The Statistics of Dimers on a Lattice

The number of ways in which a finite quadratic lattice (with edges or with periodic boundary conditions) can be fully covered with given numbers of “horizontal” and “vertical” dimers is rigorously calculated by a combinatorial method involving Pfaffians. For lattices infinite in one or two dimensions asymptotic expressions for this number of dimer configurations are derived, and as an application the entropy of a mixture of dimers of two different lengths on an infinite rectangular lattice is calculated. The relation of this combinatorial problem to the Ising problem is briefly discussed.

[1]  W. Metzler,et al.  Theory of Determinants , 1905 .

[2]  R. Fowler,et al.  An attempt to extend the statistical theory of perfect solutions , 1937 .

[3]  T. Chang.  Statistical theory of the adsorption of double molecules , 1939 .

[4]  P. Flory.  Thermodynamics of High Polymer Solutions , 1941 .

[5]  A. R. Miller,et al.  The number of configurations of a cooperative assembly , 1942, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  W. Orr.  On the calculation of certain higher-order Bethe approximations , 1944 .

[7]  L. Onsager.  Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[8]  M. Kac,et al.  A combinatorial solution of the two-dimensional Ising model , 1952 .

[9]  E. Caianiello,et al.  On the algorithm of Dirac spurs , 1952 .

[10]  Egon Ullrich,et al.  Dr. Wolfgang Gröbner (Prof. a. d. Universität Innsbruck) und Dr. Nikolaus Hofreiter (Prof. a. d. Universität Wien). Integraltafel. Zweiter Teil. Bestimmte Integrale. VI + 204S Springer‐Verlag, Wien und Innsbruck 1950. Preis 24,— DM , 1952 .

[11]  G. S. Rushbrooke,et al.  The vapour pressures of athermal mixtures , 1953 .

[12]  G. F. Newell,et al.  On the Theory of the Ising Model of Ferromagnetism , 1953 .

[13]  E. Cohen,et al.  A cell-cluster theory for the liquid state. II , 1954 .

[14]  R. B. Potts,et al.  The Combinatrial Method and the Two-Dimensional Ising Model , 1955 .

[15]  C. Berge.  Théorie des graphes et ses applications , 1958 .

[16]  ON GREEN AND LEIPNIK'S METHOD FOR SOLUTION OF THE ASSOCIATION PROBLEM , 1960 .

[17]  H. S. Green,et al.  New Solution of the Ising Problem for a Rectangular Lattice , 1960 .

[18]  Exact Solution of the Association Problem by a Matrix-Spinor Method with Applications to Statistical Mechanics , 1960 .

[19]  ASSOCIATION PROBLEM IN STATISTICAL MECHANICS--CRITIQUE OF THE TREATMENT OF H. S. GREEN AND R. LEIPNIK , 1960 .

[20]  L. Bittner.  C. Berge, Théorie des graphes et ses applications. VIII + 277 S. m. 117 Abb. Paris 1958. Dunod Editeur. Preis geb. 3400 F , 1960 .

[21]  S. Sherman.  Combinatorial Aspects of the Ising Model for Ferromagnetism. I. A Conjecture of Feynman on Paths and Graphs , 1960 .