Higher-order hyperuniformity of random measures

We study higher-order hyperuniformity through the large-scale variance of local $k$-point patterns, and write ${\text{HU}}_k$ for the resulting condition; ${\text{HU}}_1$ is ordinary hyperuniformity. The conditions ${\text{HU}}_k$ need not coincide: for every $k\ge 1$ there exists an $\mathbb{R}$-invariant weakly mixing simple point process that belongs to ${\text{HU}}_j$ for all $j\le k$ but not to ${\text{HU}}_{k+1}$. Randomly translated lattices belong to ${\text{HU}}_k$ for every $k$, whereas sufficiently small non-degenerate iid perturbations of lattices and projection determinantal point processes already belong to ${\text{HU}}_1\setminus {\text{HU}}_2$. For regular Euclidean cut-and-project processes we give an exact criterion for ${\text{HU}}_k$. For ball windows, ${\text{HU}}_k$ is equivalent to ${\text{HU}}_1$ in internal dimensions two and three, whereas in every internal dimension $m\ge 4$ we construct ball-window examples in ${\text{HU}}_1\setminus {\text{HU}}_2$. Finally, the nonperiodic Kurasov–Sarnak Fourier-quasicrystalline point process is stealthy—its first-order spectrum has a gap at the origin—yet does not belong to ${\text{HU}}_2$.