Basic properties of critical lognormal multiplicative chaos

Abstract

We study one-dimensional exact scaling lognormal multiplicative chaos measures at criticality. Our main results are the determination of the exact asymptotics of the right tail of the distribution of the total mass of the measure, and an almost sure upper bound for the modulus of continuity of the cumulative distribution function of the measure. We also find an almost sure lower bound for the increments of the measure almost everywhere with respect to the measure itself, strong enough to show that the measure is supported on a set of Hausdorff dimension $0$.

Published at http://dx.doi.org/10.1214/14-AOP931 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (7)

• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
• KPZ in one dimensional random geometry of multiplicative cascades
• Duality and KPZ in Liouville Quantum Gravity
• Hydrodynamic turbulence and intermittent random fields
• Critical Mandelbrot Cascades
• On invariant measures of stochastic recursions in a critical case

Cited by in corpus (2)

• A Review of Conjectured Laws of Total Mass of Bacry-Muzy GMC Measures on the Interval and Circle and Their Applications
• Continuum directed random polymers on disordered hierarchical diamond lattices