Chaos and Superconcentration for Poisson Functionals with Applications in Stochastic Geometry
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| Format: | Preprint |
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2026
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| _version_ | 1866910070111993856 |
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| author | Bhattacharjee, Chinmoy O'Clarey, Rowan |
| author_facet | Bhattacharjee, Chinmoy O'Clarey, Rowan |
| contents | We consider square-integrable functionals of Poisson point processes for which the variance upper bound provided by the classical Poincaré 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 $Γ$-components in random geometric graphs in the dense regime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23053 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Chaos and Superconcentration for Poisson Functionals with Applications in Stochastic Geometry Bhattacharjee, Chinmoy O'Clarey, Rowan Probability 60D05, 60H07 We consider square-integrable functionals of Poisson point processes for which the variance upper bound provided by the classical Poincaré 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 $Γ$-components in random geometric graphs in the dense regime. |
| title | Chaos and Superconcentration for Poisson Functionals with Applications in Stochastic Geometry |
| topic | Probability 60D05, 60H07 |
| url | https://arxiv.org/abs/2603.23053 |