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Combinatorics in statistical physics. (English) Zbl 0882.05014
Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 1-2. Amsterdam: Elsevier (North-Holland). 1925-1954 (1995).

A number of combinatorial problems have a counterpart in statistical mechanics and vice versa. This article surveys some of these connections.

Classical examples such as Ising or Potts models, percolation processes and various enumeration problems are discussed. Though the thermodynamic formalism is introduced and basic techniques such as transfer matrices and star-triangle transformations are explained, more powerful methods—and their recent success—are missing. Concepts of interest here are Pfaffians, Bethe’s Ansatz, and even some applications of “quantum groups” in connection with the Yang-Baxter equations.

For recent results on self-avoiding walks and polygons, see the work by A. R. Conway and A. J. Guttmann. For connections to the various branches of mathematics, including knot theory and operator algebra, see the review by V. F. R. Jones [Subfactors and knots (1991; Zbl 0743.46058)]. More on the connection to physics can be found in Vol. 1 of “Phase transitions and critical phenomena” (ed. by C. Domb and B. Green). The article also contains a short summary of spin glass systems and their ground states.

MSC:
05A99Classical combinatorial problems
05C90Applications of graph theory
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82B43Percolation (equilibrium statistical mechanics)
82B41Random walks, random surfaces, lattice animals, etc. (statistical mechanics)
05A15Exact enumeration problems, generating functions