Convergence in law for complex Gaussian multiplicative chaos in phase III

Abstract

Gaussian multiplicative chaos (GMC) is informally defined as a random measure eγXdx where X is Gaussian field on Rd (or an open subset of it) whose correlation function is of the form K(x,y)=log1|y−x|+L(x,y), where L is a continuous function of x and y and γ=α+iβ is a complex parameter. In the present paper we consider the case γ∈PIII′, where PIII′:={α+iβ:α,γ∈R,|α|< d/2,α2+β2≥d}. We prove that if X is replaced by an approximation Xε obtained by convolution with a smooth kernel, then the random distribution eγXεdx, 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 Xε and can be described as a complex Gaussian white noise with a random intensity given by a real GMC associated with parameter 2α.