{
  "abstract": "We study critical holomorphic multiplicative chaos with circular iid inputs of unit variance. Writing $R=|X_1|$, we prove that the radially normalized boundary measures converge in probability to a finite measure with almost surely positive mass if and only if $\\mathbb{E}[R^2\\log(2+R)]<\\infty$; otherwise they converge to zero. We obtain distinct sharp criteria for convergence at each fractional order $0<q<1$ for radial approximations and for the full Fourier-radial family. Convergence at every such order is equivalent to an exponential-moment radius of at least two. In the positive-limit class, coefficient vectors normalized by $(\\log n)^{1/4}$ and their analytic windows converge stably to Gaussian mixtures determined by the same critical measure. This measure is nonatomic, has full support, and is carried by a set of Hausdorff dimension zero. Coefficient periodograms recover the measure, while scalar and matrix energy normalizations give universal Gaussian limits. Exact weighted tails under $\\mathbb{E} I_0(2R)<\\infty$, where $I_0$ is the modified Bessel function, and negative mass moments distinguish uncountably many coefficient-limit laws, even modulo deterministic scalar rescaling, within a uniformly bounded input family. This disproves the common-law clause of Gu-Zhang's Question 2 at its fixed normalization. We give conditions for convergence of the original coefficient moments and show that finiteness of the limiting moments alone does not suffice. For inputs with $\\log\\mathbb{P}(R>u)\\sim-λu^α$, $0<α<1$, the coefficient normalization and Gaussian-mixture form persist, but the limit has no finite positive moments; we determine its logarithmic tails. Finally, we prove Hardy nonmembership under finite variance alone and, in the positive-limit class, sharp weighted Bergman criteria for $0<p\\le2$ and the boundary Sobolev threshold $s=-1/2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34371",
  "author_total": 1,
  "authors": [
    {
      "name": "Feng Guo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5v0yax44pn0cb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34371",
  "fulltext_url": null,
  "id": "5v0yax44pn0cb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34371",
  "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.PR"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7223943161",
  "pdf_url": "https://arxiv.org/pdf/2610.34371",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5v0yax44pn0cb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34371",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34371"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Critical measures and coefficient limits for non-Gaussian holomorphic chaos",
  "title_text": "Critical measures and coefficient limits for non-Gaussian holomorphic chaos",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/5v0yax44pn0cb",
  "venue": "arXiv",
  "versions": []
}