Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Diffusive limits for “true” (or myopic) self-avoiding random walks and self-repellent Brownian polymers in d ≥ 3
Download PDF
Download PDF
  • Published: 17 April 2011

Diffusive limits for “true” (or myopic) self-avoiding random walks and self-repellent Brownian polymers in d ≥ 3

  • Illés Horváth1,
  • Bálint Tóth1 &
  • Bálint Vető1 

Probability Theory and Related Fields volume 153, pages 691–726 (2012)Cite this article

  • 226 Accesses

  • 10 Citations

  • Metrics details

Abstract

The problems considered in the present paper have their roots in two different cultures. The `true’ (or myopic) self-avoiding walk model (TSAW) was introduced in the physics literature by Amit et al. (Phys Rev B 27:1635–1645, 1983). This is a nearest neighbor non-Markovian random walk in \({{\mathbb Z}^d}\) which prefers to jump to those neighbors which were less visited in the past. The self-repelling Brownian polymer model (SRBP), initiated in the probabilistic literature by Durrett and Rogers (Probab Theory Relat Fields 92:337–349, 1992) (independently of the physics community), is the continuous space–time counterpart: a diffusion in \({{\mathbb R}^d}\) pushed by the negative gradient of the (mollified) occupation time measure of the process. In both cases, similar long memory effects are caused by a path-wise self-repellency of the trajectories due to a push by the negative gradient of (softened) local time. We investigate the asymptotic behaviour of TSAW and SRBP in the non-recurrent dimensions. First, we identify a natural stationary (in time) and ergodic distribution of the environment (the local time profile) as seen from the moving particle. The main results are diffusive limits. In the case of TSAW, for a wide class of self-interaction functions, we establish diffusive lower and upper bounds for the displacement and for a particular, more restricted class of interactions, we prove full CLT for the finite dimensional distributions of the displacement. In the case of SRBP, we prove full CLT without restrictions on the interaction functions. These results settle part of the conjectures, based on non-rigorous renormalization group arguments (equally ‘valid’ for the TSAW and SRBP cases), in Amit et al. (1983). The proof of the CLT follows the non-reversible version of Kipnis–Varadhan theory. On the way to the proof, we slightly weaken the so-called graded sector condition.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Amit D., Parisi G., Peliti L.: Asymptotic behavior of the ‘true’ self-avoiding walk. Phys. Rev. B 27, 1635–1645 (1983)

    Article  MathSciNet  Google Scholar 

  2. Bobkov S.G., Ledoux M.: From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10, 1028–1052 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cranston M., Le Jan Y.: Self-attracting diffusions: two case studies. Math. Ann. 303, 87–93 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cranston M., Mountford T.S.: The strong law of large numbers for a Brownian polymer. Ann. Probab. 24, 1300–1323 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dobrushin, R.L., Suhov, Yu.M., Fritz, J.: A.N. Kolmogorov—the founder of the theory of reversible Markov processes. Uspekhi Mat. Nauk 43(6), 167–188 (1988) [English translation: Russ. Math. Surv. 43(6), 157–182]

  6. Durrett R.T., Rogers L.C.G.: Asymptotic behavior of Brownian polymers. Probab. Theory Relat. Fields 92, 337–349 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Funaki, T.: Stochastic interface models. In: Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol. 1869. Springer, Brelin (2005)

  8. Gordin M.I., Lifshits B.A.: Central limit theorem for stationary Markov processes. Dokl. Akad. Nauk SSSR 239, 766–767 (1978)

    MathSciNet  Google Scholar 

  9. Horváth, I., Tóth, B., Vető, B.: Diffusive limit for self-repelling Brownian polymers in d ≥ 3. http://arxiv.org/abs/0912.5174 (2009)

  10. Janson S.: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  11. Kipnis C., Varadhan S.R.S.: Central limit theorem for additive functionals of reversible Markov processes with applications to simple exclusion. Commun. Math. Phys. 106, 1–19 (1986)

    Article  MathSciNet  Google Scholar 

  12. Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation. Springer, New York (2011, to appear)

  13. Komorowski T., Olla S.: On the sector condition and homogenization of diffusions with a Gaussian drift. J. Funct. Anal. 197, 179–211 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kozlov S.M.: The method of averaging and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40, 61–120 (1885)

    Google Scholar 

  15. Landim C., Yau H.-T.: Fluctuation-dissipation equation of asymmetric simple exclusion processes. Probab. Theory Relat. Fields 108, 321–356 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mountford T.S., Tarrès P.: An asymptotic result for Brownian polymers. Ann. Inst. H. Poincaré Probab. Stat. 44, 29–46 (2008)

