Skip to main content
SpringerLink
Log in

Copolymers at Selective Interfaces: Settled Issues and Open Problems

  • Conference paper
  • First Online:
Probability in Complex Physical Systems

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 11))

Abstract

We review the literature on the localization transition for the class of polymers with random potentials that goes under the name of copolymers near selective interfaces. We outline the results, sketch some of the proofs and point out the open problems in the field. We also present in detail some alternative proofs that simplify what one can find in the literature. 2010 Mathematics Subject Classification: 60K35, 82B41, 82B44.

2010Mathematics Subject Classification: 60K35, 82B41, 82B44.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 149.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alexander, K., Zygouras, N.: Quenched and annealed critical points in polymer pinning models. Comm. Math. Phys. 291, 659–689 (2008)

    Article  MathSciNet  Google Scholar 

  2. Albeverio, S., Zhou, X.Y.: Free energy and some sample path properties of a random walk with random potential. J. Statist. Phys. 83, 573–622 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Asmussen, S.: Applied probability and queues. (2nd edn), Applications of Mathematics 51, Springer, New York (2003)

    Google Scholar 

  4. Biskup, M., den Hollander, F.: A heteropolymer near a linear interface. Ann. Appl. Probab. 9, 668–687 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bodineau, T., Giacomin, G.: On the localization transition of random copolymers near selective interfaces. J. Statist. Phys. 117, 801–818 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bodineau, T., Giacomin, G., Lacoin, H., Toninelli, F.L.: Copolymers at selective interfaces: new bounds on the phase diagram. J. Statist. Phys. 132, 603–626 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bolthausen, E., Caravenna, F., de Tilière, B.: The quenched critical point of a diluted disordered polymer model. Stoch. Process. Appl. 119, 1479–1504 (2009)

    Article  MATH  Google Scholar 

  8. Bolthausen, E., den Hollander, F.: Localization transition for a polymer near an interface. Ann. Probab. 25, 1334–1366 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caravenna, F., Giacomin, G.: The weak coupling limit of disordered copolymer models. Ann. Probab. 38, 2322–2378 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Caravenna, F., Giacomin, G., Gubinelli, M.: A numerical approach to copolymers at selective interfaces. J. Statist. Phys. 122, 799–832 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Causo, M.S., Whittington, S.G.: A Monte Carlo investigation of the localization transition in random copolymers at an interface. J. Phys. A: Math. Gen. 36, L189–L195 (2003)

    Article  Google Scholar 

  12. Derrida, B., Giacomin, G., Lacoin, H., Toninelli, F.L.: Fractional moment bounds and disorder relevance for pinning models. Commun. Math. Phys. 287, 867–887 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fitzsimmons, P.J., Fristedt, B., Maisonneuve, B.: Intersections and limits of regenerative sets. Z. Wahrscheinlichkeitstheorie verw. Gebiete 70, 157–173 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Garel, T., Huse, D.A., Leibler, S., Orland, H.: Localization transition of random chains at interfaces. Europhys. Lett. 8, 9–13 (1989)

    Article  Google Scholar 

  15. den Hollander, F.: Random Polymers, École d’Été de Probabilités de Saint-Flour XXXVII–2007, Lecture Notes in Mathematics 1974, Springer (2009)

    Google Scholar 

  16. Giacomin, G.: Random Polymer Models. Imperial College Press, World Scientific (2007)

    Book  MATH  Google Scholar 

  17. Giacomin, G., Lacoin, H., Toninelli, F.L.: Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. 63, 233–265 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Giacomin, G., Toninelli, F.L.: Estimates on path delocalization for copolymers at selective interfaces. Probab. Theory Relat. Fields 133, 464–482 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Giacomin, G., Toninelli, F.L.: The localized phase of disordered copolymers with adsorption. ALEA 1, 149–180 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Giacomin, G., Toninelli, F.L.: Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Giacomin, G., Toninelli, F.L.: On the irrelevant disorder regime of pinning models. Ann. Probab. 37, 1841–1873 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Habibzadah, N., Iliev, G.K., Martin, R., Saguia, A., Whittington, S.G.: The Order of the Localization Transition for a Random Copolymer. J. Phys. A: Math. Gen. 39, 5659–5667 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lacoin, H.: The martingale approach to disorder irrelevance for pinning models. Electron. Comm. Probab. 15, 418–427 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Monthus, C.: On the Localization of Random heteropolymers at the interface between two selective solvents. Eur. Phys. J. B 13, 111–130 (2000)

