Chaos and Superconcentration for Poisson Functionals with Applications in Stochastic Geometry

Abstract

We consider square-integrable functionals of Poisson point processes for which the variance upper bound provided by the classical Poincar\'{e} inequality is suboptimal, a phenomenon known as superconcentration. In this paper, we establish a rigorous mathematical equivalence between superconcentration and the chaotic behaviour of the functional, and certain associated random sets, under perturbations driven by the Ornstein-Uhlenbeck semigroup on the Poisson space. Leveraging the Malliavin-Stein method, we develop general variance identities and bounds for Poisson functionals, providing a unified framework to prove superconcentration, particularly for geometric functionals that can be expressed as a sum of local score functions. We apply our results to rigorously establish superconcentration and the chaotic behaviour in some models of stochastic geometry. Specifically, we analyse horizontal box-crossing indicators in certain critical continuum percolations, as well as the number of vertices with small degrees and the number of isolated $\Gamma$-components in random geometric graphs in the dense regime.