A new mathematical preprint has established that the canonical critical Gaussian multiplicative chaos on the circle has Fourier coefficients that tend to zero at high frequencies almost surely. In ordinary language, the random measure’s increasingly fine oscillatory components fade away, even though the theorem does not say how quickly they do so.
The result concerns a theoretical random measure, not an analysis of observations or an experiment. The target is the canonical critical chaos associated with the centered circle field, a construction whose limiting measure is stated to be independent of the admissible mollifier used to define it.
What the theorem says
A Fourier coefficient is a number assigned to a frequency. Calling the measure Rajchman means that these coefficients vanish as the frequency rises without bound. The preprint establishes this behavior almost surely, meaning that it holds outside a set of random outcomes with probability zero.
That conclusion is deliberately qualitative. The authors do not give an almost-sure asymptotic order for the final, or terminal, coefficients of the canonical chaos. Readers should therefore not interpret the result as a claim that those coefficients obey a particular polynomial decay law.
A proof built in stages
The argument begins with an auxiliary periodized random field whose interactions have a compact range across scales. The researchers then study its Fourier coefficients through a derivative-rooted Bessel regression. Under the resulting rooted law, the root is uniform on the circle and the rooted process is described by a three-dimensional Bessel process started from the parameter beta.
This rooted analysis controls how mass can concentrate in very small cells around typical points. For every fixed nonnegative exponent in the estimate, the proof obtains weighted small-cell summability almost surely for almost every root with respect to the random measure. That summability is used to control exceptional cells in the subsequent estimates.
At frequencies grouped into doubling bands, the terminal auxiliary coefficients are uniformly approximated by coefficients of a coarser predictable measure. The coarse object records information available at an earlier stage of the construction, where finite-range conditional independence makes concentration estimates possible.
On the probability-one events used in the argument, those coarse predictable coefficients obey an inverse-cubic bound in N for all sufficiently large N. That estimate belongs to the proof-specific coarse measure, not to the terminal coefficients of the canonical circle chaos.
From the auxiliary model to the circle field
The proof must still connect the auxiliary periodized construction to the exact circle field. To do that, it adds an independent smooth stationary Gaussian correction and an independent Gaussian constant, producing a corrected star-scale field that agrees with the exact circle field in law.
Critical-chaos uniqueness is used twice in this transfer. It first identifies the limiting law and then gives convergence in probability to the same measure on the original coupling. The periodized argument establishes the Rajchman property for the derivative limit and identifies the positive critical scale measure as a deterministic multiple of that limit.
The final constant-mode step changes the target critical measure only by an almost surely finite, strictly positive scalar. Since multiplying every Fourier coefficient by such a scalar does not change whether the coefficients vanish at infinity, the argument yields the almost-sure Rajchman result for the canonical critical chaos.
A qualitative result with a quantitative question left open
The preprint’s central achievement is the vanishing itself. It does not determine the almost-sure asymptotic order of the canonical terminal coefficients, and it offers no positive polynomial decay rate for them. The inverse-cubic estimate should be read only as an intermediate bound for the coarse predictable measure used inside the proof.
The document is an arXiv version 1 preprint dated 28 August 2026.
Paper data and sources
Original title: Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman
Authors: Yin Cai, Bonan Chen, Xiang Fang, Feng Guo
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text