Journal of Statistical Physics

, Volume 141, Issue 6, pp 909–939 | Cite as

Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings

  • Robert Shrock
  • Yan Xu


We present exact results on the partition function of the q-state Potts model on various families of graphs G in a generalized external magnetic field that favors or disfavors spin values in a subset I s ={1,…,s} of the total set of possible spin values, Z(G,q,s,v,w), where v and w are temperature- and field-dependent Boltzmann variables. We remark on differences in thermodynamic behavior between our model with a generalized external magnetic field and the Potts model with a conventional magnetic field that favors or disfavors a single spin value. Exact results are also given for the interesting special case of the zero-temperature Potts antiferromagnet, corresponding to a set-weighted chromatic polynomial Ph(G,q,s,w) that counts the number of colorings of the vertices of G subject to the condition that colors of adjacent vertices are different, with a weighting w that favors or disfavors colors in the interval I s . We derive powerful new upper and lower bounds on Z(G,q,s,v,w) for the ferromagnetic case in terms of zero-field Potts partition functions with certain transformed arguments. We also prove general inequalities for Z(G,q,s,v,w) on different families of tree graphs. As part of our analysis, we elucidate how the field-dependent Potts partition function and weighted-set chromatic polynomial distinguish, respectively, between Tutte-equivalent and chromatically equivalent pairs of graphs.


Potts model in an external field Weighted-set graph colorings 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chang, S.-C., Shrock, R.: J. Phys. A 42, 385004 (2009) CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Chang, S.-C., Shrock, R.: J. Stat. Phys. 137, 667 (2009) MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Chang, S.-C., Shrock, R.: J. Stat. Phys. 138, 496 (2010) MATHCrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Shrock, R., Xu, Y.: J. Stat. Phys. 139, 27 (2010) MATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Wu, F.Y.: J. Stat. Phys. 18, 115 (1978) CrossRefADSGoogle Scholar
  6. 6.
    Wu, F.Y.: Rev. Mod. Phys. 54, 235 (1982) CrossRefADSGoogle Scholar
  7. 7.
    Baxter, R.J.: Exactly Solved Models. Oxford Univ. Press, Oxford (1983) Google Scholar
  8. 8.
    Fortuin, C.M., Kasteleyn, P.W.: Physica 57, 536 (1972) CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Dong, F.M., Koh, K.M., Teo, K.L.: Chromatic Polynomials and Chromaticity of Graphs. World Scientific, Singapore (2005) MATHCrossRefGoogle Scholar
  10. 10.
    Biggs, N., Chang, S.C., Dong, F.M., Jackson, B., Jacobsen, J., Royle, G., Shrock, R., Sokal, A., Thomassen, C., et al.: In: Workshop on Zeros of Graph Polynomials. Newton Institute for Mathematical Sciences, Cambridge University (2008). Google Scholar
  11. 11.
    Beaudin, L., Ellis-Monaghan, J., Pangborn, G., Shrock, R.: Discrete Math. 310, 2037 (2010). arXiv:0804.2468 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field Theory. Springer, New York (1997) MATHGoogle Scholar
  13. 13.
    Lieb, E.H.: Phys. Rev. 162, 162 (1967) CrossRefADSGoogle Scholar
  14. 14.
    Wannier, G.H.: Phys. Rev. 79, 357 (1950) MATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Kim, D., Joseph, J.: J. Phys. C 7, L167 (1974) CrossRefADSGoogle Scholar
  16. 16.
    Adler, J., Brandt, A., Janke, W., Shmulyian, S.: J. Phys. A 28, 5117 (1995) MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Feldmann, H., Shrock, R., Tsai, S.-H.: J. Phys. A 30, L663 (1997) MATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Chang, S.-C., Shrock, R.: Physica A 301, 301 (2001) MATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Shrock, R., Tsai, S.-H.: Phys. Rev. E 56, 1342 (1997) CrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Harari, F.: Graph Theory. Addison-Wesley, Reading (1969) Google Scholar
  21. 21.
    Read, R.C., Wilson, R.J.: Atlas of Graphs. Oxford Univ. Press, Oxford (1998) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.C. N. Yang Institute for Theoretical PhysicsState University of New YorkStony BrookUSA

Personalised recommendations