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Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

Author(s)
Duplantier, Bertrand; Rhodes, Rémi; Vargas, Vincent; Sheffield, Scott Roger
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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.
Date issued
2014-09
URI
http://hdl.handle.net/1721.1/115337
Department
Massachusetts Institute of Technology. Department of Mathematics
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
ISSN
0091-1798

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