Advertisement

Journal of Statistical Physics

, Volume 67, Issue 1–2, pp 65–111 | Cite as

Join- and-cut algorithm for self-avoiding walks with variable length and free endpoints

  • Sergio Caracciolo
  • Andrea Pelissetto
  • AJan D. Sokal
Articles

Abstract

We introduce a new Monte Carlo algorithm for generating self-avoiding walks of variable length and free endpoints. The algorithm works in the unorthodox ensemble consisting of all pairs of SAWs such that the total number of stepsNtot in the two walks is fixed. The elementary moves of the algorithm are fixed-N (e.g., pivot) moves on the individual walks, and a novel “join- and-cut” move that concatenates the two walks and then cuts them at a random location. We analyze the dynamic critical behavior of the new algorithm, using a combination of rigorous, heuristic, and numerical methods. In two dimensions the autocorrelation time in CPU units grows as N≈1.5, and the behavior improves in higher dimensions. This algorithm allows high-precision estimation of the critical exponentγ.

Key words

Self-avoiding walk polymer Monte Carlo join- and-cut algorithm pivot algorithm critical exponent 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Lal,Molec. Phys. 17:57 (1969).Google Scholar
  2. 2.
    B. MacDonald, N. Jan, D. L. Hunter, and M. O. Steinitz,J. Phys. A 18:2627 (1985).Google Scholar
  3. 3.
    N. Madras and A. D. Sokal,J. Stat. Phys. 50:109 (1988).Google Scholar
  4. 4.
    A. T. Clark and M. Lal,Br. Polymer J. 9:92 (1977).Google Scholar
  5. 5.
    S. Caracciolo, A. Pelissetto, and A. D. Sokal,J. Stat. Phys. 60:1 (1990).Google Scholar
  6. 6.
    L. E. Dubins, A. Orlitsky, J. A. Reeds, and L. A. Shepp,IEEE Trans. Inform. Theory 34:1509 (1988).Google Scholar
  7. 7.
    N. Madras, A. Orlitsky, and L. A. Shepp,J. Stat. Phys. 58:159 (1990).Google Scholar
  8. 8.
    E. J. Janse van Rensburg, S. G. Whittington, and N. Madras,J. Phys. A 23:1589 (1990).Google Scholar
  9. 9.
    A. Berretti and A. D. Sokal,J. Stat. Phys. 40:483 (1985).Google Scholar
  10. 10.
    G. F. Lawler and A. D. Sokal,Trans. Am. Math. Soc. 309:557 (1988).Google Scholar
  11. 11.
    A. D. Sokal and L. E. Thomas,J. Stat. Phys. 54:797 (1989).Google Scholar
  12. 12.
    S. Redner and P. J. Reynolds,J. Phys. A 14:2679 (1981).Google Scholar
  13. 13.
    B. Berg and D. Foerster,Phys. Lett. 106B:323 (1981).Google Scholar
  14. 14.
    C. Aragão de Carvalho, S. Caracciolo, and J. Fröhlich,Nucl. Phys. B 215[FS7]:209 (1983).Google Scholar
  15. 15.
    C. Aragão de Carvalho and S. Caracciolo,J. Phys. (Paris)44:323 (1983).Google Scholar
  16. 16.
    A. D. Sokal and L. E. Thomas,J. Stat. Phys. 51:907 (1988).Google Scholar
  17. 17.
    S. Caracciolo, A. Pelissetto, and A. D. Sokal,J. Stat. Phys. 63:857 (1991).Google Scholar
  18. 18.
    S. Caracciolo, A. Pelissetto, and A. D. Sokal,Nucl. Phys. B (Proc. Suppl.) 20:68 (1991).Google Scholar
  19. 19.
    S. D. Silvey,Statistical Inference (Chapman and Hall, London, 1975), Chapter 4.Google Scholar
  20. 20.
    A. D. Sokal, Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms, Cours de Troisième Cycle de la Physique en Suisse Romande (Lausanne, June 1989).Google Scholar
  21. 21.
    R. G. Edwards and A. D. Sokal,Phys. Rev. D 38:2009 (1988).Google Scholar
  22. 22.
