Fourier dimension of imaginary Gaussian multiplicative chaos

TL;DR

This paper investigates the Fourier decay and regularity properties of imaginary Gaussian multiplicative chaos on the circle, revealing its Fourier dimension, limiting distribution, and white noise behavior in the high-frequency regime.

Contribution

It establishes the Fourier dimension for a broad class of log-correlated fields and proves a central limit theorem for the Fourier coefficients of imaginary Gaussian chaos.

Findings

Fourier dimension equals 1 - beta^2 for subcritical beta.

Rescaled Fourier coefficients converge to complex Gaussian variables.

High-frequency content behaves as a complex white noise with explicit intensity.

Abstract

We study the high-frequency Fourier asymptotics of imaginary Gaussian multiplicative chaos on the unit circle, a complex-valued random distribution formally given by $\mathrm M_{\mathrm i\beta}=\exp(\mathrm i\beta X)$, where $X$ is a log-correlated Gaussian field. In the subcritical phase $\beta\in(0,1)$, we prove that its Fourier dimension, defined by the optimal polynomial decay exponent of $|\widehat{\mathrm M_{\mathrm i\beta}}(n)|^2$, is almost surely equal to $1-\beta^2$. This result holds for a broad class of log-correlated fields whose covariance differs from the exact logarithmic kernel by a sufficiently regular function. For the exactly log-correlated field on the circle, we obtain the following results. We prove that the chaos almost surely fails to belong to $H^{-\beta^2/2}(\mathbb T)$, the critical Sobolev space left open by previous regularity results. We further establish a central limit theorem: the rescaled coefficients $n^{(1-\beta^2)/2}\widehat{\mathrm M_{\mathrm i\beta}}(n)$ converge in law to an isotropic complex Gaussian random variable, and finitely many consecutive coefficients converge jointly to independent copies. The high-frequency content of $\mathrm M_{\mathrm i\beta}$ behaves as a white noise: $n^{(1-\beta^2)/2}e^{\mathrm ii n\theta}\mathrm M_{\mathrm i\beta}$ converges in $H^s(\mathbb T)$, $s<-1/2$, to a complex white noise with explicit intensity $\kappa(\beta)=\frac{1}{\pi}\Gamma(1-\beta^2)\sin\big(\frac{\pi\beta^2}{2}\big)$. The proof relies on moment identities obtained from Coulomb-gas integrals and Jack-polynomial expansions. Their asymptotic analysis is governed by partitions with large gaps, where the Pieri coefficients appearing in these expansions simplify, and the leading contribution becomes explicit.