Abstract

The complex Gaussian multiplicative chaos (or complex GMC) is informally defined as a random distribution $e^{\gamma X}d<!--FUNCTION\; APPLICATION-->x$ where $X$ is a $\log <!--FUNCTION\; APPLICATION-->$-correlated Gaussian field on ${ℝ}^d$ and $\gamma =\alpha +i\beta$ is a complex parameter. The correlation function of $X$ is of the form

where $L$ is a continuous function. We consider the cases $\gamma \in {\mathcal{𝒫}}_{\mathrm{I/II}<!--FUNCTION\; APPLICATION-->}^{}$ and $\gamma \in {\mathcal{𝒫}}_{\mathrm{II/III}<!--FUNCTION\; APPLICATION-->}^{\prime }$ where

and

We prove that if $X$ is replaced by an approximation $X_{𝜀}$ obtained via mollification, then $e^{\gamma X_{𝜀}}d<!--FUNCTION\; APPLICATION-->x$, when properly rescaled, converges when $𝜀\to 0$. The limit does not depend on the mollification kernel. When $\gamma \in {\mathcal{𝒫}}_{\mathrm{I/II}<!--FUNCTION\; APPLICATION-->}^{}$, the convergence holds in probability and in $L^p$ for some value of $p\in \left[1,\sqrt{2d}∕\alpha \right)$. When $\gamma \in {\mathcal{𝒫}}_{\mathrm{II/III}<!--FUNCTION\; APPLICATION-->}^{\prime }$, the convergence holds only in law. In this latter case, the limit can be described as a complex Gaussian white noise with a random intensity given by a critical real GMC. The regions ${\mathcal{𝒫}}_{\mathrm{I/II}<!--FUNCTION\; APPLICATION-->}^{}$ and ${\mathcal{𝒫}}_{\mathrm{II/III}<!--FUNCTION\; APPLICATION-->}^{\prime }$ correspond to the phase boundary between the three different regions of the complex GMC phase diagram. These results complete previous results obtained for the GMC in phases I and III and only leave as an open problem the question of convergence in phase II.

Keywords

Mathematical Subject Classification

Milestones

Authors