Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

• Main Authors: Duplantier, Bertrand (Author), Rhodes, Rémi (Author), Vargas, Vincent (Author), Sheffield, Scott Roger (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Institute of Mathematical Statistics, 2018-05-11T18:34:54Z.
• Subjects: Article
• Online Access: Get fulltext

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.
National Science Foundation (U.S.) (Grant DMS-06-4558)