Journal of Statistical Physics

, Volume 142, Issue 2, pp 322–341 | Cite as

The Effect of Disorder on the Free-Energy for the Random Walk Pinning Model: Smoothing of the Phase Transition and Low Temperature Asymptotics



We consider the continuous time version of the Random Walk Pinning Model (RWPM), studied in (Berger and Toninelli (Electron. J. Probab., to appear) and Birkner and Sun (Ann. Inst. Henri Poincaré Probab. Stat. 46:414–441, 2010; arXiv:0912.1663). Given a fixed realization of a random walk Y on ℤ d with jump rate ρ (that plays the role of the random medium), we modify the law of a random walk X on ℤ d with jump rate 1 by reweighting the paths, giving an energy reward proportional to the intersection time \(L_{t}(X,Y)=\int_{0}^{t} \mathbf {1}_{X_{s}=Y_{s}}\,\mathrm {d}s\): the weight of the path under the new measure is exp (βL t (X,Y)), β∈ℝ. As β increases, the system exhibits a delocalization/localization transition: there is a critical value β c , such that if β>β c the two walks stick together for almost-all Y realizations. A natural question is that of disorder relevance, that is whether the quenched and annealed systems have the same behavior. In this paper we investigate how the disorder modifies the shape of the free energy curve: (1) We prove that, in dimension d≥3, the presence of disorder makes the phase transition at least of second order. This, in dimension d≥4, contrasts with the fact that the phase transition of the annealed system is of first order. (2) In any dimension, we prove that disorder modifies the low temperature asymptotic of the free energy.


Pinning/wetting models Polymer Disordered models Harris criterion Smoothing/rounding effect 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aizenman, M., Wehr, J.: Rounding effects of quenched randomness on first-order phase transitions. Commun. Math. Phys. 130, 489–528 (1990) MATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Alexander, K.S.: Ivy on the ceiling: first-order polymer depinning transitions with quenched disorder. Markov Process. Relat. Fields 13, 663–680 (2007) MATHGoogle Scholar
  3. 3.
    Alexander, K.S.: The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279, 117–146 (2008) MATHCrossRefADSGoogle Scholar
  4. 4.
    Asmüssen, S.: Applied Probabilities and Queues, 2nd edn. Springer, Berlin (2003) Google Scholar
  5. 5.
    Berger, Q., Toninelli, F.L.: On the critical point of the Random Walk Pinning Model in dimension d=3. Electron. J. Probab. 15, 654–683 (2010) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Birkner, M., Sun, R.: Annealed vs Quenched critical points for a random walk pinning model. Ann. Inst. Henri Poincaré B, Probab. Stat. 46, 414–441 (2010) MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Birkner, M., Sun, R.: Disorder relevance for the random walk pinning model in dimension 3, arXiv:0912.1663
  8. 8.
    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) MATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Fisher, M.E.: Walks, walls, wetting, and melting. J. Stat. Phys. 34, 667–729 (1984) MATHCrossRefADSGoogle Scholar
  10. 10.
    Giacomin, G.: Random Polymer Models. IC Press, London (2007) MATHCrossRefGoogle Scholar
  11. 11.
    Giacomin, G., Toninelli, F.L.: Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006) MATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Giacomin, G., Toninelli, F.L.: Force-induced depinning of directed polymers. J. Phys. A, Math. Theor. 40, 5261–5275 (2007) MATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Giacomin, G., Lacoin, H., Toninelli, F.L.: Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. 63, 233–265 (2010) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lacoin, H.: The martingale approach to disorder irrelevance for pinning models. Preprint (2010) Google Scholar
  15. 15.
    Lacoin, H., Toninelli, F.L.: A smoothing inequality for hierarchical pinning models. In: Boutet de Monvel, A., Bovier, A. (eds.) Spin Glasses: Statics and Dynamics. Progress in Probability, vol. 62, pp. 271–278. (2009) CrossRefGoogle Scholar
  16. 16.
    Spitzer, F.: Principles of Random Walks. Springer, New York (1976) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Laboratoire de Physique, ENS LyonUniversité de LyonLyonFrance
  2. 2.Università degli Studi ‘Roma Tre’RomeItaly

Personalised recommendations