Higher-order hyperuniformity of random measures
Abstract
We study higher-order hyperuniformity through the large-scale variance of local k-point patterns, and write HUk for the resulting condition; HU1 is ordinary hyperuniformity. The conditions HUk need not coincide: for every k≥1 there exists an ℝ-invariant weakly mixing simple point process that belongs to HUj for all j≤ k but not to HUk+1. Randomly translated lattices belong to HUk for every k, whereas sufficiently small non-degenerate iid perturbations of lattices and projection determinantal point processes already belong to HU1∖HU2. For regular Euclidean cut-and-project processes we give an exact criterion for HUk. For ball windows, HUk is equivalent to HU1 in internal dimensions two and three, whereas in every internal dimension m≥4 we construct ball-window examples in HU1∖HU2. 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 HU2.
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BibTeX
@misc{bjorklund2026higherorder,
title = {{Higher-order hyperuniformity of random measures}},
author = {Michael Björklund},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07502},
eprint = {2609.07502},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07502},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Higher-order hyperuniformity of random measures
AU - Michael Björklund
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07502
UR - https://arxiv.org/abs/2609.07502
AB - We study higher-order hyperuniformity through the large-scale variance of local k-point patterns, and write HU_k for the resulting condition; HU_1 is ordinary hyperuniformity. The conditions HU_k need not coincide: for every k≥1 there exists an ℝ-invariant weakly mixing simple point process that belongs to HU_j for all j≤ k but not to HU_k+1. Randomly translated lattices belong to HU_k for every k, whereas sufficiently small non-degenerate iid perturbations of lattices and projection determinantal point processes already belong to HU_1∖HU_2. For regular Euclidean cut-and-project processes we give an exact criterion for HU_k. For ball windows, HU_k is equivalent to HU_1 in internal dimensions two and three, whereas in every internal dimension m≥4 we construct ball-window examples in HU_1∖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 HU_2.
ER -