ , $B\in\mathcal{B}^d$ , and $\Psi\in\mathcal{B}({\mathcal{C}^{(d)}})$ , we let $\xi_B$ and $\xi_\Psi$ denote the restrictions of $\xi$ to $z^{-1}(B)$ and $\Psi$ , respectively. Finally, we set $\mathcal{B}_b({\mathcal{C}^{(d)}})\coloneqq \{z^{-1}(B)\mid{} B\in\mathcal{B}^d_b\}$ .

2.2. Gibbs particle processes

In this subsection we present some fundamental facts on Gibbs processes in a general setting. We base our definition of a Gibbs process on the following GNZ-equation, referring to [Reference Jansen14], for example, for a discussion of the literature.

Definition 3. Let $\kappa\colon {\mathcal{C}^{(d)}} \times{\textbf{N}} \to[0,\infty)$ be a measurable function and $\lambda>0$ . A particle process $\Xi$ is called a Gibbs process with Papangelou conditional intensity $\kappa$ and activity parameter $\lambda>0$ , if

(2.7) \begin{equation} \mathbb{E}\bigg[\int f(K,\Xi-\delta_K)\,\Xi(\textrm{d} K)\bigg] =\lambda \mathbb{E}\bigg[ \int f(K,\Xi)\kappa(K,\Xi)\,\mu(\textrm{d} K)\bigg]\end{equation}

holds for all measurable $f\colon {\mathcal{C}^{(d)}} \times{\textbf{N}} \to[0,\infty)$ , where $\delta_K$ is the Dirac measure located at K and $\mu$ is given by (2.2).

In the following, we fix a Gibbs process with Papangelou intensity $\kappa$ and activity $\lambda$ as in Definition 3. For $p\in\mathbb{N}$ , define a measurable function $\kappa_p\colon ({\mathcal{C}^{(d)}} )^p\times{\textbf{N}}\to [0,\infty)$ by

(2.8) \begin{equation} \kappa_p(K_1,\ldots,K_p,\xi) \coloneqq \kappa(K_1,\xi)\kappa(K_2,\xi+\delta_{K_1})\dotsm\kappa(K_p,\xi+\delta_{K_1}+\dotsb+\delta_{K_{p-1}}).\end{equation}

Equation (2.7) can be iterated so as to yield

(2.9) \begin{align}& \mathbb{E} \bigg[\int f(K_1,\ldots,K_p,\Xi-\delta_{K_1}-\dotsb-\delta_{K_p}) \,\Xi^{(p)}(\textrm{d}(K_1,\ldots,K_p))\bigg]\notag \\*&\quad =\lambda^p\,\mathbb{E} \bigg[\int f(K_1,\ldots,K_p,\Xi)\kappa_p(K_1,\ldots,K_p,\Xi) \,\mu^p(\textrm{d}(K_1,\ldots, K_p))\bigg],\end{align}

for each measurable $f\colon ({\mathcal{C}^{(d)}} )^p\times{\textbf{N}}\to[0,\infty)$ . Therefore $\kappa_p$ is called the conditional intensity of pth order. By [Reference Matthes, Warmuth and Mecke20, Satz 1.5], $\kappa_p$ is a symmetric function of the p particles.

Definition 4. Let $p\in\mathbb{N}$ . The pth correlation function of a Gibbs process $\Xi$ with Papangelou intensity $\kappa$ and activity $\lambda$ is the function $\rho_{{p}}\colon ({\mathcal{C}^{(d)}} )^p\to [0,\infty]$ defined by

\begin{equation*} \rho_{{p}}(K_1,\ldots,K_p) \coloneqq \lambda^p\mathbb{E}[\kappa_p(K_1,\ldots,K_p,\Xi)].\end{equation*}

Putting

\begin{equation*} f(K_1,\ldots,K_p,\xi )={\bf{1}}\{K_1\in B_1,\ldots,K_p\in B_p\},\quad B_1,\ldots,B_p\in\mathcal{B}({\mathcal{C}^{(d)}}),\end{equation*}

in (2.9), we obtain that the pth factorial moment measure of $\Xi$ is given by

(2.10) \begin{equation}\alpha^{(p)}(\!\cdot\!)=\int{\bf{1}}\{(K_1,\ldots,K_p)\in\cdot\}\rho_p (K_1,\ldots,K_p)\,\mu^p(\textrm{d}(K_1,\ldots,K_p)),\end{equation}

justifying our terminology.

We define a measurable function $H\colon{\textbf{N}}\times{\textbf{N}}\to ({-}\infty,\infty]$ , the Hamiltonian, by

\begin{equation*} H(\xi,\chi)\coloneqq \begin{cases} 0 &\text{if $\xi({\mathcal{C}^{(d)}})=0$}, \\[3pt] -\log \kappa_m(K_1,\ldots,K_m,\chi) & \text{if }\xi=\sum_{i=1}^m \delta_{K_i}\text{ with }K_1,\ldots,K_m\in {\mathcal{C}^{(d)}}, \\[3pt] \infty &\text{if $\xi({\mathcal{C}^{(d)}})=\infty$}. \end{cases}\end{equation*}

For $\Psi\in\mathcal{B}({\mathcal{C}^{(d)}})$ , let $\Pi_{\Psi,\lambda\mu}\coloneqq (\Pi_{\lambda\mu})_\Psi$ denote the restriction of the Poisson process $\Pi_{\lambda\mu}$ to $\Psi$ . The partition function $Z_\Psi\colon{\textbf{N}}\to (0,\infty]$ of $\Xi$ (on $\Psi$ ) is defined by

\begin{equation*} Z_\Psi(\chi)\coloneqq \mathbb{E} [ {\textrm{e}}^{-H(\Pi_{\Psi,\lambda\mu},\chi)} ]{.}\end{equation*}

For $\Psi\in \mathcal{B}_b({\mathcal{C}^{(d)}})$ we have $\mu(\Psi)<\infty$ and hence $Z_\Psi>0$ . It was shown in [Reference Matthes, Warmuth and Mecke20] that $Z_\Psi(\Xi_{\Psi^c})<\infty$ $\mathbb{P}$ -a.s. and that the following DLR-equations [Reference Kallenberg16, Reference Mase19, Reference Ruelle24] hold:

(2.11) \begin{equation} \mathbb{E}[f(\Xi_\Psi)\mid \Xi_{\Psi^c}=\chi] =Z_\Psi(\chi)^{-1}\mathbb{E} [f(\Pi_{\Psi,\lambda\mu})\,{\textrm{e}}^{-H(\Pi_{\Psi,\lambda\mu},\chi)}], \quad \Psi\in\mathcal{B}_b({\mathcal{C}^{(d)}} ),\end{equation}

for $\mathbb{P}(\Xi_{\Psi^c}\in\cdot)$ -a.s. $\chi \in {\textbf{N}}_{\Psi^{c}}$ and each measurable $f\colon{\textbf{N}}\to[0,\infty)$ .

Given $p\in \mathbb{N}$ , $\Psi\in\mathcal{B}({\mathcal{C}^{(d)}})$ and $\chi\in{\textbf{N}}_{\Psi^c}$ , we write $\mathbb{P}_{\Psi,\chi,K_1,\ldots,K_p}^{\,!}$ , $K_1,\ldots,K_p\in\Psi$ , for the reduced Palm distribution of $\Xi_\Psi$ with boundary condition $\chi\in{\textbf{N}}_{\Psi^c}$ . Formally, this is the Palm distribution of the conditional distribution $\mathbb{P}(\Xi_\Psi\in\cdot \mid \Xi_{\Psi^c}=\chi)$ . The corresponding Papangelou intensity is denoted as $\kappa_{\Psi,\chi,K_1,\ldots,K_p}$ .

Lemma 1. Let $p\in\mathbb{N}$ , $\Psi\in\mathcal{B}_b({\mathcal{C}^{(d)}})$ and $\chi\in{\textbf{N}}_{\Psi^c}$ . A version of $\kappa_{\Psi,\chi,K_1,\ldots,K_p}$ is given by

\begin{equation*} \kappa_{\Psi,\chi,K_1,\ldots,K_p}(K,\xi) =\kappa(K,\xi+\chi+\delta_{K_1}+\cdots+\delta_{K_p}), \quad K_1,\ldots,K_p\in\Psi,\,\xi\in{\textbf{N}}.\end{equation*}

Proof. The proof is straightforward and given here for completeness [Reference Coeurjolly, Møller and Waagepetersen4, page 17]. Without restricting generality we assume that $\lambda=1$ .

Let $g\colon{\mathcal{C}^{(d)}}\times{\textbf{N}}\to[0,\infty)$ be measurable. By the DLR-equation and the Mecke equation for $\Pi\coloneqq \Pi_{\mu}$ (see [Reference Last and Penrose18, Theorem 4.1]),

\begin{align*}& \iint g(K,\xi)\,\xi({\textrm{d}} K)\mathbb{P}(\Xi_\Psi\in\textrm{d}\xi \mid \Xi_{\Psi^c}=\chi) \\* &\quad =Z_\Psi(\chi)^{-1}\mathbb{E}\biggl[ \int_{\Psi} g(K,\Pi_\Psi+\delta_K) \,{\textrm{e}}^{- H(\Pi_\Psi+\delta_K,\chi)}\,\mu(\textrm{d} K) \biggr].\end{align*}

Since

\begin{equation*}H(\Pi_\Psi+\delta_K,\chi)=H(\Pi_\Psi,\chi)-\log \kappa(K,\Pi_\Psi+\chi){,}\end{equation*}

the above right-hand side equals (again by the DLR-equation)

\begin{equation*} \int\int_{\Psi} g(K,\xi+\delta_K)\kappa(K,\xi+\chi)\,\mu(\textrm{d} K)\, \mathbb{P}(\Xi_\Psi\in\textrm{d}\xi \mid \Xi_{\Psi^c}=\chi).\end{equation*}

Hence $\mathbb{P}(\Xi_\Psi\in\cdot \mid \Xi_{\Psi^c}=\chi)$ is a Gibbs distribution whose Papangelou kernel $\kappa_{\Psi,\chi}$ is given by

\begin{equation*}\kappa_{\Psi,\chi}(K,\xi)={\bf{1}}\{K\in\Psi\}\kappa(K,\xi+\chi).\end{equation*}

By the first step of the proof it suffices to determine the reduced Palm distributions of a Gibbs process $\Xi$ . It is convenient to write $\textbf{K}_p\coloneqq (K_1,\ldots,K_p)$ and $\delta_{\textbf{K}_p}\coloneqq \delta_{K_1}+\dotsb+\delta_{K_p}$ , for $K_1,\ldots,K_p\in {\mathcal{C}^{(d)}}$ . We proceed by induction over p. Let $g\colon ({\mathcal{C}^{(d)}})^p\to[0,\infty)$ and $f\colon {\mathcal{C}^{(d)}} \times{\textbf{N}}\to[0,\infty)$ be measurable. By (2.6) and (2.5),

\begin{align*}& \iiint g(\textbf{K}_p)f(K,\xi-\delta_K) \,\xi(\textrm{d} K)\,\mathbb{P}_{\textbf{K}_p}^{\,!}(\textrm{d} \xi) \,\rho_p(\textbf{K}_p)\,\mu^p(\textrm{d} \textbf{K}_p) \\[2pt] &\quad =\mathbb{E}\bigg[\iint g(\textbf{K}_p)f(K_{p+1},\Xi-\delta_{\textbf{K}_{p+1}})\,(\Xi-\delta_{\textbf{K}_p})(\textrm{d} K_{p+1})\, \Xi^{(p)}(\textrm{d} \textbf{K}_p)\bigg].\end{align*}

Since $(\Xi-\delta_{\textbf{K}_p})(\textrm{d} K_{p+1})\Xi^{(p)}(\textrm{d} \textbf{K}_p) =\Xi^{(p+1)}(\textrm{d} \textbf{K}_{p+1})$ , we obtain from (2.9) that the above right-hand side equals

(2.12) \begin{align} & \mathbb{E}\bigg[ \int g(\textbf{K}_p)f(K_{p+1},\Xi)\kappa_{p+1}(\textbf{K}_{p+1},\Xi) \,\mu^{p+1}(\textrm{d} \textbf{K}_{p+1}) \bigg] \notag \\[2pt] &\quad =\mathbb{E}\bigg[ \iint g(\textbf{K}_p)f(K_{p+1},\Xi) \kappa(K_{p+1},\Xi+\delta_{\textbf{K}_p})\kappa_{p}(\textbf{K}_{p},\Xi) \,\mu^{p}(\textrm{d} \textbf{K}_{p})\,\mu(\textrm{d} K_{p+1}) \bigg],\end{align}

where the identity comes from the definition of $\kappa_{p+1}$ . It follows directly from (2.5) and (2.9) that $\mathbb{P}_{\textbf{K}_p}^{\,!}$ is for $\alpha^{(p)}$ -a.e. $\textbf{K}_p$ absolutely continuous with respect to the distribution of $\Xi$ with density $\kappa_p(\textbf{K}_p,\cdot)/\rho_p(\textbf{K}_p)$ , where $a/0\coloneqq 0$ for all $a\ge 0$ . Therefore expression (2.2) equals

\begin{equation*} \iiint g(\textbf{K}_p)f(K_{p+1},\xi)\kappa(K_{p+1},\xi+\delta_{\textbf{K}_p}) \,\mathbb{P}_{\textbf{K}_p}^{\,!}(\textrm{d} \xi)\,\mu(\textrm{d} K_{p+1}) \,\rho_{p}(\textbf{K}_{p})\mu^{p}(\textrm{d} \textbf{K}_{p}).\end{equation*}

This shows that the Papangelou intensity of $\mathbb{P}_{\textbf{K}_p}^{\,!}$ is for $\alpha^{(p)}$ -a.e. $\textbf{K}_p$ given by the function $(K,\xi)\mapsto \kappa(K,\xi+\delta_{\textbf{K}_p})$ , as required.

