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New Monte Carlo method for the self-avoiding walk

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Abstract

We introduce a new Monte Carlo algorithm for the self-avoiding walk (SAW), and show that it is particularly efficient in the critical region (long chains). We also introduce new and more efficient statistical techniques. We employ these methods to extract numerical estimates for the critical parameters of the SAW on the square lattice. We findμ=2.63820 ± 0.00004 ± 0.00030γ=1.352 ± 0.006 ± 0.025νv=0.7590 ± 0.0062 ± 0.0042 where the first error bar represents systematic error due to corrections to scaling (subjective 95% confidence limits) and the second bar represents statistical error (classical 95% confidence limits). These results are based on SAWs of average length ≈ 166, using 340 hours CPU time on a CDC Cyber 170–730. We compare our results to previous work and indicate some directions for future research.

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References

  1. W. Kuhn and H. Kuhn,Helv. Chim. Acta 26:1394 (1943).

    Google Scholar 

  2. W. J. C. Orr,Trans. Faraday Soc. 43:12 (1947).

    Google Scholar 

  3. E. W. Montroll,J. Chem. Phys. 18:734 (1950).

    Google Scholar 

  4. C. Domb,Adv. Chem. Phys. 15:229 (1969).

    Google Scholar 

  5. D. S. McKenzie,Phys. Rep. 27:35 (1976).

    Google Scholar 

  6. S. G. Whittington,Adv. Chem. Phys. 51:1 (1982).

    Google Scholar 

  7. P. G. deGennes,Phys. Lett. 38A:339 (1972).

    Google Scholar 

  8. J. des Cloizeaux,J. Phys. (Paris) 36:281 (1975).

    Google Scholar 

  9. M. Daoudet al., Macromolecules 8:804 (1975).

    Google Scholar 

  10. V. J. Emery,Phys. Rev. B11:239 (1975).

    Google Scholar 

  11. C. Aragao de Carvalho, S. Caracciolo, and J. Fröhlich,Nucl. Phys. B215[FS7]:209 (1983).

    Google Scholar 

  12. A. K. Kron,Vysokomol. Soyed. 7:1228 (1965) [Polymer Sci. USSR 7:1361 (1965)].

    Google Scholar 

  13. A. K. Kronet al., Molek. Biol. 1:576 (1967) [Molec. Biol 1:487 (1967)].

    Google Scholar 

  14. F. T. Wall and F. Mandel,J. Chem. Phys. 63:4592 (1975).

    Google Scholar 

  15. F. Mandel,J. Chem. Phys. 70:3984 (1979).

    Google Scholar 

  16. B. Berg and D. Foerster,Phys. Lett. 106B:323 (1981); see also J. Fröhlich, inRecent Developments in Gauge Theories (1979 Cargèse lectures), G. 't Hooftet al., eds. (Plenum Press, New York, 1980), p. 65.

