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Concentration under scaling limits for weakly pinned Gaussian random walks
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  • Published: 04 January 2008

Concentration under scaling limits for weakly pinned Gaussian random walks

  • Erwin Bolthausen1,
  • Tadahisa Funaki2 &
  • Tatsushi Otobe2 

Probability Theory and Related Fields volume 143, pages 441–480 (2009)Cite this article

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Abstract

We study scaling limits for d-dimensional Gaussian random walks perturbed by an attractive force toward a certain subspace of \(\mathbb {R}^d\), especially under the critical situation that the rate functional of the corresponding large deviation principle admits two minimizers. We obtain different type of limits, in a positive recurrent regime, depending on the co-dimension of the subspace and the conditions imposed at the final time under the presence or absence of a wall. The motivation comes from the study of polymers or (1 + 1)-dimensional interfaces with δ-pinning.

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References

  1. Bolthausen, E., Ioffe, D.: Harmonic crystal on the wall: a microscopic approach. Commun. Math. Phys. 187, 523–566 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bolthausen, E., Velenik, Y.: Critical behavior of the massless free field at the depinning transition. Commun. Math. Phys. 223, 161–203 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Caravenna, F., Giacomin, G., Zambotti, L.: Sharp asymptotic behavior for wetting models in (1 + 1)-dimension. Elect. J. Probab. 11, 345–362 (2006)

    MathSciNet  Google Scholar 

  4. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, Second ed. Applications of Mathematics, vol. 38. Springer, Heidelberg (1998)

    Google Scholar 

  5. Deuschel, J.-D., Giacomin, G., Zambotti, L.: Scaling limits of equilibrium wetting models in (1 + 1)-dimension. Probab. Theory Relat. Fields 132, 471–500 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Feller, W.: An Introduction to Probability Theory and its Applications, vol. 1, 2nd edn. Wiley, New York (1966)

    Google Scholar 

  7. Feller, W.: An Introduction to Probability Theory and its Applications, vol. 2, 2nd edn. Wiley, New York (1971)

    Google Scholar 

  8. Funaki, T.: Stochastic interface models. In: Picard, J. (ed.) Lectures on Probability Theory and Statistics, Ecole d’Eté de Probabilités de Saint-Flour XXXIII-2003, pp. 103–274. Lecture Notes in Mathematics, vol. 1869 (2005). Springer, Heidelberg

  9. Funaki, T.: Dichotomy in a scaling limit under Wiener measure with density. Electron. Comm. Probab. 12, 173–183 (2007)

    MATH  MathSciNet  Google Scholar 

  10. Funaki, T., Hariya, Y., Yor, M.: Wiener integrals for centered powers of Bessel processes, I. Markov Proc. Relat. Fields 13, 21–56 (2007)

    MATH  MathSciNet  Google Scholar 

  11. Funaki, T., Ishitani, K.: Integration by parts formulae for Wiener measures on a path space between two curves. Probab. Theory Relat. Fields 137, 289–321 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Funaki, T., Sakagawa, H.: Large deviations for \(\nabla\varphi\) interface model and derivation of free boundary problems. In: Osada, F. (eds.) Proceedings of Shonan/Kyoto meetings “Stochastic Analysis on Large Scale Interacting Systems” (2002), Adv. Stud. Pure Math., vol. 39. Math. Soc. Japan, 2004, pp. 173–211

  13. Funaki, T., Toukairin, K.: Dynamic approach to a stochastic domination: the FKG and Brascamp-Lieb inequalities. Proc. Am. Math. Soc. 135, 1915–1922 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Giacomin, G.: Random polymer models. Imperial College Press, New York (2007)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Mogul’skii, A.A.: Large deviations for trajectories of multi-dimensional random walks. Theory Probab. Appl. 21, 300–315 (1976)

    Article  Google Scholar 

  17. Patrick, A.E.: The influence of boundary conditions on solid-on-solid models. J. Statist. Phys. 90, 389–433 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Pfister, C.-E., Velenik, Y.: Interface, surface tension and reentrant pinning transition in the 2D Ising model. Commun. Math. Phys. 204, 269–312 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Preston, C.J.: A generalization of the FKG inequalities. Commun. Math. Phys. 36, 233–241 (1974)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zürich, Switzerland

    Erwin Bolthausen

  2. Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    Tadahisa Funaki & Tatsushi Otobe

Authors
  1. Erwin Bolthausen
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  2. Tadahisa Funaki
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  3. Tatsushi Otobe
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Corresponding author

Correspondence to Tadahisa Funaki.

Additional information

Supported in part by the JSPS Grants 17654020 and (A) 18204007.

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Bolthausen, E., Funaki, T. & Otobe, T. Concentration under scaling limits for weakly pinned Gaussian random walks. Probab. Theory Relat. Fields 143, 441–480 (2009). https://doi.org/10.1007/s00440-007-0132-8

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  • Received: 30 January 2007

  • Revised: 17 December 2007

  • Published: 04 January 2008

  • Issue Date: March 2009

  • DOI: https://doi.org/10.1007/s00440-007-0132-8

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Keywords

  • Large deviation
  • Minimizers
  • Random walks
  • Pinning
  • Scaling limit
  • Concentration

Mathematics Subject Classification (2000)

  • Primary: 60K35
  • Secondary: 60F10
  • 82B41
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