2.3. Stochastic domination

For $\xi,\xi'\in{\textbf{N}}$ , we write $\xi\le\xi'$ if $\xi(\Psi)\le\xi'(\Psi)$ for all $\Psi\in{\mathcal{B}({\mathcal{C}^{(d)}})}$ . An event $E\in\mathcal{N}$ is increasing if, for all $\xi,\xi'\in{}{{\textbf{N}}_{\Psi}}$ , $E{}\ni{}\xi\le\xi'$ implies that $\xi'\in E$ . Another viewpoint is that E is closed under the addition of point measures.

A particle process $\Xi$ is stochastically dominated by another particle process $\Xi'$ if $\mathbb{P}(\Xi\in E)\le \mathbb{P}(\Xi'\in E)$ , for each increasing $E\in\mathcal{N}$ . In this case we write $\Xi\overset{d}{\le}\Xi'$ and also

\begin{equation*}\mathbb{P}(\Xi\in\cdot)\overset{d}{\le}\mathbb{P}(\Xi'\in\cdot).\end{equation*}

By the famous Strassen theorem this implies the existence of a coupling $(\tilde\Xi,\tilde \Xi')$ of $(\Xi,\Xi')$ such that $\tilde\Xi\le\tilde\Xi'$ almost surely. In this context we call $\tilde\Xi$ a thinning of $\tilde\Xi'$ . It then follows that $\mathbb{E} f(\Xi)\le \mathbb{E} f(\Xi')$ for all increasing measurable $f\colon{\textbf{N}}\to[0,\infty]$ , where f is called increasing if $f(\xi)\le f(\xi')$ whenever $\xi\le\xi'$ .

A classical example is the stochastic domination of ${\Pi_{{\alpha\mu}}}$ by ${\Pi_{{\beta\mu}}}$ for $\alpha\le{}\beta$ . Later we use the following deeper fact.

Lemma 2. Suppose that $\Xi$ is a Gibbs particle process with Papangelou intensity $\kappa\le 1$ and activity $\lambda$ . Then

\begin{equation*}\mathbb{P}(\Xi\in\cdot) \overset{d}{\le}\mathbb{P}(\Pi_{\lambda\mu}\in\cdot).\end{equation*}

Furthermore, we have for each $p\in\mathbb{N}$ that

\begin{equation*}\mathbb{P}_{K_1,\ldots,K_p}^{\,!}\overset{d}{\le}\mathbb{P}(\Pi_{\lambda\mu}\in\cdot)\quad\text{for $\alpha^{(p)}$-a.e.\ $(K_1,\ldots,K_p)$,}\end{equation*}

where $\alpha^{(p)}$ is the pth factorial moment measure of $\Xi$ .

Proof. We only prove the second assertion. The proof of the first assertion is simpler (and in fact a special case). Let $p\in\mathbb{N}$ . We use the notation of the proof of Lemma 1. By this lemma and [Reference Georgii and Küneth10, Theorem 1.1] used for finite Gibbs processes, we have that

\begin{equation*}\mathbb{P}_{\Psi,\chi,\textbf{K}_p}^{\,!}\overset{d}{\le}\mathbb{P}(\Pi_{\Psi,\lambda\mu}\in\cdot)\quad\text{for each $\Psi\in \mathcal{B}_b({\mathcal{C}^{(d)}})$}\end{equation*}

for $\mathbb{P}(\Xi_{\Psi^c}\in\cdot)$ -a.e. $\chi$ and $\alpha^{(p)}$ -a.e. $\textbf{K}_p\in\Psi^p$ . Hence the definition of Palm distributions implies, for each measurable increasing $f\colon{\textbf{N}}\to[0,\infty)$ and each measurable $g\colon{\mathcal{C}^{(d)}}\times{\textbf{N}}\to[0,\infty)$ $\mathbb{P}$ -a.s., that

\begin{align*}&\iint f(\xi-\delta_{\textbf{K}_p})g(\textbf{K}_p)\xi^{(p)}(\textrm{d} \textbf{K}_p)\,\mathbb{P}(\Xi_\Psi\in\textrm{d} \xi \mid \Xi_{\Psi^c})\\*&\quad \le \iint f(\xi)g(\textbf{K}_p)\,\mathbb{P}(\Pi_{\Psi,\lambda\mu}\in\textrm{d}\xi)\,\mathbb{E}[(\Xi_{\Psi})^{(p)}\in\textrm{d} \textbf{K}_p \mid \Xi_{\Psi^c}].\end{align*}

Taking expectations yields

\begin{equation*}\mathbb{E}\bigg[\int_{\Psi^p} f((\Xi-\delta_{\textbf{K}_p})_\Psi)g(\textbf{K}_p)\,\Xi^{(p)}(\textrm{d} \textbf{K}_p)\bigg]\le\int_{\Psi^p} \mathbb{E}[f(\Pi_{\Psi,\lambda\mu})]g(\textbf{K}_p)\,\alpha^{(p)}(\textrm{d} \textbf{K}_p),\end{equation*}

so that

\begin{equation*}\mathbb{P}_{\textbf{K}_p}^{\,!}(\{\xi\in{\textbf{N}}\mid\xi_\Psi\in\cdot\})\overset{d}{\le}\mathbb{P}(\Pi_{\Psi,\lambda\mu}\in\cdot)\quad\text{for $\alpha^{(p)}$-a.e.\ $\textbf{K}_p\in\Psi'$,}\end{equation*}

for each measurable $\Psi'\subset\Psi$ . Exactly as in the proof of [Reference Georgii and Yoo11, Corollary 3.4], we let $\Psi\uparrow{\mathcal{C}^{(d)}}$ to obtain that

\begin{equation*}\mathbb{P}_{\textbf{K}_p}^{\,!}\overset{d}{\le}\mathbb{P}(\Pi_{\lambda\mu}\in\cdot)\quad\text{for $\alpha^{(p)}$-a.e.\ $\textbf{K}_p\in\Psi'$}\end{equation*}

and hence the assertion.

2.4. Admissible Gibbs particle processes

A family $\varphi\coloneqq (\varphi_n)_{n\ge 2}$ of higher-order interaction potentials consists of measurable, symmetric, and translation-invariant functions $\varphi_n\colon (\mathcal{C}^{d})^n\to({-}\infty,\infty]$ . The potentials have finite interaction range $R_{\varphi}$ if $\varphi_n(K_1,\ldots,K_n)=0$ , for every $n\ge 2$ and all $K_1\ldots,K_n\in\mathcal{C}^{d}$ with $\max\{d_H(K_i,K_j)\mid 1\le{}i<j\le n\}>R_{\varphi}$ .

Define the Papangelou intensity $\kappa\colon {\mathcal{C}^{(d)}} \times{\textbf{N}} \to[0,\infty)$ by $\kappa(K,\xi)\coloneqq 0$ if $K\in\operatorname{supp}\xi$ , and otherwise

(2.13) \begin{equation} \kappa(K,\xi)\coloneqq \exp\biggl[ -\sum_{n=2}^\infty \dfrac{1}{(n-1)!} \int \varphi_n(K,L_1,\ldots,L_{n-1}) \,\xi^{(n-1)}(\textrm{d}(L_1,\ldots,L_{n-1})) \biggr].\end{equation}

The function $\kappa$ is measurable and translation-invariant. Here and later we make the following convention regarding the series in the exponent of (2.13). If the sum over the negative terms diverges, then the whole series is set to zero. We assume that this is not the case for all $\xi\in{\textbf{N}}$ and $\mu$ -a.e. K. We also assume that $\kappa\le 1$ . While individual potentials might be attractive (i.e. negative), their cumulative effect must be repulsive (i.e. non-negative).

Proving the existence of a Gibbs process with a given Papangelou intensity is a non-trivial task. The literature contains many existence results under varying assumptions of generality [Reference Dereudre, Drouilhet and Georgii6, Reference Flimmel and Beneš9, Reference Mase19, Reference Ruelle24, Reference Schreiber and Yukich26], none of which seems to cover our current setting. For our main findings (e.g. Theorems 2 and 6) we need to restrict the range of the activity parameter to a finite interval, the subcritical percolation regime of the associated Poisson process: see Section 3.1. In that case the Gibbs distribution is not only uniquely determined (see Corollary 1) but can be expected to exist. In the case of a non-negative pair potential we have the following result.

Remark 2. Assume that the Papangelou intensity $\kappa$ is given by a non-negative pair potential, that is, assume that $\kappa$ is given by (2.13) with $n=2$ . Assume also that $\varphi_2$ has a finite interaction range or, more generally, that

\begin{equation*}\int (1-{\textrm{e}}^{-\varphi_2(K,L)})\,\mu(\textrm{d} K)<\infty\quad\text{for $\mu$-a.e.\ \textit{L}.}\end{equation*}

(By assumption (2.4) we have that $\mu(\Psi)<\infty$ for any ball $\Psi\subset{\mathcal{C}^{(d)}}$ .) Under these assumptions it has been shown in [Reference Jansen14] that a Gibbs process exists.

We do not further address the existence problem in this paper and proceed under the assumption that the Gibbs process exists.

For all $\xi,\chi\in{\textbf{N}}$ with disjoint supports, the Hamiltonian H takes the form

\begin{align*} H(\xi,\chi)& \coloneqq \sum_{n=2}^\infty \sum_{k=1}^n \dfrac{1}{k!(n-k)!} \iint \varphi_n(K_1,\ldots,K_k,L_1,\ldots,L_{n-k}) \\[3pt] &\quad\, \times \,\xi^{(k)}(\textrm{d}(K_1,\ldots,K_k)) \,\chi^{(n-k)}(\textrm{d}(L_1,\ldots,L_{n-k})),\end{align*}

provided that $\xi$ is finite. The assumption $\kappa\le 1$ implies that $H\ge 0$ . If assumptions (2.3) and (2.4) hold, then (2.11) shows that the Gibbs process $\Xi$ has bounded particles, that is,

\begin{equation*} \int{\bf{1}}\{K\not\subseteq B(z(K),R)\}\,\Xi(\textrm{d} K)=0, \quad \text{$\mathbb{P}$-a.s.}\end{equation*}

For clarity and to avoid lengthy formulations we make the following definition.

For an admissible Gibbs particle process it follows from (2.13), (2.8), and (2.9) that

\begin{equation*} \rho_{{p}}(K_1,\ldots,K_p)\le \lambda^p,\quad K_1,\ldots,K_p\in{\mathcal{C}^{(d)}}.\end{equation*}

A classic set-up of a repulsive intersection-based pair potential arises from a measurable translation-invariant function $U\colon \mathcal{C}^{d}\to[0,\infty]$ with $U(\emptyset)=0$ and setting $\varphi_2(K,L)\coloneqq\break U(K\cap{}L)$ and $\varphi_n\coloneqq 0$ , for $n\ge 3$ . Assumption (2.4) implies an interaction range of at most 4R.

Example 1. For $d\geq 2$ , let $\mathcal{G}_d$ be the space of $(d-1)$ -dimensional linear subspaces of $\mathbb{R}^d$ . Let $R>0$ and let the measure $\mathbb{Q}$ be concentrated on

\begin{equation*} V\coloneqq \{A\cap B({\bf 0},R)\mid A\in\mathcal{G}_d\}{.}\end{equation*}

Then (2.4) holds. The particles are called facets and $\mathbb{Q}$ can be interpreted as the distribution of their normal directions. The space of facets is

\begin{equation*} \tilde{V}\coloneqq \{B+x\mid B\in V,\;x\in\mathbb{R}^d\}.\end{equation*}

Let $\mathbb{H}^m$ be the Hausdorff measure of order $m \in \{1,\ldots,d\}$ on $\mathbb{R}^d$ . For $j\in\{1,\ldots,d\}$ and $K_1,\ldots,K_j\in\tilde{V}$ define, setting $0\cdot\infty=0$ ,

(2.14) \begin{equation} Q_j(K_1,\ldots,K_j)\coloneqq \mathbb{H}^{d-j} \biggl(\bigcap_{i=1}^{j} K_i\biggr) {\bf{1}}\biggl\{\mathbb{H}^{d-j}\biggl(\bigcap_{i=1}^{j} K_i\biggr)<\infty\biggr\} .\end{equation}

A family $\varphi\coloneqq (\varphi_j)_{j\geq 2}$ of higher-order potentials is defined by

\begin{equation*} \varphi_j(K_1,\ldots,K_j)\coloneqq a_jQ_j(K_1,\ldots,K_j),\;K_1,\ldots,K_j\in\tilde{V} ,\end{equation*}

for $j\in\{2,\ldots,d\}$ , and $\varphi_j\coloneqq 0$ otherwise, where $a_2,\ldots,a_d\geq 0$ are given parameters. All these potentials have the finite range $R_\varphi=2R.$ The corresponding Gibbs particle process $\Xi$ is called the Gibbs facet process. It is admissible and its existence follows from [Reference Dereudre, Drouilhet and Georgii6, Remarks 3.7 and 3.1]. For $j\in\{2,\ldots,d\}$ , the jth submodel is the special case of only the jth potential being active, i.e. only $a_j>0$ , and we denote it by $^j\Xi$ .

3.1. Percolation

Define a symmetric relation on ${\mathcal{C}^{(d)}}$ by setting $K\sim{}L$ if and only if $K\cap{}L\not=\emptyset$ . For $\xi\in{{\textbf{N}}_{}}$ , this defines a graph $(\textrm{supp}{}\,\xi,\sim)$ . For $K,L\in{\mathcal{C}^{(d)}}$ , we say that $\xi$ connects K and L if there exists a finite path between K and L in the graph on $\xi+\delta_K+\delta_L$ . For disjoint $\Psi,\Gamma\in{\mathcal{B}({\mathcal{C}^{(d)}})}$ , we say that $\xi$ connects $\Psi$ and $\Gamma$ if there exist $K\in\Psi$ and $L\in\Gamma$ such that $\xi$ connects K and L. We write $\Psi\xleftrightarrow{{\xi}}\Gamma$ for this.

We say that $\xi$ percolates if its graph contains an infinite connected component. Because connectedness is an increasing event in ${\mathcal{N}_{}}$ , there is a critical percolation intensity

\begin{equation*} \lambda_c({d}) \equiv\lambda_c({d,\mathbb{Q}})\in[0,\infty]\end{equation*}

for percolation of ${\Pi_{{\lambda\mu}}}$ ; see [Reference Meester and Roy21]. The following consequence of a result in [Reference Ziesche31] is of crucial importance for our main results.

