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Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman — arXiv:2608.28328

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.28328
License (as archived): CC0-1.0
Archived: 2026-10-17T11:18:05.351Z
SHA-256 of the archived text: 3ee1e89a239f9ed7c3bb1946afdbefb44581cf7e60193fa2f524b7eef7f76161
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Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman — arXiv:2608.28328 — arXiv (en), https://arxiv.org/abs/2608.28328 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_986dbfffc637 (Metraipolis archive dc_986dbfffc637, sha256 3ee1e89a239f9ed7c3bb1946afdbefb44581cf7e60193fa2f524b7eef7f76161)

BibTeX

@misc{dc986dbfffc637,
  title = {Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman — arXiv:2608.28328},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.28328},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_986dbfffc637; sha256 3ee1e89a239f9ed7c3bb1946afdbefb44581cf7e60193fa2f524b7eef7f76161},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.28328. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We prove that the Fourier coefficients of the canonical critical Gaussian multiplicative chaos on the circle vanish almost surely at infinity. More precisely, let $M_\phi^{\mathrm{crit}}$ be the canonical critical chaos associated with the centered circle field $\phi$ of covariance $\mathbb{E}[\phi(\theta)\phi(\theta')] = \log\frac{1}{\lvert e^{i\theta}-e^{i\theta'}\rvert}$. Then, almost surely, $\widehat{M_\phi^{\mathrm{crit}}}(n)\longrightarrow0$ 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-31 in math.PR).
Categorie arXiv: Probability (math.PR) ; Classical Analysis and ODEs (math.CA).
Pagina del lavoro: https://arxiv.org/abs/2608.28328 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T11:05:49.683Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-31?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).