Density of imaginary multiplicative chaos via Malliavin calculus

TL;DR

This paper proves that imaginary Gaussian multiplicative chaos has a smooth density for any nonzero test function, introduces Malliavin calculus to this context, and explores implications for negative moments and analytic continuations.

Contribution

It introduces Malliavin calculus to the study of complex multiplicative chaos and develops a new decomposition theorem for log-correlated fields.

Findings

Imaginary chaos has a smooth density for nonzero test functions.

Negative moments of imaginary chaos on the unit circle do not match the analytic continuation of Fyodorov-Bouchaud formula.

Developed a new decomposition theorem for non-degenerate log-correlated fields.

Abstract

We consider the imaginary Gaussian multiplicative chaos, i.e. the complex Wick exponential $\mu_\beta := :e^{i\beta \Gamma(x)}:$ for a log-correlated Gaussian field $\Gamma$ 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 $\mu_\beta(f)$ 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.