A universality result for subcritical Complex Gaussian Multiplicative Chaos

In the present paper, we show that (under some minor technical assumption) Complex Gaussian Multiplicative Chaos defined as the complex exponential of a $\log$-correlated Gaussian field can be obtained by taking the limit of the exponential of the field convoluted with a smoothing Kernel. We consider two types of chaos: $e^{\gamma X}$ for a log correlated field $X$ and $\gamma=\alpha+i\beta$, $\alpha, \beta\in \mathbb R$ and $e^{\alpha X+i\beta Y}$ for $X$ and $Y$ two independent fields with $\alpha, \beta\in \mathbb R$. Our result is valid in the range $$\mathcal O_{\mathrm{sub}}:=\{ \alpha^2+\beta^2<d \} \cup \{ |\alpha|\in (\sqrt{d/2},\sqrt{2d} ) \text{ and } |\beta|< \sqrt{2d}-|\alpha| \},$$ which, up to boundary, is conjectured to be optimal.