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Localization Transition in Disordered Pinning Models

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Methods of Contemporary Mathematical Statistical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1970))

Summary

These notes are devoted to the statistical mechanics of directed polymers interacting with one-dimensional spatial defects. We are interested in particular in the situation where frozen disorder is present. These polymer models undergo a localization/delocalization transition. There is a large (bio)- physics literature on the subject since these systems describe, for instance, the statistics of thermally created loops in DNA double strands and the interaction between (1 + 1)-dimensional interfaces and disordered walls. In these cases the transition corresponds, respectively, to the DNA denaturation transition and to the wetting transition. More abstractly, one may see these models as random and inhomogeneous perturbations of renewal processes.

The last few years have witnessed a great progress in the mathematical understanding of the equilibrium properties of these systems. In particular, many rigorous results about the location of the critical point, about critical exponents and path properties of the polymer in the two thermodynamic phases (localized and delocalized) are now available.

Here, we will focus on some aspects of this topic—in particular, on the nonperturbative effects of disorder. The mathematical tools employed range from renewal theory to large deviations and, interestingly, show tight connections with techniques developed recently in the mathematical study of mean field spin glasses.

2000 Mathematics Subject Classification: 60K35, 82B44, 82B41, 60K05

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Authors and Affiliations

  1. Laboratoire de Physique, UMR-CNRS 5672, Technion, Israel

    Fabio Lucio Toninelli

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  1. Fabio Lucio Toninelli
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Correspondence to Fabio Lucio Toninelli .

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  1. Ctr. for Theoretical Study, Charles University, Jilska 1, Praha 1, 110 00, Czech Republic

    Roman Kotecký

  2. Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK

    Roman Kotecký

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Lucio Toninelli, F. (2009). Localization Transition in Disordered Pinning Models. In: Kotecký, R. (eds) Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics(), vol 1970. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92796-9_3

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