On the Fourier Coefficients of critical Gaussian multiplicative chaos
Louis-Pierre Arguin, Jad Hamdan
TL;DR
The paper resolves the behavior of Fourier coefficients for critical Gaussian multiplicative chaos on [0,1], proving that the coefficients c_n vanish in probability with a logarithmic scaling: (log n)|c_n| is tight. The authors develop a second-moment framework on a carefully defined good event for the star-scale invariant field X and its GMC μ, then manage oscillatory integrals by partitioning into large- and small-separation regimes, exploiting barrier estimates and Girsanov transforms. The key finding is that the dominant contribution to the second moment arises from a specific oscillatory regime, while the oscillatory range contributes negligibly, leading to c_n → 0 in probability. This work constitutes a first major step toward understanding Rajchman properties of critical GMC and motivates conjectures for subcritical regimes, where precise scaling laws and limiting laws for (log n) and n dependences are predicted. The techniques blend log-correlated field theory, barrier estimates, and oscillatory analysis to tackle a long-standing open problem in the harmonic analysis of random multifractal measures.
Abstract
We continue the study of the Fourier coefficients of Gaussian multiplicative chaos (GMC) recently initiated by Garban and Vargas. We show that the Fourier coefficients $c_n$ of critical GMC on the unit interval converge to zero in probability as the frequency $n$ tends to infinity, by establishing tightness for $\{(\log n)|c_n|\}_{n\geq 1}$.