Basic properties of critical lognormal multiplicative chaos

TL;DR

This paper investigates the fundamental properties of critical lognormal multiplicative chaos measures in one dimension, focusing on tail asymptotics, regularity, and support dimension, revealing their extreme singularity and precise distribution behavior.

Contribution

It provides the first exact asymptotics for the tail distribution and establishes the measure's support on a set of Hausdorff dimension zero, advancing understanding of critical chaos measures.

Findings

Exact tail asymptotics of the measure's total mass

Almost sure upper bounds on the measure's modulus of continuity

Support of the measure has Hausdorff dimension zero

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 $0$.