Journal of Statistical Physics

, Volume 77, Issue 5–6, pp 955–976 | Cite as

The Markov property method applied to Ising model calculations

  • George A. BakerJr.


An efficient method of computation for models possessing the Markov property is set out. We apply this method to the two-dimensional ising model and report exact computations for up to 10 by 10 models with periodic boundary conditions. We find that critical-point, finite-size rounding is quite large in the renormalized coupling constant, which is not divergent at the critical point, in contrast to the energy, which is also not divergent and has no rounding there. The difference is traced to the continuity of the energy and the discontinuity of the renormalized coupling constant at the critical point.

Key Words

Markov property Ising model critical phenomena parallel computational procdures 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. A. Baker, Jr.,J. Math. Phys. 16:1324 (1975).Google Scholar
  2. 2.
    G. A. Baker, Jr.,Phys. Rev. B 15:1552 (1977).Google Scholar
  3. 3.
    G. A. Baker, Jr.,J. Stat. Phys. 72:621 (1993).Google Scholar
  4. 4.
    M. N. Barber, inPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1983), p. 145.Google Scholar
  5. 5.
    G. Bhanot and S. Sastry,J. Stat. Phys. 60:333 (1990).Google Scholar
  6. 6.
    K. Binder,J. Phys. B. 43:119 (1981).Google Scholar
  7. 7.
    E. Brézin,J. Phys. (Paris)43:15 (1982).Google Scholar
  8. 8.
    T. W. Burkhardt and B. Derrida,Phys. Rev. B. 32:7273 (1985).Google Scholar
  9. 9.
    J. L. Cardy, ed.,Finite-Size Scaling (North Hollnd, Amsterdam, 1988).Google Scholar
  10. 10.
    F. Cooper, B. Freedman, and D. Preston,Nucl. Phys. B 210[FS6]:210 (1982).Google Scholar
  11. 11.
    J. W. Essam and D. L. Hunter,J. Phys. C. 1:392 (1968).Google Scholar
  12. 12.
    A. E. Ferdinand, Lattice statistics of finite systems, Thesis, Cornell University (1967).Google Scholar
  13. 13.
    A. E. Ferdinand and M. E. Fisher,Phys. Rev. 185:832 (1969).Google Scholar
  14. 14.
    V. Privman,Finite Size Scaling and Numerical Simulation of Statistical Systems (World Scientific, Singapore, 1990).Google Scholar
  15. 15.
    D. Ruelle,Statistical Mechanics (Benjamin, New York, 1969).Google Scholar
  16. 16.
    M. F. Sykes, J. W. Essam, B. R. Heap, and B. J. Hiley,J. Math. Phys. 7:1557 (1966).Google Scholar
  17. 17.
    C. A. Tracy and B. M. McCoy,Phys. Rev. Lett. 31:1500 (1973).Google Scholar
  18. 18.
    R. B. Griffiths,J. Math. Phys. 8:484 (1967).Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • George A. BakerJr.
    • 1
  1. 1.Theoretical Division, Los Alamos National LaboratoryUniversity of CaliforniaLos Alamos

Personalised recommendations