    Google Scholar 

  17. C. Aragão de Carvalho and S. Caracciolo,J. Phys. (Paris) 44:323 (1983).

    Google Scholar 

  18. P. H. Verdier,J. Chem. Phys. 52:5512 (1970).

    Google Scholar 

  19. A. D. Sokal, Comparative analysis of Monte Carlo methods for the self-avoiding walk (in preparation).

  20. S. Redner and P. J. Reynolds,J. Phys. A14:2679 (1981); and S. Redner, private communication.

    Google Scholar 

  21. W. W. Wood and J. J. Erpenbeck,Ann. Rev. Phys. Chem. 27:319 (1976).

    Google Scholar 

  22. H. E. Stanley,Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1971).

    Google Scholar 

  23. P. Erdös and S. J. Taylor,Acta Math. Acad. Sci. Hung. 11: 231 (1960).

    Google Scholar 

  24. G. F. Lawler,Commun. Math. Phys. 86:539 (1982).

    Google Scholar 

  25. G. Felder and J. Fröhlich,Commun. Math. Phys. 97:111 (1985).

    Google Scholar 

  26. M. Aizenman,Commun. Math. Phys. 97: 91 (1985).

    Google Scholar 

  27. D. C. Brydges, J. Fröhlich and A. D. Sokal (unpublished).

  28. J. M. Hammersley,Proc. Cambridge Philos. Soc. 53:642 (1957).

    Google Scholar 

  29. J. M. Hammersley,Proc. Cambridge Philos. Soc. 57:516 (1961).

    Google Scholar 

  30. H. Kesten,J. Math. Phys. 4:960 (1963).

    Google Scholar 

  31. H. Hesten,J. Math. Phys. 5:1128 (1964).

    Google Scholar 

  32. A. J. Guttmann,J. Phys. A16:2233 (1983).

    Google Scholar 

  33. M. Aizenman,Commun. Math. Phys. 86:1 (1982).

    Google Scholar 

  34. A. D. Sokal, unpublished.

  35. J. Fröhlich,Nucl. Phys. B200[FS4]:281 (1982).

    Google Scholar 

  36. M. Aizenman and R. Graham,Nucl. Phys. B225[FS9]:261 (1983).

    Google Scholar 

  37. J. des Cloizeaux, private communication cited by E. Brézin, inOrder and Fluctuations in Equilibrium and Nonequilibrium Statistical Mechanics (17th Solvay Conference, 1978), G. Nicolis, G. Dewel, and J. W. Turner, eds. (Wiley-Interscience, New York, 1981).

    Google Scholar 

  38. B. Mandelbrot,Fractals: Form, Chance and Dimension (Freeman, San Francisco, 1977).

    Google Scholar 

  39. D. Brydges and T. Spencer,Commun. Math. Phys. 97: 125 (1985).

    Google Scholar 

  40. K. L. Chung,Markov Chains with Stationary Transition Probabilities, 2nd ed. (Springer-Verlag, New York, 1967).

    Google Scholar 

  41. F. T. Wall and J. J. Erpenbeck,J. Chem. Phys. 30:634 (1959).

    Google Scholar 

  42. M. N. Barber, inPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1983).

    Google Scholar 

  43. G. F. Mazenko and O. T. Valls,Phys. Rev. B24:1419 (1981).

    Google Scholar 

  44. R. Bauschet al., Phys. Rev. Lett. 47:1837 (1981).

    Google Scholar 

  45. R. Friedberg and J. E. Cameron,J. Chem. Phys. 52:6049 (1970).

    Google Scholar 

  46. L. Jacobs and C. Rebbi,J. Comput. Phys. 41:203 (1981).

    Google Scholar 

  47. C. Kalle and V. Winkelmann,J. Stat. Phys. 28:639 (1982); S. Wansleben, J. G. Zabolitzky, and C. Kalle,J. Stat. Phys. 37:271 (1984).

    Google Scholar 

  48. G. O. Williams and M. H. Kalos,J. Stat. Phys. 37:283 (1984).

    Google Scholar 

  49. M. H. Kalos, in Proceedings of the Brookhaven Conference on Monte Carlo Methods and Future Computer Architectures, May 1983 (unpublished).

  50. K. E. Schmidt,Phys. Rev. Lett. 51:2175 (1983).

    Google Scholar 

  51. M. Creutz,Phys. Rev. Lett. 50:1411 (1983).

    Google Scholar 

  52. S. D. Silvey,Statistical Inference (Chapman and Hall, London, 1975).

    Google Scholar 

  53. H. Cramér,Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946).

    Google Scholar 

  54. M. Fisz,Probability Theory and Mathematical Statistics (Wiley, New York, 1963).

    Google Scholar 

  55. S. S. Wilks,Mathematical Statistics (Wiley, New York, 1962).

    Google Scholar 

  56. B. W. Lindgren,Statistical Theory, 3rd ed. (Macmillan, New York, 1976).

    Google Scholar 

  57. M. Kendall and A. Stuart,The Advanced Theory of Statistics, Vols. 1 and 2, 4th ed. (Macmillan, New York, 1977, 1979).

