N},\;{\mathbf x}_2\subset {\mathbf x}_1.$ In the case of sharp inequality we say that the process is strictly attractive, strictly repulsive, respectively.

The facet process $\mu^{(1)}$ is neither strictly attractive nor strictly repulsive for any $\nu_1\in{\mathbb R}.$ The facet processes $\mu^{(k)}$ are repulsive for $k=2,\dots ,d.$

Asymptotics with increasing intensity
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The Poisson case
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Generally on $B\subset {\mathbb R}^d,\,\lambda $ as in Section 2, for $l\geq 1$ and $i=1,\dots ,l$ let $k_i\in {\mathbb N},$ $f^{(i)}\in L_1(\lambda^{k_i})$ be symmetric functions. Consider Poisson processes $\eta_a$ with intensity measures $\lambda_a=a\lambda ,\;a\geq 1.$ Following [@RefL] $U$-statistics $$\label{poispos}F^{(i)}_a(\eta_a)=\sum_{(x_1,\dots ,x_{k_i})\in\eta^{k_i}_{a\neq }} f^{(i)}(x_1,\dots ,x_{k_i})$$ are transformed to $$\label{tri}
\hat{F}_a^{(i)}=a^{-(k_i-\frac{1}{2})}(F_a^{(i)}-{\mathbb E}F_a^{(i)}).$$ The asymptotic covariances are $$\label{asco}C_{ij}=\lim_{a\rightarrow\infty }cov(\hat{F}_a^{(i)},\hat{F}_a^{(j)})=\int_B T_1F^{(i)}(x)T_1F^{(j)}(x) \lambda ({\text{\rm d}}x),\;i,j\in\{1,\dots ,l\}.$$ The convergence under the distance between $l$-dimensional random vectors $U,Z$ $$d_3 (U,Z)=\sup_{g\in {\cal H}}|{\mathbb E}g(U)-{\mathbb E}g(Z)|,$$ where $\cal H$ is the system of functions $h\in C^3({\mathbb R}^l)$ with $$\begin{gathered}
\nonumber \max_{1\leq i_1\leq i_2\leq l}\sup_{x\in {\mathbb R}^l}\big|\frac{\partial^2h(x)}{\partial x_{i_1}\partial x_{i_2}}\big|\leq 1,\quad
\max_{1\leq i_1\leq i_2\leq i_3\leq l}\sup_{x\in {\mathbb R}^l}|\frac{\partial^3h(x)}{\partial x_{i_1}\partial x_{i_2}\partial x_{i_3}}|\leq 1\end{gathered}$$ implies convergence in distribution. Based on the multi-dimensional Malliavin-Stein inequality for the distance $d_3$ of a random vector from a centered Gaussian random vector $Z$ with covariance matrix $C=(C_{ij})_{i,j=1,\dots ,l},$ [@RefL] show that under the assumption 