Skip to main content
Log in

The Critical Curves of the Random Pinning and Copolymer Models at Weak Coupling

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study random pinning and copolymer models, when the return distribution of the underlying renewal process has a polynomial tail with finite mean. We compute the asymptotic behavior of the critical curves of the models in the weak coupling regime, showing that it is universal. This proves a conjecture of Bolthausen, den Hollander and Opoku for copolymer models (Bolthausen et al., in Ann Probab, 2012), which we also extend to pinning models.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexander K.S.: The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279, 117–146 (2008)

    Article  ADS  MATH  Google Scholar 

  2. Alexander K.S.: Excursions and local limit theorems for Bessel-like random walks. Electron. J. Prob. 16, 1–44 (2011)

    Article  ADS  MATH  Google Scholar 

  3. Alexander K.S., Zygouras N.: Quenched and annealed critical points in polymer pinning models. Commun. Math. Phys. 291, 659–689 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  4. Alexander K.S., Zygouras N.: Equality of critical points for polymer depinning transitions with loop exponent one. Ann. Appl. Prob 20, 356–366 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Bolthausen, E., den Hollander, F., Opoku, A.A.: A copolymer near a selective interface: variational characterization of the free energy. Ann. Probab. (2012, to appear). http://arxiv.org/abs/1110.1315v2 [math.PR]

  9. Caravenna F., den Hollander F.: A general smoothing inequality for disordered polymers. Electron. Commun. Probab. 18(76), 1–15 (2013)

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Caravenna, F., Giacomin, G., Toninelli, F.L.: Copolymers at selective interfaces: settled issues and open problems. In: Probability in Complex Physical Systems. In honour of Erwin Bolthausen and Jürgen Gärtner. Springer Proceedings in Mathematics, Vol. 11, Berlin-Heidelberg-New York: Springer, 2012, pp. 289–311

  13. Caravenna, F., Sun, R., Zygouras, N.: The continuum disordered pinning model. In preparation

  14. Caravenna, F., Sun, R., Zygouras, N.: Polynomial chaos and scaling limits of disordered systems. In preparation

  15. Cheliotis D., den Hollander F.: Variational characterization of the critical curve for pinning of random polymers. Ann. Probab. 41(33), 1767–1805 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications (2nd. ed.). Berlin-Heidelberg-New York: Springer, 1998

  17. 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  ADS  MATH  MathSciNet  Google Scholar 

  18. Derrida B., Hakim V., Vannimenius J.: Effect of disorder on two dimensional wetting. J. Stat. Phys. 66, 1189–1213 (1992)

    Article  ADS  MATH  Google Scholar 

  19. Forgacs G., Luck J.M., Nieuwenhuizen Th.M., Orland H.: Wetting of a disordered substrate: exact critical behavior in two dimensions. Phys. Rev. Lett. 57, 2184–2187 (1986)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  21. Giacomin, G.: Random polymer models. London: Imperial College Press, 2007

  22. Giacomin, G.: Disorder and critical phenomena through basic probability models. In: Lecture Notes from the 40th Probability Summer School held in Saint-Flour, 2010, Berlin-Heidelberg-New York: Springer, 2011

  23. Giacomin G., Lacoin H., Toninelli F.L.: Disorder relevance at marginality and critical point shift. Ann. Inst. H. Poincaré Probab. Stat. 47, 148–175 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Giacomin G., Lacoin H., Toninelli F.L.: Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. 63, 233–2650 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. den Hollander, F.: Random polymers. In: Lectures from the 37th Probability Summer School held in Saint-Flour 2007. Berlin: Springer-Verlag, 2009

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  29. Nelson D.R., Vinokur V.M.: Boson localization and correlated pinning of superconducting vortex arrays. Phys. Rev. B 48, 13060–13097 (1993)

    Article  ADS  Google Scholar 

  30. Poland, D., Scheraga, H.: Theory of helix-coil transitions in biopolymers: statistical mechanical theory of order-disorder transitions in biological macromolecules, London-New York: Academic Press, 1970

  31. Toninelli F.L.: A replica-coupling approach to disordered pinning models. Commun. Math. Phys. 280, 389–401 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rongfeng Sun.

Additional information

Communicated by F. L. Toninelli

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berger, Q., Caravenna, F., Poisat, J. et al. The Critical Curves of the Random Pinning and Copolymer Models at Weak Coupling. Commun. Math. Phys. 326, 507–530 (2014). https://doi.org/10.1007/s00220-013-1849-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-013-1849-0

Keywords

Navigation