The density of imaginary multiplicative chaos is positive

Abstract

Consider a log-correlated Gaussian field $\Gamma$ and its associated imaginary multiplicative chaos $:e^{i\beta \Gamma }:$ where $\beta$ is a real parameter. In [AJJ22], we showed that for any nonzero test function $f$, the law of $\int f:e^{i\beta \Gamma }:$ 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.

13 pages, to appear in Electronic Communications in Probability

References in corpus (1)

• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential

Cited by in corpus (1)

• Noise-like analytic properties of imaginary chaos