Communications in Mathematical Physics

, Volume 326, Issue 2, pp 507–530 | Cite as

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

  • Quentin Berger
  • Francesco Caravenna
  • Julien Poisat
  • Rongfeng Sun
  • Nikos Zygouras


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.


Partition Function Renewal Process Coarse Graining Critical Curve Fractional Moment 
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  1. 1.
    Alexander K.S.: The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279, 117–146 (2008)ADSCrossRefMATHGoogle Scholar
  2. 2.
    Alexander K.S.: Excursions and local limit theorems for Bessel-like random walks. Electron. J. Prob. 16, 1–44 (2011)ADSCrossRefMATHGoogle Scholar
  3. 3.
    Alexander K.S., Zygouras N.: Quenched and annealed critical points in polymer pinning models. Commun. Math. Phys. 291, 659–689 (2010)ADSCrossRefMathSciNetGoogle Scholar
  4. 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)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bodineau T., Giacomin G.: On the localization transition of random copolymers near selective interfaces. J. Stat. Phys. 117, 801–818 (2004)ADSCrossRefMATHMathSciNetGoogle Scholar
  6. 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)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bolthausen E., den Hollander F.: Localization transition for a polymer near an interface. Ann. Probab. 25, 1334–1366 (1997)CrossRefMATHMathSciNetGoogle Scholar
  8. 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). [math.PR]
  9. 9.
    Caravenna F., den Hollander F.: A general smoothing inequality for disordered polymers. Electron. Commun. Probab. 18(76), 1–15 (2013)MathSciNetGoogle Scholar
  10. 10.
    Caravenna F., Giacomin G.: The weak coupling limit of disordered copolymer models. Ann. Probab. 38, 2322–2378 (2010)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Caravenna F., Giacomin G., Gubinelli M.: A numerical approach to copolymers at selective interfaces. J. Stat. Phys. 122, 799–832 (2006)ADSCrossRefMATHMathSciNetGoogle Scholar
  12. 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. 289–311Google Scholar
  13. 13.
    Caravenna, F., Sun, R., Zygouras, N.: The continuum disordered pinning model. In preparationGoogle Scholar
  14. 14.
    Caravenna, F., Sun, R., Zygouras, N.: Polynomial chaos and scaling limits of disordered systems. In preparationGoogle Scholar
  15. 15.
    Cheliotis D., den Hollander F.: Variational characterization of the critical curve for pinning of random polymers. Ann. Probab. 41(33), 1767–1805 (2013)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications (2nd. ed.). Berlin-Heidelberg-New York: Springer, 1998Google Scholar
  17. 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)ADSCrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Derrida B., Hakim V., Vannimenius J.: Effect of disorder on two dimensional wetting. J. Stat. Phys. 66, 1189–1213 (1992)ADSCrossRefMATHGoogle Scholar
  19. 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)ADSCrossRefGoogle Scholar
  20. 20.
    Garel T., Huse D.A., Leibler S., Orland H.: Localization transition of random chains at interfaces. Europhys. Lett. 8, 9–13 (1989)ADSCrossRefGoogle Scholar
  21. 21.
    Giacomin, G.: Random polymer models. London: Imperial College Press, 2007Google Scholar
  22. 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, 2011Google Scholar
  23. 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)ADSCrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Giacomin G., Lacoin H., Toninelli F.L.: Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. 63, 233–2650 (2011)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Giacomin G., Toninelli F.L.: Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006)ADSCrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    den Hollander, F.: Random polymers. In: Lectures from the 37th Probability Summer School held in Saint-Flour 2007. Berlin: Springer-Verlag, 2009Google Scholar
  27. 27.
    Lacoin H.: The martingale approach to disorder irrelevance for pinning models. Electron. Commun. Probab. 15, 418–427 (2010)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Monthus C.: On the localization of random heteropolymers at the interface between two selective solvents. Eur. Phys. J. B 13, 111–130 (2000)ADSCrossRefGoogle Scholar
  29. 29.
    Nelson D.R., Vinokur V.M.: Boson localization and correlated pinning of superconducting vortex arrays. Phys. Rev. B 48, 13060–13097 (1993)ADSCrossRefGoogle Scholar
  30. 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, 1970Google Scholar
  31. 31.
    Toninelli F.L.: A replica-coupling approach to disordered pinning models. Commun. Math. Phys. 280, 389–401 (2008)ADSCrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Toninelli F.L.: Coarse graining, fractional moments and the critical slope of random copolymers. Electron. J. Probab. 14, 531–547 (2009)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Quentin Berger
    • 1
  • Francesco Caravenna
    • 2
  • Julien Poisat
    • 3
  • Rongfeng Sun
    • 4
  • Nikos Zygouras
    • 5
  1. 1.Department of Mathematics, KAP 108University of Southern CaliforniaLos AngelesUSA
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Milano-BicoccaMilanItaly
  3. 3.Mathematical InstituteLeiden UniversityLeidenThe Netherlands
  4. 4.Department of MathematicsNational University of SingaporeSingaporeSingapore
  5. 5.Department of StatisticsUniversity of WarwickCoventryUK

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