{
  "abstract": "We prove that the Fourier coefficients of the canonical critical Gaussian multiplicative chaos on the circle vanish almost surely at infinity. More precisely, let $M_ϕ^{\\mathrm{crit}}$ be the canonical critical chaos associated with the centered circle field $ϕ$ of covariance $\\mathbb{E}[ϕ(θ)ϕ(θ')] = \\log\\frac{1}{\\lvert e^{iθ}-e^{iθ'}\\rvert}$. Then, almost surely, $\\widehat{M_ϕ^{\\mathrm{crit}}}(n)\\longrightarrow0$ as $\\lvert n\\rvert\\to\\infty$. This resolves the almost-sure critical Rajchman problem for the canonical circle field. Since critical chaos has Fourier dimension zero almost surely, no positive polynomial Fourier-decay rate can hold; the theorem therefore exhibits qualitative Fourier cancellation beyond the regime of positive Fourier dimension. The proof addresses two coupled difficulties: the heavy, nonuniform cell masses of critical chaos and the need to control exponentially many frequencies in each dyadic annulus. For an auxiliary periodized compact-range star-scale field, a derivative-rooted Bessel regression yields weighted small-cell summability and moving-tail control of exceptional large cells. After conditioning at a coarse scale below the Fourier scale, finite-range independence and conditional Bernstein concentration reduce uniform control of the terminal Fourier coefficients over each dyadic annulus to a spatial-variation estimate for a coarse predictable measure. A smooth positive-definite covariance correction and critical-chaos uniqueness then transfer the Rajchman property to the canonical critical chaos of the exact circle field.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28328",
  "author_total": 4,
  "authors": [
    {
      "name": "Yin Cai"
    },
    {
      "name": "Bonan Chen"
    },
    {
      "name": "Xiang Fang"
    },
    {
      "name": "Feng Guo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/12p21sn69bvkn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28328",
  "fulltext_url": null,
  "id": "12p21sn69bvkn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28328",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7204789476",
  "pdf_url": "https://arxiv.org/pdf/2608.28328",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/12p21sn69bvkn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28328",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28328"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman",
  "title_text": "Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/12p21sn69bvkn",
  "venue": "arXiv",
  "versions": []
}