Journal of Statistical Physics

, Volume 50, Issue 1–2, pp 109–186 | Cite as

The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk

  • Neal Madras
  • Alan D. Sokal
Articles

Abstract

The pivot algorithm is a dynamic Monte Carlo algorithm, first invented by Lal, which generates self-avoiding walks (SAWs) in a canonical (fixed-N) ensemble with free endpoints (hereN is the number of steps in the walk). We find that the pivot algorithm is extraordinarily efficient: one “effectively independent” sample can be produced in a computer time of orderN. This paper is a comprehensive study of the pivot algorithm, including: a heuristic and numerical analysis of the acceptance fraction and autocorrelation time; an exact analysis of the pivot algorithm for ordinary random walk; a discussion of data structures and computational complexity; a rigorous proof of ergodicity; and numerical results on self-avoiding walks in two and three dimensions. Our estimates for critical exponents areυ=0.7496±0.0007 ind=2 andυ= 0.592±0.003 ind=3 (95% confidence limits), based on SAWs of lengths 200⩽N⩽10000 and 200⩽N⩽ 3000, respectively.

Key words

Self-avoiding walk polymer Monte Carlo pivot algorithm critical exponent 

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References

  1. 1.
    C. Domb,Adv. Chem. Phys. 15:229 (1969).Google Scholar
  2. 2.
    D. S. McKenzie,Phys. Rep. 27:35 (1976).Google Scholar
  3. 3.
    S. G. Whittington,Adv. Chem. Phys. 51:1 (1982).Google Scholar
  4. 4.
    P. G. de Gennes,Phys. Lett. 38A:339 (1972).Google Scholar
  5. 5.
    J. des Cloizeaux,J. Phys. (Paris) 36:281 (1975).Google Scholar
  6. 6.
    M. Daoudet al., Macromolecules 8:804 (1975).Google Scholar
  7. 7.
    V. J. Emery,Phys. Rev. B 11:239 (1975).Google Scholar
  8. 8.
    C. Aragão de Carvalho, S. Caracciolo, and J. Fröhlich,Nucl. Phys. B 215[FS7]:209 (1983).Google Scholar
  9. 9.
    R. Fernández, J. Fröhlich, and A. D. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory (Lecture Notes in Physics, Springer-Verlag, to appear).Google Scholar
  10. 10.
    F. T. Wall, S. Windwer, and P. J. Gans, inMethods in Computational Physics, Vol. 1, B. Alder, S. Fernbach, and M. Rotenberg, eds. (Academic Press, New York, 1963).Google Scholar
  11. 11.
    S. Redner and P. J. Reynolds,J. Phys. A 14:2679 (1981).Google Scholar
  12. 12.
    B. Berg and D. Foerster,Phys. Lett. 106B:323 (1981); C. Aragão de Carvalho and S. Caracciolo,J. Phys. (Paris) 44:323 (1983); C. Aragão de Carvalho, S. Caracciolo, and J. Fröhlich,Nucl. Phys. B 215[FS7]:209 (1983).Google Scholar
  13. 13.
    A. Berretti and A. D. Sokal,J. Stat. Phys. 40:483 (1985).Google Scholar
  14. 14.
    M. Lal,Molec. Phys. 17:57 (1969).Google Scholar
  15. 15.
    O. F. Olaj and K. H. Pelinka,Makromol. Chem. 177:3413 (1976).Google Scholar
  16. 16.
    B. MacDonald, N. Jan, D. L. Hunter, and M. O. Steinitz,J. Phys. A 18:2627 (1985).Google Scholar
  17. 17.
    D. L. Hunter, N. Jan, and B. MacDonald,J. Phys. A 19:L543 (1986); K. Kelly, D. L. Hunter and N. Jan,J. Phys. A 20:5029 (1987).Google Scholar
  18. 18.
    S. D. Stellman and P. J. Gans,Macromolecules 5:516 (1972).Google Scholar
  19. 19.
    S. D. Stellman and P. J. Gans,Macromolecules 5:720 (1972).Google Scholar
  20. 20.
    J. J. Freire and A. Horta,J. Chem. Phys. 65:4049 (1976).Google Scholar
  21. 21.
    J. M. Hammersley,Proc. Camb. Phil. Soc. 53:642 (1957).Google Scholar
