Localization, Big-Jump Regime and the Effect of Disorder for a Class of Generalized Pinning Models
- PDF / 716,336 Bytes
- 35 Pages / 439.37 x 666.142 pts Page_size
- 65 Downloads / 196 Views
Localization, Big-Jump Regime and the Effect of Disorder for a Class of Generalized Pinning Models Giambattista Giacomin1
· Benjamin Havret1
Received: 22 March 2020 / Accepted: 5 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract One dimensional pinning models have been widely studied in the physical and mathematical literature, also in presence of disorder. Roughly speaking, they undergo a transition between a delocalized phase and a localized one. In mathematical terms these models are obtained by modifying the distribution of a discrete renewal process via a Boltzmann factor with an energy that contains only one body potentials. For some more complex models, notably pinning models based on higher dimensional renewals, other phases may be present. We study a generalization of the one dimensional pinning model in which the energy may depend in a nonlinear way on the contact fraction: this class of models contains the circular DNA case considered for example in Bar et al. (Phys Rev E 86:061904, 2012). We give a full solution of this generalized pinning model in absence of disorder and show that another transition appears. In fact the systems may display up to three different regimes: delocalization, partial localization and full localization. What happens in the partially localized regime can be explained in terms of the “big-jump” phenomenon for sums of heavy tail random variables under conditioning. We then show that disorder completely smears this second transition and we are back to the delocalization versus localization scenario. In fact we show that the disorder, even if arbitrarily weak, is incompatible with the presence of a big-jump. Keywords Pinning model · Disordered systems · Critical behavior · Big-jump phenomena · Disorder relevance Mathematics Subject Classification 60K35 · 60K37 · 82B44 · 60K10
Communicated by Simone Warzel.
B 1
Giambattista Giacomin [email protected] Laboratoire de Probabilités, Statistique et Modélisation, UMR 8001, Université de Paris, 75205 Paris, France
123
G. Giacomin, B. Havret
1 Introduction of the Model and Results 1.1 Phase Transitions, Disorder and Pinning Models The pinning model, sometimes called Poland–Scheraga model, comes up in a variety of real world phenomena. For example in the context of DNA denaturation (this is the Poland– Scheraga framework [49]), for polymers in presence of a defect region [25,29,43], for one dimensional interfaces in two dimensional systems with suitable boundary conditions [52]. But pinning models have also an intrinsic and theoretical interest, due in particular to the following crucial features: • the model is solvable in its homogeneous version: with this respect we cite in particular [25], but, as pointed out in [36, App. A], the solvability mechanism is in reality just the basics of Renewal Theory developed in mathematics since the 40s with seminal contributions by J. L. Doob, P. Erd˝os, W. Feller and many others (e.g. [29, App. A] and references therein). Unless we specify otherwise,
Data Loading...
