  We consider stationary configurations of points in Euclidean space which are
marked by positive random variables called scores. The scores are allowed to
depend on the relative positions of other points and outside sources of
randomness. 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 can
often rescale positions and scores of nearby points to obtain a limiting point
process, which we call the tail configuration. Under some assumptions on
dependence between scores, this local limit determines the global asymptotics
for extreme scores within increasing windows in $\mathbb{R}^d$. The main result
establishes the convergence of rescaled positions and clusters of high scores
to a Poisson cluster process, quantifying the idea of the Poisson clumping
heuristic by D. Aldous (in the point process setting).
  We apply our results to models based on (marked) Poisson processes where the
scores depend on the distance to the $k$th nearest neighbor and where scores
are allowed to propagate through a random network of points depending on their
locations.
