Almost-sure asymptotic order of critical Fourier coefficients

Determine the almost-sure asymptotic order of the Fourier coefficients of the canonical critical Gaussian multiplicative chaos on the circle, beyond their qualitative convergence to zero.

Background

The paper proves that the Fourier coefficients of the canonical critical Gaussian multiplicative chaos associated with the exact log-correlated field on the circle converge almost surely to zero as the frequency tends to infinity, establishing the Rajchman property. This result is qualitative rather than quantitative.

Although Proposition 4.1 provides a polynomial Fourier bound for the coarse predictable measure used in the proof, that estimate does not yield a decay rate for the terminal Fourier coefficients of the critical chaos itself. The authors therefore leave unresolved the problem of identifying the almost-sure asymptotic order of those terminal coefficients.

References

Theorem~\ref{thm:main-circle-rajchman} is qualitative. The polynomial bound in Proposition~\ref{prop:predictable-annular-decay} applies only to the coarse predictable measure used in the proof and does not give a decay rate for the terminal coefficients \widehat{M_\phi{\mathrm{crit}}(n). Determining their almost-sure asymptotic order remains open.

— Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman  (2608.28328 - Cai et al., 28 Aug 2026) in Section 1, subsection “Scope and organization” (subsec:introduction-scope)