    S. Caracciolo, R. G. Edwards, A. Pelissetto, and A. D. Sokal,Nucl. Phys. B (Proc. Suppl.) 20:72 (1991); and paper in preparation.Google Scholar
  23. 23.
    P. R. Halmos,A Hilbert Space Problem Book, 2nd ed. (Springer, New York, 1982).Google Scholar
  24. 24.
    T. Kato,Perturbation Theory for Linear Operators, 2nd ed. (Springer, New York, 1976), Section I.6.10.Google Scholar
  25. 25.
    J. Keilson,Markov Chain Models-Rarity and Exponentiality (Springer, New York, 1979).Google Scholar
  26. 26.
    P. Lancaster,Theory of Matrices (Academic Press, New York, 1969), Chapter 3.Google Scholar
  27. 27.
    S. Caracciolo, G. Ferraro, and A. Pelissetto, in preparation.Google Scholar
  28. 28.
    D. E. Knuth,The Art of Computer Programming, Vol. 3 (Addison-Wesley, Reading, Massachusetts, 1973), Section 6.4.Google Scholar
  29. 29.
    S. S. Wilks,Mathematical Statistics (Wiley, New York, 1962).Google Scholar
  30. 30.
    P. J. Huber, inProceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1 (University of California Press, Berkeley, 1967), pp. 221–233.Google Scholar
  31. 31.
    I. V. Basawa and B. L. S. Prakasa Rao,Statistical Inference for Stochastic Processes (Academic Press, London, 1980), pp. 127–134, 162, 164.Google Scholar
  32. 32.
    Y. Ogata,J. Appl. Prob. 17:59 (1980).Google Scholar
  33. 33.
    A. Azzalini,Biometrika 70:381 (1983).Google Scholar
  34. 34.
    P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).Google Scholar
  35. 35.
    R. Cogburn, inProceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2 (University of California Press, Berkeley, 1972), pp. 485–512.Google Scholar
  36. 36.
    S. Niemi and E. Nummelin,Commentationes Physico-Mathematicae, No. 54 (Societas Scientiarum Fennicae, Helsinki, 1982).Google Scholar
  37. 37.
    D. Landers and L. Rogge,Z. Wahrsch. Verw. Gebiete 35:57 (1976).Google Scholar
  38. 38.
    E. Nummelin,General Irreducible Markov Chains and Non-Negative Operators (Cambridge University Press, Cambridge, 1984).Google Scholar
  39. 39.
    B. Nienhuis,Phys. Rev. Lett. 49:1062 (1982).Google Scholar
  40. 40.
    B. Nienhuis,J. Stat. Phys. 34:731 (1984).Google Scholar
  41. 41.
    A. J. Guttmann,J. Phys. A 22:2807 (1989).Google Scholar
  42. 42.
    F. J. Wegner,Phys. Rev. B 5:4529 (1972).Google Scholar
  43. 43.
    A. J. Guttmann, T. R. Osborn, and A. D. Sokal,J. Phys. A 19:2591 (1986).Google Scholar
  44. 44.
    H. Saleur,J. Phys. A 20:455 (1987).Google Scholar
  45. 45.
    S. Caracciolo, G. Ferraro, and A. Pelissetto,J. Phys. A 24:3625 (1991).Google Scholar
  46. 46.
    S. Caracciolo, A. Pelissetto, and A. D. Sokal, in preparation.Google Scholar
  47. 47.
    A. I. Larkin and D. E. Khmel'nitskii,Zh. Eksp. Teor. Fiz. 56:2087 (1969) [Sov. Phys.-JETP 29:1123 (1969)].Google Scholar
  48. 48.
    F. J. Wegner and E. K. Riedel,Phys. Rev. B 7:248 (1973).Google Scholar
  49. 49.
    E. Brézin, J.-C. LeGuillou, and J. Zinn-Justin,Phys. Rev. D 8:2418 (1973).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Sergio Caracciolo
    • 1
  • Andrea Pelissetto
    • 2
  • AJan D. Sokal
    • 3
  1. 1.Scuola Normale Superiore and INFN-Sezione di PisaPisaItaly
  2. 2.Department of PhysicsPrinceton UniversityPrinceton
  3. 3.Department of PhysicsNew York UniversityNew York

Personalised recommendations