The continuum disordered pinning model

Abstract

AbstractAny renewal processes on $${\mathbb {N}}_0$$ N 0 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 ∈ 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 $${\mathbb {R}}$$ 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 ∈ 1 2 , 1 .