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Critical Properties and Finite-Size Estimates for the Depinning Transition of Directed Random Polymers

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Abstract

We consider models of directed random polymers interacting with a defect line, which are known to undergo a pinning/depinning (or localization/delocalization) phase transition. We are interested in critical properties and we prove, in particular, finite-size upper bounds on the order parameter (the contact fraction) in a window around the critical point, shrinking with the system size. Moreover, we derive a new inequality relating the free energy F and an annealed exponent μ which describes extreme fluctuations of the polymer in the localized region. For the particular case of a (1+1)-dimensional interface wetting model, we show that this implies an inequality between the critical exponents which govern the divergence of the disorder-averaged correlation length and of the typical one. Our results are based on the recently proven smoothness property of the depinning transition in presence of quenched disorder and on concentration of measure ideas.

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References

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. K.S. Alexander and V. Sidoravicius, Pinning of polymers and interfaces by random potentials, preprint (2005). Available on: arXiv.org e-Print archive: math.PR/0501028

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. B. Derrida, V. Hakim, and J. Vannimenus, Effect of disorder on two-dimensional wetting. J. Statist. Phys. 66:1189–1213 (1992).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. W. Feller, An introduction to probability theory and its applications, vol. II, 2nd edition, New York-London-Sydney: John Wiley & Sons, Inc. (1971).

    MATH  Google Scholar 

  8. D. S. Fisher, Random transverse field Ising spin chains. Phys. Rev. Lett. 69:534–537 (1992).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    ADS  Google Scholar 

  11. G. Giacomin, Localization phenomena in random polymer models, preprint (2004). Available online: http://felix.proba.jussieu.fr/pageperso/giacomin/pub/publicat.html.

  12. G. Giacomin and F. L. Toninelli, Estimates on path delocalization for copolymers at selective interfaces. Probab. Theor. Rel. Fields 133:464–482 (2005).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. G. Giacomin and F. L. Toninelli, Smoothing of Depinning Transitions for Directed Polymers with Quenched Disorder. Phys. Rev. Lett. 96:060702 (2006).

    Article  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  16. Y. Kafri, D. Mukamel, and L. Peliti, Why is the DNA denaturation of first order? Phys. Rev. Lett. 85:4988–4991 (2000).

    Article  ADS  Google Scholar 

  17. M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs 89, American Mathematical Society (2001).

  18. R. B. Lund and R. L. Tweedie, Geometric convergence rates for stochastically ordered Markov chains. Math. Oper. Res. 20:182–194 (1996).

    Article  MathSciNet  Google Scholar 

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

    ADS  Google Scholar 

  20. D. R. Nelson and V. M. Vinokur, Phys. Rev. B 48:13060–13097 (1993).

    Article  ADS  Google Scholar 

  21. N. Petrelis, Polymer pinning at an interface, preprint (2005). Available on: arXiv.org e-Print archive: math.PR/0504464.

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Physique, UMR–CNRS 5672, ENS Lyon, 46 Allée d’Italie, 69364, Lyon Cedex 07, France

    Fabio Lucio Toninelli

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  1. Fabio Lucio Toninelli
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Correspondence to Fabio Lucio Toninelli.

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Toninelli, F.L. Critical Properties and Finite-Size Estimates for the Depinning Transition of Directed Random Polymers. J Stat Phys 126, 1025–1044 (2007). https://doi.org/10.1007/s10955-006-9123-x

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  • DOI: https://doi.org/10.1007/s10955-006-9123-x

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