Convergence for Complex Gaussian Multiplicative Chaos on phase boundaries

The complex Gaussian Multiplicative Chaos (or complex GMC) is informally defined as a random measure $e^{\gamma X} \mathrm{d} x$ where $X$ is a log correlated Gaussian field on $\mathbb R^d$ and $\gamma=\alpha+i\beta$ is a complex parameter. The correlation function of $X$ is of the form $$K(x,y)= \log \frac{1}{|x-y|}+ L(x,y),$$ where $L$ is a continuous function. In the present paper, we consider the cases $\gamma\in \mathcal P_{\mathrm{I/II}}$ and $\gamma\in \mathcal{P}'_{\mathrm{II/III}}$ where $$\mathcal P_{\mathrm{I/II}}:= \{ \alpha+i \beta \ : \alpha,\beta \in \mathbb R \ ; |\alpha|>|\beta| \ ; \ |\alpha|+|\beta|=\sqrt{2d} \},$$ and $$\mathcal{P}'_{\mathrm{II/III}}:= \{ \alpha+i \beta \ : \alpha,\beta \in \mathbb R \ ; \ |\alpha|= \sqrt{d/2} \ ; \ |\beta|>\sqrt{2d} \},$$ We prove that if $X$ is replaced by an approximation $X_\epsilon$ obtained via mollification, then $e^{\gamma X_\epsilon} \mathrm{d} x$, when properly rescaled, converges when $\epsilon\to 0$. The limit does not depend on the mollification kernel. When $\gamma\in \mathcal P_{\mathrm{I/II}}$, the convergence holds in probability and in $L^p$ for some value of $p\in [1,\sqrt{2d}/\alpha)$. When $\gamma\in \mathcal{P}'_{\mathrm{II/III}}$ the convergence holds only in law. In this latter case, the limit can be described a complex Gaussian white noise with a random intensity given by a critical real GMC. The regions $\mathcal P_{\mathrm{I/II}}$ and $\mathcal{P}'_{\mathrm{II/III}}$ correspond to phase boundary between the three different regions of the complex GMC phase diagram. These results complete previous results obtained for the GMC in phase I and III and only leave as an open problem the question of convergence in phase II.