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Zbl 0882.05014
Godsil, C.D.; Grötschel, M.; Welsh, D.J.A.
Combinatorics in statistical physics. (English)
[A] Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 1-2. Amsterdam: Elsevier (North-Holland). 1925-1954 (1995). ISBN 0-444-88002-X/set; ISBN 0-444-82346-8/vol1; ISBN 0-444-82351-4/vol2; ISBN 0-262-07169-X/set; ISBN 0-262-07170-3/vol1; ISBN 0-262-07171-1/vol2

A number of combinatorial problems have a counterpart in statistical mechanics and vice versa. This article surveys some of these connections.\par 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.\par 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 {\it 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.
[M.Baake (Tübingen)]

MSC 2000:: *05A99 Classical combinatorial problems
05C90 Appl. of graph theory
82B20 Lattice systems
82B43 Percolation
82B41 Random walks, etc. (statistical mechanics)
05A15 Combinatorial enumeration problems

Keywords: combinatorial problems; statistical mechanics; Ising or Potts models; percolation processes; enumeration problems; Pfaffians; Yang-Baxter equations; knot theory; operator algebras; spin glass systems

Citations: Zbl 0743.46058

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