Probability Theory and Related Fields

, Volume 147, Issue 3–4, pp 565–581 | Cite as

Randomized polynuclear growth with a columnar defect

  • Vincent Beffara
  • Vladas Sidoravicius
  • Maria Eulalia Vares


We study a variant of poly-nuclear growth where the level boundaries perform continuous-time, discrete-space random walks, and study how its asymptotic behavior is affected by the presence of a columnar defect on the line. We prove that there is a non-trivial phase transition in the strength of the perturbation, above which the law of large numbers for the height function is modified.


Poly-nuclear growth Interacting random walks Zero-temperature Glauber dynamics Polymer pinning 

Mathematics Subject Classification (2000)

60K35 60K37 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aldous D., Diaconis P.: Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Relat. Fields 103(2), 199–213 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arias-Castro, E., Candes, E., Helgason, H., Zeitouni, O.: Searching for a trail of evidence in a maze. Preprint (2007) arXiv:math.ST/0701668v1Google Scholar
  3. 3.
    Arratia R.: The motion of a tagged particle in the simple symmetric exclusion process on Z. Ann. Probab. 11(2), 362–373 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Baik J., Deift P., Johansson K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12(4), 1119–1178 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Baik, J., Rains, E.M.: Symmetrized random permutations. In: Random matrix models and their applications, Math. Sci. Res. Inst. Publ., vol. 40, pp. 1–19. Cambridge University Press, Cambridge (2001)Google Scholar
  6. 6.
    Beffara, V., Sidoravicius, V.: in preparation (2008)Google Scholar
  7. 7.
    Beffara, V., Sidoravicius, V., Spohn, H., Vares, M.: Polymer pinning in random medium as influence percolation. In: Den Hollander, F., Verbitsky, E. (eds.) Dynamics and Stochastics, IMS Lecture Notes Monograph Ser., vol. 48, pp. 1–15. Inst. Math. Statist (2006)Google Scholar
  8. 8.
    Chayes L., Schonmann R., Swindle G.: Lifshitz law for the volume of a two dimensional droplet at zero temperature. J. Stat. Phys. 79(4), 821–831 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Janowsky S.A., Lebowitz J.L.: Exact results for the asymmetric simple exclusion process with a blockage. J. Stat. Phys. 77(1–2), 35–51 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kesten, H.: First-passage percolation. In: From classical to modern probability. Progr. Probab., vol. 54, pp. 93–143. Birkhäuser, Basel (2003)Google Scholar
  11. 11.
    Liggett T.M.: Interacting particle systems, Grundlehren der Mathematischen Wissenschaften, vol. 276. Springer, New York (1985)Google Scholar
  12. 12.
    Myllys, M., Maunuksela, J., Merikoski, J., Timonen, J., Horvath, V.K., Ha, M., den Nijs, M.: Effect of columnar defect on the shape of slow-combustion fronts. Preprint (2003). Cond-Mat/0307231Google Scholar
  13. 13.
    Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Statist. Phys. 108(5–6), 1071–1106 (2002). Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdaysGoogle Scholar
  14. 14.
    Spohn H.: Interface motion in models with stochastic dynamics. J. Stat. Phys. 71(5–6), 1081–1132 (1993)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Vincent Beffara
    • 1
  • Vladas Sidoravicius
    • 2
    • 3
  • Maria Eulalia Vares
    • 4
  1. 1.UMPA, ENS LyonLyon Cedex 07France
  2. 2.CWIAmsterdamThe Netherlands
  3. 3.IMPARio de JaneiroBrazil
  4. 4.CBPFRio de JaneiroBrazil

Personalised recommendations