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Kinematics of multigrid Monte Carlo

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Abstract

We study the kinematics of multigrid Monte Carlo algorithms by means of acceptance rates for nonlocal Metropolis update proposals. An approximation formula for acceptance rates is derived. We present a comparison of different coarse-to-fine interpolation schemes in free field theory, where the formula is exact. The predictions of the approximation formula for several interacting models are well confirmed by Monte Carlo simulations. The following rule is found: For a critical model with fundamental Hamiltonianþ(Φ), the absence of critical slowing down can only be expected if the expansion of 〈þ(Φ +ψ)〉 in terms of the shift ψ contains no relevant (mass) term. We also introduce a multigrid update procedure for non-abelian lattice gauge theory and study the acceptance rates for gauge groupSU(2) in four dimensions.

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References

  1. K. Binder, ed.,Monte Carlo Methods in Statistical Physics (Springer, Berlin, 1986).

    Google Scholar 

  2. C. Rebbi, ed.,Lattice Gauge Theory and Monte Carlo Simulations (World Scientific, Singapore, 1983).

    Google Scholar 

  3. S. L. Adler,Phys. Rev. D 23:458 (1988); H. Neuberger,Phys. Rev. Lett. 59:1877 (1987).

    Google Scholar 

  4. U. Wolff, inProceedings of the 31. IUKT, Schladming, Austria, February 1992, H. Gausterer and C. B. Lang, eds. (Springer, Berlin, 1992).

    Google Scholar 

  5. A. D. Kennedy and B. Pendleton, in Lattice '90 (Proceedings of the International Symposium, Tallahassee, Florida, 1990), U. M. Heller, A. D. Kennedy, and S. Sanielevici, eds.,Nucl. Phys. B (Proc. Suppl.) 20:118 (1991); S.Gupta, in Lattice'91 (Proceedings of the International Symposium, Tsukuba, Japan), M. Fukugitaet al., eds.,Nucl. Phys. B (Proc. Suppl.) 26:617 (1992).

  6. R. H. Swendsen and J. S. Wang,Phys. Rev. Lett. 58:86 (1987); U. Wolff,Phys. Rev. Lett. 62:361 (1989).

    Google Scholar 

  7. G. Parisi, inProgress in Gauge Field Theory (Proceedings of the Cargèse Summer Institute, Cargèse, France, 1983), G. 'tHooftet al., eds. (Plenum Press, New York, 1984).

    Google Scholar 

  8. J. Goodman and A. D. Sokal,Phys. Rev. Lett. 56:1015 (1986).

    Google Scholar 

  9. G. Mack, inNonperturbative Quantum Field Theory (Proceedings, Cargèse, France, 1987), G. 't Hooftet al., eds. (Plenum Press, New York, 1988).

    Google Scholar 

  10. U. Wolff, inWorkshop on Fermion Algorithms (HLRZ, KFA Jülich, April 1992), H. J. Herrmann and F. Karsch, eds. (World Scientific, Singapore, 1991).

    Google Scholar 

  11. M. Hasenbusch and S. Meyer,Phys. Rev. Lett. 68:435 (1992).

    Google Scholar 

  12. M. Hasenbusch, G. Mack, and S. Meyer, in Lattice '90 (Proceedings of the International Symposium, Tallahassee, Florida, 1990), U. M. Heller, A. D. Kennedy, and S. Sanielevici, eds.,Nucl. Phys. B (Proc. Suppl.) 20:110 (1991).

  13. M. L. Laursen, J. Smit, and J. C. Vink,Phys. Lett. B 262:467 (1991).

    Google Scholar 

  14. R. G. Edwards, S. J. Ferreira, J. Goodman, and A. D. Sokal,Nucl. Phys. B 380:621 (1992).

    Google Scholar 

  15. J. Goodman and A. D. Sokal,Phys. Rev. D 40:2035 (1989).

    Google Scholar 

  16. J. Linn, Diploma thesis, Kaiserslautern (1991).

  17. M. Grabenstein and K. Pinn,Phys. Rev. D 45:4372 (1992).

    Google Scholar 

  18. M. Creutz,Phys. Rev. D 36:515 (1987); F. R. Brown and T. J. Woch,Phys. Rev. Lett. 58:2394 (1987).

    Google Scholar 

  19. S. L. Adler, in Lattice '88 (Proceedings of the International Symposium, Batavia, Illinois, 1988), A. S. Kronfeld and P. B. Mackenzie, eds.,Nucl. Phys. B (Proc. Suppl.) 9:437 (1989).

