Combinatorial Aspects of the Ising Model for Ferromagnetism. I. A Conjecture of Feynman on Paths and Graphs

Abstract

An identity on paths in planar graphs conjectured by Feynman [H] is rigorously established. This permits a complete analysis of the combinatorial approach to the two-dimensional Ising model with nearest neighbor interaction and 0 external magnetic field previously heuristically discussed by Kac and Ward [KW] and Potts and Ward [PW]. Relevant identities are established for the two-dimensional Ising model with next nearest neighbor interactions and 0 external magnetic field, for the two-dimensional Ising model with nearest neighbor interactions and positive external magnetic field, and for the three-dimensional Ising model with nearest neighbor interactions and 0 external field. For the case of a square net with an odd number of spin locations with nearest neighbor interactions and external field equal to iπ/2, it is shown that the partition function is identically zero for both plane and torus imbedding contrary to a result announced by Lee and Yang [LY; Eq. (48)], which turns out to be correct only for an even number of spin locations.