Skip to main content
SpringerLink
Log in

Simultaneous analysis of three-dimensional percolation models

  • Research Article
  • Published:
Frontiers of Physics Aims and scope Submit manuscript

Abstract

We simulate the bond and site percolation models on several three-dimensional lattices, including the diamond, body-centered cubic, and face-centered cubic lattices. As on the simple-cubic lattice [Phys. Rev. E, 2013, 87(5): 052107], it is observed that in comparison with dimensionless ratios based on cluster-size distribution, certain wrapping probabilities exhibit weaker finite-size corrections and are more sensitive to the deviation from percolation threshold p c , and thus provide a powerful means for determining p c . We analyze the numerical data of the wrapping probabilities simultaneously such that universal parameters are shared by the aforementioned models, and thus significantly improved estimates of p c are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. R. Broadbent and J. M. Hammersley, Percolation processes, Proc. Camb. Philos. Soc, 1957, 53(03): 629

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd Ed., London: Taylor & Francis, 1994

    Google Scholar 

  3. G. R. Grimmett, Percolation, 2nd Ed., Berlin: Springer, 1999

    MATH  Google Scholar 

  4. B. Bollobás and O. Riordan, Percolation, Cambridge: Cambridge University Press, 2006

    Book  MATH  Google Scholar 

  5. B. Nienhuis, Coulomb gas formulation of two-dimensional phase transitions, in: Phase Transition and Critical Phenomena, Vol. 11, edited by C. Domb, M. Green, and J. L. Lebowitz, London: Academic Press, 1987

    Google Scholar 

  6. J. L. Cardy, Conformal invariance, in: Phase Transition and Critical Phenomena, Vol. 11, edited by C. Domb, M. Green, and J. L. Lebowitz, London: Academic Press, 1987

    Google Scholar 

  7. S. Smirnov and W. Werner, Critical exponents for twodimensional percolation, Math. Res. Lett., 2001, 8(6): 729

    Article  MATH  MathSciNet  Google Scholar 

  8. J. W. Essam, Percolation and cluster size, in: Phase Transition and Critical Phenomena, Vol. 2, edited by C. Domb and M. S. Green, New York: Academic Press, 1972

    Google Scholar 

  9. X. Feng, Y. Deng, and H. W. J. Blöte, Percolation transitions in two dimensions, Phys. Rev. E, 2008, 78(3): 031136, and references therein

    Article  ADS  Google Scholar 

  10. C. Ding, Z. Fu, W. Guo, and F. Y. Wu, Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices (II): Numerical analysis, Phys. Rev. E, 2010, 81(6): 061111, and references therein

    Article  ADS  Google Scholar 

  11. G. Toulouse, Perspectives from the theory of phase transitions, Nuovo Cimento Soc. Ital. Fis. B, 1974, 23(1): 234

    Article  ADS  Google Scholar 

  12. M. Aizenman and C. M. Newman, Tree graph inequalities and critical behavior in percolation models, J. Stat. Phys., 1984, 36(1–2): 107

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions, Commun. Math. Phys., 1990, 128(2): 333

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. J. Wang, Z. Zhou, W. Zhang, T. M. Garoni, and Y. Deng, Bond and site percolation in three dimensions, Phys. Rev. E, 2013, 87(5): 052107

    Article  ADS  Google Scholar 

  15. Y. Deng and H. W. J. Blöte, Simultaneous analysis of several models in the three-dimensional Ising universality class, Phys. Rev. E, 2003, 68(3): 036125

    Article  ADS  Google Scholar 

  16. Y. Deng and H. W. J. Blöte, Monte Carlo study of the sitepercolation model in two and three dimensions, Phys. Rev. E, 2005, 72(1): 016126

    Article  ADS  Google Scholar 

  17. S. C. van der Marck, Calculation of percolation thresholds in high dimensions for FCC, BCC and diamond lattices, Int. J. Mod. Phys. C, 1998, 09(04): 529

    Article  ADS  Google Scholar 

  18. A. Silverman and J. Adler, Site-percolation threshold for a diamond lattice with diatomic substitution, Phys. Rev. B, 1990, 42(2): 1369

    Article  ADS  Google Scholar 

  19. V. A. Vyssotsky, S. B. Gordon, H. L. Frisch, and J. M. Hammersley, Critical percolation probabilities (Bond problem), Phys. Rev., 1961, 123: 1566

    Article  ADS  MathSciNet  Google Scholar 

  20. C. D. Lorenz and R. M. Ziff, Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation, J. Phys. A, 1998, 31(40): 8147

    Article  ADS  MATH  Google Scholar 

  21. C. D. Lorenz and R. M. Ziff, Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices, Phys. Rev. E, 1998, 57(1): 230

    Article  ADS  Google Scholar 

  22. R. M. Bradley, P. N. Strenski, and J. M. Debierre, Surfaces of percolation clusters in three dimensions, Phys. Rev. B, 1991, 44(1): 76

    Article  ADS  Google Scholar 

  23. D. S. Gaunt and M. F. Sykes, Series study of random percolation in three dimensions, J. Phys. A, 1983, 16(4): 783

    Article  ADS  Google Scholar 

  24. R. M. Ziff, S. R. Finch, and V. S. Adamchik, Universality of finite-size corrections to the number of critical percolation clusters, Phys. Rev. Lett., 1997, 79(18): 3447

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, 230026, China

    Xiao Xu, Junfeng Wang & Youjin Deng

  2. School of Electronic Science and Applied Physics, Hefei University of Technology, Hefei, 230009, China

    Junfeng Wang

  3. Department of Physics, China University of Mining and Technology, Xuzhou, 221116, China

    Jian-Ping Lv

Authors
  1. Xiao Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junfeng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jian-Ping Lv
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Youjin Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Jian-Ping Lv or Youjin Deng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, X., Wang, J., Lv, JP. et al. Simultaneous analysis of three-dimensional percolation models. Front. Phys. 9, 113–119 (2014). https://doi.org/10.1007/s11467-013-0403-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11467-013-0403-z

Keywords

Search

Navigation