Lemma 3. For $\lambda<\lambda_c({d})$ and $\Psi,\Gamma\in{\mathcal{B}_b({\mathcal{C}^{(d)}})}$ with $\Psi\subseteq{}\Gamma$ , there exist a monotone increasing $C_1\colon [0,\infty)\to[0,\infty)$ and $C_2\in\,(0,\infty)$ such that

(3.1) \begin{equation} \mathbb{P}\Bigl(\Psi\xleftrightarrow{{\Pi_{{\lambda\mu}}}}\Gamma^c\Bigr) \le C_1(\operatorname{diam}(\Psi))\exp({-}C_2d(\Psi,\Gamma^c)).\end{equation}

Proof. Let

\begin{equation*} \mathcal{C}_{\textbf{0}}^{(d,R)}\coloneqq \{K\in\mathcal{C}_{\textbf{0}}^{(d)}\mid K\subseteq{}B(\textbf{0},R)\}. \end{equation*}

By (2.4) we have that $\mathbb{Q}(\mathcal{C}_{\textbf{0}}^{(d,R)}) = 1$ . A slightly weakened form of the bound [Reference Ziesche31, equation (3.7)] in our notation is as follows. For $\lambda<\lambda_c({d})$ , there exists a constant $C\in\,(0,\infty)$ , such that, for all $r\ge 0$ ,

(3.2) \begin{equation} \int_{\mathcal{C}_{\textbf{0}}^{(d)}} \mathbb{P}\Bigl( K\xleftrightarrow{{\Pi_{{\lambda\mu}}}+\delta_K} B(\textbf{0},r)^c\times\mathcal{C}_{\textbf{0}}^{(d,R)} \Bigr) \,\mathbb{Q}(\textrm{d}{}K) \le {\textrm{e}}^{-C(r-2R)} .\end{equation}

The weakening is a result of a switch from the notion of connecting two sets in $\mathbb{R}^d$ used in [Reference Ziesche31] to the one connecting two sets in $\mathbb{R}^d\times\mathcal{C}_{\textbf{0}}^{(d,R)}$ used above. Thus we correct twice by R in the exponent on the right-hand side of (3.2) to account for the maximum size of K and of particles in $B(0,r)^c\times\mathcal{C}_{\textbf{0}}^{(d,R)}$ respectively.

If $\Psi=\emptyset$ , then the probability is zero in any case. Proceed by assuming that $\Psi\not=\emptyset$ .

Let $D\coloneqq \operatorname{diam}(\Psi)$ . Let $s\coloneqq d(\Psi,\Gamma^c) - D - 10R$ . If $s\le{}0$ , then choosing $C_1(D)\ge {\textrm{e}}^{C(D+10R)}$ finishes the proof. From here on, assume that $s>0$ and that all particles are deterministically bounded by R.

Choose and fix $K_\Psi\in\Psi$ and let $x\coloneqq z(K_\Psi)$ . Let $\Upsilon\coloneqq A\times\mathcal{C}_{\textbf{0}}^{(d,R)}$ be the particles with centres of circumscribed balls within the annulus $A\coloneqq B(x,D+6R)\setminus{}B(x,D+4R)$ .

For $K\in{}B(x,D+4R)\times\mathcal{C}_{\textbf{0}}^{(d,R)}=\!:\,\Upsilon^-$ and $L\in{}B(x,D+6R)^c\times\mathcal{C}_{\textbf{0}}^{(d,R)}=\!:\,\Upsilon^+$ , we have $|z(K)-z(L)|\ge 2R$ , whence $K\cap{}L=\emptyset$ . For $\xi\in{\textbf{N}}$ , let $P(\xi)$ be the particles of $\xi_{\Upsilon}$ connected in $\xi_{\Gamma\setminus{}\Upsilon^{-}}$ to $\Gamma^c$ . Then $\Psi\xleftrightarrow{{\xi}}\Gamma^c$ implies that $P(\xi)\not=\emptyset$ , because particles in $\Upsilon^-$ do not intersect particles in $\Upsilon^+$ and $\xi$ needs to contain at least one particle in $\Upsilon$ . Together with a first moment bound, this yields

\begin{equation*} \mathbb{P}\Bigl(\Psi\xleftrightarrow{{\Pi_{{\lambda\mu}}}}\Gamma^c\Bigr) \le{} \mathbb{P}(P({\Pi_{{\lambda\mu}}})\not=\emptyset) \le{} \mathbb{E}|P({\Pi_{{\lambda\mu}}})| .\end{equation*}

We apply the Mecke equation to rewrite

\begin{equation*} \mathbb{E}|P({\Pi_{{\lambda\mu}}})| ={} \mathbb{E}\int{\bf{1}}\{K\in{}P({\Pi_{{\lambda\mu}}})\}\,{\Pi_{{\lambda\mu}}}(\textrm{d}{}K) ={} \int_{\Upsilon} \mathbb{P} (K\in{}P({\Pi_{{\lambda\mu}}}+\delta_K)) \,\lambda\mu(\textrm{d}{}K) .\end{equation*}

For $K\in\Upsilon$ and $L\in\Gamma^c$ we have that

\begin{equation*} d_H(K,L) \ge d_H(K_\Psi,L) - d_H(K_\Psi,K) \ge d(\Psi,\Gamma^c) - (D + 6R + 2R) = s + 2R .\end{equation*}

In particular, we have $d_H(K,L)\ge 2R$ , so that $K\cap L=\emptyset$ . Moreover, for each $\xi\in{\textbf{N}}$ satisfying $K\xleftrightarrow{\xi}\Gamma^c$ , we obtain that $K\xleftrightarrow{\xi}B(z(K),s)^c\times\mathcal{C}_{\textbf{0}}^{(d,R)}$ . This implies that

\begin{equation*} \mathbb{P}(K\in{}P({\Pi_{{\lambda\mu}}}+\delta_K)) \le \mathbb{P}\Bigl(K\xleftrightarrow{{\Pi_{{\lambda\mu}}}+\delta_K}\Gamma^c\Bigr) \le \mathbb{P}\Bigl( K\xleftrightarrow{{\Pi_{{\lambda\mu}}}+\delta_K} B(z(K),s)^c\times\mathcal{C}_{\textbf{0}}^{(d,R)} \Bigr) .\end{equation*}

Combining these upper bounds and using the definition (2.2) of $\mu$ , we see that

\begin{align*} \mathbb{P}\Bigl(\Psi\xleftrightarrow{{\Pi_{{\lambda\mu}}}}\Gamma^c\Bigr) &\le \int_{\Upsilon} \mathbb{P}\Bigl( K\xleftrightarrow{{\Pi_{{\lambda\mu}}}+\delta_K} B(z(K),s)^c\times\mathcal{C}_{\textbf{0}}^{(d,R)} \Bigr) \,\mu(\textrm{d}{}K) \\[3pt] &= \int_{\mathcal{C}_{\textbf{0}}^{(d)}}\int_A \mathbb{P}\Bigl( K+x\xleftrightarrow{{\Pi_{{\lambda\mu}}}+\delta_{K+x}} B(x,s)^c\times\mathcal{C}_{\textbf{0}}^{(d,R)} \Bigr)\,{\textrm{d}} x \,\mathbb{Q}(\textrm{d}{}K).\end{align*}

By stationarity of ${\Pi_{{\lambda\mu}}}$ this equals

\begin{equation*} \lambda\mathcal{L}_d(A) \int_{\mathcal{C}_{\textbf{0}}^{(d)}} \mathbb{P}\Bigl( K\xleftrightarrow{{\Pi_{{\lambda\mu}}}+\delta_K} B(\textbf{0},s)^c\times\mathcal{C}_{\textbf{0}}^{(d,R)} \Bigr) \,\mathbb{Q}(\textrm{d}{}K)\le \lambda\mathcal{L}_d(A) \,{\textrm{e}}^{-C (s-2R)},\end{equation*}

where the inequality comes from (3.2). Choosing $C_1(D)\ge\lambda \mathcal{L}_d(A) \,{\textrm{e}}^{C(D+12R)}$ concludes the argument.

Monotonicity in the particle shapes allows to control the percolation threshold. In the special case of $\mathbb{Q} = \delta_{B(\textbf{0},R)}$ , the measure $\mu$ becomes $\mu_R \coloneqq \int {\bf{1}}\{B(x,R)\in \cdot\}\,\textrm{d} x$ . Assumption (2.4) implies for each $\lambda>0$ that $\Pi_{\lambda\mu}$ -a.e. $\xi$ fulfils

\begin{equation*} \bigcup_{K\in\xi} K \subseteq \bigcup_{K\in\xi} B(z(K),R).\end{equation*}

Hence we can couple ${\Pi_{{\lambda\mu}}}$ and ${\Pi_{{\lambda\mu_R}}}$ such that

\begin{equation*} \mathbb{P}\biggl( \bigcup_{K\in{\Pi_{{\lambda\mu}}}} K \subseteq \bigcup_{B\in{\Pi_{{\lambda\mu_R}}}} B \biggr)=1.\end{equation*}

A well-known lower bound (see [Reference Penrose22] and [Reference Meester and Roy21, Section 3.9]) is

(3.3) \begin{equation} \lambda_c({d,\mathbb{Q}}) \ge \lambda_c({d,\delta_{B(\textbf{0},R)}}) \ge \dfrac{1}{v_d 2^d R^d},\end{equation}

where $v_d$ is the volume of the d-dimensional unit ball.

3.2. Disagreement percolation

For $\xi,\xi'\in{}{{\textbf{N}}_{}}$ , we write $\xi\bigtriangleup{ }\xi'$ for the absolute difference measure $\max\{\xi,\xi'\}-\min\{\xi,\xi'\}$ , equivalent to $|\xi-\xi'|$ . In the relevant case of $\xi$ and $\xi'$ both being simple, there is a simpler geometric interpretation of the also simple $\xi\bigtriangleup{ }\xi'$ . Switching to the support of a simple point measure, we see that $\textrm{supp}(\xi\bigtriangleup{ }\xi')=(\textrm{supp}\,\xi)\bigtriangleup{ }(\textrm{supp}\,\xi')$ , which motivates this overloading of the set difference operator $\bigtriangleup{ }$ to point measures.

The space $({\mathcal{C}^{(d)}},d_H)$ is a complete and separable metric space. By [Reference Dudley7, Theorem 13.1.1], the spaces $({\mathcal{C}^{(d)}},{\mathcal{B}({\mathcal{C}^{(d)}})})$ and $\mathbb{R}$ equipped with the Borel $\sigma$ -algebra are Borel-isomorphic. That is, there exists a measurable bijection from ${\mathcal{C}^{(d)}}$ to $\mathbb{R}$ with measurable inverse. We use this bijection to pull back the total order from $\mathbb{R}$ to ${\mathcal{C}^{(d)}}$ and denote it by $\prec$ . Hence intervals with respect to $\prec$ are in ${\mathcal{B}({\mathcal{C}^{(d)}})}$ .

For the remainder of the section we fix an admissible Gibbs process $\Xi$ as in Definition 5.

Theorem 1. For all $\Psi\in{\mathcal{B}_b({\mathcal{C}^{(d)}})}$ and $\chi_1,\chi_2\in{{\textbf{N}}_{\Psi^c}}$ , there exists a simultaneous thinning from ${\Pi_{{\Psi,\lambda\mu}}}$ to two particle processes ${\Theta^{\Psi,\chi_1}_{}}$ and ${\Theta^{\Psi,\chi_2}_{}}$ such that ${\Theta^{\Psi,\chi_i}_{}}$ has the distribution

\begin{equation*}\mathbb{P}({\Xi_{\Psi}}\in\cdot\mid {\Xi_{\Psi^c}}=\chi_i)\quad\text{for $i\in\{1,2\}$,}\end{equation*}

and, $\mathbb{P}$ -a.s.,

(3.4) \begin{equation} \forall{}K\in\textrm{supp} ({\Theta^{\Psi,\chi_1}_{}}\bigtriangleup{ }{\Theta^{\Psi,\chi_2}_{}})\colon \{K\} \xleftrightarrow{{\Theta^{\Psi,\chi_1}_{}}\bigtriangleup{ }{\Theta^{\Psi,\chi_2}_{}}} \chi_1\bigtriangleup{ }\chi_2.\end{equation}

Proof. Using $\prec$ restricted to $\Psi$ in place of the measurable ordering of a bounded Borel subset in [Reference Hofer-Temmel and Houdebert13, Section 4.1], and the DLR-equations as formulated in (2.11), this theorem becomes a literal copy of the construction leading to [Reference Hofer-Temmel and Houdebert13, Theorem 3.3].

The term disagreement percolation comes from the fact that in the subcritical percolation regime of ${\Pi_{{\lambda\mu}}}$ , there is control of a disagreement cluster by a percolation cluster. The finiteness of the percolation clusters guarantees uniqueness of the Gibbs process.

Proof. The proof generalizes the proof of [Reference Hofer-Temmel and Houdebert13, Theorem 3.2] and Theorem 1 in a straightforward way, the only change being that the interaction range and particle size are two separate parameters. Because of the deterministic bound R from (2.4) on the particle size and the finiteness of the interaction range, the arguments remain the same.

3.3. Decorrelation of moments

With the following theorem we establish the particle counterpart of fast decay of correlations in [Reference Blaszczyszyn, Yogeshwaran and Yukich2, Definition 1.1] in the subcritical regime.

