Skip to main content

Advertisement

SpringerLink
Log in

New Bounds for the Free Energy of Directed Polymers in Dimension 1 + 1 and 1 + 2

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the free energy of the directed polymer in a random environment model in dimension 1 + 1 and 1 + 2. For dimension one, we improve the statement of Comets and Vargas in [8] concerning very strong disorder by giving sharp estimates on the free energy at high temperature. In dimension two, we prove that very strong disorder holds at all temperatures, thus solving a long standing conjecture in the field.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Albeverio S., Zhou X.: A martingale approach to directed polymers in a random environment. J. Theor. Probab. 9, 171–189 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bertin, P.: Free energy for Linear Stochastic Evolutions in dimension two. Preprint 2009

  3. Bolthausen E.: A note on diffusion of directed polymer in a random environment. Commun. Math. Phys. 123, 529–534 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Carmona P., Hu Y.: On the partition function of a directed polymer in a random Gaussian environment. Probab. Theor. Relat. Fields 124(3), 431–457 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Carmona P., Hu Y.: Strong disorder implies strong localization for directed polymers in a random environment. ALEA 2, 217–229 (2006)

    MATH  MathSciNet  Google Scholar 

  6. Comets F., Shiga T., Yoshida N.: Directed Polymers in a random environment: strong disorder and path localization. Bernouilli 9(4), 705–723 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Comets F., Shiga T., Yoshida N.: Probabilistic analysis of directed polymers in a random environment: a review. Adv. Stud. Pure Math. 39, 115–142 (2004)

    MathSciNet  Google Scholar 

  8. Comets F., Vargas V.: Majorizing multiplicative cascades for directed polymers in random media. ALEA 2, 267–277 (2006)

    MATH  MathSciNet  Google Scholar 

  9. Comets F., Yoshida N.: Directed polymers in a random environment are diffusive at weak disorder. Ann. Probab. 34(5), 1746–1770 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. 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)

    Article  MathSciNet  ADS  Google Scholar 

  11. Feller W.: An Introduction to Probability Theory and Its Applications Volume II. John Wiley & Sons, Inc, New York (1966)

    Google Scholar 

  12. Gantert N., Zeitouni O.: Large and moderate deviations for local time of a reccurent Markov chain on Z 2. Ann. Inst. H. Poincaré Probab. Statist. 34, 687–704 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Giacomin, G.: Random Polymer Models. IC press, London: World Scientific, 2007

  14. Giacomin, G., Lacoin, H., Toninelli, F.L.: Hierarchical pinning models, quadratic maps and quenched disorder. To appear in Probab. Theory. Rel. Fields, doi:10.1007/s00440-009-0205-y

  15. Giacomin, G., Lacoin, H., Toninelli, F.L.: Marginal relevance of disorder for pinning models. to appear in Commun. Pure Appl. Math., doi:10.1002/cpa.20301

  16. Giacomin, G., Lacoin, H., Toninelli, F.L.: Disorder relevance at marginality and critical point shift. http://arxiv.org/abs/0906.1942v1[math-ph], 2009

  17. Grimmett, G.: Percolation. Second Edition, Grundlehren der Mathematischen Wissenschaften 321, Berlin: Springer-Verlag, 1999

  18. Huse D.A., Henley C.L.: Pinning and roughening of domain wall in Ising systems due to random impurities. Phys. Rev. Lett. 54, 2708–2711 (1985)

    Article  ADS  Google Scholar 

  19. Imbrie J.Z., Spencer T.: Diffusion of directed polymer in a random environment. J. Stat. Phys. 52(3/4), 608–626 (1988)

    MathSciNet  ADS  Google Scholar 

  20. Kaigh W.D.: An invariance principle for random walk conditioned by a late return to zero. Ann. Probab. 4, 115–121 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lacoin, H., Moreno, G.: Directed polymer on hierarchical lattice with site disorder. http://arxiv.org/abs/0906.0992v1[math.PR], 2009

  22. Liu Q., Watbled F.: Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment. Stoch. Proc. Appl. 119(10), 3101–3132 (2009)

    Article  MATH  Google Scholar 

  23. Song R., Zhou XY.: A remark on diffusion of directed polymers in random environment. J. Stat. Phys. 85(1/2), 277–289 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. Toninelli F.L.: Coarse graining, fractional moments and the critical slope of random copolymers. To appear in Electron. Journal Probab. 14, 531–547 (2009)

    MATH  MathSciNet  Google Scholar 

  26. Vargas V.: Strong localization and macroscopic atoms for directed polymer. Probab. Theor. Relat. Fields 134(3/4), 391–410 (2008)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Paris Diderot and Laboratoire de Probabilités et Modèles Aléatoires (CNRS U.M.R. 7599), U.F.R. Mathématiques, Case 7012 (Site Chevaleret), 75205, Paris cedex 13, France

    Hubert Lacoin

Authors
  1. Hubert Lacoin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hubert Lacoin.

Additional information

Communicated by F. L. Toninelli

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lacoin, H. New Bounds for the Free Energy of Directed Polymers in Dimension 1 + 1 and 1 + 2. Commun. Math. Phys. 294, 471–503 (2010). https://doi.org/10.1007/s00220-009-0957-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0957-3

Keywords

Search

Navigation