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Fourier asymptotics of critical Gaussian multiplicative chaos on the circle — arXiv:2610.05985

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.05985
License (as archived): CC0-1.0
Archived: 2026-10-06T13:48:27.173Z
SHA-256 of the archived text: cc05b157e1cfbc4a987c0f574a6c0327edc30b7d919d043223f6380dc30a86e7
Stable URL of this page: https://metraipolis.com/documento/dc_9ef866bb64ae

How to cite

Fourier asymptotics of critical Gaussian multiplicative chaos on the circle — arXiv:2610.05985 — arXiv (en), https://arxiv.org/abs/2610.05985 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_9ef866bb64ae (Metraipolis archive dc_9ef866bb64ae, sha256 cc05b157e1cfbc4a987c0f574a6c0327edc30b7d919d043223f6380dc30a86e7)

BibTeX

@misc{dc9ef866bb64ae,
  title = {Fourier asymptotics of critical Gaussian multiplicative chaos on the circle — arXiv:2610.05985},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.05985},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_9ef866bb64ae; sha256 cc05b157e1cfbc4a987c0f574a6c0327edc30b7d919d043223f6380dc30a86e7},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.05985. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study the high-frequency Fourier coefficients $C_n$ of critical Gaussian multiplicative chaos $M$ on the circle, for logarithmic covariances with an arbitrary admissible smooth, possibly nonstationary remainder. For every fixed finite set of integer offsets, the corresponding vector of neighboring coefficients, multiplied by $\sqrt{\log n}$, converges stably relative to the full Gaussian field without additional centering. Conditionally on that field, the limit consists of Fourier coefficients of one symmetric complex stable random measure of index one with control proportional to $M$. For every $s>1$, the normalized modulated measures also converge stably in $H^{-s}(\mathbb{T})$ to this conditional stable noise. Unconditionally, the same joint approximation remains valid for offsets of size at most $(\log\log n)^{1/64}$ for every fixed admissible covariance. The scalar limit is an isotropic Cauchy mixture; the joint limiting law retains the spatial distribution of $M$, not only its total mass. We also prove an almost-sure integral test along the entire integer frequency sequence. For a class of regular deterministic gauges $H$, the limsup of $\sqrt{\log|n|}\,|C_n|/H(\log\log|n|)$ is zero or infinity according as $\int^\infty dx/H(x)$ converges or diverges. The proof combines index-one compensation with relative excursion estimates and transfers these conclusions from the canonical model by smooth covariance comparison.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.PR, nuovi lavori del giorno).
Categorie arXiv: Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2610.05985 — 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 2026-10-06T12:06:32.162Z dalla pagina https://arxiv.org/list/math.PR/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).