We prove new concentration estimates for random variables that are functionals of a Poisson measure defined on a general measure space. Our results are specifically adapted to geometric applications, and are based on a pervasive use of a powerful logarithmic Sobolev inequality proved by L. Wu [43], as well as on several variations of the so-called Herbst argument. We provide several applications, in particular to edge counting and more general length power functionals in random geometric graphs, as well as to the convex distance for random point measures recently introduced by M. Reitzner [31].

Abstract. Concentration inequalities for subgraph counts in random geometric graphs built over Poisson point processes are proved. The estimates give upper bounds for the probabilities P(N ≥ M + r) and P(N ≤ M − r) where M is either a median or the expectation of a subgraph count N . The bounds for the lower tail have a fast Gaussian decay and the bounds for the upper tail satisfy an optimality condition. A special feature of the presented inequalities is that the underlying Poisson process does not need to have finite intensity measure.The tail estimates for subgraph counts follow from concentration inequalities for more general local Poisson U-statistics. These bounds are proved using recent general concentration results for Poisson U-statistics and techniques based on the convex distance for Poisson point processes.

Upper bounds for the probabilities P(F ≥ EF + r) and P(F ≤ EF − r) are proved, where F is a certain component count associated with a random geometric graph built over a Poisson point process on R d . The bounds for the upper tail decay exponentially, and the lower tail estimates even have a Gaussian decay.For the proof of the concentration inequalities, recently developed methods based on logarithmic Sobolev inequalities are used and enhanced. A particular advantage of this approach is that the resulting inequalities even apply in settings where the underlying Poisson process has infinite intensity measure.MSC: primary 60D05; secondary 05C80, 60C05