    Article  MATH  Google Scholar 

  17. Norris J.R., Rogers L.C.G., Williams D.: Self-avoiding walk: a Brownian motion model with local time drift. Probab. Theory Relat. Fields 74, 271–287 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Obukhov S.P., Peliti L.: Renormalisation of the “true” self-avoiding walk. J. Phys. A 16, L147–L151 (1983)

    Article  MathSciNet  Google Scholar 

  19. Olla, S.: Central limit theorems for tagged particles and for Diffusions in random environment. In: Comets, F., Pardoux, É. (eds.) Milieux aléatoires Panor. Synthèses 12. Soc. Math. France, Paris (2001)

  20. Osada, H.: Homogenization of diffusion processes with random stationary coefficients. In: Probability Theory and Mathematical Statistics (Tbilisi, 1982). Lecture Notes in Mathematics, vol. 1021, pp. 507–517. Springer, Berlin (1983)

  21. Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. In: Fritz, J., Szász, D., Lebowitz, J.L. (eds.) Random Fields (Esztergom, 1979). Colloq. Math. Soc. János Bolyai, vol. 27, pp. 835–873. North-Holland, Amsterdam (1981)

  22. Peliti L., Pietronero L.: Random walks with memory. Riv. Nuovo Cimento 10, 1–33 (1987)

    Google Scholar 

  23. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vols. 1, 2. Academic Press, New York (1972–1975)

  24. Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales. Ito Calculus, vol. 2. Wiley, New York (1987)

  25. Sethuraman S., Varadhan S.R.S., Yau H.-T.: Diffusive limit of a tagged particle in asymmetric simple exclusion processes. Commun. Pure Appl. Math. 53, 972–1006 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Simon B.: The \({P(\phi)_2}\) Euclidean (Quantum) Field Theory. Princeton University Press, Princeton (1974)

    MATH  Google Scholar 

  27. Tarrès, P., Tóth, B., Valkó, B.: Diffusivity bounds for 1d Brownian polymers. Ann. Probab. (2011, to appear)

  28. Tóth B.: Persistent random walk in random environment. Probab. Theory Relat. Fields 71, 615–625 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tóth B.: The ‘true’ self-avoiding walk with bond repulsion on \({{\mathbb Z}}\): limit theorems. Ann. Probab. 23, 1523–1556 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tóth, B., Valkó, B.: Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in d=2. http://arxiv.org/abs/1012.5698 (2011, submitted)

  31. Tóth B., Vető B.: Continuous time ‘true’ self-avoiding random walk on \({{\mathbb Z}}\). ALEA, Lat. Am. J. Probab. Math. Stat. 8, 59–75 (2011)

    MathSciNet  Google Scholar 

  32. Tóth B., Werner W.: The true self-repelling motion. Probab. Theory Relat. Fields 111, 375–452 (1998)

    Article  MATH  Google Scholar 

  33. Varadhan S.R.S.: Self-diffusion of a tagged particle in equilibrium of asymmetric mean zero random walks with simple exclusion. Ann. Inst. H. Poincaré Probab. Stat. 31, 273–285 (1996)

    MathSciNet  Google Scholar 

  34. Yaglom A.M.: On the statistical treatment of Brownian motion. Dokl. Akad. Nauk SSSR 56, 691–694 (1947) (in Russian)

    MathSciNet  MATH  Google Scholar 

  35. Yaglom A.M.: On the statistical reversibility of Brownian motion. Mat. Sbornik 24, 457–492 (1949) (in Russian)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Budapest University of Technology, Egry József u. 1, Budapest, 1111, Hungary

    Illés Horváth, Bálint Tóth & Bálint Vető

Authors
  1. Illés Horváth
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bálint Tóth
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bálint Vető
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Illés Horváth.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Horváth, I., Tóth, B. & Vető, B. Diffusive limits for “true” (or myopic) self-avoiding random walks and self-repellent Brownian polymers in d ≥ 3. Probab. Theory Relat. Fields 153, 691–726 (2012). https://doi.org/10.1007/s00440-011-0358-3

Download citation

  • Received: 18 June 2010

  • Revised: 21 March 2011

  • Published: 17 April 2011

  • Issue Date: August 2012

  • DOI: https://doi.org/10.1007/s00440-011-0358-3

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Self-repelling random motion
  • Local time
  • Central limit theorem

Mathematics Subject Classification (2010)

  • 60K37
  • 60K40
  • 60F05
  • 60J55
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature