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.

Published in at http://dx.doi.org/10.1214/13-AOP890 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (8)

• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Elementary bounds on Poincare and log-Sobolev constants for decomposable Markov chains
• KPZ in one dimensional random geometry of multiplicative cascades
• Duality and KPZ in Liouville Quantum Gravity
• Conformal Fractal Geometry and Boundary Quantum Gravity
• Critical Mandelbrot Cascades
• Boundary Loop Models and 2D Quantum Gravity
• Lipschitz partition processes

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