Randomized Spectral Inference for Hyperuniformity
Abstract
We test hyperuniformity from one large realization of a stationary point process. Hyperuniformity is equivalent to the average Ar of the structure factor over Br vanishing as r↓0, so the target is a low-frequency average rather than the value of the structure factor at the origin, which need not exist. We estimate Ar from squared Fourier coefficients at frequencies drawn uniformly from Br; if these coefficients are asymptotically Gaussian at almost every fixed frequency, the squared coefficients converge to independent variables with mean Ar. Assuming Ar=s+c rα+εr with |εr|≤ Lrβ and given exponents 0<α<β, a two-radius extrapolation removes the rα term and estimates s with deterministic error of order rβ. This gives confidence bounds and a one-sided test of H0:s=0 with asymptotic level at most γ, consistent against every fixed s>0 as R→∞, then the number of sampled frequencies tends to infinity, and then r↓0. Essentially free translation actions with completely positive entropy satisfy the Fourier assumption.
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BibTeX
@misc{bjorklund2026randomized,
title = {{Randomized Spectral Inference for Hyperuniformity}},
author = {Michael Björklund},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27961},
eprint = {2609.27961},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27961},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Randomized Spectral Inference for Hyperuniformity
AU - Michael Björklund
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.27961
UR - https://arxiv.org/abs/2609.27961
AB - We test hyperuniformity from one large realization of a stationary point process. Hyperuniformity is equivalent to the average A_r of the structure factor over B_r vanishing as r↓0, so the target is a low-frequency average rather than the value of the structure factor at the origin, which need not exist. We estimate A_r from squared Fourier coefficients at frequencies drawn uniformly from B_r; if these coefficients are asymptotically Gaussian at almost every fixed frequency, the squared coefficients converge to independent variables with mean A_r. Assuming A_r=s+c r^α+ε_r with |ε_r|≤ Lr^β and given exponents 0<α<β, a two-radius extrapolation removes the r^α term and estimates s with deterministic error of order r^β. This gives confidence bounds and a one-sided test of H_0:s=0 with asymptotic level at most γ, consistent against every fixed s>0 as R→∞, then the number of sampled frequencies tends to infinity, and then r↓0. Essentially free translation actions with completely positive entropy satisfy the Fourier assumption.
ER -