Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
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Author(s)
Duplantier, Bertrand
Rhodes, Rémi
Vargas, Vincent
Sheffield, Scott Roger
Date Issued
September 2014
Journal
Annals of Probability
Publisher
Institute of Mathematical Statistics
Citation
Duplantier, Bertrand et al. “Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale.” The Annals of Probability 42, 5 (September 2014): 1769–1808 © 2014 Institute of Mathematical Statistics
Version
Final published version
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1214/13-AOP890
