title: The continuum disordered pinning model.
abstract: 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 thecontinuum disordered pinning model(CDPM). This is a random closed subset of [Formula: see text] in a white noise random environment, with subtle features:Anyfixeda.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 issingularwith 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 ofdisorder relevancefor pinning models with [Formula: see text].
authors.affiliation: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Cozzi 55, 20125 Milano, Italy.
authors.forename: Francesco
authors.initials: F
authors.lastname: Caravenna
authors.affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076 Singapore.
authors.forename: Rongfeng
authors.initials: R
authors.lastname: Sun
authors.affiliation: Department of Statistics, University of Warwick, Coventry, CV4 7AL UK.
authors.forename: Nikos
authors.initials: N
authors.lastname: Zygouras
keywords: Disorder relevance; Fell–Matheron topology; Hausdorff metric; Pinning model; Random polymer; Scaling limit; Weak disorder; Wiener Chaos expansion