Localization, big-jump regime and the effect disorder for a class of generalized pinning models

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, it has been shown that 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 in the bio-physics literature. 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.

33 pages, 5 figures

References in corpus (14)

• Large deviations for random walks under subexponentiality: the big-jump domain
• Smoothing effect of quenched disorder on polymer depinning transitions
• Fractional moment bounds and disorder relevance for pinning models
• Localization and delocalization of random interfaces
• A replica-coupling approach to disordered pinning models
• Pinning of polymers and interfaces by random potentials
• Estimates on path delocalization for copolymers at selective interfaces
• Condensation for a fixed number of independent random variables
• Instability of condensation in the zero-range process with random interaction
• An intermediate phase in DNA melting
• Condensation for random variables conditioned by the value of their sum
• On the irrelevant disorder regime of pinning models
• The Derrida--Retaux conjecture on recursive models
• On the role of mismatches in DNA denaturation

Cited by in corpus (2)

• Statistical fluctuations under resetting: rigorous results
• Scaling limit of the disordered generalized Poland--Scheraga model for DNA denaturation