Theorem 2. Assume that $\lambda<\lambda_c({d})$ . There exist $c_1,c_2\in(0,\infty)$ such that Gibbs process ${\Xi_{}}$ satisfies, for all $p,q\in\mathbb{N}$ and $\alpha^{(p+q)}$ -a.e. $(K_1,\ldots,K_{p+q})$ ,

(3.5) \begin{align}\notag& |\rho_{{p+q}}(K_1,\ldots,K_{p+q})- \rho_{{p}}(K_1,\ldots,K_p) \rho_{{q}}(K_{p+1},\ldots,K_{p+q})|\\[3pt] &\quad \le \lambda^{p+q} \min(p,q)\,c_1 \exp({-}c_2 d(\{K_1,\ldots,K_{p}\},\{K_{p+1},\ldots,K_{p+q}\})) .\end{align}

Combining Theorem 2 with the bound (3.3) on the percolation threshold gives the following constraint on the activity as a sufficient condition for the exponential decay of correlations (3.5):

(3.6) \begin{equation} \lambda <\dfrac{1}{v_d 2^d R^d}.\end{equation}

Remark 3. To compare our results with Proposition 2.1 in [Reference Schreiber and Yukich26] we consider hard spheres in equilibrium, that is, we assume that $\mathbb{Q} = \delta_{B(\textbf{0},R)}$ , $\varphi_2(K,L)=\infty\cdot{\bf{1}}\{K\cap L\ne \emptyset\}$ and $\varphi_n\equiv 0$ for $n\ge 3$ . Then $\Xi$ can be identified with a point process on $\mathbb{R}^d$ . In the language from [Reference Schreiber and Yukich26] we have that $r^\Psi=2R$ and $m^\Psi_0=0$ . Proposition 2.1 in [Reference Schreiber and Yukich26] then shows exponential decay of correlations whenever

(3.7) \begin{equation} \lambda<\dfrac{1}{v_d(1+2R)^d}.\end{equation}

This is comparable with (3.6).

Our Theorem 2 shows exponential decay of correlations for a broader range of activities. In fact, it is known from simulations that in low dimensions $\lambda_c(d)$ is considerably larger than the right-hand side of (3.6). (As $d\to\infty$ we have $\lambda_c(d)v_d2^dR^d\to 1$ ; see [Reference Penrose22].) We do not see how the methods from [Reference Schreiber and Yukich26] can be used to prove Theorem 2.

The proof of Theorem 2 is based on the following lemmas. For these lemmas and the proof of Theorem 2, fix $\lambda<\lambda_c({d})$ and let $C_1$ and $C_2$ as in Lemma 3.

Lemma 4. Let $p\in\mathbb{N}$ and let $\Psi_1,\ldots,\Psi_p\in{\mathcal{B}_b({\mathcal{C}^{(d)}})}$ be disjoint. Let $\Gamma\in{\mathcal{B}_b({\mathcal{C}^{(d)}})}$ with

\begin{equation*}\bigcup_{i=1}^p\Psi_i=\!:\,\Psi\subseteq{}\Gamma.\end{equation*}

For $\chi\in{{\textbf{N}}_{\Gamma^c}}$ and increasing $E\in\mathcal{N}$ ,

(3.8) \begin{align} & | \mathbb{P}({\Xi_{\Psi}}\in{}E\mid {\Xi_{\Gamma^c}}=\chi) -\mathbb{P}({\Xi_{\Psi}}\in{}E)|\notag \\[3pt] &\quad \le \mathbb{P}({\Pi_{{\Psi,\lambda\mu}}}\in{}E) \sum_{i=1}^p C_1(\!\operatorname{diam}\!(\Psi_i))\exp({-}C_2 d(\Psi,\Gamma^c)) .\end{align}

Proof. First we show that, for $\chi_1,\chi_2\in{{\textbf{N}}_{\Gamma^c}}$ ,

(3.9) \begin{align}& | \mathbb{P}({\Xi_{\Psi}}\in{}E\mid {\Xi_{\Gamma^c}}=\chi_1) - \mathbb{P}({\Xi_{\Psi}}\in{}E\mid {\Xi_{\Gamma^c}}=\chi_2)|\notag \\[3pt] &\quad \le \mathbb{P}({\Pi_{{\Psi,\lambda\mu}}}\in{}E) \sum_{i=1}^p C_1(\operatorname{diam}(\Psi_i))\exp({-}C_2 d(\Psi,\Gamma^c)) .\end{align}

By Theorem 1,

\begin{align*}&|\mathbb{P}({\Xi_{\Psi}}\in{}E\mid {\Xi_{\Gamma^c}}=\chi_1)-\mathbb{P}({\Xi_{\Psi}}\in{}E\mid {\Xi_{\Gamma^c}}=\chi_2) |\\[3pt] &\quad = |\mathbb{P}({\Theta^{\Gamma,\chi_1}_{\Psi}}\in{}E)-\mathbb{P}({\Theta^{\Gamma,\chi_2}_{\Psi}}\in{}E)|\\[3pt] &\quad = | \mathbb{P}({\Theta^{\Gamma,\chi_1}_{\Psi}}\in{}E ,{\Theta^{\Gamma,\chi_2}_{\Psi}}\not\in{}E)- \mathbb{P}({\Theta^{\Gamma,\chi_1}_{\Psi}}\not\in{}E ,{\Theta^{\Gamma,\chi_2}_{\Psi}}\in{}E)| \\[3pt] &\quad \le \max\{ \mathbb{P}({\Theta^{\Gamma,\chi_1}_{\Psi}}\in{}E ,{\Theta^{\Gamma,\chi_2}_{\Psi}}\not\in{}E) ,\, \mathbb{P}({\Theta^{\Gamma,\chi_1}_{\Psi}}\not\in{}E ,{\Theta^{\Gamma,\chi_2}_{\Psi}}\in{}E)\}.\end{align*}

By symmetry we only need to bound the first term in the above maximum. It follows from (3.4) that

\begin{align*} & \mathbb{P}({\Theta^{\Gamma,\chi_1}_{\Psi}}\in{}E , {\Theta^{\Gamma,\chi_2}_{\Psi}}\not\in{}E)\\[3pt] &\quad = \mathbb{P}({\Theta^{\Gamma,\chi_1}_{\Psi}}\in{}E ,{\Theta^{\Gamma,\chi_2}_{\Psi}}\not\in{}E ,\emptyset\not= {\Theta^{\Gamma,\chi_1}_{\Psi}}\bigtriangleup{ }{\Theta^{\Gamma,\chi_2}_{\Psi}} ) \\[3pt] &\quad = \mathbb{P}( {\Theta^{\Gamma,\chi_1}_{\Psi}}\in{}E ,{\Theta^{\Gamma,\chi_2}_{\Psi}}\not\in{}E ,\emptyset\not= {\Theta^{\Gamma,\chi_1}_{\Psi}}\bigtriangleup{ }{\Theta^{\Gamma,\chi_2}_{\Psi}} \xleftrightarrow{ {\Theta^{\Gamma,\chi_1}_{}} \bigtriangleup{ } {\Theta^{\Gamma,\chi_2}_{}}} \chi_1\bigtriangleup{ }\chi_2 ).\end{align*}

Since

\begin{equation*}\Bigl\{{\Theta^{\Gamma,\chi_1}_{\Psi}}\bigtriangleup{ }{\Theta^{\Gamma,\chi_2}_{\Psi}}\xleftrightarrow{{\Theta^{\Gamma,\chi_1}_{}}\bigtriangleup{ } {\Theta^{\Gamma,\chi_2}_{}}} \chi_1\bigtriangleup{ }\chi_2\Bigr\}\subseteq \Bigl\{\Psi\xleftrightarrow{{\Theta^{\Gamma,\chi_1}_{}} \bigtriangleup{ } {\Theta^{\Gamma,\chi_2}_{}}} \chi_1\bigtriangleup{ }\chi_2\Bigr\}\subseteq\Bigl\{\Psi\xleftrightarrow{{\Pi_{{\lambda\mu}}}}\Gamma^c\Bigr\},\end{equation*}

we obtain that

\begin{align*} \mathbb{P}({\Theta^{\Gamma,\chi_1}_{\Psi}}\in{}E,{\Theta^{\Gamma,\chi_2}_{\Psi}}\not\in{}E)&\le \mathbb{P}\Bigl({\Pi_{{\Psi,\lambda\mu}}}\in{}E ,\Psi\xleftrightarrow{{\Pi_{{\lambda\mu}}}}\Gamma^c\Bigr)\\*&= \mathbb{P}({\Pi_{{\Psi,\lambda\mu}}}\in{}E) \mathbb{P}\Bigl(\Psi\xleftrightarrow{{\Pi_{{\lambda\mu}}}}\Gamma^c\Bigr)\\&\le \mathbb{P}({\Pi_{{\Psi,\lambda\mu}}}\in{}E) \sum_{i=1}^p \mathbb{P}\Bigl(\Psi_i\xleftrightarrow{{\Pi_{{\lambda\mu}}}}\Gamma^c\Bigr)\\*&\le \mathbb{P}({\Pi_{{\Psi,\lambda\mu}}}\in{}E) \sum_{i=1}^p C_1(\!\operatorname{diam}(\Psi_i))\exp({-}C_2 d(\Psi,\Gamma^c)) ,\end{align*}

where the equality results from the complete independence of a Poisson process, the second inequality is a Boolean bound and the final equality uses (3.1) and the fact that $d(\Psi,\Gamma^c)\le d(\Psi_i,\Gamma^c)$ . This proves (3.9).

To prove (3.8), we use the DLR-equation (2.11) to obtain that

\begin{align*}& | \mathbb{P}({\Xi_{\Psi}}\in{}E\mid {\Xi_{\Gamma^c}}=\chi)-\mathbb{P}({\Xi_{\Psi}}\in{}E)|\\*&\quad \le\int|\mathbb{P}({\Xi_{\Psi}}\in{}E\mid {\Xi_{\Gamma^c}}=\chi)-\mathbb{P}({\Xi_{\Psi}}\in{}E\mid {\Xi_{\Gamma^c}}=\chi')|\,\mathbb{P}({\Xi_{\Gamma^c}}\in\textrm{d}{}\chi').\end{align*}

An application of (3.9) to the integrand shows (3.8).

For $\Psi\in{\mathcal{B}_b({\mathcal{C}^{(d)}})}$ and $n\in\mathbb{N}$ , let $E_{{\Psi,n}}\coloneqq \{\xi\in{\textbf{N}}\mid \xi(\Psi)\ge{}n\}$ . The event $E_{{\Psi,n}}$ is increasing. Fix $p\in\mathbb{N}$ and disjoint $\Psi_1,\ldots,\Psi_p\in{\mathcal{B}_b({\mathcal{C}^{(d)}})}$ . Let ${\Phi_{}}$ be a particle process. Then

(3.10) \begin{equation} \mathbb{E}\biggl(\prod_{i=1}^{p} {\Phi_{}}(\Psi_i)\biggr) =\sum_{\vec{n}\in\mathbb{N}^d} \mathbb{P}( \forall{}1\le{}i\le{}p\colon \Phi_{\Psi_i}\in E_{{\Psi_i,n_i}} ),\end{equation}

where we use the notation $\vec{n}=\!:\,(n_1,\ldots,n_d)$ .

Lemma 5. For $p\in\mathbb{N}$ and disjoint $\Psi_1,\ldots,\Psi_p\in{\mathcal{B}_b({\mathcal{C}^{(d)}})}$ ,

(3.11) \begin{equation} \mathbb{E}\biggl(\prod_{i=1}^p \Xi(\Psi_i)\biggr) \le \lambda^p \prod_{i=1}^p \mu(\Psi_i).\end{equation}

Proof. Applying (3.10), then stochastic domination from Lemma 2, applying (3.10) again and writing out the moment of the Poisson process yields the bound.

Lemma 6. Let $p\in\mathbb{N}$ and let $\Psi_1,\ldots,\Psi_p\in{\mathcal{B}_b({\mathcal{C}^{(d)}})}$ be disjoint. Suppose that $\Gamma\in{\mathcal{B}_b({\mathcal{C}^{(d)}})}$ satisfies $\Gamma\supseteq{}\Psi\coloneqq \bigcup_{i=1}^p \Psi_i$ and let $\chi\in{{\textbf{N}}_{\Gamma^c}}$ . Then

(3.12) \begin{equation} \biggl| \mathbb{E}\biggl(\prod_{i=1}^p {\Xi_{}}(\Psi_i)\biggr)-\mathbb{E}\biggl(\prod_{i=1}^p {\Xi_{}}(\Psi_i) \biggm| {\Xi_{\Gamma^c}} = \chi\biggr) \biggr| \le\lambda^p\biggl(\prod_{i=1}^p \mu(\Psi_i)\biggr)C_\Psi\exp({-}C_2d(\Psi,\Gamma^c)),\end{equation}

where $C_\Psi\coloneqq \sum_{i=1}^{q}C_1(\operatorname{diam}(\Psi_i))$ .

Proof. Applying (3.10) and then (3.8) gives

\begin{align*} &\biggl| \mathbb{E}\biggl(\prod_{i=1}^p {\Xi_{}}(\Psi_i)\biggr)-\mathbb{E}\biggl(\prod_{i=1}^p {\Xi_{}}(\Psi_i) \biggm| {\Xi_{\Gamma^c}} = \chi\biggr) \biggr|\\*&\quad = \sum_{\vec{n}\in\mathbb{N}^d} | \mathbb{P}( \forall{}1\le{}i\le{}p\colon{} {\Xi_{\Psi_i}}\in E_{{\Psi_i,n_i}}) -\mathbb{P}(\forall{}1\le{}i\le{}p\colon{} {\Xi_{\Psi_i}}\in E_{{\Psi_i,n_i}} \mid {\Xi_{\Gamma^c}}=\chi) |\\*&\quad \le \sum_{\vec{n}\in\mathbb{N}^d} \mathbb{P}( \forall{}1\le{}i\le{}p\colon {\Pi_{{\lambda\mu}}}\in E_{{\Psi_i,n_i}} ) \sum_{i=1}^{p}C_1(\operatorname{diam}(\Psi_i))\exp({-}C_2 d(\Psi,\Gamma^c)).\end{align*}

Conclude by another application of (3.10) and writing out the Poisson moment.

