After the introduction of the concept of lattice graph and a brief discussion of its role in the theory of the Ising model, a related combinatorial problem is discussed, namely that of the statistics of non‐overlapping dimers, each occupying two neighboring sites of a lattice graph. It is shown that the configurational partition function of this system can be expressed in terms of a Pfaffian, and hence calculated explicitly, if the lattice graph is planar and if the dimers occupy all lattice sites. By the examples of the quadratic and the hexagonal lattice, it is found that the dimer system may show a phase transition similar to that of a two‐dimensional Ising model, or one of a different nature, or no transition at all, depending on the activities of various classes of bonds. The Ising problem is then shown to be equivalent to a generalized dimer problem, and a rederivation, of Onsager's expression for the Ising partition function of a rectangular lattice graph is sketched on the basis of this equivalence.

1.
For a general review, see
C.
Domb
,
Advan. Phys.
9
,
149
(
1960
).
2.
L.
Onsager
,
Phys. Rev.
65
,
117
(
1944
).
3.
A system in a magnetic field is equivalent to a system with a more complicated lattice graph in zero field; it is this graph which will then be called the lattice graph of the original system.
4.
M.
Kac
and
J. C.
Ward
,
Phys. Rev.
88
,
1332
(
1952
);
R. B.
Potts
and
J. C.
Ward
,
Progr. Theoret. Phys. (Kyoto)
13
,
38
(
1955
);
S.
Sherman
,
J. Math. Phys.
1
,
202
(
1960
).
5.
The genus g of a surface is the maximum number of nonintersecting closed curves which one can draw on the surface without disconnecting it. A plane and a sphere have g = 0, a torus g = 1, etc.
6.
D. König, Theorie der endlichen und unendlichen Graphen (Chelsea Publishing Company, New York, 1936, and 1950), p. 198.
7.
E. W.
Montroll
,
J. Soc. Ind. Appl. Math.
4
,
241
(
1956
).
8.
H.
Davies
,
Quart. J. Math. (Oxford)
6
,
232
(
1955
).
9.
H. N. V.
Temperley
,
Discussions Faraday Soc.
25
,
92
(
1958
).
10.
E. A. Guggenheim, Mixtures (Clarendon Press, Oxford, England, 1952) Chap. X.
11.
E. G. D.
Cohen
,
J.
de Boer
, and
Z. W.
Salsburg
,
Physica
21
,
137
(
1955
).
12.
T. S.
Chang
,
Proc. Roy. Soc. (London)
A169
,
512
(
1939
).
13.
P. W.
Kasteleyn
,
Physica
27
,
1209
(
1961
).
14.
H. N. V.
Temperley
and
M. E.
Fisher
,
Phil. Mag.
6
,
1061
(
1961
);
M. E.
Fisher
,
Phys. Rev.
124
,
1664
(
1961
).
15.
For earlier references to this connection, cf.
G.
Brunel
,
Mém. Soc. Sci. Bordeaux (4)
5
,
165
(
1895
);
W. T.
Tutte
,
J. London Math. Soc.
22
,
107
(
1947
).
16.
T. Muir, A Treatise on the Theory of Determinants (Cambridge University Press, New York, 1904), p. 92.
17.
E. R.
Caianiello
,
Nuovo Cimento
10
,
1634
(
1953
).
18.
A similar behavior is found for a dimer system on a quadratic lattice graph if a suitable, but somewhat more artificial classification of bonds is introduced. This is not surprising, as the hexagonal lattice graph can be considered as a quadratic lattice graph with certain bonds removed (or having zero activity). On the other hand, there is also an (again less “natural”) classification of the bonds of the hexagonal lattice graph for which the partition function shows the analytical behavior sketched in the previous section.
19.
R. M. F.
Houtappel
,
Physica
16
,
425
(
1950
).
20.
C. A.
Hurst
and
H. S.
Green
,
J. Chem. Phys.
33
,
1059
(
1960
).
21.
Essentially the same method has been used by Fisher in his solution of the dimer problem.
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