Advertisement

Localization Transition in Disordered Pinning Models

  • Fabio Lucio ToninelliEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1970)

Summary

These notes are devoted to the statistical mechanics of directed polymers interacting with one-dimensional spatial defects. We are interested in particular in the situation where frozen disorder is present. These polymer models undergo a localization/delocalization transition. There is a large (bio)- physics literature on the subject since these systems describe, for instance, the statistics of thermally created loops in DNA double strands and the interaction between (1 + 1)-dimensional interfaces and disordered walls. In these cases the transition corresponds, respectively, to the DNA denaturation transition and to the wetting transition. More abstractly, one may see these models as random and inhomogeneous perturbations of renewal processes.

The last few years have witnessed a great progress in the mathematical understanding of the equilibrium properties of these systems. In particular, many rigorous results about the location of the critical point, about critical exponents and path properties of the polymer in the two thermodynamic phases (localized and delocalized) are now available.

Here, we will focus on some aspects of this topic—in particular, on the nonperturbative effects of disorder. The mathematical tools employed range from renewal theory to large deviations and, interestingly, show tight connections with techniques developed recently in the mathematical study of mean field spin glasses.

2000 Mathematics Subject Classification: 60K35, 82B44, 82B41, 60K05

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Aizenman, R. Sims, S. L. Starr, Extended variational principle for the Sherrington-Kirkpatrick spin-glass model, Phys. Rev. B68, 214403 (2003).CrossRefGoogle Scholar
  2. 2.
    M. Aizenman and J. Wehr, Rounding effects of quenched randomness on first–order phase transitions, Comm. Math. Phys. 130 (1990), 489–528.MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Albeverio and X. Y. Zhou, Free energy and some sample path properties of a random walk with random potential, J. Statist. Phys. 83 (1996), 573–622.MathSciNetCrossRefGoogle Scholar
  4. 4.
    K. S. Alexander, The effect of disorder on polymer depinning transitions, Commun. Math. Phys. 279, 117–146 (2008).MathSciNetCrossRefGoogle Scholar
  5. 5.
    K. S. Alexander, V. Sidoravicius, Pinning of polymers and interfaces by random potentials, Ann. Appl. Probab. 16, 636–669 (2006).MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. Asmussen, Applied Probability and Queues, 2nd ed., Springer-Verlag, New York, 2003.zbMATHGoogle Scholar
  7. 7.
    K. S. Berenhaut, R. B. Lund, Renewal convergence rates for DHR and NWU lifetimes, Probab. Engrg. Inform. Sci. 16, 67–84 (2002).MathSciNetCrossRefGoogle Scholar
  8. 8.
    N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge (1987).CrossRefGoogle Scholar
  9. 9.
    M. Biskup, F. den Hollander, A heteropolymer near a linear interface, Ann. Appl. Probab. 9, 668–687 (1999).MathSciNetCrossRefGoogle Scholar
  10. 10.
    T. Bodineau, G. Giacomin, On the localization transition of random copolymers near selective interfaces, J. Stat. Phys. 117, 801–818 (2004).MathSciNetCrossRefGoogle Scholar
  11. 11.
    E. Bolthausen, F. den Hollander, Localization transition for a polymer near an interface, Ann. Probab. 25, 1334–1366 (1997).MathSciNetCrossRefGoogle Scholar
  12. 12.
    F. Caravenna, G. Giacomin and M. Gubinelli A numerical approach to copolymers at selective interfaces, J. Stat. Phys. 122, 799–832 (2006).MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. T. Chayes, L. Chayes, D. S. Fisher and T. Spencer, Correlation Length Bounds for Disordered Ising Ferromagnets, Commun. Math. Phys. 120, 501–523 (1989).MathSciNetCrossRefGoogle Scholar
  14. 14.
    J. T. Chayes, L. Chayes, D. S. Fisher and T. Spencer, Finite-size scaling and correlation lengths for disordered systems, Phys. Rev. Lett. 57, 2999 (1986).MathSciNetCrossRefGoogle Scholar
  15. 15.
    D. Cule, T. Hwa, Denaturation of Heterogeneous DNA, Phys. Rev. Lett. 79, 2375–2378 (1997).CrossRefGoogle Scholar
  16. 16.
    B. Derrida, G. Giacomin, H. Lacoin, F. L. Toninelli, Fractional moment bounds and disorder relevance for pinning models, preprint (2007). arxiv.org/abs/0712.2515 [math.PR].Google Scholar
  17. 17.
    B. Derrida, V. Hakim and J. Vannimenius, Effect of disorder on two-dimensional wetting, J. Statist. Phys. 66 (1992), 1189–1213.MathSciNetCrossRefGoogle Scholar
  18. 18.
