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
TypeArticle accepté pour publication ou publié
External document linkhttps://arxiv.org/abs/1206.1671v3
Journal nameAnnals of Probability
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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 / Keywordsmultiplicative chaos; scale invariance; kpz; star equation; random measure
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