The density of imaginary multiplicative chaos is positive

Abstract

Consider a log-correlated Gaussian field Γ and its associated imaginary multiplicative chaos $:e^{i\mathit{\beta }\mathrm{\Gamma }}$: where β is a real parameter. In [3], we showed that for any nonzero test function f, the law of $\int f:e^{i\mathit{\beta }\mathrm{\Gamma }}$: 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.