Concentration Bounds for Geometric Poisson Functionals: Logarithmic Sobolev Inequalities Revisited

TL;DR

This paper develops new concentration bounds for Poisson functionals using logarithmic Sobolev inequalities, with applications to random geometric graphs and convex distances, enhancing probabilistic analysis in geometric probability.

Contribution

It introduces novel concentration inequalities for Poisson functionals based on logarithmic Sobolev inequalities, tailored for geometric applications.

Findings

Derived new concentration bounds for geometric Poisson functionals.

Applied bounds to edge counting and length functionals in random geometric graphs.

Extended analysis to convex distances for random point measures.

Abstract

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 (2000), 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 (2013).