A universality result for subcritical Complex Gaussian Multiplicative Chaos

Abstract

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.