    Google Scholar 

  58. M. B. Priestley,Spectral Analysis and Time Series, 2 vols. (Academic Press, London, 1981).

    Google Scholar 

  59. T. W. Anderson,The Statistical Analysis of Time Series (Wiley, New York, 1971).

    Google Scholar 

  60. G. M. Jenkins and D. G. Watts,Spectral Analysis and Its Applications (Holden-Day, San Francisco, 1968).

    Google Scholar 

  61. H. Müller-Krumbhaar and K. Binder,J. Stat. Phys. 8:1 (1973).

    Google Scholar 

  62. K. Binder, inMonte Carlo Methods in Statistical Physics, K. Binder, ed. (Springer-Verlag, Berlin, 1979), Chapter 1.

    Google Scholar 

  63. R. Zwanzig and N. K. Ailawadi,Phys. Rev. 182:280 (1969).

    Google Scholar 

  64. G. J. Daniell, A. J. G. Hey, and J. E. Mandula,Phys. Rev. D 30:2230 (1984).

    Google Scholar 

  65. IMSL Library Reference Manual, Edition 9 (IMSL, Houston, 1982), Chapter F.

  66. NAG FortranLibrary Manual, Mark 10 (Numerical Algorithms Group, Oxford, 1983), Chapter G13.

  67. J. G. Kemeny and L. J. Snell,Finite Markov Chains (Van Nostrand, Princeton, New Jersey, 1960).

    Google Scholar 

  68. F. J. Wegner,Phys. Rev. B 5:4529 (1972).

    Google Scholar 

  69. 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 

  70. Y. Ogata,J. Appl. Prob. 17:59 (1980).

    Google Scholar 

  71. A. Azzalini,Biometrika 70:381 (1983).

    Google Scholar 

  72. M. F. Sykeset al., J. Phys. A5:653 (1972).

    Google Scholar 

  73. D. E. Knuth,The Art of Computer Programming, Vol. 2, 2nd ed. (Addison-Wesley, Reading, Massachusetts, 1981).

    Google Scholar 

  74. M. H. Kalos, private communication.

  75. A. J. Guttmann,J. Phys. A17:455 (1984).

    Google Scholar 

  76. B. Derrida,J. Phys. A14:L5 (1981).

    Google Scholar 

  77. J. Adler,J. Phys. A16:L515 (1983).

    Google Scholar 

  78. M. Kolb, R. Jullien, and P. Pfeuty, J. Phys.A15:3799 (1982).

    Google Scholar 

  79. B. Nienhuis,Phys. Rev. Lett. 49:1062 (1982).

    Google Scholar 

  80. B. Nienhuis,J. Stat. Phys. 34:731 (1984).

    Google Scholar 

  81. P. Grassberger,Z. Phys. B48:255 (1982).

    Google Scholar 

  82. I. Majid, Z. V. Djordjevic, and H. E. Stanley,Phys. Rev. Lett. 51:143 (1983).

    Google Scholar 

  83. Z. V. Djordjevicet al., J. Phys. A16:L519 (1983).

    Google Scholar 

  84. K. Kremer, A. Baumgärtner, and K. Binder,Z. Phys. B40:331 (1981).

    Google Scholar 

  85. S. Havlin and D. Ben-Avraham,Phys. Rev. A 27:2759 (1983).

    Google Scholar 

  86. I. G. Enting and A. J. Guttmann,J. Phys. A (1985), to appear.

  87. B. G. Nickel, inPhase Transitions (1980 Cargèse lectures), M. Lévy, J.-C. LeGuillou, and J. Zinn-Justin, eds. (Plenum Press, New York, 1982).

