The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ in random constructions. In this paper we consider a generalised Assouad spectrum that interpolates between the quasi-Assouad to the Assouad dimension. For common models of random fractal sets we obtain a dichotomy of its behaviour by finding a threshold function where the quasi-Assouad behaviour transitions to the Assouad dimension. This threshold can be considered a phase transition and we compute the threshold for the Gromov boundary of Galton-Watson trees and one-variable random self-similar and self-affine constructions. We describe how the stochastically self-similar model can be derived from the Galton-Watson tree result.

中文翻译：

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一些随机构造的 ASOUAD 频谱阈值

度量空间的 Assouad 维度决定了它的极值标度属性。Assouad 谱的派生概念通过缩放函数固定相对比例，以获得准 Assouad 和框计数维度之间的插值行为。虽然准 Assouad 和 Assouad 维度经常重合，但它们通常在随机构造上有所不同。在本文中，我们考虑在准 Assouad 维度和 Assouad 维度之间进行插值的广义 Assouad 谱。对于随机分形集的常见模型，我们通过找到准 Assouad 行为过渡到 Assouad 维度的阈值函数来获得其行为的二分法。这个阈值可以被认为是一个相变，我们计算了 Galton-Watson 树和单变量随机自相似和自仿射构造的 Gromov 边界的阈值。我们描述了如何从 Galton-Watson 树结果导出随机自相似模型。