Concentration for Poisson functionals: component counts in random geometric graphs

الأصل · EN

Upper bounds for the probabilities P(F≥ E F + r) and P(F≤ 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 Rᵈ. 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.