The Annals of Probability

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

Bertrand Duplantier, Rémi Rhodes, Scott Sheffield, and Vincent Vargas

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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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Ann. Probab., Volume 42, Number 5 (2014), 1769-1808.

First available in Project Euclid: 29 August 2014

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Zentralblatt MATH identifier

Primary: 60G57: Random measures 60G15: Gaussian processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

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


Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent. Critical Gaussian multiplicative chaos: Convergence of the derivative martingale. Ann. Probab. 42 (2014), no. 5, 1769--1808. doi:10.1214/13-AOP890.

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