Concentration for Poisson functionals: component counts in random geometric graphs

TL;DR

This paper establishes exponential and Gaussian tail bounds for component counts in random geometric graphs built over Poisson processes, using advanced concentration inequalities that work even with infinite intensity measures.

Contribution

It introduces new concentration inequalities for Poisson functionals, extending existing methods to settings with infinite intensity measures.

Findings

Upper tail probabilities decay exponentially.

Lower tail probabilities decay Gaussian.

Applicable to Poisson processes with infinite intensity.

Abstract

Upper bounds for the probabilities $\mathbb{P}(F\geq \mathbb{E} F + r)$ and $\mathbb{P}(F\leq \mathbb{E} F - r)$ 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.