Almost sure critical convergence for Gaussian multiplicative chaos

Prove that, for Gaussian log-correlated fields with sufficiently regular covariance functions, the Gaussian multiplicative chaos total mass μ^γ/(γ_c−γ) converges almost surely as γ increases to the critical parameter γ_c.

Background

For Gaussian multiplicative chaos, the subcritical total mass divided by γ_c−γ is known to converge in probability to a critical object. The paper’s almost sure result for the analogous branching-Brownian-motion quantity suggests that this convergence may hold almost surely for Gaussian log-correlated fields under suitable covariance regularity assumptions.

References

Our Theorem~\ref{thm:as} suggests that the convergence in the third point could be reinforced into an almost sure convergence.

— Fluctuations of additive martingale limits of branching Brownian motion  (2609.10530 - Chen et al., 9 Sep 2026) in Conjecture 4, Section 3, “Related literature and further questions,” paragraph “Gaussian multiplicative chaos”