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
The diffusive phase of a model of self-interacting walks
Download PDF
Download PDF
  • Published: September 1995

The diffusive phase of a model of self-interacting walks

  • D. C. Brydges1 &
  • G. Slade2 

Probability Theory and Related Fields volume 103, pages 285–315 (1995)Cite this article

  • 113 Accesses

  • 29 Citations

  • Metrics details

Summary

We consider simple random walk onZ d perturbed by a factor exp[βT −P J T], whereT is the length of the walk and\(J_T = \sum\nolimits_{0 \leqslant i< j \leqslant T} \delta _{\omega (i),\omega (j)} \). Forp=1 and dimensionsd≥2, we prove that this walk behaves diffusively for all − ∞ < β <0, with β0 > 0. Ford>2 the diffusion constant is equal to 1, but ford=2 it is renormalized. Ford=1 andp=3/2, we prove diffusion for all real β (positive or negative). Ford>2 the scaling limit is Brownian motion, but ford≤2 it is the Edwards model (with the “wrong” sign of the coupling when β>0) which governs the limiting behaviour; the latter arises since for\(p = \frac{{4 - d}}{2}\),T −p J T is the discrete self-intersection local time. This establishes existence of a diffusive phase for this model. Existence of a collapsed phase for a very closely related model has been proven in work of Bolthausen and Schmock.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. S. Albeverio and X.Y. Zhou. A modified Domb-Joyce model in four dimensions. Preprint, (1993).

  2. P. Billingsley,Convergence of Probability Measures, John Wiley and Sons, New York (1968).

    Google Scholar 

  3. E. Bolthausen. On the construction of the three dimensional polymer measure.Probab. Theory Relat. Fields,97:81–101, (1993).

    Google Scholar 

  4. E. Bolthausen. Localization of a two-dimensional random walk with an attractive path interaction.Ann. Probab.,22:875–918, (1994).

    Google Scholar 

  5. E. Bolthausen and U. Schmock. On self-attractingd-dimensional random walks. Preprint, (1994).

  6. E. Bolthausen and U. Schmock. On self-attracting random walks. In M.C. Cranston and M.A. Pinsky, editors,Stochastic Analysis, Providence, (1995), American Mathematical Society. Proceedings of Symposia in Pure Mathematics, Volume 57.

  7. A.N. Borodin. On the asymptotic behavior of local times of recurrent random wllks with finite variance.Theory Probab. Appl.,26:758–772, (1981).

    Google Scholar 

  8. A.N. Borodin. Brownian local time.Russian Math. Surveys,44:1–51, (1989).

    Google Scholar 

  9. A. Bovier, G. Felder, and J. Fröhlich. On the critical properties of the Edwards and the self-avoiding walk model of polymer chains.Nucl. Phys. B,230 [FS10]:119–147, (1984).

    Google Scholar 

  10. R. Brak, A.J. Guttmann, and S.G. Whittington. A collapse transition in a directed walk model.J. Phys. A: Math. Gen.,25:2437–2446, (1992).

    Google Scholar 

  11. R. Brak, A.L. Owezarek, and T. Prellberg. A scaling theory of the collapse transition in geometric cluster models of polymers and vesicles.J. Phys. A: Math. Gen.,26:4565–4579, (1993).

    Google Scholar 

  12. D. Brydges, S.N. Evans, and I.Z. Imbrie. Self-avoiding walk on a hierarchical lattice in four dimensions.Ann. Probab.,20:82–124, (1992).

    Google Scholar 

  13. D.C. Brydges and G. Slade. A collapse transition for self-attracting walks.Resenhas do Instituto da Matemática e Estatística da Universidade de São Paulo,1 363–372, (1994).

    Google Scholar 

  14. S. Caracciolo, G. Parisi, and A. Pelissetto. Random walks with short-range interaction and mean-field behavior.J. Stat. Phys.,77:519–543, (1994).

    Google Scholar 

  15. J. Fröhlich and Y.M. Park. Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems.Commun. Math. Phys.,59:235–266, (1978).

    Google Scholar 

  16. A. Greven and F. den Hollander. A variational characterization of the speed of a one-dimensional self-repellent random walk.Ann. Appl. Probab.,3:1067–1099, (1993).

    Google Scholar 

  17. D. Iagolnitzer and J. Magnen. Polymers in a weak random potential in dimension four: rigorous renormalization group analysis.Commun. Math. Phys.,162:85–121, (1994).

