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Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale

Duplantier, Bertrand; Rhodes, Rémi; Sheffield, Scott; Vargas, Vincent (2014), Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale, Annals of Probability, 42, 5, p. 1769-1808. 10.1214/13-AOP890

Type
Article accepté pour publication ou publié
External document link
https://arxiv.org/abs/1206.1671v3
Date
2014
Journal name
Annals of Probability
Volume
42
Number
5
Publisher
Institute of Mathematical Statistics
Pages
1769-1808
Publication identifier
10.1214/13-AOP890
Metadata
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Author(s)
Duplantier, Bertrand
Rhodes, Rémi
Sheffield, Scott
Vargas, Vincent
Abstract (EN)
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.
Subjects / Keywords
multiplicative chaos; scale invariance; kpz; star equation; random measure

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