Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale

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

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

Number

Publisher

Institute of Mathematical Statistics

Pages

1769-1808

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