Generic point processes are intermittently hyperuniform and maximally rigid
Abstract
We study the prevalence of hyperuniformity and spatial rigidity among translation- invariant ergodic simple point processes of fixed intensity, in a topology controlling local second moments. Hyperuniform processes are dense but meagre. We prove that intermittent hyperuniformity is residual: the number variance exhibits hyperuniform behavior along some sequences of radii and hyperfluctuating behavior along others. Maximal rigidity is also residual. Thus generic processes are maximally rigid but not hyperuniform. A local-to-global principle reduces the proofs to cross-section constructions for an arbitrary fixed essentially free ergodic ℝd-action. We also prove that processes admitting revival sequences form a residual set and exhibit near-quadratic variance along suitable van Hove sequences.
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BibTeX
@misc{bjorklund2026generic,
title = {{Generic point processes are intermittently hyperuniform and maximally rigid}},
author = {Michael Björklund},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01975},
eprint = {2611.01975},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01975},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generic point processes are intermittently hyperuniform and maximally rigid
AU - Michael Björklund
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01975
UR - https://arxiv.org/abs/2611.01975
AB - We study the prevalence of hyperuniformity and spatial rigidity among translation- invariant ergodic simple point processes of fixed intensity, in a topology controlling local second moments. Hyperuniform processes are dense but meagre. We prove that intermittent hyperuniformity is residual: the number variance exhibits hyperuniform behavior along some sequences of radii and hyperfluctuating behavior along others. Maximal rigidity is also residual. Thus generic processes are maximally rigid but not hyperuniform. A local-to-global principle reduces the proofs to cross-section constructions for an arbitrary fixed essentially free ergodic ℝ^d-action. We also prove that processes admitting revival sequences form a residual set and exhibit near-quadratic variance along suitable van Hove sequences.
ER -