Gaussian multiplicative chaos (GMC) is informally defined as a random measure where X is Gaussian field on (or an open subset of it) whose correlation function is of the form , where L is a continuous function of x and y and is a complex parameter. In the present paper we consider the case , where
We prove that if X is replaced by an approximation obtained by convolution with a smooth kernel, then the random distribution , 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 and can be described as a complex Gaussian white noise with a random intensity given by a real GMC associated with parameter .
This work was realized during the author’s extended stay in Aix-Marseille University funded by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 837793.
The author is grateful to Paul Gassiat for indicating the reference  for the proof of Theorem 2.5 and letting him know about the notion of stable convergence which is the adequate framework to present the main result of this paper. He thanks J. F. Le Gall, Rémi Rhodes and Vincent Vargas for enlightening comments.
"Convergence in law for complex Gaussian multiplicative chaos in phase III." Ann. Probab. 50 (3) 950 - 983, May 2022. https://doi.org/10.1214/21-AOP1551