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Correlated Random Systems: Five Different Methods pp 1–43Cite as

Random Copolymers

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2143))

Abstract

A (directed) random polymer in d + 1 dimensions is a random walk in d dimensions whose law is transformed by a random potential. The time axis is considered as an additional dimension. The best known and most famous case is the directed polymer in random environment which has a potential given by independent random variables in space and time. Some of the basic questions are open even in 1 + 1 dimensions which is believed to be connected with the KPZ universality class. The main focus of the notes is on the so-called copolymer, first discussed in the physics literature by Garel, Huse, Leibler and Orland in 1989 which models the behavior of a polymer at an interface. Important rigorous results have first been obtained by Sinai and Bolthausen-den Hollander, and have later been developed by many authors. A basic object of interest is a critical line in the parameter space which separates a localized phase from a delocalized one. Particularly interesting is the behavior at the weak disorder limit where the phase transition is characterized by a universal critical tangent whose existence had first been proved in the Bolthausen-den Hollander paper, and whose exact value is still open. This critical tangent is discussed in detail, and new bounds are derived, partly based on large deviation techniques developed by Birkner, Greven, and den Hollander.

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Notes

  1. 1.

    As many people have over years tried without success to disprove the Monthus conjecture, this is quite a safe statement, I believe.

  2. 2.

    A hand waving computation which we have not been able to make rigorous, suggests \(\kappa \left (\alpha \right ) \leq \left (1+\alpha \right )/2\alpha\). The right hand side would be in agreement with numerical studies. Actually, the value has been conjectured to be the true value for the tangent. (Oral communication by G. Giacomin.)

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Authors and Affiliations

  1. Institut für Mathematik, University of Zürich, Winterthurerstrasse 190, 8057, Zürich, Switzerland

    Erwin Bolthausen

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  1. Erwin Bolthausen
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Correspondence to Erwin Bolthausen .

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Editors and Affiliations

  1. Institut de Mathématiques de Marseille, Aix-Marseille Université CNRS, Marseille Cedex 13, France

    Véronique Gayrard

  2. Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany

    Nicola Kistler

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Bolthausen, E. (2015). Random Copolymers. In: Gayrard, V., Kistler, N. (eds) Correlated Random Systems: Five Different Methods. Lecture Notes in Mathematics, vol 2143. Springer, Cham. https://doi.org/10.1007/978-3-319-17674-1_1

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