The density of imaginary multiplicative chaos is positive

TL;DR

The paper proves that the density of the imaginary multiplicative chaos $\mu(f)$ associated with a log-correlated Gaussian field is strictly positive everywhere, extending prior results that established the density exists and is smooth. It introduces a simple, direct strategy based on a Cameron–Martin decomposition to create a finite-dimensional, diffeomorphic map from a two-dimensional Gaussian slice to the complex plane, ensuring that small balls around any target value have positive probability. The argument hinges on a deterministic construction of a suitable oscillatory phase and a stability (probabilistic) step on a positive-probability event, plus a tail-decomposition to control high-frequency components. The circle case is treated via an explicit Fourier basis and a circle-specific deterministic diffeomorphism result, yielding analogous positivity for the total mass; collectively, these results advance understanding of densities of Gaussian functionals and have implications for related real chaos, moment behavior, and potential applications to spin models.

Abstract

Consider a log-correlated Gaussian field $Γ$ and its associated imaginary multiplicative chaos $:e^{i βΓ}:$ where $β$ is a real parameter. In [AJJ22], we showed that for any nonzero test function $f$, the law of $\int f :e^{i βΓ}:$ possesses a smooth density with respect to Lebesgue measure on $\mathbb{C}$. In this note, we show that this density is strictly positive everywhere on $\mathbb{C}$. Our simple and direct strategy could be useful for studying other functionals on Gaussian spaces.

Theorems & Definitions (14)

• Theorem 1
• Proposition 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof : Proof of Lemma \ref{['L:Brouwer']}
• Lemma 6
• proof : Proof of Lemma \ref{['L:Phi']}, assuming Proposition \ref{['P:intermediate']}