The continuum disordered pinning model

Any renewal processes on [Formula: see text] with a polynomial tail, with exponent [Formula: see text] , has a non-trivial scaling limit, known as the [Formula: see text] -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 [Formula: see text] these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subset of [Formula: see text] Any fixed a.s. property of the [Formula: see text] -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 [Formula: see text] -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 [Formula: see text] .