Geodesic
The two-dimensional Ising model and the Kac–Ward determinant
Izvestiya. Mathematics, Tome 63 (1999) no. 4, pp. 707-727 
N. P. Dolbilin; Yu. M. Zinoviev; A. S. Mishchenko; M. A. Shtan'ko; M. I. Shtogrin. The two-dimensional Ising model and the Kac–Ward determinant. Izvestiya. Mathematics, Tome 63 (1999) no. 4, pp. 707-727. http://geodesic.mathdoc.fr/item/IM2_1999_63_4_a4/
@article{IM2_1999_63_4_a4,
     author = {N. P. Dolbilin and Yu. M. Zinoviev and A. S. Mishchenko and M. A. Shtan'ko and M. I. Shtogrin},
     title = {The two-dimensional {Ising} model and the {Kac{\textendash}Ward} determinant},
     journal = {Izvestiya. Mathematics},
     pages = {707--727},
     year = {1999},
     volume = {63},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1999_63_4_a4/}
}
TY  - JOUR
AU  - N. P. Dolbilin
AU  - Yu. M. Zinoviev
AU  - A. S. Mishchenko
AU  - M. A. Shtan'ko
AU  - M. I. Shtogrin
TI  - The two-dimensional Ising model and the Kac–Ward determinant
JO  - Izvestiya. Mathematics
PY  - 1999
SP  - 707
EP  - 727
VL  - 63
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/IM2_1999_63_4_a4/
LA  - en
ID  - IM2_1999_63_4_a4
ER  - 
%0 Journal Article
%A N. P. Dolbilin
%A Yu. M. Zinoviev
%A A. S. Mishchenko
%A M. A. Shtan'ko
%A M. I. Shtogrin
%T The two-dimensional Ising model and the Kac–Ward determinant
%J Izvestiya. Mathematics
%D 1999
%P 707-727
%V 63
%N 4
%U http://geodesic.mathdoc.fr/item/IM2_1999_63_4_a4/
%G en
%F IM2_1999_63_4_a4

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove a formula that expresses the square of the statistical sum for the two-dimensional Ising model for an arbitrary plane lattice in terms of the determinant of a matrix similar to the Kac–Ward matrix.

[1] Onsager L., “Crystal statistic. I: A two-dimensional model with an order-disorder transition”, Phys. Rev., 65 (1944), 117–149 | DOI | MR | Zbl

[2] Kaufman B., “Crystal statistic. II: Partition function evaluated by Spinor Analysis”, Phys. Rev., 76 (1949), 1232–1243 | DOI | Zbl

[3] Kaufman B., Onsager L., “Crystal statistic. III: Short-range order in a binary Ising lattice”, Phys. Rev., 76 (1949), 1244–1252 | DOI | Zbl

[4] Kac M., Ward J. C., “A combinatorial solution of the two-dimensional Ising model”, Phys. Rev., 88 (1952), 1332–1337 | DOI | MR | Zbl

[5] Sherman S., “Combinatorial aspects of the Ising model for ferromagnetism. I: A conjecture of Feynman on paths and graphs”, J. Math. Phys., 1:3 (1960), 202–217 | DOI | MR

[6] Sherman S., “Addendum: Combinatorial aspects of the Ising model for ferromagnetism”, J. Math. Phys., 4:9 (1963), 1212–1214 | DOI | MR

[7] Burgoyne P. N., “Remarks on the combinatorial approach to the Ising problem”, J. Math. Phys., 4:10 (1963), 1320–1326 | DOI | MR | Zbl

[8] Vdovichenko N. V., “Vychislenie statisticheskoi summy ploskoi dipolnoi reshetki”, ZhETF, 47:8 (1964), 715–719 | MR

[9] Landau L. D., Lifshits E. M., Statisticheskaya fizika, Nauka, M., 1976 | MR

[10] Dolbilin N. P., Shtanko M. A., Shtogrin M. I., “Kombinatornye voprosy dvumernoi modeli Izinga”, Tr. MIAN, 196, Nauka, M., 1991, 51–65 | MR

[11] Van der Warden B. L., “Lange Reichweite der regelmassigen Atomanordnung in Mischkristallen”, Zeitschr. Phys., 118 (1941), 473–488 | DOI

[12] Whitney H., “On regular closed curves in the plane”, Compos. Math., 4 (1937), 276–284 | MR | Zbl