Skip to main content
SpringerLink
Log in
Book cover

Spin Glasses: Statics and Dynamics pp 235–270Cite as

Renewal Sequences, Disordered Potentials, and Pinning Phenomena

  • Conference paper

Part of the book series: Progress in Probability ((PRPR,volume 62))

Abstract

We give an overview of the state of the art of the analysis of disordered models of pinning on a defect line. This class of models includes a number of well-known and much studied systems (like polymer pinning on a defect line, wetting of interfaces on a disordered substrate and the Poland-Scheraga model of DNA denaturation). A remarkable aspect is that, in absence of disorder, all the models in this class are exactly solvable and they display a localization-delocalization transition that one understands in full detail. Moreover the behavior of such systems near criticality is controlled by a parameter and one observes, by tuning the parameter, the full spectrum of critical behaviors, ranging from first-order to infinite-order transitions. This is therefore an ideal set-up in which to address the question of the effect of disorder on the phase transition, notably on critical properties. We will review recent results that show that the physical prediction that goes under the name of Harris criterion is indeed fully correct for pinning models. Beyond summarizing the results, we will sketch most of the arguments of proof.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D.B. Abraham, Surface Structures and Phase Transitions, Exact Results, in Phase Transitions and Critical Phenomena 10, Academic Press, London (UK) (1986), 1–74.

    Google Scholar 

  2. M. Aizenman and S. Molchanov, Localization at large disorder and at extreme energies: an elementary derivation, Commun. Math. Phys. 157 (1993), 245–278.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Aizenman and J. Wehr, Rounding effects of quenched randomness on first-order phase transitions, Commun. Math. Phys. 130 (1990), 489–528.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. K.S. Alexander, Ivy on the ceiling: first-order polymer depinning transitions with quenched disorder, Markov Proc. Rel. Fields 13 (2007), 663–680.

    MATH  Google Scholar 

  6. K.S. Alexander and V. Sidoravicius, Pinning of polymers and interfaces by random Potentials, Ann. Appl. Probab. 16 (2006), 636–669.

    Article  MATH  MathSciNet  Google Scholar 

  7. K.S. Alexander and N. Zygouras, Quenched and annealed critical points in polymer pinning models, arXiv:0805.1708 [math.PR]

    Google Scholar 

  8. S. Asmussen, Applied probability and queues, Second Edition, Applications of Mathematics 51, Springer-Verlag, New York (2003).

    Google Scholar 

  9. N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular variation, Cambridge University Press, Cambridge, 1987.

    MATH  Google Scholar 

  10. M. Birkner and R. Sun, Annealed vs quenched critical points for a random walk pinning model, arXiv:0807.2752v1 [math.PR]

    Google Scholar 

  11. R. Blossey and E. Carlon, Reparametrizing the loop entropy weights: effect on DNA melting curves, Phys. Rev. E 68 (2003), 061911 (8 pages).

    Article  Google Scholar 

  12. E. Bolthausen, F. Caravenna and B. de Tilière, The quenched critical point of a diluted disordered polymer model, arXiv:0711.0141v1 [math.PR]

    Google Scholar 

  13. T.W. Burkhardt, Localization-Delocalization transition in a solid-on-solid model with a pinning potential, J. Phys. A: Math. Gen. 14 (1981), L63–L68.

    Article  MathSciNet  Google Scholar 

  14. A. Camanes and P. Carmona, Directed polymers, critical temperature and uniform Integrability, preprint (2007).

    Google Scholar 

  15. F. Caravenna and G. Giacomin, On constrained annealed bounds for pinning and wetting models, Elect. Comm. Probab. 10 (2005), 179–189.

    MATH  MathSciNet  Google Scholar 

  16. F. Caravenna, G. Giacomin and L. Zambotti, A renewal theory approach to periodic copolymers with adsorption, Ann. Appl. Probab. 17 (2007), 1362–1398.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. Cule and T. Hwa, Denaturation of Heterogeneous DNA, Phys. Rev. Lett. 79 (1997), 2375–2378.

    Article  Google Scholar 

  18. B. Coluzzi and E. Yeramian, Numerical evidence for relevance of disorder in a Poland-Scheraga DNA denaturation model with self-avoidance: scaling behavior of average quantities, Eur. Phys. Journal B 56 (2007), 349–365.

    Article  Google Scholar 

  19. B. Derrida, G. Giacomin, H. Lacoin and F.L. Toninelli, Fractional moment bounds and disorder relevance for pinning models, Commun. Math. Phys. 287 (2009), 867–887.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. R.A. Doney, One-sided local large deviation and renewal theorems in the case of infinite mean, Probab. Theory Rel. Fields 107 (1997), 451–465.

    Article  MATH  MathSciNet  Google Scholar 

  22. M.R. Evans and B. Derrida, Improved bounds for the transition temperature of directed polymers in a finite-dimensional random medium, J. Statist. Phys. 69 (1992), 427–437.

    Article  MATH  Google Scholar 

  23. W. Feller, An introduction to probability theory and its applications, Vol. I, Third edition, John Wiley & Sons, Inc., New York-London-Sydney, 1968.

    MATH  Google Scholar 

  24. M.E. Fisher, Walks, walls, wetting, and melting, J. Statist. Phys. 34 (1984), 667–729.

    Article  MATH  MathSciNet  Google Scholar 

  25. 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.

    Article  Google Scholar 

  26. S. Galluccio and and R. Graber, Depinning transition of a directed polymer by a periodic potential: a d-dimensional solution, Phys. Rev. E 53 (1996), R5584–R5587.

