Uniqueness of supercritical Gaussian multiplicative chaos
Federico Bertacco, Martin Hairer
TL;DR
This work proves that, for a broad class of log-correlated Gaussian fields regularised by convolution, the properly normalised supercritical GMC measures converge in the stable sense to a nontrivial limit that is an integrated atomic measure whose random weights are independent of the underlying field. The limit decomposes into the location of mass determined by a critical GMC and a random weighting given by a Poissonian construction, with the regularisation affecting only a multiplicative constant. The authors develop a novel decomposition of covariances into a martingale part plus a smooth independent field, enabling a transfer of convergence from a star-scale invariant model to general settings. They also extend the convergence to infinite-range correlations by combining finite-range approximations with Kahane’s convexity inequality, and they provide a detailed account of moments and multifractal properties in an appendix. Overall, the paper solidifies the universality of the supercritical GMC limit and clarifies the role of regularisation in its construction and outcome.
Abstract
We show that, for general convolution approximations to a large class of log-correlated Gaussian fields, the properly normalised supercritical Gaussian multiplicative chaos measures converge stably to a nontrivial limit. This limit depends on the choice of regularisation only through a multiplicative constant and can be characterised as an integrated atomic measure with a random intensity expressed in terms of the critical Gaussian multiplicative chaos.