  20. R. Ben-Av, H. G. Evertz, M. Marcu, and S. Solomon,Phys. Rev. D 44:2953 (1991).

    Google Scholar 

  21. R. Sinclair,Phys. Rev. D 45:2098 (1992).

    Google Scholar 

  22. M. L. Laursen and J. C. Vink, preprint HLRZ-92-14 (March 1992).

  23. G. Bhanot and S. L. Adler,Phys. Rev. Lett. 66:1806 (1991); G. Bhanot, S. L. Adler, T. Lippert, K. Schilling, and P. Ueberholz,Nucl. Phys. B 368:745 (1992).

    Google Scholar 

  24. A. Brandt, in Lattice'91 (Proceedings of the International Symposium, Tsukuba, Japan), M. Fukugitaet al., eds.,Nucl. Phys. B (Proc. Suppl.) 26:137 (1992).

  25. K. Stüben and U. Trottenberg, inMultigrid Methods (Proceedings, Köln-Porz, 1981), W. Hackbuschet al., eds. (Springer, Berlin, 1982).

    Google Scholar 

  26. S. Gupta, A. Irbäck, F. Karsch, and B. Petersson,Phys. Lett. B 242:437 (1990).

    Google Scholar 

  27. K. Gawedzki and A. Kupiainen,Commun. Math. Phys. 77:31 (1980);99:197 (1985), and references cited therein.

    Google Scholar 

  28. G. Mack and S. Meyer, in Lattice '89 (Proceedings of the International Symposium, Capri, Italy, 1989), R. Petronzioet al., eds.,Nucl. Phys. B (Proc. Suppl.) 17:293 (1990).

  29. T. Kalkreuter,Nucl. Phys. B 376:637 (1992).

    Google Scholar 

  30. G. Mack, T. Kalkreuter, G. Palma, and M. Speh, inProceedings of the 31. IUKT, Schladming, Austria, February 1992, H. Gausterer and C. B. Lang, eds. (Springer, Berlin, 1992).

    Google Scholar 

  31. M. Hasenbusch, M. Marcu, and K. Pinn, in Lattice'91 (Proceedings of the International Symposium, Tsukuba, Japan), M. Fukugitaet al., eds.,Nucl. Phys. B (Proc. Suppl.) 26:598 (1992).

    Google Scholar 

  32. R. G. Edwards, J. Goodman, and A. D. Sokal,Nucl. Phys. B 354:289 (1991); A. Hulsebos, J. Smit, and J. C. Vink,Nucl. Phys. B 356:775 (1991).

    Google Scholar 

  33. J. E. Mandula and M. Ogilvie,Phys. Lett. B 248:156 (1990); A. Hulsebos, M. L. Laursen, J. Smit, and A. J. van der Sijs, in Lattice '90 (Proceedings of the International Symposium, Tallahassee, Florida, 1990), U. M. Heller, A. D. Kennedy, and S. Sanielevici, eds.,Nucl. Phys. B (Proc. Suppl.) 20:110 (1991).

    Google Scholar 

  34. C. Michael and M. Teper,Phys. Lett. B 199:95 (1987).

    Google Scholar 

  35. A. S. Kronfeld, M. L. Laursen, G. Schierholz, and U.-J. Wiese,Phys. Lett. 198B:516 (1987).

    Google Scholar 

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Authors and Affiliations

  1. II. Institut für Theoretische Physik, Universität Hamburg, W-2000, Hamburg 50, Germany

    Martin Grabenstein

  2. Institut für Theoretische Physik I, Universität Münster, W-4400, Münster, Germany

    Klaus Pinn

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  1. Martin Grabenstein
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  2. Klaus Pinn
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Grabenstein, M., Pinn, K. Kinematics of multigrid Monte Carlo. J Stat Phys 71, 607–640 (1993). https://doi.org/10.1007/BF01058439

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  • DOI: https://doi.org/10.1007/BF01058439

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