Journal of Statistical Physics

, Volume 35, Issue 3–4, pp 193–266 | Cite as

Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities

  • George E. Andrews
  • R. J. Baxter
  • P. J. Forrester


The eight-vertex model is equivalent to a “solid-on-solid” (SOS) model, in which an integer heightl i is associated with each sitei of the square lattice. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2K, and involve a variable parameter η. Here we begin by showing that the hard hexagon model is a special case of this eight-vertex SOS model, in which η=K/5 and the heights are restricted to the range 1⩽li⩽4. We remark that the calculation of the sublattice densities of the hard hexagon model involves the Rogers-Ramanujan and related identities. We then go on to consider a more general eight-vertex SOS model, with η=K/r (r an integer) and 1⩽lir−1. We evaluate the local height probabilities (which are the analogs of the sublattice densities) of this model, and are automatically led to generalizations of the Rogers-Ramanujan and similar identities. The results are put into a form suitable for examining critical behavior, and exponentsβ, α,\(\bar \alpha \) are obtained.

Key words

Statistical mechanics lattice statistics number theory eight-vertex model solid-on-solid model hard hexagon model Rogers-Ramanujan identities 


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Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • George E. Andrews
    • 1
  • R. J. Baxter
    • 2
  • P. J. Forrester
    • 2
  1. 1.Department of Mathematics, McAllister BuildingPennsylvania State UniversityUniversity Park
  2. 2.Research School of Physical SciencesThe Australian National UniversityCanberraAustralia

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