Fourier asymptotics of critical Gaussian multiplicative chaos on the circle
Abstract
We study the high-frequency Fourier coefficients Cn 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 β(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 Hβs(π) to this conditional stable noise. Unconditionally, the same joint approximation remains valid for offsets of size at most (loglog 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 β(log|n|) |Cn|/H(loglog|n|) is zero or infinity according as β«β 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.
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BibTeX
@misc{cai2026fourier,
title = {{Fourier asymptotics of critical Gaussian multiplicative chaos on the circle}},
author = {Yin Cai and Bonan Chen and Xiang Fang and Feng Guo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05985},
eprint = {2610.05985},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05985},
}
RIS
TY - UNPB
TI - Fourier asymptotics of critical Gaussian multiplicative chaos on the circle
AU - Yin Cai
AU - Bonan Chen
AU - Xiang Fang
AU - Feng Guo
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05985
UR - https://arxiv.org/abs/2610.05985
AB - 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 β(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^βs(π) to this conditional stable noise. Unconditionally, the same joint approximation remains valid for offsets of size at most (loglog 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 β(log|n|) |C_n|/H(loglog|n|) is zero or infinity according as β«^β 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.
ER -