  22. 22.
    J. M. Hammersley,Proc. Camb. Phil. Soc. 57:516 (1961).Google Scholar
  23. 23.
    J. M. Hammersley and D. J. A. Welsh,Q. J. Math. (Oxford) Ser. 2 13:108 (1962).Google Scholar
  24. 24.
    H. Kesten,J. Math. Phys. 4:960 (1963).Google Scholar
  25. 25.
    H. Kesten,J. Math. Phys. 5:1128 (1964).Google Scholar
  26. 26.
    G. Slade,Commun. Math. Phys. 110:661 (1987).Google Scholar
  27. 27.
    J. G. Kemeny and J. L. Snell,Finite Markov Chains (Springer, New York, 1976).Google Scholar
  28. 28.
    M. Iosifescu,Finite Markov Processes and Their Applications (Wiley, Chichester, 1980).Google Scholar
  29. 29.
    K. L. Chung,Markov Chains with Stationary Transition Probabilities, 2nd ed. (Springer, New York, 1967).Google Scholar
  30. 30.
    E. Seneta,Non-Negative Matrices and Markov Chains, 2nd ed. (Springer, New York, 1981).Google Scholar
  31. 31.
    M. Hamermesh,Group Theory and Its Application to Physical Problems (Addison-Wesley, Reading, Massachusetts, 1962), Chapter 2.Google Scholar
  32. 32.
    J. Garcia de la Torre, A. Jiménez, and J. J. Freire,Macromolecules 15:148 (1982).Google Scholar
  33. 33.
    B. Nienhuis,J. Stat. Phys. 34:731 (1984).Google Scholar
  34. 34.
    A. J. Guttmann,J. Phy. A 20:1839 (1987).Google Scholar
  35. 35.
    H. Saleur,J. Phys. A 19:L807 (1986).Google Scholar
  36. 36.
    B. Duplantier,Phys. Rev. B 35:5290 (1987).Google Scholar
  37. 37.
    D. E. Knuth,The Art of Computer Programming, Vol. 3 (Addison-Wesley, Reading, Massachusetts, 1973), Section 6.4.Google Scholar
  38. 38.
    E. Horowitz and S. Sahni,Fundamentals of Data Structures (Computer Science Press, Potomac, Maryland, 1976), Section 9.3.Google Scholar
  39. 39.
    K. Suzuki,Bull. Chem. Soc. Japan 41:538 (1968).Google Scholar
  40. 40.
    Z. Alexandrowicz,J. Chem. Phys. 51:561 (1969).Google Scholar
  41. 41.
    Z. Alexandrowicz and Y. Accad,J. Chem. Phys. 54:5338 (1971).Google Scholar
  42. 42.
    N. Madras and A. D. Sokal, in preparation.Google Scholar
  43. 43.
    D. Goldsman, Ph. D. thesis, School of Operations Research and Industrial Engineering, Cornell University (1984).Google Scholar
  44. 44.
    L. Schruben,Op. Res. 30:569 (1982).Google Scholar
  45. 45.
    L. Schruben,Op. Res. 31:1090 (1983).Google Scholar
  46. 46.
    J. R. Baxter and R. V. Chacon,Ill. J. Math. 20:467 (1976).Google Scholar
  47. 47.
    D. J. Aldous,J. Lond. Math. Soc. 25:564 (1982).Google Scholar
  48. 48.
    D. Aldous, inSéminaire de Probabilités XVII (Lecture Notes in Mathematics No. 986, Springer-Verlag, Berlin, 1983).Google Scholar
  49. 49.
    D. Aldous and P. Diaconis,Am. Math. Monthly 93:333 (1986).Google Scholar
  50. 50.
    W. Feller,An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed. (Wiley, New York, 1968), pp. 224–225.Google Scholar
  51. 51.
    P. Grassberger,Z. Phys. B 48:255 (1982).Google Scholar
  52. 52.
    C. Domb and F. T. Hioe,J. Chem. Phys. 51:1915 (1969).Google Scholar
  53. 53.
    D. E. Knuth,The Art of Computer Programming, Vol. 2, 2nd ed. (Addison-Wesley, Reading, Massachusetts, 1973), pp. 102–103.Google Scholar
  54. 54.
    S. D. Silvey,Statistical Inference (Chapman and Hall, London, 1975), Chapter 3.Google Scholar
  55. 55.
    I. Majid, Z. V. Djordjevic, and H. E. Stanley,Phys. Rev. Lett. 51:1433 (1983).Google Scholar
  56. 56.
    J. Adler,J. Phys. A 16:L515 (1983).Google Scholar
  57. 57.