    Article  Google Scholar 

  27. A. Garsia and J. Lamperti, A discrete renewal theorem with infinite mean, Comment. Math. Helv. 37 (1963), 221–234.

    Article  MATH  MathSciNet  Google Scholar 

  28. P.G. de Gennes, Scaling concepts in polymer physics, Cornell University Press, Ithaca, NY (1979).

    Google Scholar 

  29. G. Giacomin, Random Polymer Models, Imperial College Press, World Scientific (2007).

    MATH  Google Scholar 

  30. G. Giacomin, Renewal convergence rates and correlation decay for homogeneous pinning models, Elect. J. Probab. 13 (2008), 513–529.

    MATH  MathSciNet  Google Scholar 

  31. G. Giacomin, H. Lacoin and F.L. Toninelli, Hierarchical pinning models, quadratic maps and quenched disorder, arXiv:0711.4649 [math.PR]

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  36. G. Giacomin and F.L. Toninelli, On the irrelevant disorder regime of pinning models, arXiv: 0707.3340v1 [math.PR]

    Google Scholar 

  37. A.B. Harris, Effect of Random Defects on the Critical Behaviour of Ising Models, J. Phys. C 7 (1974), 1671–1692.

    Article  Google Scholar 

  38. Y. Imry and S.-K. Ma, Random-field instability of the ordered state of continuous Symmetry, Phys. Rev. Lett. 35 (1975), 1399–1401.

    Article  Google Scholar 

  39. Y. Isozaki and N. Yoshida, Weakly pinned random walk on the wall: pathwise descriptions of the phase Transition, Stoch. Proc. Appl. 96 (2001), 261–284.

    Article  MATH  MathSciNet  Google Scholar 

  40. N.C. Jain and W.E. Pruitt, The range of random walk, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, 31–50. Univ. California Press, Berkeley, Calif., 1972.

    Google Scholar 

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

    Article  Google Scholar 

  42. Y. Kafri, D.R. Nelson and A. Polkovnikov, Unzipping flux lines from extended defects in type-II superconductors, Europhys. Lett. 73 (2006), 253–259.

    Article  Google Scholar 

  43. J.F.C. Kingman, Subadditive Ergodic Theory, Ann. Probab. 1 (1973), 882–909.

    Article  MathSciNet  Google Scholar 

  44. R. Kühn, Equilibrium ensemble approach to disordered systems I: general theory, exact results, Z. Phys. B 100 (1996), 231–242.

    Article  Google Scholar 

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

    MATH  Google Scholar 

  46. J.M.J. van Leeuwen and H.J. Hilhorst, Pinning of rough interface by an external Potential, Phys. A 107 (1981), 319–329.

    Article  MathSciNet  Google Scholar 

  47. D. Marenduzzo, A. Trovato and A. Maritan, Phase diagram of force-induced DNA unzipping in exactly solvable models, Phys. Rev. E 64 (2001), 031901 (12 pages).

    Article  Google Scholar 

  48. C. Monthus and T. Garel, Distribution of pseudo-critical temperatures and lack of self-averaging in disordered Poland-Scheraga models with different loop exponents, Eur. Phys. J. B 48 (2005), 393–403.

    Article  Google Scholar 

  49. N. Pétrélis, Polymer Pinning at an Interface, Stoch. Proc. Appl. 116 (2006), 1600–1621.

    Article  MATH  Google Scholar 

  50. J. Sohier, Finite size scaling for homogeneous pinning models, arXiv:0802.1040 [math.PR]

    Google Scholar 

  51. M. Talagrand, A new look at independence, Ann. Probab. 24 (1996), 1–34.

    Article  MATH  MathSciNet  Google Scholar 

  52. F.L. Toninelli, Critical properties and finite-size estimates for the depinning transition of directed random polymers, J. Statist. Phys. 126 (2007), 1025–1044.

    Article  MATH  MathSciNet  Google Scholar 

  53. F.L. Toninelli, Correlation lengths for random polymer models and for some renewal Sequences, Electron. J. Probab. 12 (2007), 613–636.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  55. F.L. Toninelli, Disordered pinning models and copolymers: beyond annealed bounds, Ann. Appl. Probab. 18 (2008), 1569–1587.

    Article  MATH  MathSciNet  Google Scholar 

  56. F.L. Toninelli, Localization transition in disordered pinning models. Effect of randomness on the critical properties, Lecture Notes from the 5th Prague Summer School on Mathematical Statistical Mechanics, September 2006, arXiv:0703912 [math.PR].

    Google Scholar 

  57. S.G. Whittington, A directed-walk model of copolymer adsorption, J. Phys. A: Math. Gen. 31 (1998), 8797–8803.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Giambattista Giacomin U.F.R. de Mathématiques, Case 7012 (site Chevaleret) Laboratoire de Probabilités et Modèles Aléatoires (LPMA, CNRS U.M.R. 7599), Université Paris Diderot (Paris 7), F-75205, Paris Cedex 13, France

    Giambattista Giacomin

Authors
  1. Giambattista Giacomin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut de Mathématique de Jussieu, Université Paris Diderot Paris 7, 175 rue du Chevaleret, 75013, Paris, France

    Anne Boutet de Monvel

  2. Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität, Wegelerstrasse 6, 53115, Bonn, Germany

    Anton Bovier

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Giacomin, G. (2009). Renewal Sequences, Disordered Potentials, and Pinning Phenomena. In: de Monvel, A.B., Bovier, A. (eds) Spin Glasses: Statics and Dynamics. Progress in Probability, vol 62. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9891-0_11

Download citation

Publish with us

Policies and ethics

Search

Navigation