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Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale

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.
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https://hal.archives-ouvertes.fr/hal-00705619
Contributor : Vincent Vargas <>
Submitted on : Sunday, July 1, 2012 - 7:46:44 PM
Last modification on : Monday, November 23, 2020 - 8:28:04 AM
Long-term archiving on: : Thursday, December 15, 2016 - 7:44:36 PM

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  • HAL Id : hal-00705619, version 2
  • ARXIV : 1206.1671

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Bertrand Duplantier, Rémi Rhodes, Scott Sheffield, Vincent Vargas. Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale. 2012. ⟨hal-00705619v2⟩

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