Chaos and Superconcentration for Poisson Functionals with Applications in Stochastic Geometry

TL;DR

This paper explores the phenomenon of superconcentration in Poisson functionals, establishing a link with chaos, and applies these insights to models in stochastic geometry such as percolation and random geometric graphs.

Contribution

It introduces a rigorous equivalence between superconcentration and chaos for Poisson functionals, and develops a unified framework using Malliavin-Stein methods for variance bounds.

Findings

Superconcentration is equivalent to chaotic behavior in Poisson functionals.

Established superconcentration in models of stochastic geometry like percolation and geometric graphs.

Provided new variance bounds and identities for Poisson functionals.

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.