有界随机测度的多重分形破缺
Multifractality Breaking from Bounded Random Measures
摘要：
多重分形系统的奇异谱通常定义在有界的Hölder指数集上。因此，其相关的多重分形标度指数预计将线性依赖于统计矩阶数在足够高的订单-一种现象称为{ \it{linearization effect} }。受湍流不稳定性模型的一般思想的启发，并专注于二维系统的情况下，我们研究了高斯乘性混沌的框架内的问题。通过蒙特卡罗模拟验证，线性化效应可以用上界标量场定义的类刘维尔随机测度来解释.此外，高斯乘性混沌的粗粒度的统计特性被发现保持在线性制度的标度指数。作为一个相关的应用程序，我们看看湍流环流统计的问题，并获得一个非常准确的评价环流统计矩，最近确定的帮助下，大量的数值模拟。
英文摘要：
Multifractal systems usually have singularity spectra defined on bounded sets of Hölder 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.
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