Lemma 7. For $p,q\in\mathbb{N}$ and disjoint $\Psi_1,\ldots,\Psi_p,\Upsilon_1,\ldots,\Upsilon_q\in\mathcal{B}_b$ ,

\begin{align*} & \biggl| \mathbb{E} \biggl(\prod_{i=1}^p {\Xi_{}}(\Psi_i)\biggr) \biggl(\prod_{j=1}^q {\Xi_{}}(\Upsilon_j)\biggr) -\mathbb{E}\biggl(\prod_{i=1}^p {\Xi_{}}(\Psi_i)\biggr) \mathbb{E}\biggl(\prod_{i=1}^q {\Xi_{}}(\Upsilon_i)\biggr) \biggr|\notag \\*&\quad \le \lambda^{p+q} \biggl(\prod_{i=1}^p \mu(\Psi_i)\biggr) \biggl(\prod_{i=1}^q \mu(\Upsilon_i)\biggr) \min(C_{\Psi},C_{\Upsilon}) \exp({-}C_2 d(\Psi,\Upsilon)) ,\end{align*}

where

\begin{equation*} \Psi \coloneqq \bigcup_{i=1}^p \Psi_i,\quad C_{\Psi} \coloneqq \sum_{i=1}^{p}C_1(\operatorname{diam}(\Psi_i)),\quad \Upsilon \coloneqq \bigcup_{i=1}^q \Upsilon_i,\quad C_{\Upsilon} \coloneqq \sum_{i=1}^{q}C_1(\operatorname{diam}(\Upsilon_i)).\end{equation*}

Proof. Let $\Gamma$ be a large sphere such that $d(\Psi\cup{}\Upsilon,\Gamma^c) > d(\Psi,\Gamma^c)$ . Hence

(3.13) \begin{equation} d(\Psi,\Upsilon\cup\Gamma^c) = d(\Psi,\Upsilon).\end{equation}

By the DLR-equation (2.11),

\begin{align*}& \biggl|\mathbb{E} \biggl(\prod_{i=1}^p {\Xi_{}}(\Psi_i)\biggr) \biggl(\prod_{j=1}^q {\Xi_{}}(\Upsilon_j)\biggr) -\mathbb{E}\biggl(\prod_{i=1}^p {\Xi_{}}(\Psi_i)\biggr) \mathbb{E}\biggl(\prod_{i=1}^q {\Xi_{}}(\Upsilon_i)\biggr) \biggr| \\* &\quad \le \int\biggl|\mathbb{E}\biggl(\prod_{i=1}^p {\Xi_{}}(\Psi_i)\biggm| {\Xi_{\Upsilon\cup\Gamma^c}}=\chi\biggr) -\mathbb{E}\biggl(\prod_{i=1}^p {\Xi_{}}(\Psi_i)\biggr) \biggr|\biggl(\prod_{j=1}^q \chi(\Upsilon_j)\biggr) \,\mathbb{P}({\Xi_{\Upsilon\cup\Gamma^c}}\in\textrm{d}\chi).\end{align*}

Applying (3.12) with $\Gamma$ replaced by $\Gamma\setminus\Upsilon$ and noting that $(\Gamma\setminus\Upsilon)^c=\Upsilon\cup\Gamma^c$ , we continue the bounds from the previous display:

\begin{align*} & \le{} \lambda^p \biggl(\prod_{i=1}^p \mu(\Psi_i)\biggr) C_{\Psi}\!\exp({-}C_2 d(\Psi,\Upsilon\cup\Gamma^c)) \int\biggl(\prod_{j=1}^q \chi(\Upsilon_j)\biggr) \,\mathbb{P}({\Xi_{\Upsilon\cup\Gamma^c}}\in\textrm{d}\chi) \quad\\ & = \lambda^p \biggl(\prod_{i=1}^p \mu(\Psi_i)\biggr) C_{\Psi}\!\exp({-}C_2 d(\Psi,\Upsilon\cup\Gamma^c)) \mathbb{E}\biggl(\prod_{j=1}^q {\Xi_{}}(\Upsilon_j)\biggr) \\* & \le \lambda^{p+q} \biggl(\prod_{i=1}^p \mu(\Psi_i)\biggr) \biggl(\prod_{j=1}^q \mu(\Upsilon_j)\biggr) C_{\Psi}\exp({-}C_2 d(\Psi,\Upsilon)),\end{align*}

where we use (3.13) and (3.11) to obtain the final inequality. The improved bound $\min(C_{\Psi},C_{\Upsilon})$ follows from the symmetry in $\Psi$ and $\Upsilon$ .

Proof of Theorem 2. Let $\Psi_{nj}$ , $n,j\in\mathbb{N}$ , be a dissection system in ${\textbf{N}}$ [Reference Kallenberg16, page 20]. Let $k\in\mathbb{N}$ and let $\alpha_k(\!\cdot\!)\coloneqq \mathbb{E} \Xi^k(\!\cdot\!)$ denote the kth moment measure of $\Xi$ . Let $\alpha_k=\alpha^{\prime}_k+\alpha^{\prime\prime}_k$ be the Lebesgue decomposition ([Reference Kallenberg16, Corollary 1.29]) of $\alpha_k$ with respect to $\mu^k$ , that is, $\alpha^{\prime}_k$ is absolutely continuous with respect to $\mu^k$ while $\alpha^{\prime\prime}_k$ and $\mu^k$ are mutually singular. Define

\begin{equation*}g_k(K_1,\ldots,K_{k})\coloneqq \limsup_{n\to\infty}\sum_{j_1,\ldots,j_{k}\in\mathbb{N}}\mu^{k}(\Psi_{n\vec{j}})^{-1}\alpha_{k}(\Psi_{n\vec{j}}){\bf{1}}\{(K_1,\ldots,K_{k})\in \Psi_{n\vec{j}}\},\end{equation*}

where we write $\vec{j}\coloneqq (j_1,\ldots,j_{k})$ and $\Psi_{n,\vec{j}}\coloneqq \Psi_{nj_1}\times\cdots\times\Psi_{nj_{k}}$ and where we set $a/0\coloneqq 0$ for all $a\in\mathbb{R}$ . Outside the generalized diagonal

\begin{equation*} D_{k}\coloneqq \{(K_1,\ldots,K_{k}) \in ({\mathcal{C}^{(d)}})^{k}\mid\text{there exist $\{i,j\}\subseteq\{1,\ldots,n\}$ with $K_i = K_j$}\}{,}\end{equation*}

the function $g_k$ coincides with

\begin{equation*}g_k^{\ne}(K_1,\ldots,K_{k})\coloneqq \limsup_{n\to\infty}\sideset{}{^{\ne}}\sum_{j_1,\ldots,j_{k}\in\mathbb{N}}\mu^{k}(\Psi_{n\vec{j}})^{-1}\alpha_{k}(\Psi_{n\vec{j}}){\bf{1}}\{(K_1,\ldots,K_{k})\in \Psi_{n\vec{j}}\},\end{equation*}

where the superscript $\ne$ indicates summation over k-tuples with distinct entries. Since $\Xi$ is simple, the measure $\alpha^{(k)}$ is the restriction of $\alpha_{k}$ to the complement of $D_{k}$ ; see [Reference Last and Penrose18, Exercise 6.9] and the proof of [Reference Last and Penrose18, Theorem 6.13]. Moreover, Lemma 5 shows that $\alpha^{(k)}$ is absolutely continuous with respect to $\mu^k$ . Therefore we obtain from the proof of [Reference Kallenberg16, Theorem 1.28] and (2.10) that the above superior limits are actually limits for $\mu^k$ -a.e. $(K_1,\ldots,K_{k})$ and, moreover, that

(3.14) \begin{equation} \rho_{{k}}(K_1,\ldots,K_{k}) = g_k(K_1,\ldots,K_{k}) ,\quad \text{$\alpha^{(k)}$-a.e.\ $(K_1,\ldots,K_{k})\in({\mathcal{C}^{(d)}})^{k}$}.\end{equation}

Let $p,q\in\mathbb{N}$ . Using (3.14) for $k\in\{p+q,p,q\}$ and combining this with Lemma 7, we obtain for $\alpha^{(p+q)}$ -a.e. $(K_1,\ldots,K_{p+q})\in({\mathcal{C}^{(d)}})^{p+q}$ that

(3.15) \begin{align}\notag &|\rho_{{p+q}}(K_1,\ldots,K_{p+q})- \rho_{{p}}(K_1,\ldots,K_p) \rho_{{q}}(K_{p+1},\ldots,K_{p+q})| \\[2pt] \notag&\quad \le\lambda^{p+q} \limsup_{n\to\infty} \\[2pt] \notag&\quad\, \quad\times \sideset{}{^{\ne}} \sum_{j_1,\ldots,j_{p+q}\in\mathbb{N}} \min(p,q)C_1(1) \exp({-}C_2 d(\cup^p_{i=1}\Psi_{nj_i},\cup^{p+q}_{i=p+1}\Psi_{nj_i})) {\bf{1}}\{(K_i)_{i=1}^{p+q}\in \Psi_{n\vec{j}}\} \\[2pt] &\quad = \lambda^{p+q}\min(p,q)C_1(1) \exp({-}C_2 d(\{K_1,\ldots,K_p\},\{K_{p+1},\ldots,K_{p+q}\})) .\end{align}

Here the inequality can be obtained from Lemma 7 as follows. Fix $(K_1,\ldots,K_{p+q})\in({\mathcal{C}^{(d)}})^{p+q}$ such that $K_i\ne K_j$ for $i\ne j$ . Then, for all sufficiently large $n\in\mathbb{N}$ , there exists a unique $\vec{j}\in \mathbb{N}^{p+q}$ with distinct entries such that $(K_1,\ldots,K_{p+q})\in\Psi_{n\vec{j}}=\!:\,\Psi_n(K_1,\ldots,K_{p+q})$ . As $n\to\infty$ we have $\Psi_n(K_1,\ldots,K_{p+q})\downarrow\{K_1,\ldots,K_{p+q}\}$ . Moreover, the diameter of $\Psi_n(K_1,\ldots,K_{p+q})$ (with respect to the product of the Hausdorff metric) tends to 0. These facts also imply the identity (3.15). Indeed, we just need to combine them with definition (2.1) of $d(\cdot,\cdot)$ and the triangle inequality.

4.1. Admissible U-statistics

A function $h\colon ({\mathcal{C}^{(d)}} )^{k} \rightarrow \mathbb{R}$ is called symmetric if, for all $K_{1},\ldots,K_{k} \in {\mathcal{C}^{(d)}}$ and every permutation $\pi$ of k elements, $h(K_{1},\ldots,K_{k}) = h(K_{\pi(1)},\ldots,K_{\pi(k)})$ . It is translation-invariant if $h(K_{1},\ldots,K_{k}) = h(\theta_{x}K_{1},\ldots,\theta_{x}K_{k})$ , for all $K_1,\ldots,K_k \in{\mathcal{C}^{(d)}} $ and $x \in \mathbb{R}^{d}$ . Given a measurable, symmetric, and translation-invariant function h, we can define

(4.1) \begin{equation} F_h(\xi)\coloneqq \dfrac{1}{k!}\int h(K_1,\ldots, K_k)\,\xi^{(k)}(\textrm{d}(K_{1},\ldots,K_{k})),\quad \xi\in{\textbf{N}}.\end{equation}

In fact, the functions $F_h(\Xi)$ and $F_h(\Xi_n)$ are U-statistics of order k; see [Reference Reitzner and Schulte23] and [Reference Last and Penrose18, Chapter 12]. Define

(4.2) \begin{equation} T(K,\xi)\coloneqq \dfrac{1}{k!}\int h(K,K_2,\ldots, K_k)\,\xi^{(k-1)}(\textrm{d}(K_{2},\ldots,K_{k})),\quad (K,\xi) \in \mathcal{C}^{(d)} \times {\textbf{N}},\end{equation}

where the case $k=1$ has to be read as $T(K)\coloneqq h(K)$ . Then

\begin{equation*}F_h(\xi)\coloneqq \int T(K,\xi)\,\xi(\textrm{d} K),\quad \xi\in{\textbf{N}}.\end{equation*}

The authors of [Reference Blaszczyszyn, Yogeshwaran and Yukich2] call T a score function.

Definition 6. Let $k\in\mathbb{N}$ and let $h\colon ({\mathcal{C}^{(d)}} )^{k} \rightarrow \mathbb{R}$ be measurable, symmetric, and translation-invariant. Then $F_h$ in (4.1) is called an admissible function (of order k) if $h(K_{1},\ldots,K_{k})=0$ , whenever either

(4.3) \begin{equation} \max_{2 \leq i \leq k} d_{H}(K_{i},K_{1}) > r,\end{equation}

for some given $r>0$ , or when $K_{i} = K_{1}$ , for some $i\in\{2,\ldots,k\}$ . If, moreover,

(4.4) \begin{equation} \|h \|_{\infty}\coloneqq \sup_{K_{1},\ldots,K_{k} \in {\mathcal{C}^{(d)}}} |h(K_{1},\ldots,K_{k})| < \infty,\end{equation}

then $F_h(\Xi_n)$ , $n\in\mathbb{N}$ , is called an admissible U-statistic of order k (of the Gibbs process $\Xi_n$ ).

Example 2. Consider Example 1 and recall that ${\textbf{N}}_{\tilde{V}}\coloneqq \{\xi\in{\textbf{N}}\mid \xi(\tilde{V}^c)=0\}$ . For $\xi\in {\textbf{N}}_{\tilde{V}}$ and $j\in\{1,\ldots,d\}$ , define using $Q_j$ from (2.14)

\begin{equation*} G_j(\xi)\coloneqq \dfrac{1}{j!}\int_{\tilde{V}^j} Q_j(K_1,\ldots,K_{j}) \,\xi^{(j)}(\textrm{d}(K_1,\ldots,K_{j})).\end{equation*}

Then $G_j$ is an admissible function with $r=2R$ ; see (2.4). In the special case of $d=2$ , facets are segments in the plane and $G_2(\xi) $ is the total number of intersections between segments in $\operatorname{supp}\xi$ having different orientation. For $d=3$ , facets are thin circular plates and $G_2(\xi )$ is the total length of intersections of pairs of facets in $\operatorname{supp}\xi$ .

Admissible functions satisfy a moment condition introduced in [Reference Blaszczyszyn, Yogeshwaran and Yukich2, Definition 1.8].

Proposition 1. If $F_h$ is an admissible function of order k and $p\in\mathbb{N}$ , then

(4.5) \begin{equation} \sup_{n\in\mathbb{N}} \sup_{1\leq q\leq p} \sup_{K_1,\ldots,K_q\in \mathcal{C}_n^d} \mathbb{E}_{K_1,\ldots,K_q} [\!\max\{|T(K_1,\Xi_n)|, 1\}^p ] < \infty,\end{equation}

where T is from (4.2) and the inner supremum is an essential supremum with respect to the qth factorial moment measure of $\Xi$ .