    Article  Google Scholar 

  25. Monthus, C., Garel, T.: Delocalization transition of the selective interface model: distribution of pseudo-critical temperatures. J. Stat. Mech. P12011 (2005)

    Google Scholar 

  26. Mourrat, J.-C.: On the delocalized phase of the random pinning model, Séminaire de Probabilités (to appear)

    Google Scholar 

  27. Pétrélis, N.: Copolymer at selective interfaces and pinning potentials: weak coupling limits. Ann. Instit. H. Poincaré 45, 175–200 (2009)

    Article  MATH  Google Scholar 

  28. Sinai, Ya.G.: A random walk with a random potential. Theory Probab. Appl. 38, 382–385 (1993)

    Google Scholar 

  29. Soteros, C.E., Whittington, S.G.: The statistical mechanics of random copolymers. J. Phys. A: Math. Gen. 37, R279–R325 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Stepanow, S., Sommer, J.-U., Erukhimovich, I.Ya.: Localization transition of random copolymers at interfaces. Phys. Rev. Lett. 81, 4412–4416 (1998)

    Google Scholar 

  31. Toninelli, F.L.: Critical properties and finite-size estimates for the depinning transition of directed random polymers. J. Stat. Phys. 126, 1025–1044 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Toninelli, F.L.: Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab. 18, 1569–1587 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Toninelli, F.L.: Coarse graining, fractional moments and the critical slope of random copolymers. Electron. J. Probab. 14, 531–547 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Trovato, A., Maritan, A.: A variational approach to the localization transition of heteropolymers at interfaces. Europhys. Lett. 46, 301–306 (1999)

    Article  Google Scholar 

  35. Vojta, T., Sknepnek, R.: Critical points and quenched disorder: From Harris criterion to rare regions and smearing. Phys. Stat. Sol. B 241, 2118–2127 (2004)

    Article  Google Scholar 

  36. Whittington, S.G.: Randomly coloured self-avoiding walks: adsorption and localization. Markov Proc. Rel. Fields 13, 761–776 (2007)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge the support of the University of Padova (F.C. under grant CPDA082105/08) and of ANR (G.G. and F.L.T. under grant SHEPI).

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Cozzi 53, 20125, Milano, Italy

    Francesco Caravenna

  2. Université Paris Diderot (Paris 7) and Laboratoire de Probabilités et Modèles Aléatoires (CNRS U.M.R. 7599), Paris, France

    Giambattista Giacomin

  3. U.F.R. Mathématiques, Case 7012 (Site Chevaleret), 75205, Paris cedex 13, France

    Giambattista Giacomin

  4. Laboratoire de Physique and CNRS, UMR 5672, Ecole Normale Supérieure de Lyon, 46 Allée d’Italie, 69364, Lyon Cedex 07, France

    Fabio Lucio Toninelli

Authors
  1. Francesco Caravenna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giambattista Giacomin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fabio Lucio Toninelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Caravenna .

Editor information

Editors and Affiliations

  1. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Jean-Dominique Deuschel

  2. Faculty of Mathematics, University of Bielefeld, P.O. Box 10 01 31, Bielefeld, 33501, Germany

    Barbara Gentz

  3. for Applied Analysis and Stochastics, Weierstrass Institute Berlin, Mohrenstr. 39, Berlin, 10117, Germany

    Wolfgang König

  4. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany

    Max von Renesse

  5. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany

    Michael Scheutzow

  6. Institute for Mathematical, Methods in Economics, Vienna University of Technology, Wiedner Hauptstraße 8-10/105-1, Vienna, 1040, Austria

    Uwe Schmock

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Caravenna, F., Giacomin, G., Toninelli, F.L. (2012). Copolymers at Selective Interfaces: Settled Issues and Open Problems. In: Deuschel, JD., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23811-6_12

Download citation

Publish with us

Policies and ethics

Search

Navigation