    Google Scholar 

  18. T. Kennedy. Ballistic behavior in a 1D weakly self-avoiding walk with decaying energy penalty.J. Stat. Phys.,77:565–579, (1994).

    Google Scholar 

  19. G.F. Lawler.Intersections of Random Walks. Birkhäuser, Boston, (1991).

    Google Scholar 

  20. J.-F. Le Gall. Sur le temps local d'intersection du mouvement brownien plan et la methode de renormalization de Varadhan. In J. Azéma and M. Yor, editors.Séminaire de Probabilités XIX.Lecture Notes in Mathematics #1123. Berlin, (1985), Springer.

    Google Scholar 

  21. J.-F. Le Gall. Propriétés d'intersection des marches aléatoires I. Convergence vers le temps local d'intersection.Commun. Math. Phys.,104:471–507, (1986).

    Google Scholar 

  22. J.-F. Le Gall. Exponential moments for the renormalized self-intersection local time of planar Brownian motion. In J. Azéma, P.A. Meyer, and M. Yor, editors,Séminaire de Probabilités XXVIII.Lecture Notes in Mathematics #1583, Berlin, (1994). Springer.

    Google Scholar 

  23. J.L. Lebowitz, H.A. Rose, and E.R. Speer. Statistical mechanics of the nonlinear Schrödinger equation.J. Stat. Phys.,50:657–687, (1988).

    Google Scholar 

  24. Y. Oono, On the divergence of the perturbation series for the excluded-volume problem in polymers.J. Phys. Soc. Japan,39:25–29, (1975).

    Google Scholar 

  25. Y. Oono. On the divergence of the perturbation series for the excluded-volume problem in polymers. II. Collapse of a single chain in poor solvents.J. Phys. Soc. Japan,41:787–793, (1976).

    Google Scholar 

  26. E. Perkins. Weak invariance principles for local time.Z. Wahrsch. verw. Gebiete,60:437–451, (1982).

    Google Scholar 

  27. J. Rosen, Self-intersections of random fields.Ann. Probab.,12:108–119, (1984).

    Google Scholar 

  28. J. Rosen. Random walks and intersection local time.Ann. Probab.,18:959–977, (1990).

    Google Scholar 

  29. U. Schmock. Convergence of the normalized one-dimensional Wiener sausage path measures to a mixture of Brownian taboo processes.Stochastics and Stochastic Reports,29:171–183, (1989).

    Google Scholar 

  30. F. Spitzer,Principles of Random Walk. Springer, New York, 2nd edition, (1976).

    Google Scholar 

  31. A. Stoll. Invariance principles for Brownian intersection local time and polymer measures.Math. Scand.,64:133–160, (1989).

    Google Scholar 

  32. A.-S. Sznitman. On the confinement property of two-dimensional Brownian motion among Poissonian obstacles.Commun. Pure Appl. Math.,44:1137–1170, (1991).

    Google Scholar 

  33. S.R.S. Varadhan. Appendix to: Euclidean quantum field theory, by K. Symanzik. In R. Jost, editor,Local Quantum Field Theory. New York, (1969). Academic Press.

    Google Scholar 

  34. J. Westwater. On Edwards' model for long polymer chains.Commun. Math. Phys.,72:131–174, (1980).

    Google Scholar 

  35. J. Westwater. On Edwards' model for long polymer chains III. Borel summability.Commun. Math. Phys.,84:459–470, (1982).

    Google Scholar 

  36. J. Westwater. On Edwards' model for long polymer chains. In S. Albeverio and P. Blanchard, editors,Trends and Developments in the Eighties. Bielefeld Encounters in Mathematical Physics IV/V. World Scientific, Singapore, (1985).

    Google Scholar 

  37. H. Zoladek. One-dimensional random walk with self-interaction.J. Stat. Phys.,47:543–550, (1987).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Virginia, 22903-3199, Charlottesville, VA, USA

    D. C. Brydges

  2. Department of Mathematics and Statistics, McMaster University, L8S 4K1, Hamilton, Ont., Canada

    G. Slade

Authors
  1. D. C. Brydges
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Slade
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Brydges, D.C., Slade, G. The diffusive phase of a model of self-interacting walks. Probab. Th. Rel. Fields 103, 285–315 (1995). https://doi.org/10.1007/BF01195476

Download citation

  • Received: 25 August 1994

  • Revised: 18 April 1995

  • Issue Date: September 1995

  • DOI: https://doi.org/10.1007/BF01195476

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

Mathematics Subject Classification

  • 82B41
  • 60K35
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