Convergence in law for Complex Gaussian Multiplicative Chaos in phase III

Gaussian Multiplicative Chaos (GMC) is informally defined as a random measure $e^{\gamma X} \mathrm{d} x$ where $X$ is Gaussian field on $\mathbb R^d$ (or an open subset of it) whose correlation function is of the form $K(x,y)= \log \frac{1}{|y-x|}+ L(x,y),$ where $L$ is a continuous function $x$ and $y$ and $\gamma=\alpha+i\beta$ is a complex parameter. In the present paper, we consider the case $\gamma\in \mathcal P'_{\mathrm{III}}$ where $$\mathcal P'_{\mathrm{III}}:= \{ \alpha+i \beta \ : \alpha,\gamma \in \mathbb R , \ |\alpha|<\sqrt{d/2}, \ \alpha^2+\beta^2\ge d \}.$$ We prove that if $X$ is replaced by the approximation $X_\varepsilon$ obtained by convolution with a smooth kernel, then $e^{\gamma X_\varepsilon} \mathrm d x$, when properly rescaled, has an explicit non-trivial limit in distribution when $\varepsilon$ goes to zero. This limit does not depend on the specific convolution kernel which is used to define $X_{\varepsilon}$ and can be described as a complex Gaussian white noise with a random intensity given by a real GMC associated with parameter $2\alpha$.