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September 2014 Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
Bertrand Duplantier, Rémi Rhodes, Scott Sheffield, Vincent Vargas
Ann. Probab. 42(5): 1769-1808 (September 2014). DOI: 10.1214/13-AOP890


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.


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Bertrand Duplantier. Rémi Rhodes. Scott Sheffield. Vincent Vargas. "Critical Gaussian multiplicative chaos: Convergence of the derivative martingale." Ann. Probab. 42 (5) 1769 - 1808, September 2014.


Published: September 2014
First available in Project Euclid: 29 August 2014

zbMATH: 1306.60055
MathSciNet: MR3262492
Digital Object Identifier: 10.1214/13-AOP890

Primary: 60D05 , 60G15 , 60G57

Keywords: Gaussian multiplicative chaos , Liouville quantum gravity , maximum of log-correlated fields

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 5 • September 2014
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