Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs

Concentration bounds for the probabilities $\mathbb{P}(N\ge M+r)$ and $\mathbb{P}(N\le M?r)$ are proved, where $M$ is a median or the expectation of a subgraph count $N$ associated with a random geometric graph built over a Poisson process. The lower tail bounds have a Gaussian decay and the upper tail inequalities satisfy an optimality condition. A remarkable feature is that the underlying Poisson process can have a.s. infinitely many points.The estimates for subgraph counts follow from tail inequalities for more general local Poisson U-statistics. These bounds are proved using recent general concentration results for Poisson U-statistics and techniques involving the convex distance for Poisson processes.