    Google Scholar 

  88. B. Derrida and L. De Seze,J. Phys. (Paris) 43:475 (1982), Appendix A.

    Google Scholar 

  89. V. Privman and M. E. Fisher,J. Phys. A16:L295 (1983).

    Google Scholar 

  90. V. Privman,Physica 123A:428 (1984).

    Google Scholar 

  91. H. E. Stanleyet al., inReal-Space Renormalization, T. W. Burkhardt and J. M. J. van Leeuwen, eds. (Springer-Verlag, Berlin, 1982), Chapter 7.

    Google Scholar 

  92. S. Havlin and D. Ben-Avraham,Phys. Rev. A 26:1728 (1982).

    Google Scholar 

  93. D. C. Rapaport,J. Phys. A18:L39 (1985);J. Phys. A18:113 (1985);Phys. Rev. 530:2906 (1984).

    Google Scholar 

  94. B. Shapiro,J. Phys. C11:2829 (1978).

    Google Scholar 

  95. M. Napiórkowski, E. H. Hauge, and P. C. Hemmer,Phys. Lett. 72A:193 (1979).

    Google Scholar 

  96. A. Malakis,Physica 104A:427 (1980).

    Google Scholar 

  97. S. L. A. de Quieroz and C. M. Chaves,Z. Phys. B40:99 (1980).

    Google Scholar 

  98. F. Family,J. Phys. A13:L325 (1980).

    Google Scholar 

  99. K. Kremer and M. N. Barber,J. Phys. A17:L215 (1984).

    Google Scholar 

  100. R. H. Swendsen, inReal-Space Renormalization, T. W. Burkhardt and J. M. J. van Leeuwen, eds. (Springer-Verlag, Berlin, 1982), Chapter 3.

    Google Scholar 

  101. A. J. Guttmann and M. F. Sykes,Austral. J. Phys. 26:207 (1973).

    Google Scholar 

  102. A. J. Guttmann, T. Osborn, and A. D. Sokal, in preparation.

  103. L. R. Shenton,Proc. Cambridge Philos. Soc. 51:442 (1955).

    Google Scholar 

  104. D. Revuz,Markov Chains (North-Holland, Amsterdam, 1975).

    Google Scholar 

  105. Z. Šidák,Czechoslovak Math. J. 14:438 (1964).

    Google Scholar 

  106. D. Vere-Jones,Pacific J. Math. 22:361 (1967).

    Google Scholar 

  107. D. Vere-Jones,Pacific J. Math. 26:601 (1968).

    Google Scholar 

  108. E. Nelson,Duke Math. J. 25:671 (1958).

    Google Scholar 

  109. D. G. Kendall, inProbability and Statistics (Cramér Memorial Volume), U. Grenader, ed. (Amlqvist and Wiskell, Stockholm/New York, 1959).

    Google Scholar 

  110. D. G. Kendall,Proc. London Math. Soc. 9:417 (1959).

    Google Scholar 

  111. E. Seneta,Non-Negative Matrices and Markov Chains, 2nd ed. (Springer-Verlag, New York, 1981), pp. 195, 218.

    Google Scholar 

  112. D. E. Knuth,The Art of Computer Programming, Vol. 3 (Addison-Wesley, Reading, Massachusetts, 1973).

    Google Scholar 

  113. F. L. McCrackin,J. Chem. Phys. 47:1980 (1967).

    Google Scholar 

  114. J. Mazur and F. L. McCrackin,J. Chem. Phys. 49:648 (1968).

    Google Scholar 

  115. P. H. Verdier,J. Comput. Phys. 4:204 (1969).

    Google Scholar 

  116. P. C. Jurs and J. E. Reissner,J. Chem. Phys. 55:4948 (1971).

    Google Scholar 

  117. R. Grishman,J. Chem. Phys. 58:220 (1973).

    Google Scholar 

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, 10012, New York, New York

    Alan D. Sokal

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  1. Alberto Berretti
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Berretti, A., Sokal, A.D. New Monte Carlo method for the self-avoiding walk. J Stat Phys 40, 483–531 (1985). https://doi.org/10.1007/BF01017183

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