Precise aysmptotics for functionals of point processes having fast decay of correlations

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The project considers a large class of geometric functionals of point processes on the d-dimensional Euclidean space having fast decay of correlations. In many application the functionals are expressible as a sum of spatially dependent terms (score functions), which present the interaction of points with respect to locally finite point sets. The sumstypically describe a global geometric feature of a structure of finite point sets in terms of local contributions. Possible applications include statistics of simplicial complexes, statistics of high-dimensional data sets, statistics of germ-grain models and statistics of percolation models. For Poisson and binomial point processes even general square-integrable functionals are considered. Our main goal is to study precise large and moderate deviations and rates of normal convergence for these functionals. We are planing to use and to develop further the method of cumulants (and factorial moments and cluster measures) as well as to use and to develop further Stein’s method.Summarizing our goals comprise the rigorous description and analysis of geometric functionals of random structures and geometric systems driven by correlated spatial structures.