Stretched Polymers in Random Environment

  • Dmitry Ioffe
  • Yvan Velenik
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 11)


We survey recent results and open questions on the ballistic phase of stretched polymers in both annealed and quenched random environments.


Partition Function Lyapunov Exponent Random Environment Annealed Model Simple Random Walk 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The research of DI was supported by the Israeli Science Foundation (grant No. 817/09). YV is partially supported by the Swiss National Science Foundation. The authors would like to thank the anonymous referee for a very careful reading and useful remarks.


  1. 1.
    Antal, P.: Enlargement of obstacles for the simple random walk. Ann. Probab. 23(3), 1061–1101 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bolthausen, E.: A note on diffusion of directed polymers in random environment. Comm. Math. Phys. 123(4), 529–534 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bolthausen, E., Sznitman, A.-S.: On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9(3), 345–375 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Carmona, P., Hu, Y.: On the partition function of a directed polymer in a random environment. Probab. Theory Rel. Fields 124(3), 431–457 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Comets, F., Yoshida, N.: Proabilistic analysis of directed polymers in random environment: a review. Stochastic analysis of large scale interacting systems. Adv. Stud. Pure. Math. 39, 115–142 (2004)MathSciNetGoogle Scholar
  6. 6.
    Comets, F., Yoshida, N.: Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34(5), 1746–1770 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Comets, F., Shiga, T., Yoshida, N.: Directed polymers in a random environment: path localization and strong disorder. Bernoulli 9(4), 705–723 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Flury, M.: Large deviations and phase transition for random walks in random nonnegative potentials. Stochastic Process. Appl. 117(5), 596–612 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Flury, M.: A note on the ballistic limit of random motion in a random potential. Electron. Commun. Probab. 13, 393–400 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Flury, M.: Coincidence of Lyapunov exponents for random walks in weak random potentials. Ann. Probab. 36(4), 1528–1583 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Huse, D.A., Henley, C.: Pinning and roughening of domain walls in Ising systems due to random impurities. Phys. Rev. Lett. 54(25), 2708–2711 (1985)CrossRefGoogle Scholar
  12. 12.
    Ioffe, D., Velenik, Y.: In preparationGoogle Scholar
  13. 13.
    Ioffe, D., Velenik, Y.: Ballistic phase of self-interacting random walks. In: Analysis and Stochastics of Growth Processes and Interface Models, pp. 55–79. Oxford University Press, Oxford (2008)Google Scholar
  14. 14.
    Ioffe, D., Velenik, Y.: Crossing random walks and stretched polymers at weak disorder. Ann. Probab; arXiv:1002.4289 (to appear, 2012)Google Scholar
  15. 15.
    Kardar, M.: Roughening by impurities at finite temperatures. Phys. Rev. Lett. 55(26), 2923 (1985)CrossRefGoogle Scholar
  16. 16.
    Kosygina, E., Mountford, T., Zerner, M.: Lyapunov exponents of Green’s functions for random potentials tending to zero. Prob. Theory Rel. Fields. 150(1–2), 43–59 doi:10.1007/s00440-010-0266-yGoogle Scholar
  17. 17.
    Lacoin, H.: New bounds for the free energy of directed polymers in dimension 1 + 1 and 1 + 2. Comm. Math. Phys. 294(2), 471–503 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ledoux, M.: The concentration of measure phenomenon, vol. 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2001)Google Scholar
  19. 19.
    Martin, P., Zerner, W.: Directional decay of the Green’s function for a random nonnegative potential on Z d. Ann. Appl. Probab. 8(1), 246–280 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    McLeish, D.L.: A maximal inequality and dependent strong laws. Ann. Probab. 3(3), 829–839 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Sinai, Y.G.: A remark concerning random walks with random potentials. Fund. Math 147, 173 –180 (1995)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Song, R., Zhou, X.Y.: A remark on diffusion of directed polymers in random environments. J. Statist. Phys. 85(1–2), 277–289 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Sznitman, A.-S.: Brownian motion, obstacles and random media. Springer Monographs in Mathematics. Springer, Berlin (1998)zbMATHCrossRefGoogle Scholar
  24. 24.
    Trachsler, M.: Phase Transitions and Fluctuations for Random Walks with Drift in Random Potentials. PhD thesis, Universität Zürich (1999)Google Scholar
  25. 25.
    Vargas, V.: A local limit theorem for directed polymers in random media: the continuous and the discrete case. Ann. Inst. H. Poincaré Probab. Statist. 42(5), 521–534 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Vargas, V.: Strong localization and macroscopic atoms for directed polymers. Probab. Theory Related Fields 138(3–4), 391–410 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Wouts, M.: Surface tension in the dilute Ising model. The Wulff construction. Comm. Math. Phys. 289(1), 157–204 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Zygouras, N.: Lyapounov norms for random walks in low disorder and dimension greater than three. Probab. Theory Related Fields 143(3–4), 615–642 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Zygouras, N.: Strong disorder in semidirected random polymers. preprint, arXiv:1009.2693 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Industrial Engineering and ManagementTechnionHaifaIsrael
  2. 2.Department of MathematicsUniversity of GenevaGenève 4Switzerland

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