{
  "abstract": "We study Gaussian fluctuations of differential observables of centered stationary random fields on the discrete $d$-dimensional torus of mesh $1/N$. Our main result is a Central Limit Theorem for the fields $f\\mapsto N^{-d/2-m}\\sum_xϕ_x^N(Pf)(x/N)$, where $P$ is a scalar homogeneous constant-coefficient differential operator of order $m\\geq0$ with Fourier symbol $p$, e.g., $P=Δ$ and $m=2$. Under suitable assumptions on the spectral scaling limit $γ$ of the covariance and the normalized cumulants of order $r\\geq3$, we prove convergence in law in a negative Sobolev space to a centered Gaussian random distribution with covariance multiplier $|p(k)|^2γ(k)$. The two factors separate the effects of the operator and the field and, notably, differentiation can compensate a low-frequency spectral singularity. For the lattice Gaussian Free Field, this gives a projected white-noise limit for the gradient. The proofs use Fourier analysis and the method of moments and cumulants, together with a tightness argument. Finally, we give two sufficient criteria for the cumulant assumption, namely a tree--graph bound and the Dobrushin uniqueness condition for finite-range Gibbs fields, and recover the white-noise limits of the centered Ising magnetization and Potts colour fluctuation fields at high temperature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04452",
  "author_total": 2,
  "authors": [
    {
      "name": "Fabio Coppini"
    },
    {
      "name": "Wioletta M. Ruszel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2nr1sc1tx9vyp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04452",
  "fulltext_url": null,
  "id": "2nr1sc1tx9vyp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04452",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7220649898",
  "pdf_url": "https://arxiv.org/pdf/2610.04452",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2nr1sc1tx9vyp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.04452",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04452"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Gaussian fluctuations of differential observables of stationary lattice fields",
  "title_text": "Gaussian fluctuations of differential observables of stationary lattice fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/2nr1sc1tx9vyp",
  "venue": "arXiv",
  "versions": []
}