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 on statistical moment orders at high-enough orders—a phenomenon referred to as the 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.

中文翻译：

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打破有界随机测度的多重分形

多重分形系统通常具有定义在 Hölder 指数有界集上的奇异谱。因此，它们相关的多重分形标度指数预计将线性依赖于足够高阶的统计矩阶——这种现象称为线性化效应. 受来自湍流间歇性模型的一般思想的启发，并关注二维系统的情况，我们在高斯乘法混沌的框架内研究了这个问题。通过蒙特卡罗模拟验证，结果表明线性化效果可以通过根据上限标量场定义的类似 Liouville 的随机测度来解释。此外，还发现高斯乘法混沌的粗粒度统计特性保留在缩放指数的线性范围内。作为相关应用，我们研究了湍流环流统计问题，并获得了最近在大规模数值模拟的帮助下确定的环流统计矩的非常准确的评估。