Proof. Let $q\in\{1,\ldots,p\}$ . Since h is admissible, property (4.4) implies that

\begin{equation*} \max \{|T(K,\xi)|,1 \}^{p} \leq g_2(K,\xi), \quad (K,\xi)\in {\mathcal{C}^{(d)}}\times{\textbf{N}},\end{equation*}

where

\begin{equation*} g_2(K,\xi)\coloneqq c\max \{1, (\xi(B(K,r))^{k-1})^{p}\}\end{equation*}

with $c\coloneqq \max\{1,\|h \|_{\infty}/k!\}$ . In the following we argue for $\mu^q$ -a.e. $(K_1,\ldots,K_q)\in ({\mathcal{C}^{(d)}})^q$ . Since $\kappa\le 1$ , Lemma 2 shows that $\mathbb{P}_{K_1,\ldots,K_q}^{\,!}$ is stochastically dominated by the distribution of $\Pi_{\lambda\mu}$ . Therefore

\begin{align*} \mathbb{E}_{K_1,\ldots,K_q} [\!\max\{|T(K_1,\Xi_n)|, 1\}^p ] &\le \mathbb{E}[g_2(K_1,\Pi_{\lambda\mu}+\delta_{K_1}+\dotsb+\delta_{K_q})] \\[2pt] &\le c\, \mathbb{E}[(q+\Pi_{\lambda\mu}(B(K_1,r)))^{p(k-1)}].\end{align*}

Using the inequality $(a+b)^{p(k-1)}\le 2^{p(k-1)-1}(a^{p(k-1)}+b^{p(k-1)})$ , for $a,b>0$ , we obtain that

\begin{equation*} \mathbb{E}_{K_1,\ldots,K_q} [\!\max\{|T(K_1,\Xi_n)|, 1\}^p ] \le c2^{p(k-1)-1}\, (q^{p(k-1)-1}+\mathbb{E}[\Pi_{\lambda\mu}(B(K_1,r))^{p(k-1)}]).\end{equation*}

The random variable $\Pi_{\lambda\mu}(B(K_1,r))$ has a Poisson distribution with parameter

\begin{align*}\mathbb{E}[\Pi_{\lambda\mu}(B(K_1,r))]&=\lambda\int {\bf{1}}\{d_H(K,K_1)\le r\}\,\mu(\textrm{d} K)\\[2pt] &=\lambda\iint {\bf{1}}\{d_H(K+x,K_1)\le r\}\,\mathbb{Q}(\textrm{d} K)\, \textrm{d} x.\end{align*}

By (2.3) and (2.4) we may assume that $K_1\subset B(z(K_1),R)$ . Assume that $K\in {\mathcal{C}^{(d)}}$ satisfies $K\subset B(\textbf{0},R)$ and that $\|x-z(K_1)\|>R$ . It follows from the definition of the Hausdorff distance that $d_H(K+x,K_1)\ge \|x-z(K_1)\|-R$ . Hence, uniformly in $K_1$ under our assumptions, we obtain the finite bound

\begin{align*} &\mathbb{E}[\Pi_{\lambda\mu}(B(K_1,r))]\\[2pt] &\quad \le \lambda\int {\bf{1}}\{\|x-z(K_1)\|\le R\}\, \textrm{d} x +\lambda\int {\bf{1}}\{R<\|x-z(K_1)\|\le r+R\}\, \textrm{d} x\\[2pt] &\quad = \lambda\int {\bf{1}}\{\|x\|\le r+R\}\, \textrm{d} x.\end{align*}

Thus the assertion follows from the moment properties of a Poisson random variable.

4.2. Factorization of weighted mixed moments

In this subsection we study an admissible pair $(\Xi ,T)$ defined as follows.

Given $n,p,k_1, \ldots,k_{p}\in\mathbb{N}$ and $K_1,\ldots,K_{p}\in {\mathcal{C}^{(d)}}$ , we define the weighted mixed moment

(4.6) \begin{equation} m_n^{(k_1,\ldots,k_{p})}(K_1,\ldots,K_{p}) \coloneqq \rho_{{p}}(K_1,\ldots,K_{p}) \int_{\textbf{N}} ( T(K_1,\xi_{n})^{k_1}\dots T(K_{p},\xi_{n})^{k_{p}} ) \,\mathbb{P}_{K_1,\ldots,K_{p}}(\textrm{d}\xi).\end{equation}

In the following all equations and inequalities involving Palm distributions and correlation functions are to be understood in the a.e. sense with respect to the appropriate factorial moment measures of $\Xi$ .

Definition 8. We adapt the terminology of [Reference Blaszczyszyn, Yogeshwaran and Yukich2] and say that weighted mixed moments have fast decay of correlations if there exist constants $a_l,b_l>0$ , $l\in\mathbb{N}$ , such that

(4.7) \begin{align} &|m_n^{(k_1,\ldots,k_{p+q})}(K_1,\ldots,K_{p+q}) - m_n^{(k_1,\ldots,k_{p})}(K_1,\ldots,K_{p}) m_n^{(k_{p+1},\ldots,k_{p+q})}(K_{p+1},\ldots,K_{p+q})|\notag \\[2pt] &\quad \le b_t \exp({-}a_t d(\{K_1,\ldots,K_{p}\},\{K_{p+1},\ldots,K_{p+q}\})),\end{align}

for all $n,p,q,k_1, \ldots,k_{p+q}\in\mathbb{N}$ and for all $K_1,\ldots,K_{p+q}\in z^{-1}(W_n)$ , where $t\coloneqq \sum_{i=1}^{p+q}k_{i}$ .

We intend to use Theorem 2 to show that (4.7) holds in our context. The method from [Reference Blaszczyszyn, Yogeshwaran and Yukich2] is used and transformed step by step from point processes on $\mathbb{R}^{d}$ to particle processes. Recall that $\prec$ is the total order of ${\mathcal{C}^{(d)}}$ introduced in Section 3.2 and that intervals with respect to $\prec$ are in ${\mathcal{B}({\mathcal{C}^{(d)}})}$ . In particular, for $K\in{\mathcal{C}^{(d)}}$ , $({-}\infty,K)=\{L\in{\mathcal{C}^{(d)}}\mid L \prec K\}$ .

Let o be the zero-measure, i.e. $o(\Psi)=0$ , for all $\Psi \in \mathcal{B}({\mathcal{C}^{(d)}} )$ . Abbreviate $[l] \coloneqq \{1,\ldots,l\}$ , for $l\in\mathbb{N}$ . We define a difference operator for a measurable function $\psi\colon {\textbf{N}} \rightarrow \mathbb{R}$ , $l \in \mathbb{N} \cup \{ 0\}$ and $K_{1},\ldots,K_{l} \in {\mathcal{C}^{(d)}} $ by

(4.8) \begin{equation} D^{l}_{K_{1},\ldots,K_{l}} \psi(\xi) \coloneqq \begin{cases} \sum_{J \subseteq [l]}({-}1)^{l-|J|} \psi(\xi_{({-}\infty,K_{\ast})} + \sum_{j \in J}\delta_{K_{j}}) &\text{if $l>0$,} \\[3pt] \psi(o) &\text{if $l=0$,} \end{cases}\end{equation}

where $K_{\ast} \coloneqq \min \{K_{1},\ldots,K_{l} \}$ with respect to $\prec$ . We say that $\psi$ is $\prec$ -continuous at $\infty$ if $\lim_{K \uparrow \mathbb{R}^{d}}\psi(\xi_{({-}\infty,K)}) = \psi(\xi)$ , for all $\xi \in {\textbf{N}}$ .

We use the following factorial moment expansion (FME) proved in [Reference Blaszczyszyn, Merzbach and Schmidt1, Theorem 3.1] on a general Polish space. For stronger results in the special case of a Poisson particle process we refer to [Reference Last17] and [Reference Last and Penrose18, Chapter 19]. The notation from the proof of Lemma 1 is extended: $\textbf{K}_{p+1,p+q}\coloneqq (K_{p+1},\ldots,K_{p+q})$ and $\delta_{\textbf{L}_J}\coloneqq \sum_{j\in J}\delta_{L_{j}}$ .

Theorem 3. Let $\psi\colon{\textbf{N}}\rightarrow \mathbb{R}$ be $\prec$ -continuous at $\infty$ . Assume that, for all $l \in \mathbb{N}$ ,

(4.9) \begin{align} \int_{({\mathcal{C}^{(d)}} )^{l}} \mathbb{E}^{!}_{\textbf{K}_{l}} [|D^{l}_{\textbf{K}_{l}}\psi(\Xi)| ] \rho_{{l}}(\textbf{K}_{l}) \,\mu^l( \textrm{d}\textbf{K}_{l}) < \infty\end{align}

and

(4.10) \begin{align} \lim_{l \rightarrow \infty} \dfrac{1}{l!} \int_{({\mathcal{C}^{(d)}} )^{l}} \mathbb{E}^{!}_{\textbf{K}_{l}} [D^{l}_{\textbf{K}_{l}}\psi(\Xi) ] \rho_{{l}}(\textbf{K}_{l}) \,\mu^l( \textrm{d} \textbf{K}_{l} ) = 0.\end{align}

Then $\mathbb{E} [\psi(\Xi)]$ has the FME

\begin{equation*} \mathbb{E} [\psi(\Xi)] = \psi(o) + \sum_{l=1}^{\infty} \dfrac{1}{l!} \int_{({\mathcal{C}^{(d)}} )^{l}} D^{l}_{\textbf{K}_{l}}\psi(o) \rho_{{l}}(\textbf{K}_{l}) \,\mu^l( \textrm{d} \textbf{K}_{l} ).\end{equation*}

For an admissible pair $(T, \Xi )$ , $K_{1},\ldots,K_{p} \in {\mathcal{C}^{(d)}} $ and $ \xi \in {\textbf{N}} $ , set

(4.11) \begin{align}\psi_{k_{1},\ldots,k_{p}}(\textbf{K}_{p};\,\xi) & \coloneqq \prod_{i=1}^{p}T(K_{i},\xi)^{k_{i}}, \end{align}

(4.12) \begin{align} \psi^{!}_{k_{1},\ldots,k_{p}}(\textbf{K}_{p};\,\xi) & \coloneqq \prod_{i=1}^{p} T (K_{i},\xi+\delta_{\textbf{K}_{p}})^{k_{i}},\end{align}

with $k_{1},\ldots,k_{p} \geq 1$ . It holds that $\mathbb{E}_{\textbf{K}_{p}}[\psi(\Xi_n)]=\mathbb{E}^{!}_{\textbf{K}_{p}}[\psi^{!}(\Xi_n)]$ . Given $p\in\mathbb{N}$ and $K_1,\ldots,K_p\in {\mathcal{C}^{(d)}}$ we let $\rho_{{l}}^{\textbf{K}_{p}}$ denote the lth correlation function of $\mathbb{P}^{!}_{\textbf{K}_{p}}$ . Further, we let $(\mathbb{P}^{!}_{\textbf{K}_{p}})^{!}_{\textbf{L}_{l}}$ , $L_1,\ldots,L_l\in {\mathcal{C}^{(d)}}$ denote the reduced Palm distributions of $\mathbb{P}^{!}_{\textbf{K}_{p}}\!$ . It is easy to show that

(4.13) \begin{equation}\rho_{{p}}(\textbf{K}_{p}) \rho_{{l}}^{\textbf{K}_{p}}(\textbf{L}_{l}) = \rho_{{p+l}}(\textbf{K}_{p},\textbf{L}_{l})\end{equation}

and

(4.14) \begin{equation}(\mathbb{P}^{!}_{\textbf{K}_{p}})^{!}_{\textbf{L}_{l}}=\mathbb{P}^{!}_{\textbf{K}_{p},\textbf{L}_{l}}.\end{equation}

Lemma 8. For distinct $K_{1},\ldots,K_{p} \in \mathcal{C}^{(d)}$ , $k_{1},\ldots,k_{p}$ , $n\in\mathbb{N}$ , $t_p\coloneqq \sum_{i=1}^pk_i$ and k the order of the U-statistic, the functional $\psi^{!}$ admits the FME

\begin{align*}&\mathbb{E}^{!}_{\textbf{K}_{p}} [\psi^{!}_{k_{1},\ldots,k_{p}}(\textbf{K}_{p};\,\Xi_n)]\\[3pt] &\quad = \psi^{!}_{k_{1},\ldots,k_{p}}(\textbf{K}_{p};\,o) + \sum_{l=1}^{t_p(k-1)} \dfrac{1}{l!} \int_{({\mathcal{C}^{(d)}} )^{l}} D^{l}_{\textbf{L}_{l}} \psi^{!}_{k_{1},\ldots,k_{p}}(\textbf{K}_{p};\,o) \rho_{{l}}^{\textbf{K}_{p}}(\textbf{L}_{l}) \,\mu^l(\textrm{d} \textbf{L}_{l}).\end{align*}

Proof. We abbreviate $\psi_{k_{1},\ldots,k_{p}}(\textbf{K}_{p};\,\Xi)$ by $\psi(\textbf{K}_{p};\,\Xi)$ . The radius bound r from (4.3) for the function h implies that $\psi^{!}$ is $\prec$ -continuous at $\infty$ . In [Reference Blaszczyszyn, Yogeshwaran and Yukich2, Lemma 5.1] it is shown that $\psi^{!}$ is the sum of U-statistics of orders no larger than $t_p(k-1)$ . Thus, for $l \in (t_p(k-1),\infty)$ and all $L_{1},\ldots,L_{l} \in {\mathcal{C}^{(d)}}$ , we have

(4.15) \begin{equation} D^{l}_{\textbf{L}_{l}} \psi^{!}(\textbf{K}_{p};\, \xi) = 0.\end{equation}

This implies that (4.9), for $l \in (t_p(k-1),\infty)$ , and (4.10) are satisfied for $\psi^{!}$ from (4.12). We need to verify (4.9), for $l \in [1,t_p(k-1)]$ . For $L_{1},\ldots,L_{l} \in {\mathcal{C}^{(d)}} $ , $\xi\in{\textbf{N}}$ and $J \subseteq [l]$ , set

\begin{equation*} \xi_{J} \coloneqq \xi_{({-}\infty,L_{\ast})} + \delta_{\textbf{L}_{J}},\end{equation*}

where $L_{\ast}\coloneqq \min \{L_{1},\ldots,L_{l} \}$ and $({-}\infty,L_{\ast})$ are with respect to the order $\prec$ .

