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 $\mathrm M_{\mathrm iβ}=\exp(\mathrm iβX)$, where $X$ is a log-correlated Gaussian field. In the subcritical phase $β\in(0,1)$, we prove that its Fourier dimension, defined by the optimal polynomial decay exponent of $|\widehat{\mathrm M_{\mathrm iβ}}(n)|^2$, is almost surely equal to $1-β^2$. 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 $H^{-β^2/2}(\mathbb T)$, the critical Sobolev space left open by previous regularity results. We further establish a central limit theorem: the rescaled coefficients $n^{(1-β^2)/2}\widehat{\mathrm M_{\mathrm iβ}}(n)$ 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 $\mathrm M_{\mathrm iβ}$ behaves as a white noise: $n^{(1-β^2)/2}e^{\mathrm ii nθ}\mathrm M_{\mathrm iβ}$ converges in $H^s(\mathbb T)$, $s<-1/2$, to a complex white noise with explicit intensity $κ(β)=\frac{1}πΓ(1-β^2)\sin\big(\frac{πβ^2}{2}\big)$. 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.