{
  "abstract": "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 $k\\geq1$ there exists an $\\mathbb R$-invariant weakly mixing simple point process that belongs to $\\HU_j$ for all $j\\leq k$ 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 $\\HU_1\\setminus\\HU_2$. 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 $m\\geq4$ we construct ball-window examples in $\\HU_1\\setminus\\HU_2$. 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 $\\HU_2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07502",
  "author_total": 1,
  "authors": [
    {
      "name": "Michael Björklund"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fct0w7e3pr06m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07502",
  "fulltext_url": null,
  "id": "fct0w7e3pr06m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07502",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7212042813",
  "pdf_url": "https://arxiv.org/pdf/2609.07502",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fct0w7e3pr06m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07502",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07502"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Higher-order hyperuniformity of random measures",
  "title_text": "Higher-order hyperuniformity of random measures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/fct0w7e3pr06m",
  "venue": "arXiv",
  "versions": []
}