The difference operator $D^{l}_{\textbf{L}_{l}}$ vanishes like in (4.15) as soon as

\begin{equation*}L_{m} \notin \bigcup_{i=1}^{p}B(K_{i},2r),\end{equation*}

for some $m \in [l]$ . To prove this, expand (4.8) to obtain

\begin{equation*} D^{l}_{\textbf{L}_{l}} \psi^{!}(\textbf{K}_{p};\,\xi) = \sum_{J \subseteq [l], m \notin J} ({-}1)^{l-|J|} \psi^{!}(\textbf{K}_{p};\,\xi_{J}) + \sum_{J \subseteq [l], m \notin J} ({-}1)^{l-|J|-1} \psi^{!}(\textbf{K}_{p};\,\xi_{J\cup \{m \}}) = 0,\end{equation*}

since, for fixed $J\subseteq [l]$ and $m \notin J $ , $\psi^{!}(\textbf{K}_{p};\,\xi_{J})=\psi^{!}(\textbf{K}_{p};\,\xi_{J\cup\{m \}})$ . Then we have

\begin{align*}\psi^{!}(\textbf{K}_{p};\,\xi_{J})&\leq \prod_{i=1}^{p} \Vert h \Vert^{k_{i}}_{\infty} \biggl( \xi \biggl( \bigcup_{i=1}^{p}B(K_{i},2r)\biggr) + |J| + p \biggr)^{k_{i}(k-1)}\\[3pt]&\leq \Vert h \Vert^{t_p}_{\infty} \biggl( \xi \biggl( \bigcup_{i=1}^{p}B(K_{i},2r)\biggr) + |J| + p \biggr)^{t_p(k-1)}.\end{align*}

Using this for the difference operator, we get

(4.16) \begin{align}\notag| D^{l}_{\textbf{L}_{l}} \psi^{!}(\textbf{K}_{p};\,\xi) |&\leq \Vert h \Vert^{t_p}_{\infty} \sum_{J \subseteq [l]} \biggl( \xi\biggl(\bigcup_{i=1}^{p}B(K_{i},2r)\biggr) + |J| + p \biggr)^{t_p(k-1)}\\[3pt] &\leq \Vert h \Vert^{t_p}_{\infty} 2^{l} \biggl( \xi\biggl(\bigcup_{i=1}^{p}B(K_{i},2r)\biggr) + l + p \biggr)^{t_p(k-1)}.\end{align}

Using (4.14), the defining equation (2.5) and (4.16) results in

\begin{align*}&\dfrac{1}{l!} \int_{({\mathcal{C}^{(d)}} )^{l}} (\mathbb{E}^{!}_{\textbf{K}_{p}})^{!}_{\textbf{L}_{l}} [|D^{l}_{\textbf{L}_{l}} \psi^{!}(\textbf{K}_{p};\,\Xi_{n}) |] \rho_{{l}}^{\textbf{K}_{p}}(\textbf{L}_{l}) \,\mu^l(\textrm{d} \textbf{L}_{l})\\[3pt]&\quad =\dfrac{1}{l!} \int_{({\mathcal{C}^{(d)}} )^{l}} \mathbb{E}^{!}_{\textbf{K}_{p},\textbf{L}_{l}} [|D^{l}_{\textbf{L}_{l}}\psi^{!}(\textbf{K}_{p};\,\Xi_{n}) |] \rho_{{l}}^{\textbf{K}_{p}}(\textbf{L}_{l}) \,\mu^l(\textrm{d} \textbf{L}_{l})\\[3pt]&\quad \leq\Vert h \Vert^{t_p}_{\infty} 2^{l} \mathbb{E}_{\textbf{K}_{p}} \biggl[ \Xi \biggl( \bigcup_{i=1}^{p}B(K_{i},r) \biggr)^{l} \biggl( \Xi \biggl( \bigcup_{i=1}^{p}B(K_{i},r) \biggr)+l+p \biggr)^{t_p(k-1)} \biggr].\end{align*}

Since $\Xi$ has all moments under the Palm distribution, the finiteness of the last term and hence the validity of the condition for $l \in [1,t_p(k-1)]$ follows. This justifies the FME expansion.

Proof. Let $p,q,k_1,\ldots,k_{p+q}\in\mathbb{N}$ be fixed. Let $u\coloneqq \max(4R+R_\varphi,r)$ , with $R_\varphi$ being the finite interaction range of the admissible particle process as outlined in Definition 5 and taking into account the particle size from (2.4), as well as (4.3). Given $n\in\mathbb{N}$ , $K_{1},\ldots,K_{p+q}\in z^{-1}(W_{n})$ , we set

\begin{equation*}s \coloneqq d(\{K_{1},\ldots,K_{p} \}, \{K_{p+1},\ldots, K_{p+q} \}) .\end{equation*}

Without loss of generality we assume that $s \in (8u,\infty)$ . Put t as in Definition 8 and $t_p$ as in Lemma 8 respectively, and let $t_q\coloneqq \sum_{i=p+1}^{p+q}k_{i}$ .

Then, using Lemma 8, (4.15) and (4.13), we obtain

\begin{align*} m_n^{(k_{1},\ldots,k_{p+q})}(\textbf{K}_{p+q})&= \mathbb{E}^{!}_{\textbf{K}_{p+q}} [\psi^{!}(\textbf{K}_{p+q};\,\Xi_{n})] \rho_{{p+q}}(\textbf{K}_{p+q})\\[2pt]& = \sum_{l=0}^{t(k-1)} \dfrac{1}{l!} \int_{(\mathcal{C}_{n}^{d})^{l}} D^{l}_{\textbf{L}_{l}} \psi^{!}(o) \rho_{{l+p+q}}(\textbf{K}_{p+q},\textbf{L}_{l}) \,\mu^l(\textrm{d} \textbf{L}_{l}) \\[2pt]& = \sum_{l=0}^{t(k-1)} \dfrac{1}{l!} \int_{(\bigcup_{i=1}^{p+q}B(K_{i},2u))^{l}} D^{l}_{\textbf{L}_{l}}\psi^{!}(o) \rho_{{l+p+q}}(\textbf{K}_{p+q},\textbf{L}_{l})\, \mu^l(\textrm{d} \textbf{L}_{l}).\end{align*}

Let

\begin{equation*}\Psi_{j,l}\coloneqq \biggl(\bigcup_{i=1}^{p}B(K_{i},2u)\biggr)^{j} \times\biggl(\bigcup_{i=1}^{q}B(K_{p+i},2u)\biggr)^{l}.\end{equation*}

Then, using the FME from Lemma 8,

\begin{align*} &m_n^{(k_{1},\ldots,k_{p+q})}(\textbf{K}_{p+q})\\[2pt]&\quad = \sum_{l=0}^{t(k-1)} \dfrac{1}{l!} \sum_{j=0}^{l} \dfrac{l!}{j!(l-j)!} \int_{\Psi_{j,l-j}} D^{l}_{\textbf{L}_{l}} \psi^{!}(\textbf{K}_{p+q};\,o) \rho_{{l+p+q}}(\textbf{K}_{p+q},\textbf{L}_{l}) \,\mu^l(\textrm{d} \textbf{L}_{l})\\[2pt]&\quad = \sum_{l=0}^{t(k-1)} \sum_{j=0}^{l} \dfrac{1}{j!(l-j)!} \int_{\Psi_{j,l-j}} \sum_{J \subseteq [l]} ({-}1)^{l-|J|} \psi^{!} (\textbf{K}_{p+q};\, \delta_{\textbf{L}_{J}} ) \rho_{{l+p+q}}(\textbf{K}_{p+q},\textbf{L}_{l}) \,\mu^l(\textrm{d} \textbf{L}_{l}).\end{align*}

To compare the $(p+q)$ th mixed moment with the product of the pth and qth mixed moments, we use a factorization that holds for

\begin{equation*}L_{1},\ldots,L_{j} \in \bigcup_{i=1}^{p}B(K_{i},2u)\;\;\text{and}\;\;L_{j+1},\ldots,L_{l} \in \bigcup_{i=1}^{q}B(K_{p+i},2u).\end{equation*}

If

\begin{equation*} K\in \bigcup_{i=1}^{p}B(K_{i},2u),\quad L\in \bigcup_{i=1}^{q}B(K_{p+i},2u),\end{equation*}

then $K\cap L = \emptyset$ . Hence

(4.17) \begin{equation} \psi^{!}(\textbf{K}_{p+q};\,\delta_{\textbf{L}_{l}}) = \psi^{!} (\textbf{K}_{p};\,\delta_{\textbf{L}_{j}}) \psi^{!} (\textbf{K}_{p+1,p+q};\,\delta_{\textbf{L}_{j+1,l}}).\end{equation}

Using (4.17) and similar steps as in the case of the $(p+q)$ th mixed moment, we express the product of pth and qth mixed moments:

\begin{align*} & m_n^{(k_{1},\ldots,k_{p})} (\textbf{K}_{p})m_n^{(k_{p+1},\ldots,k_{p+q})}(\textbf{K}_{p+1,p+q})\\[2pt] &\quad = \mathbb{E}^{!}_{\textbf{K}_{p}} [\psi^{!}(\textbf{K}_{p},\Xi_{n})] \mathbb{E}^{!}_{\textbf{K}_{p+1,p+q}} [\psi^{!}(\textbf{K}_{p+1,p+q},\Xi_{n})] \rho_{{p}}(\textbf{K}_{p}) \rho_{{q}}(\textbf{K}_{p+1,p+q})\\[2pt] &\quad = \sum_{l=0}^{t(k-1)}\sum_{j=0}^{l} \dfrac{1}{j!(l-j)!}\int_{\Psi_{j,l-j}} \sum_{J \subseteq [l]}({-}1)^{l-|J|} \psi^{!} (\textbf{K}_{p+q};\,\delta_{\textbf{L}_{J}}) \\[2pt] &\quad\quad\, \times \rho_{{j+p}}(\textbf{K}_{p},\textbf{L}_{j}) \rho_{{l-j+q}}(\textbf{K}_{p+1,p+q},\textbf{L}_{j+1,l}) \,\mu^l(\textrm{d} \textbf{L}_{l}).\end{align*}

Altogether we have, using $c_1,c_2$ from (3.5), that

\begin{align*}& |m_n^{(k_{1},\ldots,k_{p+q})}(\textbf{K}_{p+q})- m_n^{(k_{1},\ldots,k_{p})}(\textbf{K}_{p}) m_n^{(k_{p+1},\ldots,k_{p+q})}(\textbf{K}_{p+1,p+q})|\\[2pt]&\quad \leq \sum_{l=0}^{t(k-1)}\sum_{j=0}^{l}\sum_{J \subseteq [l]} \dfrac{({-}1)^{l-|J|}}{j!(l-j)!} \int_{\Psi_{j,l-j}} \psi^{!} (\textbf{K}_{p+q};\,\delta_{\textbf{L}_{J}})\\[2pt] &\quad\quad\, \times |\rho_{{j+p}}(\textbf{K}_{p},\textbf{L}_{j}) \rho_{{l-j+q}}(\textbf{K}_{p+1,p+q},\textbf{L}_{j+1,l})- \rho_{{l+p+q}}(\textbf{K}_{p+q},\textbf{L}_{l})| \,\mu^l(\textrm{d} \textbf{L}_{l})\\[3pt] &\quad \leq \sum_{l=0}^{t(k-1)} \sum_{j=0}^{l} \sum_{J \subseteq [l]} \dfrac{({-}1)^{l-|J|}}{j!(l-j)!} \int_{\Psi_{j,l-j}} \psi^{!}(\textbf{K}_{p+q};\,\delta_{\textbf{L}_{J}})\\&\quad\quad\, \times \exp({-}c_2d(\{K_{1},\ldots,K_{p},L_{1},\ldots,L_{j}\}, \{K_{p+1},\ldots,K_{p+q},L_{j+1},\ldots,L_{l}\})\\*&\quad\quad\, \times \lambda^{l+p+q}\min(j+p,l-j+q)c_1 \mu^l(\textrm{d} \textbf{L}_{l}).\end{align*}

Using

\begin{equation*} T(K,\xi){\bf{1}}\{\xi(\mathcal{C}^{(d)})=n\} \leq \dfrac{n^{k-1}}{k} \Vert h \Vert_{\infty},\end{equation*}

we have

\begin{equation*}\sum_{J \subseteq [l]} \biggl|\psi^{!}\biggl(\textbf{K}_{p+q};\,\sum_{i \in J}\delta_{L_{i}}\biggr)\biggr| \leq 2^{l} \biggl( \Vert h \Vert_{\infty}\dfrac{|p+q+l|^{k-1}}{k} \biggr)^{t} .\end{equation*}

The difference of weighted mixed moments is finally bounded by

\begin{align*}&|m_n^{(k_{1},\ldots,k_{p+q})}(\textbf{K}_{p+q})- m_n^{(k_{1},\ldots,k_{p})}(\textbf{K}_{p}) m_n^{(k_{p+1},\ldots,k_{p+q})}(\textbf{K}_{p+1,p+q})|\\*&\quad \leq \sum_{l=0}^{t(k-1)}\sum_{j=0}^{l} \biggl( \Vert h \Vert_{\infty} \dfrac{|p+q+l|^{k-1}}{k} \biggr)^{t} \dfrac{\exp({-}8u)2^{l}({-}1)^{l-|J|}} {j!(l-j)!} \mu(B(\{\textbf{0}\},3u))^{l}\\* &\quad\quad\, \times \lambda^{l+p+q}\min(j+p,l-j+q)c_1 \exp({-}c_2d( \{K_{1},\ldots,K_{p}\},\{K_{p+1},\ldots,K_{p+q}\}).\end{align*}

As $\min(j+p,l-j+q)\le{}l+p+q\le{}kt$ , we obtain the desired constants for exponential decay depending only on t and the attributes of the admissible pair.

