Density of imaginary multiplicative chaos via Malliavin calculus
Juhan Aru, Antoine Jego, Janne Junnila
TL;DR
This work establishes that the imaginary Gaussian multiplicative chaos $\mu_\beta$ associated with a non-degenerate log-correlated field possesses a smooth, Schwartz density for $\mu_\beta(f)$ when tested against nonzero compactly supported $f$. The authors introduce and implement Malliavin calculus in this complex setting, complemented by a novel decomposition theorem that enables a transfer to almost $\star$-scale invariant components and facilitates small-ball Sobolev estimates. Key contributions include precise bounds for the Malliavin determinant and related functionals, leading to the density result and implications for moments in the Fyodorov-Bouchaud framework. The techniques illuminate a pathway for analyzing densities and tail behavior of complex multiplicative chaos and suggest broader applicability to sine-Gordon-type models and related log-correlated systems.
Abstract
We consider the imaginary Gaussian multiplicative chaos, i.e. the complex Wick exponential $μ_β:= :e^{iβΓ(x)}:$ for a log-correlated Gaussian field $Γ$ in $d \geq 1$ dimensions. We prove a basic density result, showing that for any nonzero continuous test function $f$, the complex-valued random variable $μ_β(f)$ has a smooth density w.r.t. the Lebesgue measure on $\mathbb{C}$. As a corollary, we deduce that the negative moments of imaginary chaos on the unit circle do not correspond to the analytic continuation of the Fyodorov-Bouchaud formula, even when well-defined. Somewhat surprisingly, basic density results are not easy to prove for imaginary chaos and one of the main contributions of the article is introducing Malliavin calculus to the study of (complex) multiplicative chaos. To apply Malliavin calculus to imaginary chaos, we develop a new decomposition theorem for non-degenerate log-correlated fields via a small detour to operator theory, and obtain small ball probabilities for Sobolev norms of imaginary chaos.