    R. A. Doney, One-sided large deviation and renewal theorems in the case of infinite mean, Probab. Theory Rel. Fields. 107, 451–465 (1997).MathSciNetCrossRefGoogle Scholar
  19. 19.
    W. Feller, An introduction to probability theory and its applications, vol. 1, 2nd ed., John Wiley & Sons Inc. (1966).Google Scholar
  20. 20.
    D. S. Fisher, Random transverse field Ising spin chains, Phys. Rev. Lett. 69 (1992), 534–537.CrossRefGoogle Scholar
  21. 21.
    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 (1986), 2184–2187.CrossRefGoogle Scholar
  22. 22.
    G. Giacomin, Random polymer models, Imperial College Press, Imperial College Press, World Scientific (2007).Google Scholar
  23. 23.
    G. Giacomin, Renewal convergence rates and correlation decay for homogeneous pinning models, Elect. J. Probab. 13, 513–529 (2008).MathSciNetCrossRefGoogle Scholar
  24. 24.
    G. Giacomin, F. L. Toninelli, Estimates on path delocalization for copolymers at selective interfaces, Probab. Theory Rel. Fields. 133, 464–482 (2005).MathSciNetCrossRefGoogle Scholar
  25. 25.
    G. Giacomin and F. L. Toninelli, The localized phase of disordered copolymers with adsorption, ALEA. 1, 149–180 (2006).MathSciNetzbMATHGoogle Scholar
  26. 26.
    G. Giacomin and F. L. Toninelli, Smoothing effect of quenched disorder on polymer depinning transitions, Commun. Math. Phys. 266, 1–16 (2006).MathSciNetCrossRefGoogle Scholar
  27. 27.
    G. Giacomin, F. L. Toninelli, Smoothing of Depinning Transitions for Directed Polymers with Quenched Disorder, Phys. Rev. Lett. 96, 060702 (2006).CrossRefGoogle Scholar
  28. 28.
    G. Giacomin, F. L. Toninelli, On the irrelevant disorder regime of pinning models, preprint (2007). arXiv:0707.3340v1 [math.PR]Google Scholar
  29. 29.
    F. Guerra, Replica Broken Bounds in the Mean Field Spin Glass Model, Commun. Math. Phys. 233, 1–12 (2003).CrossRefGoogle Scholar
  30. 30.
    F. Guerra, Sum rules for the free energy in the mean field spin glass model, in Mathematical Physics in Mathematics and Physics: Quantum and Operator Algebraic Aspects, Fields Inst. Commun. 30, AMS, 2001.Google Scholar
  31. 31.
    F. Guerra, F. L. Toninelli, Quadratic replica coupling for the Sherrington-Kirkpatrick mean field spin glass model, J. Math. Phys. 43, 3704–3716 (2002).MathSciNetCrossRefGoogle Scholar
  32. 32.
    F. Guerra, F. L. Toninelli, The Thermodynamic Limit in Mean Field Spin Glass Models, Commun. Math. Phys. 230, 71–79 (2002).MathSciNetCrossRefGoogle Scholar
  33. 33.
    A. B. Harris, Effect of Random Defects on the Critical Behaviour of Ising Models, J. Phys. C 7, 1671–1692 (1974).Google Scholar
  34. 34.
    N. C. Jain, W. E. Pruitt, The Range of Rando Walk, in Proceedings of the Sixth Berkeley Simposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/71), Vol. III: Probability Theory, pp. 31–50, Univ. California Press, Berkeley, Cakuf., 1972.Google Scholar
  35. 35.
    Y. Kafri, D. Mukamel and L. Peliti Why is the DNA denaturation transition first order?, Phys. Rev. Lett. 85, 4988–4991 (2000).CrossRefGoogle Scholar
  36. 36.
    R. B. Lund, R. L. Tweedie, Geometric convergence rates for stochastically ordered Markov chains, Math. Oper. Res. 21, 182–194 (1996).MathSciNetCrossRefGoogle Scholar
  37. 37.
    C. Monthus, On the localization of random heteropolymers at the interface between two selective solvents, Eur. Phys. J. B 13, 111–130 (2000).CrossRefGoogle Scholar
  38. 38.
    P. Ney, A refinement of the coupling method in renewal theory, Stochastic Process. Appl. 11, 11–26 (1981).MathSciNetCrossRefGoogle Scholar
  39. 39.
    M. Talagrand, The Parisi Formula, Ann. Math. 163, 221–263 (2006).MathSciNetCrossRefGoogle Scholar
  40. 40.
    M. Talagrand, Spin glasses, a Challenge for Mathematicians, Springer-Verlag (2003).Google Scholar
  41. 41.
    F. L. Toninelli, Critical properties and finite-size estimates for the depinning transition of directed random polymers, J. Stat. Phys. 126, 1025–1044 (2007).MathSciNetCrossRefGoogle Scholar
  42. 42.
    F. L. Toninelli, Correlation lengths for random polymer models and for some renewal sequences, Electron. J. Probab. 12, 613–636 (2007).MathSciNetCrossRefGoogle Scholar
  43. 43.
    F. L. Toninelli, A replica-coupling approach to disordered pinning models, Commun. Math. Phys. 280, 389–401 (2008).MathSciNetCrossRefGoogle Scholar
  44. 44.
    F. L. Toninelli, Disordered pinning models and copolymers: beyond annealed bounds, to appear on Ann. Appl. Probab. arXiv:0709.1629v1 [math.PR].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1. Laboratoire de Physique, UMR-CNRS 5672TechnionIsrael

Personalised recommendations