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Chaos and Superconcentration for Poisson Functionals with Applications in Stochastic Geometry

Chinmoy Bhattacharjee, Rowan O'Clarey

Abstract

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.

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 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 -components in random geometric graphs in the dense regime.
Paper Structure (14 sections, 16 theorems, 160 equations, 1 figure)

This paper contains 14 sections, 16 theorems, 160 equations, 1 figure.

Table of Contents

  1. Introduction and Main Results
  2. Main results
  3. Equivalence of Chaos and Superconcentration
  4. General Variance Estimations
  5. Superconcentration for sums of scores
  6. Applications
  7. Horizontal box-crossings in continuum percolations
  8. Vertices with small degrees in random geometric graphs
  9. $\Gamma$-components in random geometric graphs
  10. Proofs of main results
  11. Poisson space preliminaries
  12. Proof of results in Section \ref{['sec:chaosresults']}
  13. Proofs of Results in Section \ref{['Sec:VarEst']}
  14. Proof of \ref{['thm:supcon_sos']}

Key Result

Theorem 2.1

Let $\eta_s$ be as in def:supconc. Suppose that a functional $F_s \in L^2_{\eta_s}$ is $\varepsilon_s$-Superconcentrated with $\varepsilon_s \to 0$ as $s \to \infty$. Then for any $\delta_s>0$ with $\delta_s \to 0$ and $\varepsilon_s/\delta_s \to 0$ as $s \to \infty$, we have $F_s$ is $(\varepsilon_

Figures (1)

  • Figure 1: Illustration of the lower bound in \ref{['eq:vollb']} in 2D. The point $e^*$ is an extreme point in $x_{k+1:2k}$, while $e_1,e_2,e_3$ are points in $x_{1:k}$. Since $x_{1:k}$ lie below the separating hyperplane and $d(x_{1:k},e^*) \ge r_s$, even in the worst case $\cup_{x \in x_{1:k}} B(x,r_s)$ cannot cover the coloured region, which has volume $\Omega(r_s^d)$.

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1: Consequence of Theorem 3.5 in Chatterjee
  • Theorem 2.2
  • Corollary 2.3
  • Remark 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Theorem 2.8
  • ...and 22 more