    Z. V. Djordjevic, I. Majid, H. E. Stanley, and R. J. dos Santos,J. Phys. A 16:L519 (1983).Google Scholar
  58. 58.
    V. Privman,Physica 123A:428 (1984).Google Scholar
  59. 59.
    A. J. Guttmann,J. Phys. A 17:455 (1984).Google Scholar
  60. 60.
    D. C. Rapaport,J. Phys. A 18:L201 (1985).Google Scholar
  61. 61.
    D. C. Rapaport,J. Phys. A 18:113 (1985).Google Scholar
  62. 62.
    S. Havlin and D. Ben-Avraham,Phys. Rev. A 27:2759 (1983).Google Scholar
  63. 63.
    D. C. Rapaport,J. Phys. A 18:L39 (1985).Google Scholar
  64. 64.
    J. W. Lyklema and K. Kremer,Phys. Rev. B 31:3182 (1985).Google Scholar
  65. 65.
    F. T. Wall and J. J. Erpenbeck,J. Chem. Phys. 30:637 (1959).Google Scholar
  66. 66.
    F. Mandel,J. Chem. Phys. 70:3984 (1979).Google Scholar
  67. 67.
    F. T. Wall and J. J. Erpenbeck,J. Chem. Phys. 30:634 (1959).Google Scholar
  68. 68.
    A. K. Kron,Vysokomol. Soyed. 7:1228 (1965) [Polymer Sci. USSR 7:1361 (1965)].Google Scholar
  69. 69.
    A. K. Kronet al., Molek. Biol. 1:576 (1967) [Molec. Biol. 1:487 (1967)].Google Scholar
  70. 70.
    F. T. Wall and F. Mandel,J. Chem. Phys. 63:4592 (1975).Google Scholar
  71. 71.
    N. Madras and A. D. Sokal,J. Stat. Phys. 47:573 (1987).Google Scholar
  72. 72.
    E. Brézin, J.-C. LeGuillou, and J. Zinn-Justin, inPhase Transitions and Critical Phenomena, Vol. 6, C. Domb and M. S. Green, eds. (Academic Press, London, 1976).Google Scholar
  73. 73.
    R. M. Karp and M. Luby, in24th Annual Symposium on Foundations of Computer Science (IEEE, New York, 1983), pp. 56–64.Google Scholar
  74. 74.
    R. M. Karp, M. Luby, and N. Madras, Monte-Carlo Approximation Algorithms for Enumeration Problems, submitted toJ. Algorithms.Google Scholar
  75. 75.
    A. Birnbaum and W. C. Healy Jr.,Ann. Math. Stat. 31:662 (1960).Google Scholar
  76. 76.
    S. Caracciolo and A. D. Sokal,J. Phys. A 19:L797 (1986).Google Scholar
  77. 77.
    A. D. Sokal and L. E. Thomas, in preparation.Google Scholar
  78. 78.
    S. Caracciolo, U. Glaus, and A. D. Sokal, in preparation.Google Scholar
  79. 79.
    S. Caracciolo and A. D. Sokal,J. Phys. A 20:2569 (1987).Google Scholar
  80. 80.
    D. S. Gaunt and A. J. Guttmann, inPhase Transitions and Critical Phenomena, Vol. 3, C. Domb and M. S. Green, eds. (Academic Press, London, 1974).Google Scholar
  81. 81.
    A. J. Guttmann, in preparation, to appear inPhase Transitions and Critical Phenomena, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York).Google Scholar
  82. 82.
    I. C. Enting and A. J. Guttmann,J. Phys. A 18:1007 (1985).Google Scholar
  83. 83.
    M. B. Priestley,Spectral Analysis and Time Series (Academic Press, London, 1981).Google Scholar
  84. 84.
    T. W. Anderson,The Statistical Analysis of Time Series (Wiley, New York, 1971).Google Scholar
  85. 85.
    J. Goodman and A. D. Sokal,Phys. Rev. Lett. 56:1015 (1986).Google Scholar
  86. 86.
    M. Benhamou and G. Mahoux,J. Physique Lett. 46:L-689 (1985).Google Scholar
  87. 87.
    T. A. Witten and L. Schäfer,J. Phys. A 11:1843 (1978).Google Scholar
  88. 88.
    J. des Cloizeaux,J. Physique 42:635 (1981).Google Scholar
  89. 89.
    M. K. Kosmas,J. Phys. A 14:2779 (1981).Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • Neal Madras
    • 1
  • Alan D. Sokal
    • 2
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Department of PhysicsNew York UniversityNew York

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