Convergence in law for complex Gaussian multiplicative chaos in phase III

Abstract

Gaussian multiplicative chaos (GMC) is informally defined as a random measure ${\mathit{e}}^{\mathit{\gamma }\mathit{X}}\mathrm{d}\mathit{x}$ where X is Gaussian field on ${\mathbb{R}}^{\mathit{d}}$ (or an open subset of it) whose correlation function is of the form $\mathit{K}(\mathit{x},\mathit{y})=log\frac{1}{|\mathit{y}-\mathit{x}|}+\mathit{L}(\mathit{x},\mathit{y})$, where L is a continuous function of x and y and $\mathit{\gamma }=\mathit{\alpha }+\mathit{i}\mathit{\beta }$ is a complex parameter. In the present paper we consider the case $\mathit{\gamma }\in {\mathcal{P}}_{\mathrm{III}}^{\prime }$, where

$${\mathcal{P}}_{\mathrm{III}}^{\prime }:=\{\mathit{\alpha }+\mathit{i}\mathit{\beta }:\mathit{\alpha },\mathit{\gamma }\in \mathbb{R},|\mathit{\alpha }|<\sqrt{\mathit{d}/2},{\mathit{\alpha }}^2+{\mathit{\beta }}^2\ge \mathit{d}\}.$$

We prove that if X is replaced by an approximation ${\mathit{X}}_{\mathit{\epsilon }}$ obtained by convolution with a smooth kernel, then the random distribution ${\mathit{e}}^{\mathit{\gamma }{\mathit{X}}_{\mathit{\epsilon }}}\mathrm{d}\mathit{x}$, when properly rescaled, has an explicit nontrivial limit in law when ε goes to zero. This limit does not depend on the specific convolution kernel which is used to define ${\mathit{X}}_{\mathit{\epsilon }}$ and can be described as a complex Gaussian white noise with a random intensity given by a real GMC associated with parameter $2\mathit{\alpha }$.