Any renewal processes on $${\mathbb {N}}_0$$N0 with a polynomial tail, with exponent $$\alpha \in (0,1)$$α∈(0,1), has a non-trivial scaling limit, known as the $$\alpha $$α-stable regenerative set. In this paper we consider Gibbs transformations of such renewal processes in an i.i.d. random environment, called disordered pinning models. We show that for $$\alpha \in \left( \frac{1}{2}, 1\right) $$α∈12,1 these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subset of $${\mathbb {R}}$$R in a white noise random environment, with subtle features:Any fixed a.s. property of the $$\alpha $$α-stable regenerative set (e.g., its Hausdorff dimension) is also an a.s. property of the CDPM, for almost every realization of the environment.Nonetheless, the law of the CDPM is singular with respect to the law of the $$\alpha $$α-stable regenerative set, for almost every realization of the environment. The existence of a disordered continuum model, such as the CDPM, is a manifestation of disorder relevance for pinning models with $$\alpha \in \left( \frac{1}{2}, 1\right) $$α∈12,1.

中文翻译：

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连续无序钉扎模型

$${\mathbb {N}}_0$$N0 上的任何更新过程都具有多项式尾，指数 $$\alpha \in (0,1)$$α∈(0,1)，具有非平凡缩放限制，称为$$\alpha $$α-stable regeneration set。在本文中，我们考虑了 iid 随机环境中此类更新过程的吉布斯变换，称为无序钉扎模型。我们证明对于 $$\alpha \left( \frac{1}{2}, 1\right) $$α∈12,1 这些模型有一个通用的缩放限制，我们称之为连续无序钉扎模型 ( CDPM）。这是在白噪声随机环境中 $${\mathbb {R}}$$R 的随机封闭子集，具有微妙的特征：任何固定为 $$\alpha $$α-stable regeneration set 的属性（例如，它的 Hausdorff 维度）也是 CDPM 的一个属性，几乎适用于环境的每一个实现。尽管如此，对于几乎所有的环境实现，CDPM 定律相对于 $$\alpha $$α-稳定再生集定律是单数的。无序连续统模型（例如 CDPM）的存在是与 $$\alpha \in \left( \frac{1}{2}, 1\right) $$α∈12 的钉扎模型的无序相关性的表现,1。