4.3. Limit theorems

In this subsection we prove mean and variance asymptotics of admissible U-statistics $F_n\coloneqq F_h(\Xi_n)$ , $n\in\mathbb{N}$ , as well as a central limit theorem. The approaches to the proofs come from [Reference Blaszczyszyn, Yogeshwaran and Yukich2] and they can be generalized from point processes in $\mathbb{R}^d$ to particle processes. In Theorem 5 this generalization is possible thanks to our version of the p-moment condition in Proposition 1. The method of the proof of Theorem 6 is standard, and its step-by-step generalization to particle processes is omitted.

To proceed, we need good versions of the correlation functions $\rho_1$ and $\rho_2$ and the first- and second-order Palm distributions. The measure $\mathbb{E}[\int {\bf{1}}\{(K,\Xi)\in\cdot\}\,\Xi(\textrm{d} K)]$ is invariant under (joint) translations. By [Reference Kallenberg16, Theorem 7.6] there exists an translation-invariant version of the correlation function $\rho_1$ . Moreover, the first-order Palm distributions can be chosen in an invariant way, that is,

(4.18) \begin{equation}\mathbb{P}_{K-x}(\theta_x\Xi\in\cdot)=\mathbb{P}_{K}(\Xi\in\cdot),\quad(K,x)\in {\mathcal{C}^{(d)}}\times\mathbb{R}^d.\end{equation}

By the same argument we can assume that $\rho_2$ is translation-invariant and

(4.19) \begin{equation}\mathbb{P}_{K-x,L-x}(\theta_x\Xi\in\cdot)=\mathbb{P}_{K,L}(\Xi\in\cdot),\quad(K,L,x)\in {\mathcal{C}^{(d)}}\times{\mathcal{C}^{(d)}}\times\mathbb{R}^d.\end{equation}

For the weighted mixed moments in (4.6) we denote some special cases as follows. For $K,L\in\mathcal{C}^{(d)}$ we set

\begin{align*} m_{(1)}(K) &\coloneqq \mathbb{E}_K[T(K,\Xi)]\rho_1(K),\\[2pt] m_{(2)}(K,L)&\coloneqq \mathbb{E}_{K,L}[T(K,\Xi)T(L,\Xi)]\rho_2(K,L),\\[2pt] m_{(1,2)}(K) &\coloneqq \mathbb{E}_K[T^2(K,\Xi)]\rho_1(K).\end{align*}

Further, for $n\in\mathbb{N},\,x\in\mathbb{R}^d$ we abbreviate

\begin{align*}m_{(1n)}(K;\,x)& \coloneqq \mathbb{E}_K[T(K,\Xi_n^x)]\rho_1(K),\\[2pt] m_{(2n)}(K,L;\,x)& \coloneqq \mathbb{E}_{K,L}[T(K,\Xi_n^x)T(L,\Xi_n^x)]\rho_2(K,L),\end{align*}

where $\Xi_n^x\coloneqq \Xi\cap (W_n-n^{{{1}/{d}}}x)$ .

Theorem 5. Let $(T,\Xi )$ be an admissible pair. Then it holds that

(4.20) \begin{align}\lim_{n\rightarrow\infty}\dfrac{1}{n} \mathbb{E} F_n & = \int_{\mathcal{C}^d_0} m_{(1)}(K)\,\mathbb{Q}(\textrm{d} K),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \end{align}

(4.21) \begin{align} \lim_{n\rightarrow\infty}\dfrac{1}{n}\operatorname{var}{F_n} & = \int_{\mathcal{C}^d_0}m_{(1,2)}(K)\,\mathbb{Q}(\textrm{d} K)\nonumber\\[3pt] &\quad + \int_{(\mathcal{C}^d_0)^2}\int_{\mathbb{R}^d} (m_{(2)}(K,L+x)-m_{(1)}(K)m_{(1)}(L)) \,\textrm{d} x\,\mathbb{Q}(\textrm{d} K)\,\mathbb{Q}(\textrm{d} L) <\infty .\end{align}

Proof. From (2.5), (2.10), and (2.2), we deduce that

\begin{align*} \mathbb{E} F_n &=\int_{\mathcal{C}_n^d}\mathbb{E}_K T(K,\Xi_n)\rho_1(K)\,\mu(\textrm{d} K) \\[3pt] &=\int_{\mathcal{C}_{\textbf{0}}^{(d)}}\int_{W_n} \mathbb{E}_{K+x}[T(K+x,\Xi_n)]\rho_1(K+x)\,\textrm{d} x\,\mathbb{Q}(\textrm{d} K).\end{align*}

By equation (4.18) and translation invariance of T,

\begin{equation*}\mathbb{E}_{K+x}[T(K+x,\Xi)]=\mathbb{E}_{K}[T(K,\Xi)].\end{equation*}

Together with the translation invariance of $\rho_1$ this gives

\begin{equation*} \int_{\mathcal{C}_{\textbf{0}}^{(d)}}\int_{W_n}\mathbb{E}_{K+x}[T(K+x,\Xi)]\rho_1(K+x)\,\textrm{d} x\,\mathbb{Q}(\textrm{d} K) = n\int_{\mathcal{C}^{(d)}}m_{(1)}(K)\,\mathbb{Q}(\textrm{d} K).\end{equation*}

To prove (4.20) it remains to show that

\begin{equation*} \dfrac{1}{n} \int_{\mathcal{C}_{\textbf{0}}^{(d)}} \int_{W_n} | \mathbb{E}_{K+x}[T(K+x,\Xi_n)] -\mathbb{E}_{K+x}[T(K+x,\Xi)] | \,\textrm{d} x \,\rho_1(K)\,\mathbb{Q}(\textrm{d} K)\end{equation*}

tends to zero as $n\rightarrow\infty$ . The function $\rho_1$ is bounded. For $K\in\mathcal{C}^d_0$ , $K\subseteq B(\textbf{0},R)$ fixed and $x\in W_n$ , we use (4.3) to obtain $T(K+x,\Xi_n)=T(K+x,\Xi)$ whenever $d(x,\partial W_n)\geq 2R$ for the distance from x to the boundary of $W_n$ holds. The first moment condition (4.5) implies the existence of some $0<a<\infty$ such that $\mathbb{E}_{K+x}[|T(K+x,\Xi_n)|]\leq a$ and $\mathbb{E}_{K+x}[|T(K+x,\Xi)|]\leq a$ , uniformly in $n\in\mathbb{N}$ and $K+x\in\mathcal{C}_n^d$ . Since

\begin{equation*} \lim_{n\rightarrow\infty} \dfrac{1}{n}\mathcal{L}_d \{x\in W_n\mid d(x,\partial W_n)\leq 2R\} = 0,\end{equation*}

the first assertion of the theorem is proved.

By a standard point process calculation we obtain for the second moment

\begin{align*} \mathbb{E} F_n^2& = J_1+J_2\coloneqq \int_{\mathcal{C}_{\textbf{0}}^{(d)}} \int_{W_n} \mathbb{E}_{K+u} T^2(K+u,\Xi_n) \rho_1(K+u) \,\textrm{d} u \,\mathbb{Q}(\textrm{d} K)\\[3pt]&\quad\, + \int_{(\mathcal{C}_{\textbf{0}}^{(d)})^2} \int_{(W_n)^2} \mathbb{E}_{K+u,L+v}[T(K+u,\Xi_n)T(L+v,\Xi_n)]\\[3pt] &\quad\, \quad \times \rho_2(K+u,L+v) \,\textrm{d} u\,\textrm{d} v \,\mathbb{Q}(\textrm{d} K)\,\mathbb{Q}(\textrm{d} L).\end{align*}

Then

\begin{equation*} \lim_{n\rightarrow\infty}\dfrac{J_1}{n} = \int_{\mathcal{C}^{(d)}}m_{(1,2)}(K)\,\mathbb{Q}(\textrm{d} K)\end{equation*}

is obtained analogously to the mean value asymptotics above using the second moment condition (4.5).

In the second term $J_2$ we use the substitutions $x=n^{-{{1}/{d}}}u$ and $z=v-u$ , obtaining

\begin{align*}\dfrac{J_2}{n}&= \int_{(\mathcal{C}_{\textbf{0}}^{(d)})^2} \int_{W_1} \int_{W_n-n^{{{1}/{d}}}x} \mathbb{E}_{K+n^{{{1}/{d}}}x,L+z+n^{{{1}/{d}}}x} [T(K+n^{{{1}/{d}}}x,\Xi_n) T(L+z+n^{{{1}/{d}}}x,\Xi_n)]\\[3pt] &\quad\, \times \rho_2(K+n^{{{1}/{d}}}x,L+z+n^{{{1}/{d}}}x) \,\textrm{d} z\,\textrm{d} x\,\mathbb{Q}(\textrm{d} K)\,\mathbb{Q}(\textrm{d} L)\\[3pt] &= \int_{(\mathcal{C}_{\textbf{0}}^{(d)})^2} \int_{W_1} \int_{W_n-n^{{{1}/{d}}}x} \mathbb{E}_{K,L+z} [T(K,\Xi^x_n) T(L+z,\Xi^x_n)]\\[3pt] &\quad\, \times \rho_2(K,L+z) \,\textrm{d} z\,\textrm{d} x \,\mathbb{Q}(\textrm{d} K)\,\mathbb{Q}(\textrm{d} L),\end{align*}

where we have used the invariance of $\rho_2$ , (4.19), the invariance of T, and $(\Xi+n^{{{1}/{d}}}x)_n-n^{{{1}/{d}}}x=\Xi^x_n.$ Since $\operatorname{var}{F_n}=\mathbb{E} F_n^2-(\mathbb{E} F_n)^2$ , we investigate the expression

\begin{equation*}\dfrac{1}{n}(J_2-(\mathbb{E} F_n)^2).\end{equation*}

It takes the form

(4.22) \begin{equation} \int_{(\mathcal{C}^d_0)^2} \int_{W_1} \int_{W_n-n^{{{1}/{d}}}x} (m_{(2n)}(K,L+z;\,x)- m_{(1n)}(K;\,x) m_{(1n)}(L+z;\,x)) \,\textrm{d} z\,\textrm{d} x \,\mathbb{Q}(\textrm{d}{}K) \,\mathbb{Q}(\textrm{d}{}L).\end{equation}

Splitting the innermost integral in (4.21) into the two terms

(4.23) \begin{equation} \int_{W_n-n^{{{1}/{d}}}x} ({\dots}) \,\textrm{d} z= \int_{W_n-n^{{{1}/{d}}}x} ({\dots}){\bf 1} \{|z|\leq M\} \,\textrm{d} z + \int_{W_n-n^{{{1}/{d}}}x} ({\dots}){\bf 1} \{|z|> M\}\, \textrm{d} z,\end{equation}

for an arbitrary $M>0$ , we observe that the part of (4.22) corresponding to the first term of (4.23), that is,

\begin{align*} & \int_{(\mathcal{C}^d_0)^2} \int_{W_1} \int_{W_n-n^{{{1}/{d}}}x} (m_{(2n)}(K,L+z;\,x)- m_{(1n)}(K;\,x) m_{(1n)}(L+z;\,x)) \\[2pt] &\quad\, \times {\bf 1}\{|z|\leq M\} \,\textrm{d} z\,\textrm{d} x \,\mathbb{Q}(\textrm{d}{}K) \,\mathbb{Q}(\textrm{d}{}L) ,\end{align*}

converges to

\begin{equation*} \int_{(\mathcal{C}_{0}^{d})^2} \int_{\mathbb{R}^d} (m_{(2)}(K,L+x)-m_{(1)}(K)m_{(1)}(L)) \,\textrm{d} x \,\mathbb{Q}(\textrm{d} K)\,\mathbb{Q}(\textrm{d} L) ,\end{equation*}

when first $n\rightarrow\infty$ and then $M\rightarrow\infty$ . Using (4.7), the absolute value of the second term in (4.23) can be bounded uniformly in n by

\begin{equation*}b_2\int_{|z|>M}\exp({-}a_2d_H(K,L+z))\textrm{d} z,\end{equation*}

which tends to zero when $M\rightarrow\infty$ . Thus the part of (4.22) corresponding to the second term of (4.23), that is,

\begin{align*}& \int_{(\mathcal{C}^d_0)^2} \int_{W_1} \int_{W_n-n^{{{1}/{d}}}x} (m_{(2n)}(K,L+z;\,x)- m_{(1n)}(K;\,x) m_{(1n)}(L+z;\,x)) \\[2pt]&\quad\, \times {\bf 1}\{|z|> M\} \,\textrm{d} z\,\textrm{d} x \,\mathbb{Q}(\textrm{d}{}K) \,\mathbb{Q}(\textrm{d}{}L) ,\end{align*}

converges to zero. We can justify these limits analogously to Lemma 4.1 in [Reference Blaszczyszyn, Yogeshwaran and Yukich2]. The boundedness in (4.21) follows from the second moment condition (4.5) for the first term and from (4.7) for the second term.

Theorem 6. Let $(T,\Xi )$ be an admissible pair. If, for some $\beta\in (0,\infty)$ ,

(4.24) \begin{equation} \liminf_{n\rightarrow\infty} \dfrac{\operatorname{var}\,F_n} {n^\beta} >0, \end{equation}

then we have the CLT

(4.25) \begin{equation} \dfrac{{F_n}-\mathbb{E} {F_n}} { (\!\operatorname{var}{F_n})^{1/2}} \xrightarrow[n\rightarrow\infty]{d} N(0,1).\end{equation}

Proof. Denote $\bar{F}_n\coloneqq F_n-\mathbb{E} F_n$ . The idea is to prove that the lth-order cumulants of

\begin{equation*} (\!\operatorname{var}{F_n})^{-1/2}\bar{F}_n \end{equation*}

vanish as $n\rightarrow\infty $ and l is sufficiently large. This follows by showing that (4.5) and (4.7) imply volume order growth (i.e. of order O(n)) for the lth-order cumulant of $\bar{F}_n$ , $l\geq 2$ , and using the assumption (4.24). Then (4.25) holds. The details are analogous to [Reference Blaszczyszyn, Yogeshwaran and Yukich2, pages 881–886], with the difference that it deals with measures defined on $\mathcal{C}_n^d$ .