Boundary behavior of complex Gaussian multiplicative chaos

Characterize the behavior of complex Gaussian multiplicative-chaos measures as the parameter z approaches the boundary of P_1, including analogues of the almost sure convergence, continuity, and P_1-P_3 fluctuation statements formulated for complex additive martingales.

Background

The paper notes that complex Gaussian multiplicative chaos has been studied in several phases and on several boundaries, but that the behavior when parameters in P_1 approach the boundary has not been established in general. The authors expect analogues of their conjectures for almost sure convergence, continuity, and Gaussian fluctuations near P_{1,3}.

References

The behavior of complex GMC measures as the parameter $z \in P_1$ approaches the boundary of $P_1$ is also open and should be given by statements similar to Conjectures~\ref{conj:as}, \ref{conj:continuity} and \ref{conj:CLT_13}.

— Fluctuations of additive martingale limits of branching Brownian motion  (2609.10530 - Chen et al., 9 Sep 2026) in Section 3, “Related literature and further questions,” paragraph “Gaussian multiplicative chaos”