The density of imaginary multiplicative chaos is positive

Consider a log-correlated Gaussian field Gamma and its associated imaginary multiplicative chaos : e i beta Gamma : where beta is a real parameter. In [3], we showed that for any nonzero test function f , the law offf f f : e i beta Gamma : possesses a smooth density with respect to Lebesgue measure on C. In this note, we show that this density is strictly positive everywhere on C. Our simple and direct strategy could be useful for studying other functionals on Gaussian spaces.