Large deviations for the volume of k-nearest neighbor balls

Abstract

We prove a large deviation principle for the point process associated to k-element connected components in ℝd with respect to the connectivity radii rn→∞. The random points are generated from a homogeneous Poisson point process, so that (rn)n≥1 satisfies nkrd(k−1)n→∞ and nrdn→0 as n→∞ (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.