Concentration for Poisson functionals: component counts in random geometric graphs

Abstract

Upper bounds for the probabilities $\mathbb{P}(F\ge \mathbb{E}F+r)$ and $\mathbb{P}(F\le \mathbb{E}F-r)$ are proved, where $F$ is a certain component count associated with a random geometric graph built over a Poisson point process on ${\mathbb{R}}^d$. 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.

26 pages, 1 figure

References in corpus (1)

• Random geometric complexes in the thermodynamic regime

Cited by in corpus (2)

• Modified log-Sobolev inequalities, Beckner inequalities and moment estimates
• Restricted hypercontractivity on the Poisson space