Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.02627

Asymptotic fluctuations of smooth linear statistics of independently perturbed lattices

Gabriel Mastrilli
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gabriel Mastrilli. 2025-03-04. Asymptotic fluctuations of smooth linear statistics of independently perturbed lattices. https://arxiv.org/abs/2503.02627

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Distribution-uniform strong laws of large numbers

We revisit the question of whether the strong law of large numbers (SLLN) holds uniformly in a rich family of distributions, culminating in a distribution-uniform generalization of the Marcinkiewicz-Zygmund SLLN. These results can be viewed as extensions of Chung's distribution-uniform SLLN to random variables with uniformly integrable $q^\text{th}$ absolute central moments for $0 < q < 2$. Furthermore, we show that uniform integrability of the $q^\text{th}$ moment is both sufficient and necessary for the SLLN to hold uniformly at the Marcinkiewicz-Zygmund rate of $n^{1/q - 1}$. These proofs centrally rely on novel distribution-uniform analogues of some familiar almost sure convergence results including the Khintchine-Kolmogorov convergence theorem, Kolmogorov's three-series theorem, a stochastic generalization of Kronecker's lemma, and the Borel-Cantelli lemmas. We also consider the non-identically distributed case.

math.PR↗
arXiv2026

Malliavin Calculus for rough stochastic differential equations

In this work we show that rough stochastic differential equations (RSDEs), as introduced by Friz, Hocquet, and Lê (2021), are Malliavin differentiable. We use this to prove existence of a density when the diffusion coefficients satisfies standard ellipticity assumptions. Moreover, when the coefficients are smooth and the diffusion coefficients satisfies a Hörmander condition, the density is shown to be smooth. The key ingredient is to develop a comprehensive theory of linear rough stochastic differential equations, which could be of independent interest.

math.PR↗
arXiv2026

Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation

We present nonasymptotic concentration inequalities for sums of independent and identically distributed random variables that yield asymptotic strong Gaussian approximations of Komlós, Major, and Tusnády (KMT) [1975,1976]. The constants appearing in our inequalities are either universal or explicit, and thus as corollaries, they imply distribution-uniform generalizations of the aforementioned KMT approximations. In particular, it is shown that uniform integrability of a random variable's $q^{\text{th}}$ moment is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $o(n^{1/q})$ for $q > 2$ and that having a uniformly lower bounded Sakhanenko parameter -- equivalently, a uniformly upper-bounded Bernstein parameter -- is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $O(\log n)$. Instantiating these uniform results for a single probability space yields the analogous results of KMT exactly.

math.PR↗