Randomized Spectral Inference for Hyperuniformity

Abstract

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 $r\downarrow0$, 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 $A_r=s+c r^\alpha+\varepsilon_r$ with $|\varepsilon_r|\leq Lr^\beta$ and given exponents $0<\alpha<\beta$, a two-radius extrapolation removes the $r^\alpha$ term and estimates $s$ with deterministic error of order $r^\beta$. This gives confidence bounds and a one-sided test of $H_0:s=0$ with asymptotic level at most $\gamma$, consistent against every fixed $s>0$ as $R\to\infty$, then the number of sampled frequencies tends to infinity, and then $r\downarrow0$. Essentially free translation actions with completely positive entropy satisfy the Fourier assumption.