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We consider stationary configurations of points in Euclidean space which aremarked by positive random variables called scores. The scores are allowed todepend on the relative positions of other points and outside sources ofrandomness. Such models have been thoroughly studied in stochastic geometry,e.g. in the context of random tessellations or random geometric graphs.  It turns out that in a neighbourhood of a point with an extreme score one canoften rescale positions and scores of nearby points to obtain a limiting pointprocess, which we call the tail configuration. Under some assumptions ondependence between scores, this local limit determines the global asymptoticsfor extreme scores within increasing windows in $\mathbb{R}^d$. The main resultestablishes the convergence of rescaled positions and clusters of high scoresto a Poisson cluster process, quantifying the idea of the Poisson clumpingheuristic by D. Aldous (in the point process setting).  We apply our results to models based on (marked) Poisson processes where thescores depend on the distance to the $k$th nearest neighbor and where scoresare allowed to propagate through a random network of points depending on theirlocations.

Extremal behavior of stationary marked point processes