Convergence for complex Gaussian multiplicative chaos on phase boundaries

The complex Gaussian Multiplicative Chaos (or complex GMC) is informally defined as a random measure e γX dx where X is a log correlated Gaussian field on R d and γ " α `iβ is a complex parameter.The correlation function of X is of the form Kpx, yq " log 1 |x ´y| `Lpx, yq, where L is a continuous function.In the present paper, we consider the cases γ P P I{II and γ P P 1 II{III where P I{II :" tα `iβ : α, β P R ; |α| ą |β| ; |α| `|β| " ?2du, and P 1 II{III :" tα `iβ : α, β P R ; |α| " a d{2 ; |β| ą ?2du, We prove that if X is replaced by an approximation Xε obtained via mollification, then e γXε dx, when properly rescaled, converges when ε Ñ 0. The limit does not depend on the mollification kernel.When γ P P I{II , the convergence holds in probability and in L p for some value of p P r1, ?2d{αq.When γ P P 1 II{III the convergence holds only in law.In this latter case, the limit can be described a complex Gaussian white noise with a random intensity given by a critical real GMC.The regions P I{II and P 1 II{III correspond to phase boundary between the three different regions of the complex GMC phase diagram.These results complete previous results obtained for the GMC in phase I [18] and III [16] and only leave as an open problem the question of convergence in phase II.