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Asymptotic fluctuations of smooth linear statistics of independently perturbed lattices

Gabriel Mastrilli

TL;DR

This work precisely characterizes the large-scale fluctuations of smooth linear statistics on hyperuniform lattices perturbed by i.i.d. shifts, revealing a rich dimension- and tail-dependent landscape. By expressing cumulants in the Fourier domain and leveraging Poisson summation, it proves universal central limit theorems for $d\ge 3$, and, in $d=2$ and $d=1$, identifies nuanced Gaussian, non-Gaussian, and $\alpha$-stable limits controlled by the perturbations’ tail behavior and the characteristic function near zero. Notably, in dimension one, the limiting law can be Gaussian, non-Gaussian with all moments finite, or $\alpha$-stable with $\alpha\in(1,2]$, including a non-standard transition at $\alpha=d=1$. The results extend to stationary perturbed lattices and illuminate how hyperuniformity interacts with smoothing function properties to shape large-scale statistics, providing rigorous grounding for physics-inspired observations.

Abstract

We consider the hyperuniform model of d-dimensional integer lattice perturbed by independent random variables and we investigate the large scale asymptotic fluctuations of smoothed versions of the usual counting statistics, specifically of linear statistics associated to a smooth function with rapid decay at infinity. We highlight three distinct classes of limit, depending on the dimension d and on the tails of the perturbations. On the one hand, we establish that for dimensions larger than two, central limit theorems hold under mild assumptions on the perturbations. This confirms numerical observations from physics, suggesting that even for highly correlated hyperuniform models, large dimensions favor asymptotic normality. On the other hand, in dimension one, the limiting distribution can be Gaussian, non-Gaussian with finite moments of all orders, or stable with parameter strictly between one and two. These two latter results represent rare examples of non-Gaussian limits for smooth linear statistics of hyperuniform point processes of Classes I and II.

Asymptotic fluctuations of smooth linear statistics of independently perturbed lattices

TL;DR

This work precisely characterizes the large-scale fluctuations of smooth linear statistics on hyperuniform lattices perturbed by i.i.d. shifts, revealing a rich dimension- and tail-dependent landscape. By expressing cumulants in the Fourier domain and leveraging Poisson summation, it proves universal central limit theorems for , and, in and , identifies nuanced Gaussian, non-Gaussian, and -stable limits controlled by the perturbations’ tail behavior and the characteristic function near zero. Notably, in dimension one, the limiting law can be Gaussian, non-Gaussian with all moments finite, or -stable with , including a non-standard transition at . The results extend to stationary perturbed lattices and illuminate how hyperuniformity interacts with smoothing function properties to shape large-scale statistics, providing rigorous grounding for physics-inspired observations.

Abstract

We consider the hyperuniform model of d-dimensional integer lattice perturbed by independent random variables and we investigate the large scale asymptotic fluctuations of smoothed versions of the usual counting statistics, specifically of linear statistics associated to a smooth function with rapid decay at infinity. We highlight three distinct classes of limit, depending on the dimension d and on the tails of the perturbations. On the one hand, we establish that for dimensions larger than two, central limit theorems hold under mild assumptions on the perturbations. This confirms numerical observations from physics, suggesting that even for highly correlated hyperuniform models, large dimensions favor asymptotic normality. On the other hand, in dimension one, the limiting distribution can be Gaussian, non-Gaussian with finite moments of all orders, or stable with parameter strictly between one and two. These two latter results represent rare examples of non-Gaussian limits for smooth linear statistics of hyperuniform point processes of Classes I and II.

Paper Structure

This paper contains 19 sections, 31 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Motivation and related works
  3. Summary of the results
  4. Main ideas of the proofs
  5. General notations
  6. Cumulants of smooth linear statistics of perturbed lattices
  7. Gaussian limits
  8. Central limit theorem in dimension $d \geq 3$
  9. Central limit theorem in dimension $d = 2$
  10. The case of square integrable perturbations
  11. The case of heavy tailed perturbations
  12. The case of perturbations having a regularly varying characteristic function
  13. Non-Gaussian limits
  14. Lack of a general central limit theorem when the variance is bounded in dimension 1
  15. Convergence toward $\alpha$-stable distributions when $d = 1$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $d \geq 1$ and $f$ be a Schwartz function. We consider i.i.d. random variables $(\xi_x)_{x \in \mathbb{Z}^d}$ with characteristic function $\varphi$. We assume that $f$ is non-null and that the perturbations are not a.s. constant and we consider for $i \in \{0, 1\}$ the linear statistics $T_r^{i For the next results, we suppose that $1 - \varphi(x) \sim L(|x|)|x|^{\alpha}$ as $|x| \to 0$, wher

Theorems & Definitions (67)

  • Theorem 1.1
  • Remark 1
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.1
  • proof
  • Corollary 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 57 more