Multifractality Breaking from Bounded Random Measures

TL;DR

This paper investigates the linearization effect in multifractal systems, particularly Gaussian multiplicative chaos, revealing how bounded measures influence scaling exponents and applying findings to turbulent circulation statistics.

Contribution

It demonstrates that the linearization effect can be explained by Liouville-like bounded measures and shows the preservation of statistical properties in the linear regime.

Findings

Linearization effect explained by bounded Gaussian measures

Statistical properties preserved in the linear regime

Accurate evaluation of turbulent circulation moments

Abstract

Multifractal systems usually have singularity spectra defined on bounded sets of H\"older exponents. As a consequence, their associated multifractal scaling exponents are expected to depend linearly upon statistical moment orders at high enough orders -- a phenomenon referred to as the {\it{linearization effect}}. Motivated by general ideas taken from models of turbulent intermittency and focusing on the case of two-dimensional systems, we investigate the issue within the framework of Gaussian multiplicative chaos. As verified by means of Monte Carlo simulations, it turns out that the linearization effect can be accounted for by Liouville-like random measures defined in terms of upper-bounded scalar fields. The coarse-grained statistical properties of Gaussian multiplicative chaos are furthermore found to be preserved in the linear regime of the scaling exponents. As a related application, we look at the problem of turbulent circulation statistics, and obtain a remarkably accurate evaluation of circulation statistical moments, recently determined with the help of massive numerical simulations.