the case of non-Poisson processes we restrict to models with finitely many orientations. Detailed analysis of correlation functions exhibits various asymptotics for different combination of $U$-statistics and submodels of the facet process.

Keywords: asymptotics, correlation function, facet process, moments, $U$-statistics.

Mathematical Subject Classification: 60D05, 60G55

Introduction
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Modeling of particle systems with interactions by means of finite point processes presents an interesting field of stochastic geometry the research of which is not yet completed. The work of [@RefK] concerning area-interaction models of planar discs was later developed by [@RefM] to models of interacting discs with more general densities (with respect to the Poisson process) from exponential families. The authors also developed sophisticated simulation procedures of the models. The motivation for the modeling was to serve for statistical purposes in the evaluation of real data.

In the present paper we develop similar interaction processes in a bounded window in arbitrary Euclidean dimension. The difference in comparison to [@RefM] is that the particles (called facets) are lower dimensional, they form compact subsets of hyperplanes. The interactions arise from intersections of facets and the their global amounts form $U$-statistics [@RefR]. In 2D and 3D these models may also serve to real data evaluation 