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$isacontinuousfunction$x$and$y$and$\gamma=\alpha+i\beta$ is a complex parameter. In the present paper, we consider the case $\gamma \in {\mathcal{P}}_{\mathrm{I}\mathrm{I}\mathrm{I}}^{\mathrm{\prime }}$ where $${\mathcal{P}}_{\mathrm{I}\mathrm{I}\mathrm{I}}^{\mathrm{\prime }}:=\{\alpha +i\beta :\alpha ,\gamma \in \mathbb{R},\mathrm{\mid }\alpha \mathrm{\mid }<\sqrt{d\mathrm{/}2},{\alpha }^2+{\beta }^2\ge d\}\mathrm{.}$$ We prove that if $X$ is replaced by the approximation $X_{\epsilon }$ obtained by convolution with a smooth kernel, then $e^{\gamma X_{\epsilon }}\mathrm{d}x$, when properly rescaled, has an explicit non-trivial limit in distribution when $\epsilon$ goes to zero. This limit does not depend on the specific convolution kernel which is used to define $X_{\epsilon }$ and can be described as a complex Gaussian white noise with a random intensity given by a real GMC associated with parameter $2\alpha$.