Mathematical and Computer Modelling

Volume 26, Issues 8–10, October–November 1997, Pages 269-274

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The infinite-state potts model and restricted multidimensional partitions of an integer ☆

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F.Y. Wu fywu@neu.edu
Department of Physics, Northeastern University Boston, MA 02115, U.S.A.

Available online 14 May 1998

doi:10.1016/S0895-7177(97)00208-2

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Abstract

It is shown that the partition function of the q-state Potts model on a finite ddimensional hypercubic lattice in the q → ∞ limit is precisely the generating function of (d − 1)-dimensional restricted partitions of an integer. For d = 2,3, this equivalence leads to closed-form expressions of the q = ∞ Potts partition function. Our discussion also establishes symmetry and reciprocal properties for the generating function of restricted partitions in higher dimensions.

Keywords

  • The Potts model;
  • Restricted partitions
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References

  • 1. 

    R.B. Potts

    Some generalized order-disorder transformations

    Proc. Camb. Phil. Soc. (Second Edition), 48 (1952), pp. 106–109

  • 2. 

    F.Y. Wu

    The Potts model

    Rev. Mod. Phys., 54 (1982), pp. 235–268

  • 3. 

    L. Onsager

    Crystal statistics I. A two-dimensional model with an order-disorder transition

    Phys. Rev., 65 (1944), pp. 117–149

  • 4. 

    G.E. Andrews

    The theory of partitions

    G.-C. Rota (Ed.), Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, MA (1976) Chaps. 3 and 11

  • 5. 

    C.F. Gauss

    Werke (Second Edition), Vol. 2, Königliche Gesellschaft der Wissenschaften, Göttingen (1870)

  • 6. 

    P.A. MacMahon

    Combinatory Analysis (Second Edition), Vol. 2, Cambridge University Press, London (1960) Reprinted by Chelsea, New York

  • 7. 

    F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.K. Hu, C.N. Chen

    Directed compact lattice animals, restricted partitions of an integer and the infinite-state Potts model

    Phys. Rev. Lett., 76 (1996), pp. 173–176

  • 8. 

    R.J. Baxter

    Potts model at the critical temperature

    J. Phys., C6 (1973), pp. L445–L448

  • 9. 

    P.A. Pearce, R.B. Griffiths

    Potts model in the many-component limit

    J. Phys., A13 (1980), pp. 2143–2148

  • 10. 

    H.Y. Huang, F.Y. Wu

    The infinite-state Potts model and solid partitions of an integer

    Int. J. Mod. Phys., 11 (1997), pp. 121–126

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This work is supported in part by NSF Grant DMR-9313648.

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