Advertisement

Communications in Mathematical Physics

, Volume 291, Issue 3, pp 659–689 | Cite as

Quenched and Annealed Critical Points in Polymer Pinning Models

  • Kenneth S. AlexanderEmail author
  • Nikos Zygouras
Article

Abstract

We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential u + V n which the chain encounters when it visits a special state 0 at time n. The disorder (V n ) is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when u exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length n has the form \({n^{-c}\varphi(n)}\) with c ≥  1 and φ slowly varying. Comparing to the corresponding annealed system, in which the V n are effectively replaced by a constant, it was shown in [1,4,13] that the quenched and annealed critical points differ at all temperatures for 3/2 < c < 2 and c > 2, but only at low temperatures for c < 3/2. For high temperatures and 3/2 < c < 2 we establish the exact order of the gap between critical points, as a function of temperature. For the borderline case c = 3/2 we show that the gap is positive provided \({\varphi(n) \to 0}\) as n → ∞, and for c > 3/2 with arbitrary temperature we provide an alternate proof of the result in [4] that the gap is positive, and extend it to c = 2.

Keywords

Partition Function Coarse Graining Exponential Moment Contact Fraction Dense Return 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alexander K.S.: The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279, 117–146 (2008)zbMATHCrossRefADSGoogle Scholar
  2. 2.
    Alexander K.S., Sidoravicius V.: Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16, 636–669 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bodineau T., Giacomin G.: On the localization transition of random copolymers near selective interfaces. J. Stat. Phys. 117, 801–818 (2004)zbMATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    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)CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Derrida B., Hakim V., Vannimenus J.: Effect of disorder on two-dimensional wetting. J. Stat. Phys. 66, 1189–1213 (1992)zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Forgacs G., Luck J.M., Nieuwenhuizen Th.M., Orland H.: Exact critical behavior of two-dimensional wetting problems with quenched disorder. J. Stat. Phys. 51, 29–56 (1988)zbMATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Giacomin G.: Random Polymer Models. Imperial College Press, London (2007)zbMATHGoogle Scholar
  8. 8.
    Giacomin, G., Lacoin, H., Toninelli, F.L.: Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. (2009, to appear)Google Scholar
  9. 9.
    Giacomin G., Toninelli F.L.: Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006)zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Imry Y., Ma S.-K.: Random-field instability of the ordered state of continuous symmetry. Phys. Rev. Lett. 35, 1399–1401 (1975)CrossRefADSGoogle Scholar
  11. 11.
    Naidenov A., Nechaev S.: Adsorption of a random heteropolymer at a potential well revisited: location of transition point and design of sequences. J. Phys. A: Math. Gen. 34, 5625–5634 (2001)zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Seneta, E.: Regularly Varying Functions. Lecture Notes in Math. 508, Berlin: Springer-Verlag, 1976Google Scholar
  13. 13.
    Toninelli F.L.: Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab. 18(4), 1569–1587 (2007)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Toninelli F.L.: A replica-coupling approach to disordered pinning models. Commun. Math. Phys. 280, 389–401 (2008)zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematics KAP 108University of Southern CaliforniaLos AngelesUSA

Personalised recommendations