Fourier dimension of imaginary Gaussian multiplicative chaos

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: $n^{(1-{\beta }^2)/2}{e}^{\mathrm{i}in\theta }{\mathrm{M}}_{\mathrm{i}\beta }$ converges in $H^s(\mathbb{T})$, $s<-1/2$, to a complex white noise with explicit intensity $\kappa (\beta )=\frac{1}{\pi }\Gamma (1-{\beta }^2)\sin \left(\frac{\pi {\beta }^2}{2}\right)$. 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.

Extension of the Fourier dimension to general log-correlated fields