2014/06/19 by Francesco Caravenna, Caravenna, Francesco, Rongfeng Sun +3 · 1 citation
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60K35 #msc:82B44 #msc:82D60
paper · pdf · doi:10.48550/arxiv.1406.5088
35 pages. Minor changes, explanations added. To appear in PTRF
arxiv created 2014/12/02 · arxiv updated 2014/12/03
Any renewal processes on ℕ with a polynomial tail, with exponent α∈ (0,1), has a non-trivial scaling limit, known as the 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 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 ℝ in a white noise random environment, with subtle features: -Any fixed a.s. property of the 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 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 α∈ (1/2, 1).