The continuum disordered pinning model

Abstract

Any renewal processes on $\mathbb{N}$ with a polynomial tail, with exponent $\alpha \in (0,1)$, has a non-trivial scaling limit, known as the $\alpha$-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 $\alpha \in (1/2, 1)$ these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subset of $\mathbb{R}$ in a white noise random environment, with subtle features: -Any fixed a.s. property of the $\alpha$-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 $\alpha$-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 $\alpha \in (1/2, 1)$.