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Monte Carlo study of the ∵-point for collapsing trees

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Abstract

We investigate the collapse transition of lattice trees with nearest neighbor attraction in two and three dimensions. Two methods are used: (1) A stochastic optimization process of the Robbins-Monro type, which is designed solely to locate the maximum value of the specific heat; and (2) umbrella sampling, which is designed to sample data over a wide temperature range, as well as to combat the quasiergodicity of Metropolis algorithms in the collapsed phase. We find good evidence that the transition is second order with a divergent specific heat, and that the divergence of the specific heat coincides with the metric collapse.

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References

  1. P. J. Flory,Principles of Polymer Chemistry (Cornell University Press, Ithaca, 1953).

    Google Scholar 

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

    Google Scholar 

  3. J. Mazur and D. McIntyre,Macromolecules 8:464 (1975).

    Google Scholar 

  4. B. Derrida and H. J. Herrmann,J. Phys. (Paris) 44:1365 (1983).

    Google Scholar 

  5. R. Dickman and W. C. Schieve,J. Phys. 45:1727 (1984).

    Google Scholar 

  6. R. Dickman and W. C. Schieve,J. Stat. Phys. 44:465 (1986).

    Google Scholar 

  7. P. M. Lam,Phys. Rev. B 36:6988 (1987).

    Google Scholar 

  8. P. M. Lam,Phys. Rev. B 38:2813 (1988).

    Google Scholar 

  9. I. S. Chang and Y. Shapir,Phys. Rev. B 38:6736 (1988).

    Google Scholar 

  10. D. S. Gaunt and S. Flesia,Physica A 168:602 (1990).

    Google Scholar 

  11. D. S. Gaunt and S. Flesia,J. Phys. A: Math. Gen. 24:3655 (1991).

    Google Scholar 

  12. S. Flesia and D. S. Gaunt,J. Phys. A: Math. Gen. 25:2127 (1992).

    Google Scholar 

  13. N. Madras, C. E. Soteros, S. G. Whittington, J. L. Martin, M. F. Sykes, S. Flesia, and D. S. Gaunt,J. Phys. A: Math. Gen. 23:5327 (1990).

    Google Scholar 

  14. D. C. Rapaport,J. Phys. A: Math. Gen. 10:637 (1977).

    Google Scholar 

  15. H. A. Lim, A. Guha, and Y. Shapir,Phys. Rev. A 38:371 (1988).

    Google Scholar 

  16. E. J. Janse van Rensburg and N. Madras,J. Phys. A: Math. Gen. 25:303 (1992).

    Google Scholar 

  17. G. M. Torrie and J. P. Valleau,J. Comput. Phys. 23:187 (1977).

    Google Scholar 

  18. S. G. Whittington, C. E. Soteros, and N. Madras,J. Math. Chem. 7:87 (1991).

    Google Scholar 

  19. S. Flesia, D. S. Gaunt, C. E. Soteros, and S. G. Whittington,J. Phys. A: Math. Gen. 26:3515 (1992).

    Google Scholar 

  20. S. Flesia, D. S. Gaunt, C. E. Soteros, and S. G. Whittington,J. Phys. A: Math. Gen. 26:L993 (1992).

    Google Scholar 

  21. R. Brak, A. Guttmann, and S. G. Whittington,J. Phys. A: Math. Gen. 23:4581 (1992).

    Google Scholar 

  22. A. L. Owczarek,J. Stat. Phys. 72:737 (1993).

    Google Scholar 

  23. D. Dhar,Phys. Rev. Lett. 51:853 (1983).

    Google Scholar 

  24. D. Dhar,Physica A 140:210 (1986).

    Google Scholar 

  25. K. De'Bell and S. G. Whittington, Unpublished (1992).

  26. R. Brak, Unpublished (1994).

  27. H. Robbins and S. Monro,Ann. Math. Stat. 22:400 (1951).

    Google Scholar 

  28. G. Kersting,Ann. Prob. 5:954 (1977).

    Google Scholar 

  29. P. W. Glynn, InProceedings of the 1986 Winter Simulation Conference, J. Wilson, J. Henriksen, and S. Roberts, eds., p. 52.

  30. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller,J. Chem. Phys. 21:1087 (1953).

    Google Scholar 

  31. J. M. Hammersley and D. C. Handscomb,Monte Carlo Methods (Chapman and Hall, London, 1964).

    Google Scholar 

  32. J. P. Valleau,J. Comput. Phys. 96:193 (1991).

    Google Scholar 

  33. A. M. Ferrenberg and R. H. Swendsen,Phys. Rev. Lett. 61:2635 (1988).

    Google Scholar 

  34. A. M. Ferrenberg and R. H. Swendsen,Phys. Rev. Lett. 63:1195 (1989).

    Google Scholar 

  35. C. J. Geyer and E. A. Thompson,J. Royal Stat. Soc. B 54:657 (1992).

    Google Scholar 

  36. E. Marinari and G. Parisi,Europhys. Lett. 19:451 (1992).

    Google Scholar 

  37. N. Madras and M. Piccioni, Preprint (1994).

  38. C. J. Geyer and E. A. Thompson,J. Am. Stat. Assoc. 90:909 (1995).

    Google Scholar 

  39. B. A. Berg and T. Neuhaus,Phys. Lett. B 267:249 (1991).

    Google Scholar 

  40. C. J. Geyer, Preprint (1993).

  41. G. Parisi and N. Sourlas,Phys. Rev. Lett. 46:871 (1981).

    Google Scholar 

  42. C. Vanderzande,Phys. Rev. Lett. 70:3595 (1993).

    Google Scholar 

  43. S. Flesia, D. S. Gaunt, C. E. Soteros, and S. G. Whittington,J. Phys. A: Math. Gen. 27:5831 (1994).

    Google Scholar 

  44. S. G. Whittington, Private communication (1994).

  45. F. Seno and C. Vanderzande, Preprint (1995).

  46. A. L. Stella, E. Orlandini, I. Beichl, F. Sullivan, M. C. Tesi, and T. L. Einstein,Phys. Rev. Lett. 69:3650 (1992).

    Google Scholar 

  47. N. Madras and A. D. Sokal,J. Stat. Phys. 50:109 (1988).

    Google Scholar 

  48. D. R. Cox and D. V. Hinkley,Theoretical Statistics (Chapman and Hall, London, 1974).

    Google Scholar 

  49. C. J. Geyer,J. R. Stat. Soc. B 56:261 (1994).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, M3J 1P3, North York, Ontario, Canada

    N. Madras & E. J. Janse van Rensburg

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  1. N. Madras
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  2. E. J. Janse van Rensburg
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Madras, N., Janse van Rensburg, E.J. Monte Carlo study of the ∵-point for collapsing trees. J Stat Phys 86, 1–36 (1997). https://doi.org/10.1007/BF02180197

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  • DOI: https://doi.org/10.1007/BF02180197

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