Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman
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 Mϕcrit be the canonical critical chaos associated with the centered circle field ϕ of covariance 𝔼[ϕ(θ)ϕ(θ')] = log1/(lvert eiθ−eiθ'rvert). Then, almost surely, Mϕcrit̂(n)⟶0 as lvert nrvert→∞. 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.
Related
- Fourier asymptotics of critical Gaussian multiplicative chaos on the circle
- Power variations of critical Gaussian multiplicative chaos and their spectral applications
- Phase II Complex Gaussian Multiplicative Chaos
- Derivatives of Gaussian multiplicative chaos
- Fourier decay of Gaussian multiplicative chaos and boundary geometry
BibTeX
@misc{cai2026critical,
title = {{Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman}},
author = {Yin Cai and Bonan Chen and Xiang Fang and Feng Guo},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28328},
eprint = {2608.28328},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28328},
}
RIS
TY - UNPB
TI - Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman
AU - Yin Cai
AU - Bonan Chen
AU - Xiang Fang
AU - Feng Guo
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28328
UR - https://arxiv.org/abs/2608.28328
AB - 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^crit be the canonical critical chaos associated with the centered circle field ϕ of covariance 𝔼[ϕ(θ)ϕ(θ')] = log1/(lvert e^iθ−e^iθ'rvert). Then, almost surely, M_ϕ^crit̂(n)⟶0 as lvert nrvert→∞. 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.
ER -